Happy New Year's Eve - Ring in 2018 at midnight. |

## Sunday, December 31, 2017

## Saturday, December 30, 2017

## Friday, December 29, 2017

### Visualizing Square Roots

One of the things I have to work on with my students is the concept of squares and square roots. One of the better ways for them to deal with this topic is to physically play with visual representations.

A good way to introduce the topic of squared numbers is to show this video where Elvis sings "Think I'm squared baby" a nice 1.25 min in which the perfect squares are sung.

This hands on activity for learning more about perfect squares uses Cheez-its (small edible square cheese crackers) to help students explore what numbers make a square and which ones do not. For instance, you might ask if 8 crackers make a perfect square. Students can physically move them around to check. This is a kinesthetic method of determining if the number of Cheez-its make a perfect square.

Once students have created a shape as close to a square, they use a specially created number line to figure out the length of each sides so they can estimate the square root of the shape. They then check their answer on a calculator. Yes it is a decimal approximation but they are getting a chance to see how they find the square root approximation for shapes that are not perfect squares.

I like having them use geo-boards either real or digital to create perfect squares as a way of representing squared numbers such as 2^2 or 3^2. They can "see" that the base gives the length of each side of the square. With discussion, they can conclude the square root is the length of the side of the square.

In other words is 2^2 means a square that is 2 units by 2 units, the area is four. Therefore, the square root of four is 2. The geo-board makes it so easy to see the relationship. Without the geo-boards, my students often struggle to relate the two topics.

The next step in the process is to crate squares that are not perfectly oriented both vertically and horizontally. In other words, its time to create squares that cut across the rows and columns so the slope is not zero or undefined. When the sides are done this way, the Pythagorean theorem has to be employed to find the length of the sides.

Desmos has a lovely activity where students put cards on the same square or square root together so at the end, they will have 5 stacks. The trick to connect cards is to place one on top of the other so they connect and make a stack. It took me a bit to figure out how to do it but I really enjoyed it. The only thing to let students know ahead of time if they give up easily is that not all stacks are the same height.

So I have additional activities to begin the next semester in my pre-algebra class and any class I need to review squares and square roots. Let me know what you think. I'd love to know.

A good way to introduce the topic of squared numbers is to show this video where Elvis sings "Think I'm squared baby" a nice 1.25 min in which the perfect squares are sung.

This hands on activity for learning more about perfect squares uses Cheez-its (small edible square cheese crackers) to help students explore what numbers make a square and which ones do not. For instance, you might ask if 8 crackers make a perfect square. Students can physically move them around to check. This is a kinesthetic method of determining if the number of Cheez-its make a perfect square.

Once students have created a shape as close to a square, they use a specially created number line to figure out the length of each sides so they can estimate the square root of the shape. They then check their answer on a calculator. Yes it is a decimal approximation but they are getting a chance to see how they find the square root approximation for shapes that are not perfect squares.

I like having them use geo-boards either real or digital to create perfect squares as a way of representing squared numbers such as 2^2 or 3^2. They can "see" that the base gives the length of each side of the square. With discussion, they can conclude the square root is the length of the side of the square.

In other words is 2^2 means a square that is 2 units by 2 units, the area is four. Therefore, the square root of four is 2. The geo-board makes it so easy to see the relationship. Without the geo-boards, my students often struggle to relate the two topics.

The next step in the process is to crate squares that are not perfectly oriented both vertically and horizontally. In other words, its time to create squares that cut across the rows and columns so the slope is not zero or undefined. When the sides are done this way, the Pythagorean theorem has to be employed to find the length of the sides.

Desmos has a lovely activity where students put cards on the same square or square root together so at the end, they will have 5 stacks. The trick to connect cards is to place one on top of the other so they connect and make a stack. It took me a bit to figure out how to do it but I really enjoyed it. The only thing to let students know ahead of time if they give up easily is that not all stacks are the same height.

So I have additional activities to begin the next semester in my pre-algebra class and any class I need to review squares and square roots. Let me know what you think. I'd love to know.

## Thursday, December 28, 2017

### Ways to Introduce Variables.

The other day, at the family Christmas gathering with people from 2 to 93 years old present, the topic of math always comes up because I'm there. One of my nieces told me she loved her last math class while another one told me she was ok with one letter but not two. She couldn't even use the term variable.

I know that there is a recent trend to use emoji's or characters instead of variables in three equations so the first equation is one equation, the second equation is two variables and the third equation has three variables. Usually there is a fourth equation that students use the values from the earlier three equations.

Another way to introduce multiple variables into equations is using Marilyn Burns Math Solutions lesson by Carrie DeFrancisco who created a lovely lesson based on Dr Suess's Green Eggs and Ham. The idea is using a menu based on certain choices from the book such as green eggs, regular eggs, bacon, ham, etc. Each item is represented by a single letter so green eggs is G, the special is x, ham is H, etc.

The lesson has students calculate the cost for various orders such as G + B + S =? or 2G + H =? at first but the orders become more complex as they move through the lesson. Students are encouraged to explain their thinking throughout the whole process. Although this lesson is listed for grades 6 to 8, I plan to use it with my pre-algebra class in January. I need new ways to present the information. This looks like a fun way to do it.

The Utah Education network has a nice lesson designed to introduce the concept of variables to students. It begins having students determine whether certain equations such as 5 + 7 = 12 are true or false. They indicate their choice using a thumbs up or thumbs down. The lesson has several activities which take a student from translating verbal equations into numerical equations which are then solved, while the second activity helps them set up expressions using variables with shapes first before using the actual letters. The final step is to create verbal sentences for an equation such as x + 4.

NCTM Illuminations has a nice variable machine activity using a code machine which aligns the alphabet to numbers to find the value of their first and last names. They get to calculate the difference between the two names. This is extended by having students find words equal to certain values.

All of these activities have an element giving a total which they have to find the equation which will give that number or give part of an equation where they have to find the missing value. I love activities with open ended questions so they have to think more.

I love using whiteboards, either digital or real, with students because they can do their work, hold it up and with a glance, I know if they understand the concept. I can also go around the class with a marker and draw a smiley or frowny face that provides an immediate feedback. If its a frowny face, they have to figure out why its incorrect and make corrections. They can ask table mates to help them.

Let me know what you think. I'd love to hear.

I know that there is a recent trend to use emoji's or characters instead of variables in three equations so the first equation is one equation, the second equation is two variables and the third equation has three variables. Usually there is a fourth equation that students use the values from the earlier three equations.

Another way to introduce multiple variables into equations is using Marilyn Burns Math Solutions lesson by Carrie DeFrancisco who created a lovely lesson based on Dr Suess's Green Eggs and Ham. The idea is using a menu based on certain choices from the book such as green eggs, regular eggs, bacon, ham, etc. Each item is represented by a single letter so green eggs is G, the special is x, ham is H, etc.

The lesson has students calculate the cost for various orders such as G + B + S =? or 2G + H =? at first but the orders become more complex as they move through the lesson. Students are encouraged to explain their thinking throughout the whole process. Although this lesson is listed for grades 6 to 8, I plan to use it with my pre-algebra class in January. I need new ways to present the information. This looks like a fun way to do it.

The Utah Education network has a nice lesson designed to introduce the concept of variables to students. It begins having students determine whether certain equations such as 5 + 7 = 12 are true or false. They indicate their choice using a thumbs up or thumbs down. The lesson has several activities which take a student from translating verbal equations into numerical equations which are then solved, while the second activity helps them set up expressions using variables with shapes first before using the actual letters. The final step is to create verbal sentences for an equation such as x + 4.

NCTM Illuminations has a nice variable machine activity using a code machine which aligns the alphabet to numbers to find the value of their first and last names. They get to calculate the difference between the two names. This is extended by having students find words equal to certain values.

All of these activities have an element giving a total which they have to find the equation which will give that number or give part of an equation where they have to find the missing value. I love activities with open ended questions so they have to think more.

I love using whiteboards, either digital or real, with students because they can do their work, hold it up and with a glance, I know if they understand the concept. I can also go around the class with a marker and draw a smiley or frowny face that provides an immediate feedback. If its a frowny face, they have to figure out why its incorrect and make corrections. They can ask table mates to help them.

Let me know what you think. I'd love to hear.

## Wednesday, December 27, 2017

### Can Frequent Testing Help Students Learn Better?

Unfortunately, when any one starts talking about testing, we immediately think of the once a year, multiple choice, computer based tests organized by the state government or unit end tests but that is not the type of testing I refer to in this blog entry.

As research indicates testing, when done properly, does help learning. The key is a well designed test paired with equally well designed activities before and after the actual test because these items can help students recall information better, develop a deeper and more complex learning compared to no tests. However, it does not look like the standardized tests both teachers and students dread.

The best testing is designed to create a situation where students practice retrieving the information. Retrieval practice is not used to assess, it is used to create a learning atmosphere so students learn during the testing situation. These situations are used to force the brain to recall information from the brain. Every time this happens, new memories are changed and made stronger, more accessible, and easier to access.

The brain cannot possibly remember everything it is exposed to daily so the brain is selective only remembering the information that has been demonstrated to be important and needed for the future. So when we retrieve the information in a testing situation, we tell the brain, this information is important enough to be accessible again.

Research indicates that recalling information is actually more effective then restudying the material. The more active certain parts of the brain are, the more likely the brain will be able to recall the information. In addition to helping students the material they are studying, it also helps them remember information not directly being tested.

Furthermore, retrieval practice helps the brain separate the material currently being learned from prior knowledge and prepares the brain to absorb more material after testing. Thus we need to incorporate methods to help students practice retrieval so they learn better.

First are flash cards or something similar used by the students to quiz themselves. The key here is quiz themselves rather than just studying the material. A study found that students who quizzed themselves on vocabulary remembered 80 percent of the material while students who only read the words repeatedly, remembered about 30 percent. This is where those flash card apps come in handy. The other thing about flash cards is they say a student must get the concept right three times for the information to be remembered.

Another methods is to make it into a game such as Kahoot or Jeopardy. Kahoot if you have not used it allows you to choose a premade game or create your own. My students love it because they can play individually, get immediate feedback, and call out wrong answers in the hope they will cause others to make the wrong choice. I add a commentary of who is on the scoreboard and who goes up or down on the board.

As far as Jeopardy, I have students work in groups because the game is only set up for a few players. The students in the group, all have to work the answer and come up with an answer written on a white board I look at but everyone shares at the end. Every group who gets the correct answer gets the points, not just the first ones done. If I don't give points to all the groups, the ones who struggle give up because one group gets all the points. If they get points too, they are willing to keep trying.

Its also easy to sneak in a few quick quiz problems in the warm-up/ bell ringer or as an exit ticket so students can practice. I usually have a qr code by the door so students can check their answers at the end for immediate feedback.

Be sure to share with students that the standard practice of rereading and highlighting information has proven to be the least effective form of retrieval practice.

Let me know what you think. I'd love to hear.

As research indicates testing, when done properly, does help learning. The key is a well designed test paired with equally well designed activities before and after the actual test because these items can help students recall information better, develop a deeper and more complex learning compared to no tests. However, it does not look like the standardized tests both teachers and students dread.

The best testing is designed to create a situation where students practice retrieving the information. Retrieval practice is not used to assess, it is used to create a learning atmosphere so students learn during the testing situation. These situations are used to force the brain to recall information from the brain. Every time this happens, new memories are changed and made stronger, more accessible, and easier to access.

The brain cannot possibly remember everything it is exposed to daily so the brain is selective only remembering the information that has been demonstrated to be important and needed for the future. So when we retrieve the information in a testing situation, we tell the brain, this information is important enough to be accessible again.

Research indicates that recalling information is actually more effective then restudying the material. The more active certain parts of the brain are, the more likely the brain will be able to recall the information. In addition to helping students the material they are studying, it also helps them remember information not directly being tested.

Furthermore, retrieval practice helps the brain separate the material currently being learned from prior knowledge and prepares the brain to absorb more material after testing. Thus we need to incorporate methods to help students practice retrieval so they learn better.

First are flash cards or something similar used by the students to quiz themselves. The key here is quiz themselves rather than just studying the material. A study found that students who quizzed themselves on vocabulary remembered 80 percent of the material while students who only read the words repeatedly, remembered about 30 percent. This is where those flash card apps come in handy. The other thing about flash cards is they say a student must get the concept right three times for the information to be remembered.

Another methods is to make it into a game such as Kahoot or Jeopardy. Kahoot if you have not used it allows you to choose a premade game or create your own. My students love it because they can play individually, get immediate feedback, and call out wrong answers in the hope they will cause others to make the wrong choice. I add a commentary of who is on the scoreboard and who goes up or down on the board.

As far as Jeopardy, I have students work in groups because the game is only set up for a few players. The students in the group, all have to work the answer and come up with an answer written on a white board I look at but everyone shares at the end. Every group who gets the correct answer gets the points, not just the first ones done. If I don't give points to all the groups, the ones who struggle give up because one group gets all the points. If they get points too, they are willing to keep trying.

Its also easy to sneak in a few quick quiz problems in the warm-up/ bell ringer or as an exit ticket so students can practice. I usually have a qr code by the door so students can check their answers at the end for immediate feedback.

Be sure to share with students that the standard practice of rereading and highlighting information has proven to be the least effective form of retrieval practice.

Let me know what you think. I'd love to hear.

## Tuesday, December 26, 2017

### The Style of Font and Learning.

I'd like to start today's entry with a huge thank you to Adam Liss for the idea. He shared a link on the style of font and learning which was quite interesting and something I'd not heard about before.

After reading up on the topic, my first response was wow followed by "That makes sense." So here is information on the topic.

Most people think that if you change the size of the font,

What they are finding is that it is better to use a font that is more difficult to read if you want people to remember the information. If you want to learn the information, we learn better if we have to struggle a bit so by using an unfamiliar difficult to read font, we are adding that bit of struggle.

If the fonts are familiar, we become over confident in thinking that we have retained the information when in reality we've only skimmed it for the essence of the material and really do not spend time learning it. When we have to slow down our reading speed, we are more likely to spend time really reading the piece while processing and learning the information.

According to a summary published by the University of Washington, researchers at Princeton conducted one experiment to see if the type of font would improve memory. Volunteers were given material printed in different fonts for 90 seconds. 15 minutes later, they were tested and the researchers discovered that 72.8% remembered the material printed with easy to read font while 86.5% remembered material printed with harder to read fonts such as Comic Sans.

A second experiment took place in several high school classes with the same level of abilities. The researchers changed the fonts on worksheets and power points. The results showed higher scores for all students who used the worksheets and power points written in a harder to read fonts. The thing about this experiment is that it was conducted at a school with teachers teaching at least two sections of each class. One class got materials using the normal fonts while the other class had the unfamiliar fonts.

The researchers concluded that this one simple change could improve learning. Their explanation for these results is that people must employ deeper processing strategies when using the harder to read fonts. So they have to think more about the material they are reading when its an unfamiliar font. This disfluency causes people to process information more deeply, abstractly, carefully, and better.

Just to let you know magazines and newspapers from the New York Times, to Wired, to the Harvard Business Review felt this information is significant enough to publish something on it. This study made an impact in the education field. They conclude that this is one easy thing to do to help students learn more.

I plan to use this with my students beginning in January. Let me know what you think. Thank you Adam for this. Have a good holiday.

After reading up on the topic, my first response was wow followed by "That makes sense." So here is information on the topic.

Most people think that if you change the size of the font,

*italicize it,***Make it Bold,**or other adjustment, it is going to help people retain the knowledge better. Research indicates this is not correct.What they are finding is that it is better to use a font that is more difficult to read if you want people to remember the information. If you want to learn the information, we learn better if we have to struggle a bit so by using an unfamiliar difficult to read font, we are adding that bit of struggle.

If the fonts are familiar, we become over confident in thinking that we have retained the information when in reality we've only skimmed it for the essence of the material and really do not spend time learning it. When we have to slow down our reading speed, we are more likely to spend time really reading the piece while processing and learning the information.

According to a summary published by the University of Washington, researchers at Princeton conducted one experiment to see if the type of font would improve memory. Volunteers were given material printed in different fonts for 90 seconds. 15 minutes later, they were tested and the researchers discovered that 72.8% remembered the material printed with easy to read font while 86.5% remembered material printed with harder to read fonts such as Comic Sans.

A second experiment took place in several high school classes with the same level of abilities. The researchers changed the fonts on worksheets and power points. The results showed higher scores for all students who used the worksheets and power points written in a harder to read fonts. The thing about this experiment is that it was conducted at a school with teachers teaching at least two sections of each class. One class got materials using the normal fonts while the other class had the unfamiliar fonts.

The researchers concluded that this one simple change could improve learning. Their explanation for these results is that people must employ deeper processing strategies when using the harder to read fonts. So they have to think more about the material they are reading when its an unfamiliar font. This disfluency causes people to process information more deeply, abstractly, carefully, and better.

Just to let you know magazines and newspapers from the New York Times, to Wired, to the Harvard Business Review felt this information is significant enough to publish something on it. This study made an impact in the education field. They conclude that this is one easy thing to do to help students learn more.

I plan to use this with my students beginning in January. Let me know what you think. Thank you Adam for this. Have a good holiday.

## Monday, December 25, 2017

## Sunday, December 24, 2017

## Saturday, December 23, 2017

## Friday, December 22, 2017

### Music and Math

While researching learning styles, I came across the idea that it is possible to connect a visual representation of a fraction to the value of various notes. The teacher encouraged clapping, drumming, and chanting to provide another avenue of instruction. The third graders who went through this experience improved their understanding of basic fractions significantly.

I wish I'd known about this because it would have added a layer to the fraction unit I just taught in Pre-Algebra. I've found quite a few resources I'm sharing with you so if you want to extend your tools, you can include these.

First is the Math Science Music group who have quite a few resources for the math classroom. Remember in the first paragraph, I talked about connecting time fractions with music? This is the group who created the program used. They have Academic Music which is the program described earlier. They also have Groove Pizza which uses angles, shapes and groove, or Indian Rhythms using rhythm, patterns, and music. Check out OIID with music, composing, and math.

Each unit comes complete with everything needed for the teacher to plan a lesson or set of lessons. They've included a list of resources, information on the grade level, concepts and processes. I am impressed with their offerings. I have a few students who are into music and this would appeal to them.

Teacher Vision has a wonderful list of resources on their Mathematics in Music web page. There is one of the Fibonacci sequence and music while the next link down focuses on patterns in both music and math. In addition, there are links on permutations and improvisation in jazz, the use of fractions in music, connecting music and math via Mozart, or taping out different rhythms.

This 176 page PDF on Mathematics and Music written by David Wright in 2009 contains a wide variety of connections between the two topics. It was originally designed as a textbook for college freshmen and as such each chapter focuses on a different topic. The first chapter lays the foundation looking at the basic concepts in math and in music so as a student progresses through the book, they have a foundation for the topic.

The second chapter focuses on note values and time signatures while the third chapter looks at chords and chord identification. By the fourth chapter, the author discusses the mathematical ratios of intervals. Each chapter covers a higher level of math from logarithms and musical intervals, integers as intervals, and periodic functions. Next semester, I plan to have my Algebra II class explore logs and natural logs. This material will be perfect to use in that class.

Finally for today's list, I include a set of lesson plans from Thirteen.org on the Math in Music. It comes with a couple of different videos including one where two people discuss how math is used to produce hip-hop music. It comes with everything needed to conduct a cool lesson or two.

Let me know what you think. I think this is another cool way to make math easier for the student who hates math or thinks they are not good at it to relate to.

I wish I'd known about this because it would have added a layer to the fraction unit I just taught in Pre-Algebra. I've found quite a few resources I'm sharing with you so if you want to extend your tools, you can include these.

First is the Math Science Music group who have quite a few resources for the math classroom. Remember in the first paragraph, I talked about connecting time fractions with music? This is the group who created the program used. They have Academic Music which is the program described earlier. They also have Groove Pizza which uses angles, shapes and groove, or Indian Rhythms using rhythm, patterns, and music. Check out OIID with music, composing, and math.

Each unit comes complete with everything needed for the teacher to plan a lesson or set of lessons. They've included a list of resources, information on the grade level, concepts and processes. I am impressed with their offerings. I have a few students who are into music and this would appeal to them.

Teacher Vision has a wonderful list of resources on their Mathematics in Music web page. There is one of the Fibonacci sequence and music while the next link down focuses on patterns in both music and math. In addition, there are links on permutations and improvisation in jazz, the use of fractions in music, connecting music and math via Mozart, or taping out different rhythms.

This 176 page PDF on Mathematics and Music written by David Wright in 2009 contains a wide variety of connections between the two topics. It was originally designed as a textbook for college freshmen and as such each chapter focuses on a different topic. The first chapter lays the foundation looking at the basic concepts in math and in music so as a student progresses through the book, they have a foundation for the topic.

The second chapter focuses on note values and time signatures while the third chapter looks at chords and chord identification. By the fourth chapter, the author discusses the mathematical ratios of intervals. Each chapter covers a higher level of math from logarithms and musical intervals, integers as intervals, and periodic functions. Next semester, I plan to have my Algebra II class explore logs and natural logs. This material will be perfect to use in that class.

Finally for today's list, I include a set of lesson plans from Thirteen.org on the Math in Music. It comes with a couple of different videos including one where two people discuss how math is used to produce hip-hop music. It comes with everything needed to conduct a cool lesson or two.

Let me know what you think. I think this is another cool way to make math easier for the student who hates math or thinks they are not good at it to relate to.

## Thursday, December 21, 2017

### Learning Styles Is A Myth.

The other evening I ran across a new instructional technique or at least one that is new to me and in it, the author of the article made a comment about knowing your student's learning style.

I've seen tests for students to take and books on ways to teach to meet the needs of the various learning styles but recently, I've come across information that this particular view is wrong.

Evidence is coming out that teaching based on student learning style is a myth. In other words, the belief that student performance increases if instruction is tailored to a students preferred way of learning is totally wrong.

Although there is a significant amount of literature available on learning styles, very little has been carried out using an experimental methodology designed to test its validity in regard to education. The studies that did use a proper methodology, produced results contrary to the prevailing view point.

There are indications that students with good working memories do better in all subjects regardless of preferred learning style. This could be because students with good working memories are able to adapt to different learning styles no matter how the information is presented.

It also appears that our best learning is not via our preferred style but is based instead on the type of material being presented. For instance, its harder to learn a foreign language using only pictures while its hard to teach geometry using only verbal language. What research does say is that novices learn better from examples while those who are more advanced learn better by solving problems themselves.

One scientist postulated the idea of learning styles came from research indicating that different parts of the cortex play a different role in visual, auditory, and sensory processing. This birthed the theory that students should learn better depending on what part of their brain works better, however, this unsound assumption does not sit well due to the interconnectivity of the brain.

So over the past few years, this particular educational theorem blossomed. This is not to say that teachers should present information in only one form. All learners benefit from having the information presented in a variety of ways that engage a students senses. By presenting the material using a variety of methods, it keeps students from getting bored and they learn better. As long as the activity furthers a students knowledge of the subject, it doesn't matter if they are dancing, acting, or singing a song.

Furthermore, when people discuss learning styles, they are actually saying cognitive ability is learning style but its not. Cognitive ability effects a student's ability to learn but at learning style does not. So from very current reputable sources, we now know that learning styles is a myth that has been debunked. I am glad to know this because I get tired of looking up information and finding it based on this rather than on the best ways to present material to my students.

I'd love to hear from everyone on their thoughts on this topic. Have a great day.

I've seen tests for students to take and books on ways to teach to meet the needs of the various learning styles but recently, I've come across information that this particular view is wrong.

Evidence is coming out that teaching based on student learning style is a myth. In other words, the belief that student performance increases if instruction is tailored to a students preferred way of learning is totally wrong.

Although there is a significant amount of literature available on learning styles, very little has been carried out using an experimental methodology designed to test its validity in regard to education. The studies that did use a proper methodology, produced results contrary to the prevailing view point.

There are indications that students with good working memories do better in all subjects regardless of preferred learning style. This could be because students with good working memories are able to adapt to different learning styles no matter how the information is presented.

It also appears that our best learning is not via our preferred style but is based instead on the type of material being presented. For instance, its harder to learn a foreign language using only pictures while its hard to teach geometry using only verbal language. What research does say is that novices learn better from examples while those who are more advanced learn better by solving problems themselves.

One scientist postulated the idea of learning styles came from research indicating that different parts of the cortex play a different role in visual, auditory, and sensory processing. This birthed the theory that students should learn better depending on what part of their brain works better, however, this unsound assumption does not sit well due to the interconnectivity of the brain.

So over the past few years, this particular educational theorem blossomed. This is not to say that teachers should present information in only one form. All learners benefit from having the information presented in a variety of ways that engage a students senses. By presenting the material using a variety of methods, it keeps students from getting bored and they learn better. As long as the activity furthers a students knowledge of the subject, it doesn't matter if they are dancing, acting, or singing a song.

Furthermore, when people discuss learning styles, they are actually saying cognitive ability is learning style but its not. Cognitive ability effects a student's ability to learn but at learning style does not. So from very current reputable sources, we now know that learning styles is a myth that has been debunked. I am glad to know this because I get tired of looking up information and finding it based on this rather than on the best ways to present material to my students.

I'd love to hear from everyone on their thoughts on this topic. Have a great day.

## Wednesday, December 20, 2017

### 3D Graphic Organizers for Math.

Research indicated that graphic organizers can increase learning through brainstorming, organizing and communicating ideas, finding patterns, connections and relationships, accessing prior knowledge, classify concepts, develop vocabulary, focus on main ideas, and improve collaboration.

Most graphic organizers you find on the internet are nothing more than flat papers with the organizer already on it, ready to be filled out, much like a worksheet.

Three dimensional graphic organizers are known as foldables among other names add a dimension to the equation. Foldables is a term used to describe organizers which you fold, cut, and write on to create the final product.

Foldables integrate reading and writing with math, encourages critical thinking, allows for creativity, encourages individual learning. In fact, foldables have a physical element for students who love to make paper airplanes during class. These items use lots paper. Plain white paper, colored paper, graph paper, construction paper, even card stalk. Add in scissors, glue sticks, and stapler and you are set to go.

Both the teacher and student can use it. The teacher shows students how to put it together while teaching the material and the student uses it time after time to reinforce learning. Foldables can be books with one part fold to four part folds, pop-ups, mobiles, standing cubes, tab books, accordion folds, folded charts, graphs, even a concept map book.

One of the big names in creating 3D Graphic Organizers is Dinah Zike who has published quite a few books on the topic and is one of the pioneers. I've got one of her books dealing with math foldables. I love it because she shows how to use various ones with different topics. In addition, the class set of the textbook comes with instructions to create foldables for every chapter. The directions are in the book, so students get a chance to work independently.

I use them with my students. Each time I have them create one, I require they glue it into their notebook so they don't loose it and so the notebook becomes interact. I find the foldables make it easier for students find the information because of its form. Instead of being written on a page in note form, the information is organized into something that is much easier to read and find.

If you aren't sure where to start, check out the internet for ideas on how to use them. I strongly recommend their use because my students love making them. Unfortunately due to everything that happened this past semester, I let it fall by the wayside but I have plans to implement them for often next semester.

Let me know what you think. I'd love to hear.

Most graphic organizers you find on the internet are nothing more than flat papers with the organizer already on it, ready to be filled out, much like a worksheet.

Three dimensional graphic organizers are known as foldables among other names add a dimension to the equation. Foldables is a term used to describe organizers which you fold, cut, and write on to create the final product.

Foldables integrate reading and writing with math, encourages critical thinking, allows for creativity, encourages individual learning. In fact, foldables have a physical element for students who love to make paper airplanes during class. These items use lots paper. Plain white paper, colored paper, graph paper, construction paper, even card stalk. Add in scissors, glue sticks, and stapler and you are set to go.

Both the teacher and student can use it. The teacher shows students how to put it together while teaching the material and the student uses it time after time to reinforce learning. Foldables can be books with one part fold to four part folds, pop-ups, mobiles, standing cubes, tab books, accordion folds, folded charts, graphs, even a concept map book.

One of the big names in creating 3D Graphic Organizers is Dinah Zike who has published quite a few books on the topic and is one of the pioneers. I've got one of her books dealing with math foldables. I love it because she shows how to use various ones with different topics. In addition, the class set of the textbook comes with instructions to create foldables for every chapter. The directions are in the book, so students get a chance to work independently.

I use them with my students. Each time I have them create one, I require they glue it into their notebook so they don't loose it and so the notebook becomes interact. I find the foldables make it easier for students find the information because of its form. Instead of being written on a page in note form, the information is organized into something that is much easier to read and find.

If you aren't sure where to start, check out the internet for ideas on how to use them. I strongly recommend their use because my students love making them. Unfortunately due to everything that happened this past semester, I let it fall by the wayside but I have plans to implement them for often next semester.

Let me know what you think. I'd love to hear.

## Tuesday, December 19, 2017

### Do They Really Know How To Use It?

I freely admit I am one of the people enrolled in the Ditch The Textbook Summit this year. During the first presentation, the comment that students can use mobile devices to play games, use Social Media, etc but when you ask them to create a presentation using Google Slides, Power Point, or Keynote and they can't.

I mentioned this to another teacher because I've discovered that and she agreed with the statement. She said her last principal said they have to teach students how to use these tools.

We know they live on their devices but they do not focus on the tools they have access too. Its kind of funny because I get a new app on my tablet and I play with it until I know how to use it. My students are unwilling to do that. They want me to show them how to do everything.

I checked the internet for articles on this topic but didn't find many. Most articles are recommendations for which apps to use or why mobile devices are good for education but not on teaching students to use apps.

From personal experience, my students can get the app open but that is as far as it goes. So I have to take them step by step through the app in order for them to use it. I often prepare a slide show with pictures to learn the basics but I let them create the specifics.

After I wrote this paragraph, I remembered that you tube has videos on how to use most any app but you tube is not open during school hours so I have to download them after hours, then up load them to another site so my students can access them. This is a school wide rule to cut down on streaming which slows our internet down to a crawl.

For instance, if I have them use a slide presentation app, I have them open the app. Then with the slide show I've prepared, I take them step by step setting up titles, adding pictures, etc. Rather than take a class to do this, I think I need to begin creating videos with pauses so students can create at the same time.

One reason my students have this problem is because of a cultural influence. In this culture, you don't do anything on your own. You have to wait for someone to show you how to do it and you have to prove you are doing it "correctly". This sometimes makes it harder for my students to step outside the cultural learning style.

Let me know what you think! I'd love to hear about this from others. Have a good day.

I mentioned this to another teacher because I've discovered that and she agreed with the statement. She said her last principal said they have to teach students how to use these tools.

We know they live on their devices but they do not focus on the tools they have access too. Its kind of funny because I get a new app on my tablet and I play with it until I know how to use it. My students are unwilling to do that. They want me to show them how to do everything.

I checked the internet for articles on this topic but didn't find many. Most articles are recommendations for which apps to use or why mobile devices are good for education but not on teaching students to use apps.

From personal experience, my students can get the app open but that is as far as it goes. So I have to take them step by step through the app in order for them to use it. I often prepare a slide show with pictures to learn the basics but I let them create the specifics.

After I wrote this paragraph, I remembered that you tube has videos on how to use most any app but you tube is not open during school hours so I have to download them after hours, then up load them to another site so my students can access them. This is a school wide rule to cut down on streaming which slows our internet down to a crawl.

For instance, if I have them use a slide presentation app, I have them open the app. Then with the slide show I've prepared, I take them step by step setting up titles, adding pictures, etc. Rather than take a class to do this, I think I need to begin creating videos with pauses so students can create at the same time.

One reason my students have this problem is because of a cultural influence. In this culture, you don't do anything on your own. You have to wait for someone to show you how to do it and you have to prove you are doing it "correctly". This sometimes makes it harder for my students to step outside the cultural learning style.

Let me know what you think! I'd love to hear about this from others. Have a good day.

## Monday, December 18, 2017

### Brain Dump

I was reading something the other day which suggested incorporating brain dumps into your daily routine to help students improve understanding and provides a quick assessment. It is an activity that does not provide a grade.

Brain dump is simply having students write down or talk about everything that they remember about a topic. The request for a brain dump could be on what they already know, what they remember from the day before, or what they remember from the lesson so far.

If you go the way of having students write down what they remember, the next step would be having them share the information with others in groups. They can ask questions of each other or make comments to clarify ideas.

Using a brain dump or download helps students transfer information to long term memory storage so its easily recalled. This is important because it helps eliminate the blank mind during a test or the "But I understood it yesterday!"

The specifics for doing a well run brain dump is as follows:

1. Set a specific time limit for the activity such as 10 minutes about the essentials of the unit. For my students, I would start with less time to get them used to writing or talking.

2. Let them know the information should be done in free flow. Let students know they do not have to organize the notes. Just write.

3. Do not save them. Let them struggle because it helps them transfer the information from short term to long term memory. Practicing should be hard because it helps students learn better.

4. Practice generation, elaboration, and reflection. Generation refers to writing down the main ideas, restate key ideas in your own words and create questions. Elaboration is taking the ideas and expanding them with details, connecting facts, and show causes. Finally reflection has students writing about what they learned and how they learned.

In math, the brain dump could also include having students work together to make corrections so they learn doing it correctly. The corrections should be made in a different color so they can "see" the difference and know which one is the corrections by just looking.

So if you aren't doing it yet, take a bit of time to incorporate brain dumps into your daily routine to help students move the information from short term to long term memory.

Let me know what you think, I'd love to hear.

Brain dump is simply having students write down or talk about everything that they remember about a topic. The request for a brain dump could be on what they already know, what they remember from the day before, or what they remember from the lesson so far.

If you go the way of having students write down what they remember, the next step would be having them share the information with others in groups. They can ask questions of each other or make comments to clarify ideas.

Using a brain dump or download helps students transfer information to long term memory storage so its easily recalled. This is important because it helps eliminate the blank mind during a test or the "But I understood it yesterday!"

The specifics for doing a well run brain dump is as follows:

1. Set a specific time limit for the activity such as 10 minutes about the essentials of the unit. For my students, I would start with less time to get them used to writing or talking.

2. Let them know the information should be done in free flow. Let students know they do not have to organize the notes. Just write.

3. Do not save them. Let them struggle because it helps them transfer the information from short term to long term memory. Practicing should be hard because it helps students learn better.

4. Practice generation, elaboration, and reflection. Generation refers to writing down the main ideas, restate key ideas in your own words and create questions. Elaboration is taking the ideas and expanding them with details, connecting facts, and show causes. Finally reflection has students writing about what they learned and how they learned.

In math, the brain dump could also include having students work together to make corrections so they learn doing it correctly. The corrections should be made in a different color so they can "see" the difference and know which one is the corrections by just looking.

So if you aren't doing it yet, take a bit of time to incorporate brain dumps into your daily routine to help students move the information from short term to long term memory.

Let me know what you think, I'd love to hear.

## Sunday, December 17, 2017

### Warm-up

A turkey shrinks about 25% when cooked. How much does a cooked turkey weigh if it weighed

15.5 pounds before being put in the oven?

## Saturday, December 16, 2017

## Friday, December 15, 2017

### Why Use Exit Tickets

Exit tickets are a quick way to check student understanding on the topic they are studying in class. This activity can be done one a week or daily depending on the material being taught.

If the exit ticket is well written, the resulting information can give the teacher enough data to determine if the students are ready to move on or if they need to be retaught the material.

When designing an exit ticket, it should be linked to the lesson objective and focus only on the skill or concept for that day. Multiple choice questions, or short answers are good types of questions for exit tickets. Each ticket should have three to five questions that can be answered in a few minutes.

Do not use yes or no questions because they give no real information. The questions should assess understanding, demonstrate the concept, or apply the concept so the teacher really knows what the student understands. It is best to prepare the questions the day before. Some teachers write the exit ticket first, then create the lesson so the material is covered properly.

The exit ticket can be prepared to pass out to students or it could be written on the board so students write their answers on a 3 x 5 card, or on an email or post the answer in Google Classroom. One can also up load the questions to a Google form, or a program designed to do exit tickets. If you use a Google form, you can upload the results to a Google Doc and get the information into a spread sheet.

It is important to let students know the information on the exit ticket is not being graded but is being used to determine their level of understanding. Once you have the results you can determine if you need to reteach the concept, move on, or work on tweaking their understanding. You can decide if half the class needs more independent practice while you do some small group instruction to strengthen their understanding.

Exit tickets can be used to preview a topic you are planning to teach. It gives you a chance to ascertain their previous knowledge or activate it. If you don't have time to create a written exit ticket you can ask them a question and have them answer it on their way out of the room. Either way, exit tickets need to be analyzed as soon as possible so the information can be used.

From the students point of view, exit tickets require them to synthesize the day's lesson and challenge them to apply the concept because they have to think about the material and it also highlights any confusion they still have.

There are four types of exit tickets that can be used in the classroom. The first is a prompt that provides formative assessment. The second, stimulates self analysis when the teacher asks "How hard did you work?" or "What could you have done better today?". Although neither question asks about the material, they do address student effort.

The third prompt asks how certain instructional strategies helped students learn such as asking if the group work helped the student understand the material better. The last type is the least common type of prompt in which the teacher asks what can they do to help the students learn the material better.

I'd love to hear what you think. I like the idea and know that I know more about exit tickets, I plan to use them in class a bit more often than I use them.

If the exit ticket is well written, the resulting information can give the teacher enough data to determine if the students are ready to move on or if they need to be retaught the material.

When designing an exit ticket, it should be linked to the lesson objective and focus only on the skill or concept for that day. Multiple choice questions, or short answers are good types of questions for exit tickets. Each ticket should have three to five questions that can be answered in a few minutes.

Do not use yes or no questions because they give no real information. The questions should assess understanding, demonstrate the concept, or apply the concept so the teacher really knows what the student understands. It is best to prepare the questions the day before. Some teachers write the exit ticket first, then create the lesson so the material is covered properly.

The exit ticket can be prepared to pass out to students or it could be written on the board so students write their answers on a 3 x 5 card, or on an email or post the answer in Google Classroom. One can also up load the questions to a Google form, or a program designed to do exit tickets. If you use a Google form, you can upload the results to a Google Doc and get the information into a spread sheet.

It is important to let students know the information on the exit ticket is not being graded but is being used to determine their level of understanding. Once you have the results you can determine if you need to reteach the concept, move on, or work on tweaking their understanding. You can decide if half the class needs more independent practice while you do some small group instruction to strengthen their understanding.

Exit tickets can be used to preview a topic you are planning to teach. It gives you a chance to ascertain their previous knowledge or activate it. If you don't have time to create a written exit ticket you can ask them a question and have them answer it on their way out of the room. Either way, exit tickets need to be analyzed as soon as possible so the information can be used.

From the students point of view, exit tickets require them to synthesize the day's lesson and challenge them to apply the concept because they have to think about the material and it also highlights any confusion they still have.

There are four types of exit tickets that can be used in the classroom. The first is a prompt that provides formative assessment. The second, stimulates self analysis when the teacher asks "How hard did you work?" or "What could you have done better today?". Although neither question asks about the material, they do address student effort.

The third prompt asks how certain instructional strategies helped students learn such as asking if the group work helped the student understand the material better. The last type is the least common type of prompt in which the teacher asks what can they do to help the students learn the material better.

I'd love to hear what you think. I like the idea and know that I know more about exit tickets, I plan to use them in class a bit more often than I use them.

## Thursday, December 14, 2017

### Instant Feedback.

I know I should be giving immediate feedback but I don't always have the time to do it and I need ways to provide it so students don't go too long doing the same thing before they are corrected.

I also need to help students learn to use the feedback as a learning tool rather than something they ignore.

One way I've been using recently is through the use of the game site Kahoot. After a problem is done and shows the answer, students have already been notified if they got it right. I take time to explain how the problem should be done if most of the students miss it. It provides a great review.

Feedback needs to be specific on what they didn't do correctly. Rather than saying "Good job!" or "Almost there!" take time to say specifically what they did right and what they didn't get quite right. When they are learning a skill, they need the explicit information. Research also indicates students who receive immediate feedback are more likely to do better than those who receive delayed feedback.

In addition, instant feedback helps students improve their technique and increases their motivation to complete the work. Furthermore, if they are consistently getting the wrong answer because of a lack of immediate feedback, they feel as if math is too hard or they can't do it. It is quite reasonable to use a technology based game to provide instant feedback. I sometimes have students work on IXL which allows a certain number of free questions each day. If a student gets the answer wrong, the program shows them how to do it.

I also will play Jeopardy with my students. I divide them into groups and they must work together to get the answer. I must see everyone's attempt at the problem. Every group that gets the correct answer will receive the points. If I just gave it to the first correct answer, those who struggle the most will give up and shut down. I also give them several minutes per round so everyone has a chance to calculate the answer.

One activity I enjoy assigning students is "Search and Rescue" which is low tech but can be done using QR codes. The idea is that you place 10 questions around the room. Each question has an answer but the answer is with a different question. Students begin by calculating the answer to one question. Once they have the answer, they go looking around to find the paper with that answer. If they don't find the answer, they know they are wrong and can get help from me to learn to do it correctly. If they find the answer, they do the problem on the paper, find the answer and go find the answer. Eventually they will end up where they started. It often takes a full period but they have fun.

Another possible way to check for understanding so you can give immediate feedback is to provide a problem for students to solve about 5 minutes before the end of class. They write the answer on an exit ticket so you can glance at it and let them know if its right or if they need to try again.

These are just a few ways other than wandering around the classroom, checking student work to see if its right or wrong. Students often need a fun way to give them a break and make them willing to listen to the feedback.

Let me know what you think, I'd love to hear.

I also need to help students learn to use the feedback as a learning tool rather than something they ignore.

One way I've been using recently is through the use of the game site Kahoot. After a problem is done and shows the answer, students have already been notified if they got it right. I take time to explain how the problem should be done if most of the students miss it. It provides a great review.

Feedback needs to be specific on what they didn't do correctly. Rather than saying "Good job!" or "Almost there!" take time to say specifically what they did right and what they didn't get quite right. When they are learning a skill, they need the explicit information. Research also indicates students who receive immediate feedback are more likely to do better than those who receive delayed feedback.

In addition, instant feedback helps students improve their technique and increases their motivation to complete the work. Furthermore, if they are consistently getting the wrong answer because of a lack of immediate feedback, they feel as if math is too hard or they can't do it. It is quite reasonable to use a technology based game to provide instant feedback. I sometimes have students work on IXL which allows a certain number of free questions each day. If a student gets the answer wrong, the program shows them how to do it.

I also will play Jeopardy with my students. I divide them into groups and they must work together to get the answer. I must see everyone's attempt at the problem. Every group that gets the correct answer will receive the points. If I just gave it to the first correct answer, those who struggle the most will give up and shut down. I also give them several minutes per round so everyone has a chance to calculate the answer.

One activity I enjoy assigning students is "Search and Rescue" which is low tech but can be done using QR codes. The idea is that you place 10 questions around the room. Each question has an answer but the answer is with a different question. Students begin by calculating the answer to one question. Once they have the answer, they go looking around to find the paper with that answer. If they don't find the answer, they know they are wrong and can get help from me to learn to do it correctly. If they find the answer, they do the problem on the paper, find the answer and go find the answer. Eventually they will end up where they started. It often takes a full period but they have fun.

Another possible way to check for understanding so you can give immediate feedback is to provide a problem for students to solve about 5 minutes before the end of class. They write the answer on an exit ticket so you can glance at it and let them know if its right or if they need to try again.

These are just a few ways other than wandering around the classroom, checking student work to see if its right or wrong. Students often need a fun way to give them a break and make them willing to listen to the feedback.

Let me know what you think, I'd love to hear.

## Wednesday, December 13, 2017

### Grading Using Understanding and Not Points.

There is currently a move out there to eliminate point based grading and replace it with a mastery scale so students are being graded on their mastery of various standards or skills.

It is a radical change from the current grading system and actually has some aspects which I think are needed. Lets look at some of the ideas for this alternative form of grading.

1. All assignments must be completed to pass the class. Zeros are not given. Assignments may be repeated as often as necessary until they show a level of mastery.

2. Rather than using A to F, teachers use a more accurate mastery scale such as far below, below, proficient, or advanced to give a more accurate view of their understanding and ability.

3. Instead of dividing assignments into classwork, testing, quizzes, etc, students should be graded on specific skills such as applying all four operations to integers, finding the slope, etc.

4. Do not grade everything they do, only grade a final culminating assignment to determine if they mastered the skill. This gives them time to work on properly mastering the skill rather than focusing on the points. If they are not proficient, the work can be used to determine what the student needs scaffolding or reteaching in.

5. When students are not working towards accumulating enough points to get an A, they can spend more time focused on learning the material. It also means that a student who gets it quickly can prove they have mastered the skill and move on while another student can take a bit more time.

6. It has been suggested one use a four point scale to grade the work. 1 is below basic, 2 is basic, 3 is proficient or 4 is advanced which translates to 0 is didn't do it. 1 means that with help they managed some work or they sort of have an idea but not quite. A 2 means they can do simpler problems of the type taught. A 3 means they met the learning goal and learned what was taught while a 4 means the student can do more complex problems or they went beyond what was expected.

7. It is possible to convert these units to a 4.0 scale used by most schools. The 4.0 is a 100 percent or an A. So any average between a 3 and 4 is an A, while anything between a 2.5 and 2.99 is a B, a 2.0 to a 2.49 is a C. An average between a 1.0 and a 1.99 is a D while anything below a 1.0 is a F. I include this information because most schools still want the averages with letter grades.

To summarize, each learning goal gets one grade rather than each paper. Each student is graded on their proficiency for meeting the goal. In addition, it uses the most recent evidence of mastery rather than everything.

I'd love to hear what you think. I admit, I am leaning towards this for next semester because I get tired of seeing students refuse to do anything unless it gets a grade. It will also decrease the amount of grading I have to do for each of 6 periods. I think I'll be able to work with more students and provide better and more immediate feedback.

I'm interested in people's thoughts on this. Thank you ahead of time.

It is a radical change from the current grading system and actually has some aspects which I think are needed. Lets look at some of the ideas for this alternative form of grading.

1. All assignments must be completed to pass the class. Zeros are not given. Assignments may be repeated as often as necessary until they show a level of mastery.

2. Rather than using A to F, teachers use a more accurate mastery scale such as far below, below, proficient, or advanced to give a more accurate view of their understanding and ability.

3. Instead of dividing assignments into classwork, testing, quizzes, etc, students should be graded on specific skills such as applying all four operations to integers, finding the slope, etc.

4. Do not grade everything they do, only grade a final culminating assignment to determine if they mastered the skill. This gives them time to work on properly mastering the skill rather than focusing on the points. If they are not proficient, the work can be used to determine what the student needs scaffolding or reteaching in.

5. When students are not working towards accumulating enough points to get an A, they can spend more time focused on learning the material. It also means that a student who gets it quickly can prove they have mastered the skill and move on while another student can take a bit more time.

6. It has been suggested one use a four point scale to grade the work. 1 is below basic, 2 is basic, 3 is proficient or 4 is advanced which translates to 0 is didn't do it. 1 means that with help they managed some work or they sort of have an idea but not quite. A 2 means they can do simpler problems of the type taught. A 3 means they met the learning goal and learned what was taught while a 4 means the student can do more complex problems or they went beyond what was expected.

7. It is possible to convert these units to a 4.0 scale used by most schools. The 4.0 is a 100 percent or an A. So any average between a 3 and 4 is an A, while anything between a 2.5 and 2.99 is a B, a 2.0 to a 2.49 is a C. An average between a 1.0 and a 1.99 is a D while anything below a 1.0 is a F. I include this information because most schools still want the averages with letter grades.

To summarize, each learning goal gets one grade rather than each paper. Each student is graded on their proficiency for meeting the goal. In addition, it uses the most recent evidence of mastery rather than everything.

I'd love to hear what you think. I admit, I am leaning towards this for next semester because I get tired of seeing students refuse to do anything unless it gets a grade. It will also decrease the amount of grading I have to do for each of 6 periods. I think I'll be able to work with more students and provide better and more immediate feedback.

I'm interested in people's thoughts on this. Thank you ahead of time.

## Tuesday, December 12, 2017

### The Importance of Finding Mistakes.

My students hate looking back at any problems they made a mistake on. As far as they are concerned, they answered the problem and its done. Who cares if its correct. I don't spend enough time on helping them learn to identify mistakes and verbalize the reason for making the mistake.

One common error I constantly see is in using the distributive property, they only multiply the first term in parenthesis. If they remember to do something with the second term, they usually add the numbers rather than multiplying.

Unfortunately, my students have it ingrained in their heads that anything they do must be graded or its not worth anything. They do not want to correct mistakes because there are no points associated with the activity, therefore its worthless in their minds.

One of the first steps I have to do is convince them that it is important for them to find their basis of their misconception which causes them to make the same mistake over and over again. The important thing about making mistakes is that it is part of the learning process. To identify the cause of the error is a step to gaining understanding and deeper learning.

I don't do this enough but I plan to do more of this activity next semester. During warm-ups, I plan to put a problem with mistakes and ask students to identify what was done incorrectly and how should the problem have been done. This way, they get practice identifying mistakes and correcting them.

Here are some reasons why making mistakes is a good thing and what they help you learn.

1. Making mistakes show everyone you tried. The student may not fully understand how to do it but when they don't try, there is no chance of learning.

2. Trying the problems help exercise their brain because the brain likes solving challenging problems.

3. When they understand why certain mistakes occur, it means that they are able to do that type of problem using a deeper understanding.

4. When a student finds the error and is able to correct it, experiences personal satisfaction. It also helps build the persistence necessary to work through more complex problems. It also helps them increase their motivation to learn.

5. Teachers can mistakes as a form of assessment because analyzing the type of mistake can give the teacher information for scaffolding or reteaching a topic if the majority of students are making the same error. Mistakes are multifaceted and offer so many possibilities for assessment.

6. Create an environment where it is acceptable to make mistakes and where students are not afraid to make them. Make them aware that it is fine to make a mistake but once the mistake is identified, they need to continue working until they have a correct answer.

7. Provide timely feedback on the mistake. As part of the process, it is important to determine if the mistake is a misconception, systematic error, or careless mistake so students know what to look for the next time they work a problem.

8. It is equally important to provide students the chance to correct mistakes on their own. By looking to find the root cause of the mistake, it helps develop conceptual understanding.

9. Use technology that helps support mistakes by providing immediate feedback. I've discovered I need to train my students to look at the feedback, otherwise they ignore it and move on.

These are some reasons students need to be allowed to make mistakes and to make corrections including a short comment on why the mistake was made.

Let me know what you think.

One common error I constantly see is in using the distributive property, they only multiply the first term in parenthesis. If they remember to do something with the second term, they usually add the numbers rather than multiplying.

Unfortunately, my students have it ingrained in their heads that anything they do must be graded or its not worth anything. They do not want to correct mistakes because there are no points associated with the activity, therefore its worthless in their minds.

One of the first steps I have to do is convince them that it is important for them to find their basis of their misconception which causes them to make the same mistake over and over again. The important thing about making mistakes is that it is part of the learning process. To identify the cause of the error is a step to gaining understanding and deeper learning.

I don't do this enough but I plan to do more of this activity next semester. During warm-ups, I plan to put a problem with mistakes and ask students to identify what was done incorrectly and how should the problem have been done. This way, they get practice identifying mistakes and correcting them.

Here are some reasons why making mistakes is a good thing and what they help you learn.

1. Making mistakes show everyone you tried. The student may not fully understand how to do it but when they don't try, there is no chance of learning.

2. Trying the problems help exercise their brain because the brain likes solving challenging problems.

3. When they understand why certain mistakes occur, it means that they are able to do that type of problem using a deeper understanding.

4. When a student finds the error and is able to correct it, experiences personal satisfaction. It also helps build the persistence necessary to work through more complex problems. It also helps them increase their motivation to learn.

5. Teachers can mistakes as a form of assessment because analyzing the type of mistake can give the teacher information for scaffolding or reteaching a topic if the majority of students are making the same error. Mistakes are multifaceted and offer so many possibilities for assessment.

6. Create an environment where it is acceptable to make mistakes and where students are not afraid to make them. Make them aware that it is fine to make a mistake but once the mistake is identified, they need to continue working until they have a correct answer.

7. Provide timely feedback on the mistake. As part of the process, it is important to determine if the mistake is a misconception, systematic error, or careless mistake so students know what to look for the next time they work a problem.

8. It is equally important to provide students the chance to correct mistakes on their own. By looking to find the root cause of the mistake, it helps develop conceptual understanding.

9. Use technology that helps support mistakes by providing immediate feedback. I've discovered I need to train my students to look at the feedback, otherwise they ignore it and move on.

These are some reasons students need to be allowed to make mistakes and to make corrections including a short comment on why the mistake was made.

Let me know what you think.

## Monday, December 11, 2017

### Chunking When Reading.

Most of us when we were in school, never really learned to read a textbook. The only thing I learned was to read the questions at the end of the section before reading so I could find the answer. This technique does not work as well in mathematics.

What I learned later in life after leaving college and working for a few years were things I wished I'd known about then. Over time, its become clear that chunking material, even in a textbook is important.

Too many of our students have a limited attention span and are not willing to read through the whole textbook. If a student reads below grade level or is an English Language Learner, they find the assignment to be difficult but if we teach them to chunk the material, it can make it easier. In addition, these are things everyone should know.

Chunking by its nature, breaks the material down into smaller pieces so students are able to concentrate on one block at a time. Here are a few suggestions to make chunking easier.

1. Break the reading down into paragraphs or sentences. Have the student read one section and then record the important point or points in a graphic organizer you prepare in advance. At least until they've learned what information should go into the graphic organizer.

2. Before you have them paraphrase the text review specific decoding techniques so they have some tools at their hands. I'd make copies of the text so they can write on the copy rather than the book.

Techniques include, circling unfamiliar words, use context clues to determine their meaning before looking them up, replace the words with synonyms to make it easier to read, underline important information, read aloud, and read multiple times.

3. Begin by you deciding how the material should be chunked so as to provide modeling for the students. Over time, have them do more and more so eventually they can do it on their own.

4. Take time to teach students how to paraphrase the text. Many of my students think that writing it in their own words mean they should copy it but change one or two words.

5. Have students compare their paraphrased text to see similarities and differences.

Variations on the above activity:

1. First step have students identify and define key words in their own words.

2. Create a visual with pictures or symbols. Let them choose some visual that helps them remember.

3. To help students become proficient at paraphrasing, give them 10 words or less to summarize the material.

4. Ask them if they see connections to anything they've learned before so they call on prior learning.

5. Break the text into parts and use the jigsaw strategy to assign parts to different groups so they can share back the main information.

The idea behind chunking is that with the information broken into smaller pieces, the brain finds it easier to learn and remember it.

Let me know what you think. I'd love to hear. Have a good day.

What I learned later in life after leaving college and working for a few years were things I wished I'd known about then. Over time, its become clear that chunking material, even in a textbook is important.

Too many of our students have a limited attention span and are not willing to read through the whole textbook. If a student reads below grade level or is an English Language Learner, they find the assignment to be difficult but if we teach them to chunk the material, it can make it easier. In addition, these are things everyone should know.

Chunking by its nature, breaks the material down into smaller pieces so students are able to concentrate on one block at a time. Here are a few suggestions to make chunking easier.

1. Break the reading down into paragraphs or sentences. Have the student read one section and then record the important point or points in a graphic organizer you prepare in advance. At least until they've learned what information should go into the graphic organizer.

2. Before you have them paraphrase the text review specific decoding techniques so they have some tools at their hands. I'd make copies of the text so they can write on the copy rather than the book.

Techniques include, circling unfamiliar words, use context clues to determine their meaning before looking them up, replace the words with synonyms to make it easier to read, underline important information, read aloud, and read multiple times.

3. Begin by you deciding how the material should be chunked so as to provide modeling for the students. Over time, have them do more and more so eventually they can do it on their own.

4. Take time to teach students how to paraphrase the text. Many of my students think that writing it in their own words mean they should copy it but change one or two words.

5. Have students compare their paraphrased text to see similarities and differences.

Variations on the above activity:

1. First step have students identify and define key words in their own words.

2. Create a visual with pictures or symbols. Let them choose some visual that helps them remember.

3. To help students become proficient at paraphrasing, give them 10 words or less to summarize the material.

4. Ask them if they see connections to anything they've learned before so they call on prior learning.

5. Break the text into parts and use the jigsaw strategy to assign parts to different groups so they can share back the main information.

The idea behind chunking is that with the information broken into smaller pieces, the brain finds it easier to learn and remember it.

Let me know what you think. I'd love to hear. Have a good day.

## Sunday, December 10, 2017

## Saturday, December 9, 2017

## Friday, December 8, 2017

### Magic Squares

Magic squares are those wonderful mathematical constructions where the sums vertically, horizontally, and diagonally are all the same.

There is a lot of math behind creating a 3 x 3 magic square. One way is to place the basic number in the center square but it should be greater than four because the numbers should be positive.

The grid to the right shows the mathematics involved in creating a 3 by 3 square. The lowest number n equals is 5 so none of the squares are zero.

In addition, if you want larger numbers you simply have to add a value to n for all squares so the square remains a magic square.

For a 4 x 4 magic square, its not that difficult. Begin by creating a 4 x 4 square. Starting at the upper left most square, write one, then count each square till you get to four which should be at the upper right corner. Continue counting and placing numbers which fall along the diagonal. This means you have used the numbers, 1, 4, 6, 7, 10, 11, 13, and 16. Then start at the bottom and work upwards, filling in the unused numbers so 2 is next to 16 and 3 is next to 13. 5 is above the 16 and 9 is between the 5 and the 4 while the 8 is above the 13 and the 12 between the 8 and the 1. That leaves the 15 in the first row next to the 1 and 14 in the top row next to the four. So it all shows up as

1, 15, 14, 4

12, 6, 7, 9

8, 10, 11, 5

13, 3, 2, 16

The cool thing is that you can move the rows or columns around so the digits are in the same order in the rows and the columns in order to get the same squares.

Have fun letting your students attempt to find the method behind the 3 x 3 squares first before showing them the mathematics. Then let them move on to the 4 x 4 square.

Let me know what you think. I'd love to hear. Have a great weekend.

There is a lot of math behind creating a 3 x 3 magic square. One way is to place the basic number in the center square but it should be greater than four because the numbers should be positive.

The grid to the right shows the mathematics involved in creating a 3 by 3 square. The lowest number n equals is 5 so none of the squares are zero.

In addition, if you want larger numbers you simply have to add a value to n for all squares so the square remains a magic square.

For a 4 x 4 magic square, its not that difficult. Begin by creating a 4 x 4 square. Starting at the upper left most square, write one, then count each square till you get to four which should be at the upper right corner. Continue counting and placing numbers which fall along the diagonal. This means you have used the numbers, 1, 4, 6, 7, 10, 11, 13, and 16. Then start at the bottom and work upwards, filling in the unused numbers so 2 is next to 16 and 3 is next to 13. 5 is above the 16 and 9 is between the 5 and the 4 while the 8 is above the 13 and the 12 between the 8 and the 1. That leaves the 15 in the first row next to the 1 and 14 in the top row next to the four. So it all shows up as

1, 15, 14, 4

12, 6, 7, 9

8, 10, 11, 5

13, 3, 2, 16

The cool thing is that you can move the rows or columns around so the digits are in the same order in the rows and the columns in order to get the same squares.

Have fun letting your students attempt to find the method behind the 3 x 3 squares first before showing them the mathematics. Then let them move on to the 4 x 4 square.

Let me know what you think. I'd love to hear. Have a great weekend.

## Thursday, December 7, 2017

### Mind Mapping in Math

As you know a mind map is a visual representation of information going from the big topic to smaller topics. Its a visual structuring of ideas, easily read.

Since its a type of graphical organizer, it can help clarify material for students. Mind maps can be in the traditional form or in more of a sketch note form.

If you choose to have them create a mind map by hand, you need to provide students a large blank paper placed landscape. To begin, they need to place the topic in the center inside a box. Then write the main ideas or aspects around the topic and link them by lines. Then expand these branches into sub branches much like you see on a tree. Be sure to have them use color, numbers, and arrows to show relationships and make things stand out.

If you are not sure what a mind map for math should look like, check out this presentation because it has so many different styles and mathematical topics so you can share them with your students to help them learn to create their own.

The other way is to send students to either an online site or an app to create a more linear mind map. The process is the same only its done using items reminiscent of flow charts from the early days of computer programming. Again the presentation has at least one done using technology so your students know what that looks like.

In addition, mind mapping can be used by students to help organize their brainstorming when they need to solve a problem or a performance task. The way to do this is to place the problem in the center but don't make the problem too narrow. Next is to use the IDEAL strategy to solve it.

IDEAL stands for:

I - identify the problem

D - define and represent the problem

E - explore possible strategies or solutions

A - act on selected strategy or solution

L - look back and evaluate.

No matter whether you have the students use mind maps as a way of putting the material in perspective or for solving problems, it is important we teach them how to do it because a student who knows how to use one in English may have difficulty applying the same technique to math. We should not assume they can look at an example and figure it out themselves. We must provide guided instruction to help them learn.

Let me know what you think. I'd love to hear.

second have them create one using either an online site or an app on the iPad.

Since its a type of graphical organizer, it can help clarify material for students. Mind maps can be in the traditional form or in more of a sketch note form.

If you choose to have them create a mind map by hand, you need to provide students a large blank paper placed landscape. To begin, they need to place the topic in the center inside a box. Then write the main ideas or aspects around the topic and link them by lines. Then expand these branches into sub branches much like you see on a tree. Be sure to have them use color, numbers, and arrows to show relationships and make things stand out.

If you are not sure what a mind map for math should look like, check out this presentation because it has so many different styles and mathematical topics so you can share them with your students to help them learn to create their own.

The other way is to send students to either an online site or an app to create a more linear mind map. The process is the same only its done using items reminiscent of flow charts from the early days of computer programming. Again the presentation has at least one done using technology so your students know what that looks like.

In addition, mind mapping can be used by students to help organize their brainstorming when they need to solve a problem or a performance task. The way to do this is to place the problem in the center but don't make the problem too narrow. Next is to use the IDEAL strategy to solve it.

IDEAL stands for:

I - identify the problem

D - define and represent the problem

E - explore possible strategies or solutions

A - act on selected strategy or solution

L - look back and evaluate.

No matter whether you have the students use mind maps as a way of putting the material in perspective or for solving problems, it is important we teach them how to do it because a student who knows how to use one in English may have difficulty applying the same technique to math. We should not assume they can look at an example and figure it out themselves. We must provide guided instruction to help them learn.

Let me know what you think. I'd love to hear.

second have them create one using either an online site or an app on the iPad.

## Wednesday, December 6, 2017

### The Math of Car Insurance.

If you own a car, you have to pay insurance on it. If you drive the family car because you do not own your own car, someone has to pay for the insurance.

Have you wondered how the cost of car insurance is calculated? Have you ever questioned why the cost is not the same across the country? Have you wondered why girls are not charged a premium when they turn 21 while boys have to pay the premium till they are 25?

Calculating premiums for car insurance is extremely complex because they take into account potential loss based on type of car, zip code, location of where car is parked, how far its driven every day, driving history including DWI's, tickets, accidents, and other things. In addition, each insurance company uses slightly different factors to calculate rates and all rates are approved by the state.

This activity, from Canada, explains about the terms involved in car insurance and has students predict which items can effect the cost of their insurance. It lists things like type of car, house, job, so students have to think and decide. At the end, the teacher can go through to discuss all the factors.

This power point presentation covers all sorts of insurance but has an excellent section on using tables to calculate the cost of insurance and goes into detail explaining things like 10/30/25. Although the information may be a bit out of date, it gives a good feel on figuring that out. The slides walks students through the process of calculating premiums.

Both of the above are actual lesson plans designed to help the student learn more about calculating car insurance premiums but this paper is more scientifically based. It looks at using the generalized linear models used to calculate auto insurance premiums based on the factors of the insured. This paper actually shows the mathematics involved which is actually more complex than most high school students can do but its nice to show them.

One last resource is a unit from the state of Nebraska which lists a ton of resources to help with the unit. It looks at premiums, types of coverage, calculating premiums for mandatory coverage, calculating premiums for optional insurance, and a project. Admitted, the information is from Nebraska but the process is the same everywhere.

One of the homework assignments that could be done is to contact an insurance agent to get a current quote to share with the class.

Let me know what you think. Have a great day.

Have you wondered how the cost of car insurance is calculated? Have you ever questioned why the cost is not the same across the country? Have you wondered why girls are not charged a premium when they turn 21 while boys have to pay the premium till they are 25?

Calculating premiums for car insurance is extremely complex because they take into account potential loss based on type of car, zip code, location of where car is parked, how far its driven every day, driving history including DWI's, tickets, accidents, and other things. In addition, each insurance company uses slightly different factors to calculate rates and all rates are approved by the state.

This activity, from Canada, explains about the terms involved in car insurance and has students predict which items can effect the cost of their insurance. It lists things like type of car, house, job, so students have to think and decide. At the end, the teacher can go through to discuss all the factors.

This power point presentation covers all sorts of insurance but has an excellent section on using tables to calculate the cost of insurance and goes into detail explaining things like 10/30/25. Although the information may be a bit out of date, it gives a good feel on figuring that out. The slides walks students through the process of calculating premiums.

Both of the above are actual lesson plans designed to help the student learn more about calculating car insurance premiums but this paper is more scientifically based. It looks at using the generalized linear models used to calculate auto insurance premiums based on the factors of the insured. This paper actually shows the mathematics involved which is actually more complex than most high school students can do but its nice to show them.

One last resource is a unit from the state of Nebraska which lists a ton of resources to help with the unit. It looks at premiums, types of coverage, calculating premiums for mandatory coverage, calculating premiums for optional insurance, and a project. Admitted, the information is from Nebraska but the process is the same everywhere.

One of the homework assignments that could be done is to contact an insurance agent to get a current quote to share with the class.

Let me know what you think. Have a great day.

## Tuesday, December 5, 2017

### The Math Of Roller Coasters.

The other day my students were discussing the seniors and if they'd be going on a trip. In past years they've gone to Hawaii or to Los Angeles. The last group who went to Los Angeles visited six flags and did a bit of sight seeing.

The nice thing about roller coasters is you can find them as huge creatures or a more portable one that although small is nice.

The other thing is these amusement park landmarks require a lot of math. The designer needs to know the maximum velocity a train can go while still thrilling people but not hurting them to the maximum height determined by the building materials. You can build a higher one with steel but it might be cheaper using wood. On the other hand, if you go for height you have to decide if you want a launch coaster or a crank coaster. If the coaster is designed to be too extreme, the designer might put a loop in the middle. This site has a great introduction on the topic.

Just from the introduction, its possible to see that it isn't only the shape of the ride but way more factors. In addition, information and projects can be found for almost any level of mathematics. The Mathematical Association of America has a wonderful project on designing roller coasters using derivatives from Calculus. The project is designed to have students work their way through several modules from start to finish and has everything needed.

NCTM has a nice unit on this topic which begins with students applying a standard equation to a coaster to determine the height of the coaster at a certain point in time. The results are used to find average velocity. The second activity has students matching results with possible graphs so they have to turn the initial results into ordered pairs. The final activity has them take their new knowledge to design their own coaster.

They have a second activity on coasters in which they compare several roller coasters from around the country based on pictures from the web. Once they've made predictions on fastest, highest, etc, they go to another site to check their predictions. They repeat this activity a couple of times working to improve their extimates.

The futures channels has a lovely technology based activity which begins with the student being given and equation to work on either a graphing calculator or on a spread sheet. The activity has students change various coefficients on the 6th degree polynomial and record their results. Most activities us a second degree equation so this is more complex.

For the upper level math classes Mathspace has an activity which has students choosing the coefficients for a series of polynomials. They are expected to identify domain, range, zeros, max and min, increasing and decreasing, zeros and other things after graphing each of the four functions. The nice thing about this activity is that it can be done without a knowledge of Calculus so Algebra classes can do it.

RAFT or Resource Area For Teaching has a lovely final hands on project to end this by having students build their own roller coaster using slope, rates, ratios, velocity, and speed while working in groups of four. Students will carry out experiments once the roller coaster is completed.

I think I'm going to have my Algebra II class use the activity from Mathspace to finish out the semester. I think they'll like knowing how to use the graph information from this unit. Let me know what you think. I'd love to hear.

The nice thing about roller coasters is you can find them as huge creatures or a more portable one that although small is nice.

The other thing is these amusement park landmarks require a lot of math. The designer needs to know the maximum velocity a train can go while still thrilling people but not hurting them to the maximum height determined by the building materials. You can build a higher one with steel but it might be cheaper using wood. On the other hand, if you go for height you have to decide if you want a launch coaster or a crank coaster. If the coaster is designed to be too extreme, the designer might put a loop in the middle. This site has a great introduction on the topic.

Just from the introduction, its possible to see that it isn't only the shape of the ride but way more factors. In addition, information and projects can be found for almost any level of mathematics. The Mathematical Association of America has a wonderful project on designing roller coasters using derivatives from Calculus. The project is designed to have students work their way through several modules from start to finish and has everything needed.

NCTM has a nice unit on this topic which begins with students applying a standard equation to a coaster to determine the height of the coaster at a certain point in time. The results are used to find average velocity. The second activity has students matching results with possible graphs so they have to turn the initial results into ordered pairs. The final activity has them take their new knowledge to design their own coaster.

They have a second activity on coasters in which they compare several roller coasters from around the country based on pictures from the web. Once they've made predictions on fastest, highest, etc, they go to another site to check their predictions. They repeat this activity a couple of times working to improve their extimates.

The futures channels has a lovely technology based activity which begins with the student being given and equation to work on either a graphing calculator or on a spread sheet. The activity has students change various coefficients on the 6th degree polynomial and record their results. Most activities us a second degree equation so this is more complex.

For the upper level math classes Mathspace has an activity which has students choosing the coefficients for a series of polynomials. They are expected to identify domain, range, zeros, max and min, increasing and decreasing, zeros and other things after graphing each of the four functions. The nice thing about this activity is that it can be done without a knowledge of Calculus so Algebra classes can do it.

RAFT or Resource Area For Teaching has a lovely final hands on project to end this by having students build their own roller coaster using slope, rates, ratios, velocity, and speed while working in groups of four. Students will carry out experiments once the roller coaster is completed.

I think I'm going to have my Algebra II class use the activity from Mathspace to finish out the semester. I think they'll like knowing how to use the graph information from this unit. Let me know what you think. I'd love to hear.

## Monday, December 4, 2017

### The 12 Days of Christmas.

Its the Christmas season again. I often see news pieces about how much these gifts would cost now. Honestly, I'm not sure where I'd find some of the gifts because one does not find partridges in Alaska unless its at the zoo.

We do have cows for milking but no calling birds. I've always wondered how they determined the cost of the gifts. Do they count one of each day or do they count it so you give 12 patridges in 12 pear trees, with eleven groups of two turtle doves, etc.

Fortunately, there are several lessons on the internet to help students learn to calculate the cost. I did see a quote for somewhere between $35,000 and $250,000 determined by the unit or by the group.

Cpalms has a nice one designed for upper elementary children complete with items and prices. Although it is a worksheet, it has lots of pictures and requires students to add a sales tax at the end. Another site provides individual costs so students have a starting point.

Many math teachers still prefer giving out worksheets which is fine but this would be a great exercise to use technology with. Imagine having the students create a movie or documentary? Perhaps they could create a interactive book? What about a collage for each day? The possibilities are endless.

I can see taking the worksheet from the Cpalms activity and using the information in a news report video so each day, the newscaster discusses the ongoing cost of the gift. I could see some of my more musically inclined students creating a music video or those who are not great artists might create a book.

This is a good group project because it would allow students with different talents to collaborate to create a final project. The rules for creating the project should be gone through in advance with the rubric set up so everyone is graded for their contributions.

When I was in school, we had group projects but only the people who were concerned about grades actually did the work. Everyone got the same grade regardless of their contribution. That is why its important to set the rubric so everyone is graded on their contribution and on the final project. I've heard of teachers giving two grades for any project. One for the individual contribution to the project, while the other is for the completed project.

i know many folks discourage the use of worksheets with digital devices but I see them as a starting point such as with the 12 days of Christmas. The sheets provide prices for the projects. Its always good to keep in mind what the worksheet is for.

Note, there are a couple Canadian versions, an Australian version, and a Hawaiian version which might work better in certain areas such as Alaska because we can find the moose, caribou, and dog sleds way easier here.

One more thing. The 12 days of Christmas actually runs from December 25th to January 6th but since school is seldom in session then, most teachers count the 12 days prior as being associated with the song.

I hope you enjoyed this. Let me know what you think.

We do have cows for milking but no calling birds. I've always wondered how they determined the cost of the gifts. Do they count one of each day or do they count it so you give 12 patridges in 12 pear trees, with eleven groups of two turtle doves, etc.

Fortunately, there are several lessons on the internet to help students learn to calculate the cost. I did see a quote for somewhere between $35,000 and $250,000 determined by the unit or by the group.

Cpalms has a nice one designed for upper elementary children complete with items and prices. Although it is a worksheet, it has lots of pictures and requires students to add a sales tax at the end. Another site provides individual costs so students have a starting point.

Many math teachers still prefer giving out worksheets which is fine but this would be a great exercise to use technology with. Imagine having the students create a movie or documentary? Perhaps they could create a interactive book? What about a collage for each day? The possibilities are endless.

I can see taking the worksheet from the Cpalms activity and using the information in a news report video so each day, the newscaster discusses the ongoing cost of the gift. I could see some of my more musically inclined students creating a music video or those who are not great artists might create a book.

This is a good group project because it would allow students with different talents to collaborate to create a final project. The rules for creating the project should be gone through in advance with the rubric set up so everyone is graded for their contributions.

When I was in school, we had group projects but only the people who were concerned about grades actually did the work. Everyone got the same grade regardless of their contribution. That is why its important to set the rubric so everyone is graded on their contribution and on the final project. I've heard of teachers giving two grades for any project. One for the individual contribution to the project, while the other is for the completed project.

i know many folks discourage the use of worksheets with digital devices but I see them as a starting point such as with the 12 days of Christmas. The sheets provide prices for the projects. Its always good to keep in mind what the worksheet is for.

Note, there are a couple Canadian versions, an Australian version, and a Hawaiian version which might work better in certain areas such as Alaska because we can find the moose, caribou, and dog sleds way easier here.

One more thing. The 12 days of Christmas actually runs from December 25th to January 6th but since school is seldom in session then, most teachers count the 12 days prior as being associated with the song.

I hope you enjoyed this. Let me know what you think.

## Sunday, December 3, 2017

## Saturday, December 2, 2017

## Friday, December 1, 2017

### Scaffolding and Justifications.

As you know most of my students are classified ELL (English Language Learners) who often have trouble explaining their thinking process in regard to answering questions.

For the majority of students, we have to teach them how to solve performance tasks. Unfortunately, it is a fine line between telling them every step vs scaffolding the process so they learn to do it on their own.

It is also important to provide opportunities for students to justify their thinking when solving an open ended performance task. The idea is to have students do the same task but focusing on a different aspect so they learn to justify at each step.

The task in this case might be exploring how scaling the sides of a two dimensional figure changes the area of the new figure. The first time through, students should focus on patterns between the scaling sides and the resulting area. They are expected to explore the relationship using numerical observations. it is important to ask for justification at this point so students can express their thoughts and understanding of the patterns. Setting up a table for students to record their findings like they do in science provides a way to organize the data.

If they start with a square and move on to doing the same activity with rectangles, circles, triangles and trapezoids it allows students to generalize their findings to other figures. The next time through, students are expected to explain why the relationship is true using algebraic and geometric representations rather than the numerical. The students are again asked to justify what they discovered.

The things to keep in mind when scaffolding a task includes designing it so students focus on one facet of it in detail before connecting it to a more generalized view. Next its important to ask students to justify their reasoning early and often. Furthermore, students should be prompted to create multiple representations of their thinking.

In addition, it is important to show how to use different strategies because the use of different strategies can increase student understanding. Take time to help students understand that there is a difference between identifying a pattern and understanding why it works.

This meets the idea that as students learn to explain and justify their thinking, they have taken steps towards being in charge of their own learning. In addition, it is necessary to create learning activities based on prior knowledge or adjust help tailored to student need as expressed above by focusing on one aspect, then moving towards the more generalized math.

These two activities show the interrelations between concepts and it is something that should be done when scaffolding learning. It is important to do this so students can see that all of mathematics is related.

Let me know what you are thinking. I'd love to hear. Have a great weekend.

For the majority of students, we have to teach them how to solve performance tasks. Unfortunately, it is a fine line between telling them every step vs scaffolding the process so they learn to do it on their own.

It is also important to provide opportunities for students to justify their thinking when solving an open ended performance task. The idea is to have students do the same task but focusing on a different aspect so they learn to justify at each step.

The task in this case might be exploring how scaling the sides of a two dimensional figure changes the area of the new figure. The first time through, students should focus on patterns between the scaling sides and the resulting area. They are expected to explore the relationship using numerical observations. it is important to ask for justification at this point so students can express their thoughts and understanding of the patterns. Setting up a table for students to record their findings like they do in science provides a way to organize the data.

If they start with a square and move on to doing the same activity with rectangles, circles, triangles and trapezoids it allows students to generalize their findings to other figures. The next time through, students are expected to explain why the relationship is true using algebraic and geometric representations rather than the numerical. The students are again asked to justify what they discovered.

The things to keep in mind when scaffolding a task includes designing it so students focus on one facet of it in detail before connecting it to a more generalized view. Next its important to ask students to justify their reasoning early and often. Furthermore, students should be prompted to create multiple representations of their thinking.

In addition, it is important to show how to use different strategies because the use of different strategies can increase student understanding. Take time to help students understand that there is a difference between identifying a pattern and understanding why it works.

This meets the idea that as students learn to explain and justify their thinking, they have taken steps towards being in charge of their own learning. In addition, it is necessary to create learning activities based on prior knowledge or adjust help tailored to student need as expressed above by focusing on one aspect, then moving towards the more generalized math.

These two activities show the interrelations between concepts and it is something that should be done when scaffolding learning. It is important to do this so students can see that all of mathematics is related.

Let me know what you are thinking. I'd love to hear. Have a great weekend.

## Thursday, November 30, 2017

### Flipgrid in the Math Classroom.

In case you haven't heard of Flipgrid, it's a video discussion app/program that allows the teacher to post a topic and students to respond via video. I've seen it used in a class I took but it was not a math class.

I'm always interested in finding ways my students can work on using their mathematical vocabulary and improving their ways of explaining or justifying their work. Flipgrid has a wonderful pdf filled with ideas for grades kindergarten to seniors.

Some ways to use Flipgrid in the math classroom are:

1. Have students explain how they solved the weekly problem and providing the answer. They could also explain how they solved any problem out of a list of possible problems.

2. Allow students to post problems for other students to solve.

3. Post a find the mistake picture and have students discuss where the mistake was made.

4. Post a would you rather math problem and have them discuss their choice and justification.

5. Create a math help line where students post questions and others answer those questions.

6. Ask students to compare and contrast mathematical ideas such as adding and subtracting, rational and irrational numbers, linear and exponential models.

7. Have students develop word problems based on topics that matter to them.

8. Ask students to take current events and turn them into word problems.

9. Have students collaborate on data collecting problems.

10. Ask students to provide current contexts on positive and negative numbers, fractions or any other math topic. Or they could model solutions to determine the amount of emergency supplies needed after being hit by a hurricane.

11. Have students research a mathematician and share the information via Flipgrid.

12. Have students report back on their thoughts after completing an activity either at Geogebra.com or Desmos.com.

Flipgrid provides an opportunity for students to develop their ability to discuss mathematical topics while doing it in a way they are comfortable. My students love taking selfies of themselves, record videos, create snapchats and Flipgrid uses this to become part of the classroom.

Let me know what you think. I'd love to hear. Thank you for reading. Have a good day.

## Wednesday, November 29, 2017

### The Mathematics of Pyramids.

The cool things about pyramids is that they are not found only in Egypt anymore. You find them in Las Vegas, and other places around the world. The even cooler thing about this topic is that you can use Google to bring up actual pictures to give students a better feel for the actual size.

Photographs are nice but they are usually taken from a distance so you get the whole picture but when viewed using google, you get a much better feel for its size.

As you know, a pyramid is basically a square with four triangular sides also known as a square pyramids. The most famous one is the Pyramid at Giza but others include The Nima Sand Museum in Japan which is a cluster of 6 different pyramids, the tallest of which is over 69 feet tall with a 56 foot base. The tallest one was built to house the worlds most functional sand timer measuring a year span and is flipped every year at midnight of the last day of the year.

In addition, the Louvre has three pyramids, the tallest of which is 71 feet tall with a 115 foot base. It opened in 1991 and provides an interesting juxtaposition to the older architectural style of the Louvre itself. Long Beach University has its Walter Pyramid, a monument to athletics as its 18 stories has seating for 4,500 but can hold up to 7,000. In addition, there are two other pyramids in Russia, one designed for a religious gathering while the other is for entertainment.

We all know the the formula for the volume of a pyramid is 1/3 x base x height where as the formula for the surface area of a pyramid with regular sides is base area + 1/2 x perimeter x slant length. In regard to the Pyramid at Giza has some fascinating math associated with it.

First is the golden ratio (Phi) which is the the only mathematical number whose square is one more than its original number. Its the number found in nature, its ratio being pleasant to the human eye. Apply the Pythagorean theorem to these numbers and you get the values which are similar to the golden triangle.

Another idea is that the pyramid could have been based on Pi because the theoretical numbers are within 0.1% different from the Great Pyramid. Either way the mathematics involved in these claims is fascinating.

Just think with a little work, students could design a Google Tour focusing on these and other pyramid shaped buildings including their volume, surface area and other pieces of information. They could include a map with pins for each site. So many possibilities. In February, I'll be giving a talk on integrating Google into the math classroom, so this would be perfect as an example.

When I get this project done over Christmas break, I'll share it with you . Have a great day and let me know what you think.

Photographs are nice but they are usually taken from a distance so you get the whole picture but when viewed using google, you get a much better feel for its size.

As you know, a pyramid is basically a square with four triangular sides also known as a square pyramids. The most famous one is the Pyramid at Giza but others include The Nima Sand Museum in Japan which is a cluster of 6 different pyramids, the tallest of which is over 69 feet tall with a 56 foot base. The tallest one was built to house the worlds most functional sand timer measuring a year span and is flipped every year at midnight of the last day of the year.

In addition, the Louvre has three pyramids, the tallest of which is 71 feet tall with a 115 foot base. It opened in 1991 and provides an interesting juxtaposition to the older architectural style of the Louvre itself. Long Beach University has its Walter Pyramid, a monument to athletics as its 18 stories has seating for 4,500 but can hold up to 7,000. In addition, there are two other pyramids in Russia, one designed for a religious gathering while the other is for entertainment.

We all know the the formula for the volume of a pyramid is 1/3 x base x height where as the formula for the surface area of a pyramid with regular sides is base area + 1/2 x perimeter x slant length. In regard to the Pyramid at Giza has some fascinating math associated with it.

First is the golden ratio (Phi) which is the the only mathematical number whose square is one more than its original number. Its the number found in nature, its ratio being pleasant to the human eye. Apply the Pythagorean theorem to these numbers and you get the values which are similar to the golden triangle.

Another idea is that the pyramid could have been based on Pi because the theoretical numbers are within 0.1% different from the Great Pyramid. Either way the mathematics involved in these claims is fascinating.

Just think with a little work, students could design a Google Tour focusing on these and other pyramid shaped buildings including their volume, surface area and other pieces of information. They could include a map with pins for each site. So many possibilities. In February, I'll be giving a talk on integrating Google into the math classroom, so this would be perfect as an example.

When I get this project done over Christmas break, I'll share it with you . Have a great day and let me know what you think.

## Tuesday, November 28, 2017

### The Mathematics Of A Pyramid Scheme.

As math teachers we spend so much time explaining the basics of mathematics as determined by the curriculum but what if one day, we took a break and explained the mathematics behind various money making schemes such as the pyramid scheme or a Ponzi scheme.

I've occasionally had students come in with letters or emails from lawyers or countries claiming they were due to inherit a ton of money or they won some sweepstakes they had not entered. Those are easy to expose but the other money making schemes are harder but they still make a great topic in math classes.

It turns out the pyramid scheme area covers several types of scams including one that is not as common as it used to be. The chain letter, the one where you'd get a letter from a friend or relative stating if you send this out to x number of other people, you'd get so much money in a certain amount of time. Each level brings in twice as many as the previous level, if two people send back money.

Another more common one is the modified 8 ball model in which a person does not get paid until they have recruited enough people to have established several levels. The idea is that you recruit two people, who recruit two more each who then recruit two more people each so there are three layers with 8 people involved. At this point, their participation fee is given to the first person. So if the fee is $1000, you just made $8000.

Both schemes use a geometric progression to grow but there is a fine line between legal and illegal in that the people who join have to get merchandise equal to their investment. The actual math formula is well stated here but it boils down to about 88% of the people will loose their money because those past a certain level.

A new scheme is the two up model. The sales earnings from the first two people you recruited goes to the person who recruited you. It isn't until the third person, you start making a profit. This involves a geometric progression of three times rather than the two of the previous two. Unfortunately, those at the very bottom usually do not make money. In fact about 67% of the people involved in this one, do not make any money.

The formula used to calculate the above averages is the geometric series formula from calculus.

Now for the Ponzi Scheme. This is the people are more likely to be familiar with due to some of those shows that focus on con men who earn millions of dollars before getting caught. The most well known is Bernard L. Madoff who conned people out of over $50 million dollars but it was named after Carlo Ponzi the originator of the scheme. In 1920, he collected almost $10 million from 10,550 people but only paid out about $8 million.

The idea behind the Ponzi scheme is that you play upon the greed of people by offering extremely high rates of return which cannot be met so you take the money from later investors and give it to the earlier ones. The mathematics is rather complex but shown here in wonderful detail.

Lets just say several calculus equations are involved in creating a mathematical model of the ponzi schemes. The author of the paper, indicates that based on the zeros, it is possible to determine what is going on with the fund. If there are no positive zeros, the fund has a positive balance and is solvent. If there is one positive zero, it means the fund has collapsed and two positive zeros indicates the fund has become negative but will become positive later on with a bailout.

This explanation is a bit easier to follow but still involves a certain amount of calculus.

Let me know what you think. I'd love to hear. I find the mathematics of pyramid and Ponzi schemes fascinating. Tomorrow, its the mathematics of regular pyramids.

I've occasionally had students come in with letters or emails from lawyers or countries claiming they were due to inherit a ton of money or they won some sweepstakes they had not entered. Those are easy to expose but the other money making schemes are harder but they still make a great topic in math classes.

It turns out the pyramid scheme area covers several types of scams including one that is not as common as it used to be. The chain letter, the one where you'd get a letter from a friend or relative stating if you send this out to x number of other people, you'd get so much money in a certain amount of time. Each level brings in twice as many as the previous level, if two people send back money.

Another more common one is the modified 8 ball model in which a person does not get paid until they have recruited enough people to have established several levels. The idea is that you recruit two people, who recruit two more each who then recruit two more people each so there are three layers with 8 people involved. At this point, their participation fee is given to the first person. So if the fee is $1000, you just made $8000.

Both schemes use a geometric progression to grow but there is a fine line between legal and illegal in that the people who join have to get merchandise equal to their investment. The actual math formula is well stated here but it boils down to about 88% of the people will loose their money because those past a certain level.

A new scheme is the two up model. The sales earnings from the first two people you recruited goes to the person who recruited you. It isn't until the third person, you start making a profit. This involves a geometric progression of three times rather than the two of the previous two. Unfortunately, those at the very bottom usually do not make money. In fact about 67% of the people involved in this one, do not make any money.

The formula used to calculate the above averages is the geometric series formula from calculus.

Now for the Ponzi Scheme. This is the people are more likely to be familiar with due to some of those shows that focus on con men who earn millions of dollars before getting caught. The most well known is Bernard L. Madoff who conned people out of over $50 million dollars but it was named after Carlo Ponzi the originator of the scheme. In 1920, he collected almost $10 million from 10,550 people but only paid out about $8 million.

The idea behind the Ponzi scheme is that you play upon the greed of people by offering extremely high rates of return which cannot be met so you take the money from later investors and give it to the earlier ones. The mathematics is rather complex but shown here in wonderful detail.

Lets just say several calculus equations are involved in creating a mathematical model of the ponzi schemes. The author of the paper, indicates that based on the zeros, it is possible to determine what is going on with the fund. If there are no positive zeros, the fund has a positive balance and is solvent. If there is one positive zero, it means the fund has collapsed and two positive zeros indicates the fund has become negative but will become positive later on with a bailout.

This explanation is a bit easier to follow but still involves a certain amount of calculus.

Let me know what you think. I'd love to hear. I find the mathematics of pyramid and Ponzi schemes fascinating. Tomorrow, its the mathematics of regular pyramids.

## Monday, November 27, 2017

### The Brain and Videos

Last week I wrote about a teacher who is telling students that people cannot learn from videos and that teachers in college never use videos. I suspect her attitude came from the fact she has never learned to be an active watcher.

I found information on how the brain processes the information from a video. The first element to consider is cognitive load. It has been suggested the memory is made up of several parts. The sensory memory which is transient and it collects information from the environment.

Information from the sensory memory may end up in the temporary storage or processed in the working memory which has limited space. The processing is a precursor to encoding the information into long term memory which has unlimited space. Due to the limitations of the working memory, the viewer must be selective about what they choose to remember.

Cognitive load is composed of three parts. The first part is the intrinsic load which is determined by the amount of connectivity felt by the viewer. The second is the germane load is the amount of cognitive activity needed to reach the learning outcome and the third part is extraneous load or the material that is not directly associated with the topic.

In response to this, four practices are recommended to make video learning more effective. The first is signaling or cuing which uses on screen text or symbols to indicate important information. Signaling may be by a few key words, a change in color, or a symbol to draw attention to a part of the screen. Next is segmenting which is the chunking of information so they have control over the flow of new information. This can be accomplished by breaking the video into small pieces with questions sprinkled throughout so they cannot move forward until they've answered the question.

Another recommendation is weeding or getting rid of any extra material such as music when the person is talking, or extra animation. It means to eliminate any extra things that can make it harder for the listener to decide if its material they should learn. The final is matching modality or using both audio/verbal with visual/pictorial to share the new information. An example of this would be to show a process while explaining it so they have both channels engaged.

So when choosing videos, keep them short of no longer than six minutes. Make sure the voice over is done in a casual conversational style rather than formal language because the conversational style makes students feel as if they are partners. The narrator should speak at a normal speed with enthusiasm.

For students to get the most out of videos we need to teach them active learning skills such as guiding questions so they know what is important and to help them pay attention rather than switching into a passive watching style. Make sure the video has interactive controls so the student can rewatch segments as needed and so they have control of the speed at which they move through the video. Integrate questions into the video. Research shows that embedded questions increase the understanding and retention of the material in the video.

So videos can be an effective part of the classroom as long as they meet the above criteria so a to meet student learning needs.

Let me know what you think. I'd love to hear.

I found information on how the brain processes the information from a video. The first element to consider is cognitive load. It has been suggested the memory is made up of several parts. The sensory memory which is transient and it collects information from the environment.

Information from the sensory memory may end up in the temporary storage or processed in the working memory which has limited space. The processing is a precursor to encoding the information into long term memory which has unlimited space. Due to the limitations of the working memory, the viewer must be selective about what they choose to remember.

Cognitive load is composed of three parts. The first part is the intrinsic load which is determined by the amount of connectivity felt by the viewer. The second is the germane load is the amount of cognitive activity needed to reach the learning outcome and the third part is extraneous load or the material that is not directly associated with the topic.

In response to this, four practices are recommended to make video learning more effective. The first is signaling or cuing which uses on screen text or symbols to indicate important information. Signaling may be by a few key words, a change in color, or a symbol to draw attention to a part of the screen. Next is segmenting which is the chunking of information so they have control over the flow of new information. This can be accomplished by breaking the video into small pieces with questions sprinkled throughout so they cannot move forward until they've answered the question.

Another recommendation is weeding or getting rid of any extra material such as music when the person is talking, or extra animation. It means to eliminate any extra things that can make it harder for the listener to decide if its material they should learn. The final is matching modality or using both audio/verbal with visual/pictorial to share the new information. An example of this would be to show a process while explaining it so they have both channels engaged.

So when choosing videos, keep them short of no longer than six minutes. Make sure the voice over is done in a casual conversational style rather than formal language because the conversational style makes students feel as if they are partners. The narrator should speak at a normal speed with enthusiasm.

For students to get the most out of videos we need to teach them active learning skills such as guiding questions so they know what is important and to help them pay attention rather than switching into a passive watching style. Make sure the video has interactive controls so the student can rewatch segments as needed and so they have control of the speed at which they move through the video. Integrate questions into the video. Research shows that embedded questions increase the understanding and retention of the material in the video.

So videos can be an effective part of the classroom as long as they meet the above criteria so a to meet student learning needs.

Let me know what you think. I'd love to hear.

## Sunday, November 26, 2017

## Saturday, November 25, 2017

## Friday, November 24, 2017

### Actively Watching Videos

I include annotated videos as part of my instruction because I believe students need to learn to gather information when they watch videos.

Unfortunately, I am fighting the attitude of my students who believe they cannot learn from videos and a teacher who told them point blank that people cannot learn from videos.

She even spoke to me about returning to straight lecturing because professors in college do not use videos, they lecture and our students shouldn't use videos. I decided the other teacher is wrong because I've taken distance classes which incorporated videos as part of the regular classes. Furthermore, I've gone to YouTube to watch videos to learn to do things like soldier.

The other day, I spoke with my students about videos. I began asking students if they ever watch a You Tube video to learn something. They answered no. I asked if they ever watched videos for fun. To be entertained. Of course they said yes. This lead to my explaining about mind set. When they say they are unable to learn using videos, they have a closed mind. I told them, they can learn but they do not know how to do anything other than passively watch videos because they watch videos with the expectation of being entertained.

The videos I assign are focused on a specific topics with questions sprinkled throughout so they have to stop and answer the questions before moving on. I took time to go through a video, explaining how I would watch it more actively. I talked about what I would write down as notes, which material wasn't as important, and how to review the notes I took to find the answers to a question. I told them we'd have another lesson after Thanksgiving to help them learn to watch actively, rather than passively.

KQED has a column on the topic in which the author discusses how her students entered "TV mode" when watching videos. In other words, they didn't pay attention to the content, they focused on the accents, or hairstyle or voice. She had to do things to help them learn to look at the content.

She recommends a teacher pre-watch the video, edit it so it is in smaller chunks, insert questions, comments, or commentaries, and prepare guided notes before showing the video. The guided notes have blanks so students know what is important to write down.

Before students watch the video, activate prior knowledge and give them a reason to watch it. During the video, its good to have the video pause often so students can process information and so they can fill out the guided notes. If watching the video as a group, teach students to Watch, Think, Write. They watch the video, think about it while possibly discussing it, and then writing the information down.

When the video is over, students can create concept maps, use the information to answer a question or solve a problem, find a video clip that clarifies something from the original video and share it with the class.

The BBC recommends the instructor introduce the video to give them a purpose for watching the video. In addition, it is suggested that videos should not be any longer than 15 minutes due to student attention span. It is important to include questions so students know what to focus on when watching the video. These questions turn the passive watching into active watching. Also let students know they can rewatch a video if they didn't get all the information the first time.

It is good to know that my conclusion of passive versus active learning is correct and that I need to instruct students in active learning when watching videos. Please let me know what you think. I'd love to hear.

Unfortunately, I am fighting the attitude of my students who believe they cannot learn from videos and a teacher who told them point blank that people cannot learn from videos.

She even spoke to me about returning to straight lecturing because professors in college do not use videos, they lecture and our students shouldn't use videos. I decided the other teacher is wrong because I've taken distance classes which incorporated videos as part of the regular classes. Furthermore, I've gone to YouTube to watch videos to learn to do things like soldier.

The other day, I spoke with my students about videos. I began asking students if they ever watch a You Tube video to learn something. They answered no. I asked if they ever watched videos for fun. To be entertained. Of course they said yes. This lead to my explaining about mind set. When they say they are unable to learn using videos, they have a closed mind. I told them, they can learn but they do not know how to do anything other than passively watch videos because they watch videos with the expectation of being entertained.

The videos I assign are focused on a specific topics with questions sprinkled throughout so they have to stop and answer the questions before moving on. I took time to go through a video, explaining how I would watch it more actively. I talked about what I would write down as notes, which material wasn't as important, and how to review the notes I took to find the answers to a question. I told them we'd have another lesson after Thanksgiving to help them learn to watch actively, rather than passively.

KQED has a column on the topic in which the author discusses how her students entered "TV mode" when watching videos. In other words, they didn't pay attention to the content, they focused on the accents, or hairstyle or voice. She had to do things to help them learn to look at the content.

She recommends a teacher pre-watch the video, edit it so it is in smaller chunks, insert questions, comments, or commentaries, and prepare guided notes before showing the video. The guided notes have blanks so students know what is important to write down.

Before students watch the video, activate prior knowledge and give them a reason to watch it. During the video, its good to have the video pause often so students can process information and so they can fill out the guided notes. If watching the video as a group, teach students to Watch, Think, Write. They watch the video, think about it while possibly discussing it, and then writing the information down.

When the video is over, students can create concept maps, use the information to answer a question or solve a problem, find a video clip that clarifies something from the original video and share it with the class.

The BBC recommends the instructor introduce the video to give them a purpose for watching the video. In addition, it is suggested that videos should not be any longer than 15 minutes due to student attention span. It is important to include questions so students know what to focus on when watching the video. These questions turn the passive watching into active watching. Also let students know they can rewatch a video if they didn't get all the information the first time.

It is good to know that my conclusion of passive versus active learning is correct and that I need to instruct students in active learning when watching videos. Please let me know what you think. I'd love to hear.

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