## Friday, December 30, 2016

### In Transit

I am in transit right now, traveling home.  I sit in a small airport terminal where I'm catching a small puddle jumper whose maximum number of passengers is at a huge 9.  I'll be back to regular columns tomorrow.

I hope everyone has had a great holiday and will have a safe New Years

## Thursday, December 29, 2016

### Using Newspapers in High School Math

I love the idea of using newspapers in high school math but where I live, we have access to a weekly one which arrives Wednesday afternoon if the weather is good or Thursday otherwise.

The good news is that many newspapers can be found online so access is quite reasonable if you are not near a daily one. Some newspapers do offer complementary copies to schools if wanted.

Newspapers offer the perfect way to connect classroom learning with the real world.  So what are ways you can use newspapers in your classroom?

1.  Use the weather maps to find mean, median, and mode.  Most newspapers have wonderful weather maps which may cover the area or the nation.

2. Take the temperatures and have students convert them from Fahrenheit to Celsius or have students convert miles per gallon to kilometers per gallon.

3.  Look at mpg for various cars and have students compare models to determine which model has better gas mileage in percents.

4. Check out sales flyers to compare prices to determine better prices and again use percentages.

5. Use inequalities to find the best buys for used vehicles < \$3000 or >\$35,000.  Apply inequalities to mileage for cars or perhaps use the same type of criteria for appliances, or even sports.

6. Have students write word problems to go with an assortment of graphs and tables.

7. Look at clothing, food, or other ads to write functions based on buying combinations such as jeans and shirts.  Or you could have them write functions for inequalities based on not spending more than a certain amount for your new jeans and shirts at the beginning of school.

8. Check out the help wanted ads so students can take a yearly salary and figure out the monthly or weekly gross pay.  If they find a weekly or monthly amount, students can calculate yearly salary.

9.  Create posters with real world examples of math to hand around the classroom.

10.  Go house hunting.  Find a house, then calculate the mortgage payment for 15 and 30 years using current rates.

11. Rank the colleges using the Academic Index which uses algebra to rank colleges. Change the parameters to see how that changes college rankings.

12. Check out this article which has lots of different ways to use the New York Times in Algebra.  Ideas 10 and 11 are from the article but they have several more great ways.  In addition, they include the appropriate standards and mathematical practices these activities meet.

If you want to include this article from 1922 which looks at the use of Algebra in Tariffs.  Image, even that long ago, they asked the same question regarding its use.

Enjoy, have a good day and let me know what you think.

## Wednesday, December 28, 2016

### Teaching Division

Division seems to be as difficult for students to master as subtraction.  I have students who struggle with it if its a division problem.  If I change it into a multiplication problem, they struggle less but its still a challenge.

According to two different articles I read,  division is introduced to young children by using the idea of sharing, such as there are 12 pieces of candy shared between 3 children. How many pieces will they each get.

The second stage of teaching division has students looking at division as grouping rather than sharing.  It also incorporates relating it to division.   Add into the mix the fact that the division algorithm makes less sense to most students than any other operation because its a left to right rather than the usual right to left process.

Most of the students who are successful using the division algorithm have memorized the steps without understanding the concepts behind each step.  I am the first to admit, I am one of those.  I was told how to do it, I followed the steps and got the answer,  What more did I need to know?

The students who struggle are the ones who do not know their multiplication tables well and who missed the step of placing a zero in when they cannot divide the number.  These same students  struggle when division is taken from using straight numbers such as 28/7 to dividing polynomials such as x^2 + 5x + 6/x+2.  The algorithm is the same but it uses variables.

In my opinion, division is really just a way of finding the second factor of a number when you know one factor.  Yes, sort of the reverse of multiplication.  When my students struggle with 28/7, I'll often ask them "7 times what gives you 28? because it is sometimes easier for them.

It has been suggested students should not even attempt the division algorithm until they have their multiplication facts down cold, otherwise it is difficult for them to learn.  Its also suggested students have division down as both sharing and grouping because they need both interpretations.  With grouping if you have a problem like 900/30 it could be interpreted as 900 items split into 30 groups or 900 items split into groups with 30 items.

Add into this the idea of remainders.  I don't think the elementary teachers or books have taken time to explain remainders properly.  My students see it as remainders, not as so many items left over which are not enough to make a group itself.

I'd love to hear from you all on ideas you have to help make division easier for high school students who struggle.  Hope you are having a good break.

Then comes the question of when will I ever use polynomial division?  It is used in most crypto and error correcting algorithms which are used in most modern digital devices.

## Tuesday, December 27, 2016

### Learning Math Facts

If you teach middle school or high school, you have a few students who never got their multiplication and division facts down.  Unfortunately this makes it harder for them as they struggle to do well in their math class.

Unfortunately, the lack of having their math facts solidly entrenched means they will not do as well on standardized tests and it becomes harder as they take Algebra, Geometry, and other classes.

As they struggle, they develop the self-image of being dumb or unable to do the mathematics. Sometimes they develop behavior issues but they can learn, even if they are in high school.

There are three stages involved in learning math facts.  The first stage is for students to figure out the math concepts such as skip counting to figure out the answers.  There is evidence that students who struggle with mathematics have not mastered this step.

The second stage is developing strategies to remember basic facts.  If a student has developed his own strategies to remember facts, they do better but if a student has not, it is important the teacher teach the facts in a logical manner which emphases relationships makes it easier for students to learn.  One relationship is 2 x 6 has the same answer as 6 x 2.

The final stage is developing the ability to automatically give answers when asked a fact.  This means they do not hesitate when responding.  They know them all and don't have to think about it.  It is important to know these stages so when you work with older students you have a better idea where they are in their development.

Since I work with older students, I need to help many of them learn their math facts.  Fortunately, there are ways to help older students learn their facts without a lot of work on your or their part.

1.  Help them find out how many facts they already know cold.  Rather than facing all 100 or 144 facts, it will be much less.

2.  Do a few facts each week.  Only add more facts as they master the ones they already have.  Do not move on until they have the new facts cold.

3. Show how some of the facts are really the same such as 2 x 6 is the same as 6 x 2 so that means even fewer facts to know.

4.  Show the hand trick for the 9's which eliminates another set of facts.

5. It is important to combine already learned facts with the new facts so practice is cummlative.

6. Have students practice with a verbal component so they might say "Eight times nine is seventy two" so they have the whole both the problem and answer associated.  Since most students like music, look for musical videos on You Tube they can listen to as a way of helping them learn their facts better.

7.  Have regular practice schedules which include a method or immediate correction if they make a mistake.  The goal is to have them automatically respond with answers to each math fact.

8. Use a timed test occasionally to check their progress but do not do math minutes everyday.

I plan to use some of these suggestions when the new semester begins to help some of my students learn their multiplication tables.  Let me know what you think.

## Monday, December 26, 2016

### Better Ways to Teach Math

As a math teacher, I am always looking for ways to improve my instruction and as I've been reading research on the brain and the way it works, I'm discovering ways I've never thought of to use in my teaching.

There are suggestions out there which can help us but honestly if you are like me, you don't always have time to implement everything you want to.  I've found the best way is just to start with one or two things, get them implemented and then add another one or two things.

Here are a few easily implemented suggestions to help students with their learning.

1. Ask "Why?"  Ask them why something worked at least once a day. Ask why it would work with other numbers.  The why is important.

2.  Rather than looking at whether the problem is right or wrong, look for what students did correctly and build on that because it shows what they understood.  You can build on it during the next discussion or task.

3. Use the textbook to find meaningful tasks or tasks that can be easily adjusted to be more meaningful.

4.  Provide at least one opportunity for students to solve problems mentally.  This helps increase their ability to determine if the answer makes sense and allows their creativity in solving problems to develop.

It is said the students who struggle most with math have difficulty remembering their basic math facts, have difficulty solving word problems, and completing multi-step equations.  One way to help them learn the subject is to break the steps into micro steps and access their understanding at each micro step.

In addition, add in something they can relate to without the math.  For instance asking them to add a negative and a positive number often results in a guess for the answer.  Instead ask them if you lost that amount of money and someone gave you money is it good or bad.  Such as for -8 + 3, you tell them they lost \$8 and were given \$3, is it a good day or a bad day?  No answer involved but there is a context.

Let me know what you think about this.  I know these look like ones, I plan to include in my teaching in the future.  Have a good day.

## Friday, December 23, 2016

### Interesting Math Thought

The above picture is one we might see at this time of year.  We hear the story that he travels around the world in 24 hours while dropping presents off with all the children in the world.  Its a wonderful tale but have you every wondered what that would entail?

To figure out the mathematics of such an endeavor you need the following information:

1.  The area of the land mass is 57,259 square miles.

2. There are about 57 million children world wide under the age of 12.

3.  24 hours will need to be converted to seconds so as to make the distribution rate more accurate.

With this information, you can have students calculate how fast Santa has to distribute gifts to cover the whole world in the time period.

If you wanted to add another element to it, have students figure out where Santa would be delivering gifts first and the direction he would have to travel.  If you want to add another layer to the calculations, have them figure out a path that would work for the actual world wide distribution.

Have a wonderful holiday weekend.

## Thursday, December 22, 2016

### Reticular Activating System

Recently we've been learning more about the brain, how it stores information, and how it functions.  As teachers, we need to keep track of these advances so we can help them learn better.

First of all, what is the reticular activating system and what does it actually do?  Well this system is found at the base of the brain by the top of the spine and sends information upwards.

This is where thoughts, internal feelings, and outside influences meet.  In other words, it works to filter internal thoughts and external information that bombards us.  It decides what we are aware of and what we ignore.

So it likes surprises and new things or things we are interested which is why we drift off during long boring lectures or cannot concentrate when we are hungry or thirsty.  So if you want to keep the attention of your students, you need to tap into both the creative and logical parts of the brain.

Some of the suggestions made to help create lessons where students learn rather than tune out are as follows:

1. Help students remember the material by connecting the critical information to positive emotional experiences in the classroom.

2. Begin the lesson with the large concept, invite predictions, KWL, graphic organizers as a way to preview and give an overview of the lesson.

3. Try to set it up so students are not dividing their attention.  Allow pauses so students can focus on taking down notes. Contrary to the multitasking myth, students should only be doing one activity at a time.

4. Students focus better if they know there is a follow up activity such as a think-pair-share.

5. Do crazy things like sing, speak in a different voice, or hang a dollar bill to provide the surprise the brain likes.

6. Pause to build anticipation before you say something important.

7.  Use color for fun and differentiation. If you write the most important point in one color and key points in a second color, it can increase recall.

8. Ask questions which make them think a bit such as do you want one half of quarter of a hamburger or  a quarter of a half of a hamburger.  Explain your answer.

9. Every 15 minutes or so add in some physical activity such as have them get something, change your position in the room, so they have to move.

10.  Change the arrangement of the room every so often or change the seating, or bulletin boards so that there is a bit of surprise.

11. Use your students names in problems to personalize them.

The above suggestions can help students stay focused on the lessons so as to learn better rather than fading out and focusing on other things.  Let me know what you think!

## Wednesday, December 21, 2016

### Changing Brains

We all know that there is the idea that if students can get past the notion they are good or bad at mathematics and their ability is set, they can learn.  It is often referred to as mindset and we know there are both parents and students who are convinced they cannot learn math.  They just aren't good at it.

So what is it about our brains that allow us to  continually learn even if you are older? The brain has a certain amount of plasticity which means a person's intelligence is not fixed.  Instead, the brain is changing and growing throughout a lifetime.

Apparently as a person learns or accesses new knowledge, the neural networks as a cluster and form themselves according to the activity or memory.  When you quit practicing a skill, your brain eventually eliminates that node so it goes away.   Its like roads, if the road has lots of traffic, it will be wider to allow for the number of cars but if its not used much, it can be smaller and if its not used at all, it disappears.

It has been suggested we take time to explain how the brain is open to learning so students know things are not fixed.  If they realize they can change their brains, it can improve their self-esteem and be willing to learn.  Furthermore, it is important to have them practice because when they repeat an activity that has them retrieving information, they create stronger neural connections.

If at all possible, connect the new material to prior knowledge to provide a context so students see a relationship. When they see the relationship, their brains display a greater amount of activity which helps them make better long term memories.

Understand that changes occur in larger networks rather than in just a single synapse. Learning occurs when different parts of the larger networks are strengthened.  It also appears that both the neurons and brain gather information from any sources so they can change their function as needed.

Furthermore changes occur in both the synapses and the brain circuit. In fact, the changes occur in all the connected neurons in the brain circuit.  When learning is happening, changes occur in many different places at once using different mechanisms and different sequences.

So how does this information help in teaching mathematics?  It has been discovered if you include hand gestures such as pointing to both sides of the equations, it helps increase student learning.  Tomorrow I'll be talking more about learning, teaching, and the brain.

## Tuesday, December 20, 2016

### Homework

As teachers, we are all aware of the debate over homework.  Is it effective?  Is it worth it?  If we give it, what is the best way to do it?  Certainly not like it was when I was in school.  I suspect many of you encountered the same type of assignments.  The page whatever, every 3rd one so you practiced the same type of thing 30 times.

That is not an effective way to assign homework.  So what is the best way to assign homework so it is effective and has real value?

There are several factors to keep in mind when creating homework so it is effective and not a waste of time.  Homework is best used to build and maintain proficiency of skills.

1.  Create smaller assignments which are assigned more often so practice is distributed over time. It is important so students maintain their knowledge and retain the information better.  These smaller practices are also less intimidating than larger ones.

2. Set a purpose for the assignment and share it with students so they have a reason for the homework.

3.  Make sure the homework does not take very long to do - no more than 30 minutes or so.

4. Provide answers for the homework you have assigned so students know if their answers are correct.  This provides immediate feedback to students so they can rework problems before turning them in.

5. Personalize the assignment so students take ownership of it.

6.  Make sure they feel they can do the assignment otherwise they may give up before even trying.

Think of homework as part of the learning process in which it provides practice while checking for understanding, or improving their ability to process information.

I do give homework once a week in all my classes.  The homework assignment has 10 problems which cover the material already taught, even from the beginning of the year.  I provide answers either in the QR code attached to a corner or I place the answers in the back of the room.  When I put the answers on the wall, I show how each and every problem is worked so they can check it out if they get stuck.

The homework problems are only for review and I do not expect them to apply the material to new situations  without actually talking about it and showing a couple examples first. Students need these examples to check as they do the assignment.  The homework is only worth about 10% of the grade so if they do not turn it in, their over all grade does not go down too much.

I'd be interested in hearing what you all think about assigning homework.

## Monday, December 19, 2016

### The Moon, Mars, and Slope

I meet a geophysicist at the SeaTac Airport and she introduced me to two great websites filled with wonderful information that could easily be used in the math classroom.

The first is Moon Trek put together by JPL and NASA.  this site has great pictures of the Moon which you can explore in a variety of ways.  The two tools I can use in Math is the elevation profile and distance.  Yes, you can find the distance but then as the instructor, you can set up some great problems using the distance with a variety of speeds to see how long it might take to drive from Point A to Point B.

The elevation profile creates a wonderful cross section complete in meters with elevations using both positive and negative numbers.  The line I chose started at -3075 meters and jumped up to positive 3000 meters so students can see what is happening.  With that information, its not that much further to having students calculate the gradient for the profile.

JPL and NASA have also created Mars Trek, which offers the same tools to use with Mars.  You can calculate distance, create elevation profiles just as you did for mars.  We know that slope when dealing with the real world is often the overall taking it into account.

Both sites offer a person the chance to create a 3D printer file so students can print out models of any part of the maps.  Furthermore, the program will calculate sun angle.  Just think what activities you could create so students have the opportunity to analyze real life data!

The maps are great and it is easy to zoom in or out on any parcel of land.

I realize this is not a long entry today but I've been traveling for a couple of days and I am totally exhausted.  Tomorrow its going to be back to normal.

## Sunday, December 18, 2016

### In transit

I I

I am intransit and will be back tomorrow.

## Friday, December 16, 2016

### Integrating Videos Into Instruction.

Up until recently, I just showed a video on my smart board before doing a lecture but after reading some things I've discovered there are ways to make video watching more effective.

When I grew up, you just watched the movie  before filling out a worksheet associated with the movie and you hoped you remembered enough to fill it out.

There are suggestions as to how to prepare videos so they are more effective than just watching it and hoping the students learn the content the way you hope.

Step One:  Watch the video or video clip to make sure the material on it matches the objective fully.  Why show something that does not illustrate the topic.

Step Two: Use a program to cut the clip down to cover exactly what you need.

Step Three: Add in questions, comments, and commentaries so students have a way of knowing what the teacher considers important.

Step Four: Create a set of guided notes to accompany the video with blanks for students to fill in with key concepts or other important information.

Before you have the students watch the video ask a few questions to help activate prior knowledge so they establish a connection.  In addition, give them a reason for watching the video.  If you are showing the video in class, pause it often so they can fill out the guided reading other wise they try to scribble down answers verbatim and they miss other important information.

If they watch the video outside of school, suggest they pause the video as needed to complete the worksheet.  When the movie is done, have them complete another activity such as journaling, completing graphic organizer.

There are lots of different websites you can go to in order to add voice over, quizzes, questions, etc to help students identify the material that you as a teacher feel they need to know.  Most students need guidance and this is a way to do it.  Give it some thought and let me know what you think.

## Thursday, December 15, 2016

### Creating a Good Quiz.

Have you ever thought seriously about how to create a proper quiz?  I haven't up to know because I use a quiz just to check out if they can do the problems but it turns out there is more to writing a quiz than that.

I'm sure others out there were taught to either take the all ready created quiz from the textbook or to write their own.

By good quiz, I mean you are able to analyze the results to provide usable data to help create better lesson plans.

This leads to the question of "How do you write a good quiz?"  The suggestions are more like things to keep in mind and many of them make sense.

1.  Tie each question to a goal or standard.  That way you know if the standard is being met.

3. If you write multiple choice questions, make sure the wrong answers represent a misunderstanding of the concept.  This allows you to determine how the student processes the information and helps you decide if you need to reteach a concept.

4. Write questions using different levels of Blooms Taxonomy.  Have some that are simple recall while others might be a real life application of the material.

5. Test your questions.  If they are not useful, get rid of them.  You want to create a bank of good questions you can pull from as needed.

6. Keep the questions clear because you are not trying to trick the students.

7. True and False questions are not good to use because students have a 50% chance of getting it right.

8. Rather than grading the quiz, evaluate it to see if the student has mastered the material.

9. Let students know the reason for the quiz.

10. If you use an online quiz program, make sure its one that can provide information on all the results so you can just look at it and know what students need scaffolding on.

11.  If it is a topic you could use a video quiz on, you might want to use it so  you use other types of learning.

Give it some thought when you create your next quiz.  Are you writing one to help students and you discern what they need to work on or do you just quiz to quiz.  Let me know what you think.

## Wednesday, December 14, 2016

### What Works In Teaching

The other day, I came across this great article from the government which summarizes nine ways to improve student learning.  Some of the nine we've seen before if you follow Josh Fisher or the Learning Scientists.  Some you may never have seen before.

1.  Space learning over time.  This includes reviewing key concepts several weeks to several months later so students retain the material better.

2. Weave worked examples complete with solutions among the problems so they can refer to examples while trying to solve new problems on their own.

3. Combine illustrations or graphical representations of key processes with verbal descriptions.

4. Create a connection between abstract and concrete representations of the concept.

5. Integrate more quizzes to help promote learning and use pre-questions when introducing new material.  In addition, use quizzes to expose students to already taught material.

6. Teach students to allocate study time and allocate the material to increase student learning.  In addition, teach students to use the results of the quizzes to determine what to study.

7. It is recommended that both teachers use deep-level questions which require students to respond with explanations that support deep learning.

Numbers five and six actually have two interrelated parts but still it works out to 9 different suggestions.  All of these help students learn the material better and do well.  I like the part about helping students analyze their own quiz and test results to determine what areas they need to work on.

In addition, this site provides a 63 page guide on Organizing Instruction and Study to Improve Student Learning with some very good ideas and information to implement these suggestions in your classroom.  Furthermore, the guide offers possible roadblocks and solutions for each roadblock for each suggestion along with a checklist for implementing each suggestion.

I have already downloaded this guide which I plan to read over the holidays. I like a lot of the suggestions I've seen in it.  Let me know what you think.

Sometimes I think that as teachers we spend so much time trying to find the best ways to work with our students

## Tuesday, December 13, 2016

### Participating vs Engagement.

According to an article published recently by KQED, participating is not the same thing as being engaged although they might look similar.

The big difference is that participation is when a student is doing it but if they do not make meaning of the material, they are not actively engaged.

Researchers identified eight qualities to determine if students are engaged in the activity.

1. Does the activity allow students to personalize their response

2. Are the expectations clear and well modeled?

3. Do the students feel they have an audience beyond the teacher?

4. Are students able to socially interact so they are discussing the material?

5. Are mistakes valued as part of the discussion.  Are students allowed to make mistakes in a safe environment.

6. Are students allowed a choice within the activity?

7. Does the activity always connect to the real world?

8. Is the activity new and still cool for the students?

The researchers admit it is hard to have all eight qualities for every activity but if a teacher is able to have three for any activity, it increases cognitive engagement to between 84 and 86 percent of the time.

In addition, the three chosen qualities should be visible to anyone who walks into the room.  All students should be able to explain what they are working on, what the teacher's expectations are and how the activity will lead to learning.

It is also important to set up immediate feedback for the activity so students increase their learning.  If you decrease the amount of assignments and increase the feedback so its more immediate, students will increase their learning.  Feedback does not necessarily  mean handing fully corrected work back.  Instead it can mean giving examples of proficient, below proficient, and above proficient work.

So when you create an activity for students keep in mind the eight qualities designed to help increase engagement rather than just participation.

Let me know what you think!  I'd love to hear back from people.

## Monday, December 12, 2016

### Mathemagics

Yes you read the title of this entry correctly.  Mathemagics is the title of a book written by Margaret Ball back in 1996.  Its still possible to obtain copies of it but its out of print otherwise I think.

You probably wonder why I'm discussing a fantasy book but there is a point so please hang in with me for a bit longer ok?

This book is preceded by several short stories in the same world.  It involves a female sword fighter who lives in another world.  She is not a mage because she cannot seem to master the system of magic there. When she tries, accidents happen and she's never sure how to undo the damage.

Yes I can hear you muttering about getting to the point.  Here is the point.  The magic system in this book is based on calculus.  You read that right!  Calculus.  Every spell is a derivative and every counter spell the antiderivative to the original spell.

As I said earlier, she may not be great at math but a man from our universe who just happens to be a math teacher, ends up in hers.  They fall in love and of course they marry but in the process, he ends up a great mage because he can do both derivatives and antiderivatives.

The book takes up when the family living in this world where she is the author of several sword and sorceress books.  Her husband teaches Mathematics but when their world is invaded from hers so out comes the spells and they are off.

I found the idea of calculus as the basis of a magic system.  Yes I did check each and every spell and counter spell to make sure they were correct and yes they were.  In addition, every chapter title is also a mathematical expression such as i^4 for chapter 1 or sqroot 4 for chapter 2.

I loved the book and have reread it numerous times since then and continue to do so because it is funny and the math is so well integrated you don't mind it.  Give it a shot and enjoy.

## Friday, December 9, 2016

### Is Group Work Collaboration?

As noted yesterday, group work and collaboration were considered the same thing when I was growing up but it is not, really.

Group work is defined as grouping students together to complete a specific assignment but there may or may not be collaboration involved.

In addition, groups may not even be cooperating let alone collaborating.  Do you think of cooperating as collaborating?  I did up until a couple years ago because they referred to group work as cooperative learning.

Cooperative learning is defined working together towards a shared goal by dividing the work up while collaborating is working and thinking together towards a shared goal. So collaboration requires a more complex interaction than cooperative learning.

Cooperative learning often has a leader or manager who assigns tasks so that each person is expected to complete a part.  The leader controls the rate at which things are done and members do not need to like each other or be able to work together because they can work alone on their part.

In collaboration, members must learn to work together because they are all working on the end product.  There is no one leader so the group as a whole must learn to make sure the work is completed because they are sharing power.  Although collaboration is great for creativity, people must also be flexible because they are working together rather than alone.  If a member is not flexible, they may find it difficult to collaborate.

So which is better?  It depends on the circumstances because both types of learning are important to use.  Cooperative learning teaches students how to work in a project based situation with a manager who is in charge and each participant has a specific role designed to help create the final product much like a sports team.  So there is accountability for the individual parts and for the final product.

In other situations, collaborative learning is important because it teaches them to work together and communicate at every stage of the process to create the final product.  It requires negotiation, the ability to let go of your idea because it may not be the best one to use, communication, and a shared vision step by step much like people who work together to produce a commercial.

In reality, graduates need to work under both circumstances because work does not require people to one or the other all the time.  I've worked places where I needed to do both at various times depending on the situation?

In answer to the question asked in the heading, yes it can be depending on the requirements of the situation but without the requirements being set, then no, its nothing more than a free for all because both collaboration and cooperation need parameters to work.

## Thursday, December 8, 2016

### Teaching Collaboration.

When I was in school, collaboration was called "working in groups".  At that time, members contributed an unequal amount of work but everyone got the same grade.  I usually ended up in a group with one of the high achieving girls who didn't bother trying to get everyone to contribute but just did it and we all got the grade.

Over the years, people  have developed methods to get everyone working such as assigning a team leader, materials person, the transcriber, etc so everyone has an assigned job but that didn't always work well.   So what is the current thought on collaboration and using it in the classroom?

According to an article in Edutopia one of the first things students should do is to create a working agreement so everyone knows what the rules are they've chosen.  The rules might include things like "One student speaks at a time."

The second thing they recommend is help students learn to listen and hear what is being said.  This means that they need to practice restraint and not rush into the conversation.   It is sometimes helpful to add in something that requires students to wait until two or three other students comment before they speak again otherwise someone might dominate the whole conversation.

In addition, students need to learn the art of good questioning.  Teachers can sometimes provide question starters to help students learn to phrase their question properly.  Instead of saying "What do you think?", the question might be "What comes to mind when I say _______?".

Furthermore, they need to be taught to negotiate because negotiation is an art in and of itself.  It is suggested a teacher models listening, questioning, and negotiation so students are regularly exposed to it and learn how it works.

Another article suggests teachers should use conflict within the groups to help reinforce negotiation.  with direction, students learn the art of working with each other to solve problems.  In addition, the assignment should contain problems which require students to use complex problem solving and deep thinking.

It is also suggested that the whole class reflect on the activities and how well the groups worked together.  These reflection times help everyone identify both good and bad examples of collaboration. It is important to identify examples so students know what they should be doing.

There are books out there filled with "collaboration" activities but not all of these require deep thinking or need complex problem solving to complete.  When you choose activities ask yourself if they have these.  More on this tomorrow.

## Wednesday, December 7, 2016

### Mastering Vocabulary

In math, its harder to teach vocabulary because there are three levels of language involved.
1. General words such as hello, bye, or good.

2.  Words with both mathematical and general definitions such as plane, or product.

3. Words that are primarily mathematical such as Torus (doughnut shaped), or Polygon.

When we work with students, especially those who are ELL, we need ways to make sure they understand the vocabulary.  I've developed several ways to work on vocabulary.

1.  A word wall - I have a spare place in the room I use for a word wall.  A word wall is a place where you post both words and definitions that students help create.  I came across this 8 page pdf with tons of ideas on using the word wall in class.  The suggestions include having a pairs of students select two words and discuss how they are related.  I love this because it gives suggestions for actively integrating the word wall into teaching.

2. Word cloud - a way of placing all the associated words together such as types of numbers, addition, or subtraction.

3.  Graphic organizers with the word, definitions both mathematical and general, pictures, examples and non-examples.

4. Graphic organizers showing all the words relating for a specific section such as transformations would have translations, dilations, reflections, etc.

5. Flash cards - students create their own digitally or on 3 by 5 cards so they can practice them as needed.

6. Create posters for each word or set of associated words.

7.  Write a short paragraph using the vocabulary words.

8.  Write word problems using assigned numbers and words.

9. Create lists of homophones and then use them in sentences.

This is a short list of possibilities but each one helps students create the knowledge they need to do well in math.

Let me know what your think.

## Tuesday, December 6, 2016

### Vocabulary - Putting It In Your Own Words!

When a student studies mathematics, they are learning a new language.  Many of the words are familiar such as product, plane, or less than but the meanings are not.

All through school beginning in Kindergarten, teachers work on helping students acquire the necessary vocabulary to succeed in math but we don't always teach students how to write definitions in their own words.

When they see a definition such as "It is a decimal number that cannot be expressed as the ratio of two integers and neither terminates nor repeats", if they do not have a solid foundation in mathematical terminology, they will not understand it.  If you ask them to define it in their own words, they might try to copy it and just change out one or two words or they might ask the teacher to tell them what to write.

The other day, I realized I'd never broken down the process of taking a mathematical definition and putting it into their own words.  In fact, they made it to high school without learning the process and it appears teachers in earlier grades are allowing them to just copy material down verbatim without learning the process.  I do not blame the teachers because many students will take the easier path and just copy rather than struggle to learn to use their own words.

I'm not sure if these students have a good vocabulary so they don't have enough words to put it into their own words.  So I took time to demonstrate how I turn a definition into something more understandable.  It was enlightening because some of the students paid close attention to the process.

Wednesdays are my word problem day.  I take something out of the NCTM journals to work on with my students. This last Wednesday, the activity required students to know irrational, rational, integer, whole and natural numbers.  I realized I needed to take time to help them learn to translate the definitions from math into common English.

So I took them through the above definition step by step to look at what each part meant.  By the time we were done with the definition for irrational, they had a great definition, they understood.  I do believe it is important for them to be exposed to the mathematical definitions but its also know what they mean in understandable English.

I admit, I assume they have learned this vocabulary in elementary and middle school but I'm finding they are being taught the processes but not the vocabulary that accompanies it.  I've been working on finding ways to help with that.

Check back tomorrow for ideas on how to increase their understanding of mathematical words and expand their vocabulary.

## Monday, December 5, 2016

### Sleep and Learning

I work with students who often do not go to bed till 3, 4, or perhaps never made it to bed the night before.  They arrive yawning and by lunch, they've placed their heads on the table and are sound asleep.

I've seen students so tired, they slept through the bells ringing and the changing of classes.  Why does this happen?  Lots of reasons from alcohol, to fighting, to no one making sure they have gone to bed.

I'm sure many of you have the same happening at your school.  The students assure me they don't need much sleep but I know a lack of sleep can interfere significantly with learning.  There are two areas in which a lack of sleep can be detrimental to learning.  First, it can make it difficult for a person to focus on the material and learn it effectively.  Second, sleep gives the brain time to process information.

The lack of sleep can leave your brain feeling foggy, messes with your judgement, and it interferes ith your fine motor skills.  In addition, it can put you at risk for certain health issues such as diabetes, obesity, and other diseases.

As for your memory, scientists believe sleep is needed so the brain has time to consolidate information.  It is believed that the hippocampus and neocortex are the actual parts of the brain involved in the process.  While you sleep the hippocampus relays the events of the day (including what you've learned) to the neocortex where these memories are reviewed and processed and stored as long term memories.

In addition, it appears that some memories become more stable during REM (Rapid Eye Movement), while other memories become more secure during deep sleep.  Sleep allows your brain to sort through memories to determine the important ones and strengthens information.

Apparently when you sleep, your brain cells shrink some to allow fluid to flow around and between the cells.  The fluid washes out toxins and cleans it up so you can think better. Furthermore when you get enough sleep your ability to tackle harder problems and understand problems increase.

One reason it is good to go over material before you take a nap or go to sleep for the night increases your chances of dreaming about it.  When you dream about the material, it helps you understand the material better and remember it better.

So anytime you have a teen who tells you they do well on 3 or 4 hours a sleep each night, let them know how more sleep will help them in school.

## Friday, December 2, 2016

### Animation in the Math Classroom

I am always looking for and playing around with ways to use animation, stop motion, and movie making in my math classroom because I have several students who would rather do something like that.  In addition, they enjoy watching animated clips showing how to do math.

So today I decided to look specifically at animation in the classroom today because the tools for creating animated videos has become more and more available.

According to an article on Creative Educator,  students who spend their time just memorizing material rather than understanding and internalizing it are less likely to create links between their prior knowledge and the new knowledge. Allowing students to create digital animation can help them increase their understanding of the topic.

Animation allows students to create their own unique meaningful connections to the material.  When they create animation, they have to make their own graphics which helps them implement their own visual mathematical representations.  In addition, they learn, understand, and internalize better.

Animation can meet the four conditions needed for effective learning by providing active engagement, group participation, frequent feedback, and connections to real world context.

I know from personal experience that you don't need to download anything special if you want to animate objects.  That can be done simply by using keynote or power point.  I generally use keynote to animate equations and such.

This is an example of animation I made using keynote for congruent triangles.

I hope this works as I don't usually upload quicktime movies.  It is done keynote so it is animated.  Todo this, students have to create a storyboard, write a script, make it and export it.

As for using animation like cartoons, there are lots of free apps out there that provide the student with the ability to create characters who show each other how to do a problem.  I don't have one done at the moment but I have used animation to show the process.  I'll find those and share them with you at a later date.

## Thursday, December 1, 2016

### Keeping Up With Research

As a teacher, I try to keep up with current research so I am able to do the best job I can but its hard.  Its hard because it means I have to change the way I do things, I have to convince the admin that what I'm doing is much better than the way I'm doing it all while trying to prep students for state tests, etc.

I know I'm trying to incorporate more technology since my students live, breath, and sleep technology.  I included sleep because many students take their devices to bed to text, read, or play games.  I am trying to capitalize on that while making it a worthwhile assignment and not just something to use because its all the rage.

I want to incorporate choice boards, menus, etc but I'm still trying to figure out how to set up choices so they further student learning.  I've got some ideas I'll be working on over the holidays but its hard.  I also admit that when I'm on holidays, I do not want to think  about school because I need the down time.

I just read something stating we need to quit teaching students math shortcuts because when they learn the shortcut, there is less chance of learning the actual concept associated with the problem.  This makes sense with the newer standardized tests that require the student to justify their work.  Why not have them learn the concept and then show why the shortcut works.  This gives them a chance to develop understanding first.

One concept my students have had trouble with is the one where anything to the zero power is one.  Once I showed how by using the rules of exponents and division you get one when you divide 5^1/5^1 because 5^1-1 = 5^0. This actually made sense to them.

One site I like to read on a regular basis is the Learning Scientists Blog which has great posts for teachers interested in current research and its applications.   Its where I learned about the six strategies for effective learning. It gave me something to look at in terms of its application to Math.

Another place I go is to Facebook and the Education Blogging: Working Out What Works group because people report on what they've read and tried applying to the classroom.  It is a place I get ideas from.

An of course, I love the Mathematics Education (K-12) group on Google Plus because this is a place people share what they are doing so I get even more Ideas.  Josh is great at including current research so I don't have to surf the web for information.

I'd like to know how others keep up with current research and how you apply it to your classroom.  Drop me a line and let me know.

## Wednesday, November 30, 2016

### Transference

This is a topic many of us reflect on in regard to our students.  There are times we think they have it and should be able to apply the procedural knowledge to a similar task and they can't.  One place I see this is in adding numerical fractions where students have to make sure they have a common denominator to do this.  They learn it but when they apply the same principal to adding algebraic fractions, they have no idea what to do.

So what is going on?.  According to an article I read there are several reasons students have difficulty with transference.

1. Initial learning is necessary for transfer but it is not known about the type of learning needed to promote transference.

2. Abstract representations of knowledge helps improve transference.

3.  Transference is considered to be an active, dynamic process.

4. All new learning is based on transfer of previous learning.

It now appears that transference is based on the degree to which people understand the material rather than memorizing procedures and facts. In addition, it takes a lot of time to really learn the material, more than we usually schedule in class for any one topic.  It has been found that students need to know the when, where, and why to use the new knowledge.  In other words, build a connection.

According to another study done to see if students could transfer the mathematical processes to other subjects, it was found students had a difficult time because it required them to translate a problem stated in words into a math problem.  A second study claims the reason students are unable to transfer their learning from high school over to work because they do not have enough authentic based and project based learning.

This leads me to believe there are two types of transference:
1.  Transference of a skill from a simple problem to a more complex one such as requiring a common denominator for regular fractions and algebraic fractions.

2.  Transference of the skill from the classroom to other subjects or even to work.

In addition, there are two types of transference which might explain why I see the two versions.  The first type is near transfer where students are able to apply their knowledge to problems and situations similar to what they learned.  The other is far transfer where students apply their knowledge to a situation quite different from the context they learned it in.

It is noted that schools offer more opportunities for near transfer.  Students find far transfer more difficult because they have to seriously consider the situation in order to remember the rules and concepts needed to solve the new problem.

Furthermore, there are three factors involved in transference, the person's representation of the problem, their background experiences, and their understanding of the problem.  Representation refers to how the person mentally solves the problem which is related to their knowledge of the content of the problem while experiences is often based on prior knowledge and understanding is linked to representation.

This gives us some understanding on why many of our students are unable to transfer math as easily as we think they should.  I know my eyes have been opened.  Let me know what you think!

## Tuesday, November 29, 2016

### Deep Learning.

I ran across the term deeper learning in the computer field but wondered if it had applications to mathematics. In education, deeper learning refers to using presenting the material in innovative ways so students learn the material and apply what they have learned.

It is also a way to encourage students to take control of their own learning because they are expected to combine communication, collaboration, with in-depth academic knowledge.

Many of the currently recommended practices such as project based learning,  assessments accumulated over time, etc help with deep learning but are not necessarily being practiced.  My school changed to a modified block schedule one year but it was done without putting a proper foundation in place so it failed miserably.

The collaboration requires students to work in small groups so everyone has a chance to participate.  Small groups also allows for peer teaching because students will ask each other for help when the teacher is unable to get over to answer questions.  Students are often able to explain a topic to each other in ways that are more easily understood. In addition, digital devices can help promote deeper learning because it allows a more personalized education.

The research is showing that students who graduate after using deeper learning often score higher on tests, graduate on time and are more likely to enroll in college than those that don't.

Some of the ways to promote deeper learning in the classroom include:
1. Master academic content in addition to providing real world applications so students see cross curricular applications.

2. Require students to think critically while solving more complex tasks. Integrate researching, brainstorming, and design thinking.

3.  Have students collaborate because it helps students gain a skill that is important in today's society.

4. Require effective communication because students need to communicate effectively and persuasively.

5. Students need to learn how to learn by learning to set and make goals, track progress, reflect on their own strengths and weaknesses.

6.  Students need to develop a life long growth mindset so they are willing to take the initiative and develop persistence.

These are excellent strategies, many of which are not that difficult to implement but often when we want to start using new methods in the classroom, we run across opposition because policy is often behind what research shows we need to do.

I know I have to do some of these things but I need to do it in a way my administration is comfortable with.  Let me know what you think about deeper learning.

## Monday, November 28, 2016

### Note-taking in Mathamatics

I grew up in an age where you wrote down everything the professor wrote on the board in the hopes you got it all, it made sense later, and you could decipher those same notes when you read them over.

I never learned to take notes properly but with today's technology, I can open an app, record, and go back later to take notes at my own pace.

The other day, I was working with my seniors and realized they do not know who to take notes as they read the textbook.  So  we took a class period to go over looking for information you want and how to go through examples.  I showed them my thought process involved in working through examples.  Most of them sat there and tried to write everything down but one girl watched and listened and heard what I was saying.

I found two videos on note-taking on you tube.  I went straight to the videos because these students can watch for themselves or you can show in class to give students a better idea of how to do it.

1. Math notetaking from the textbook.  The creator of this video uses colored sticky notes to designate vocabulary, key concept, direct quote, equations and she writes on these notes, sticks them to the pages as she reads the book. This is the first step in note taking.

2.  Next is the lecture notes from class.  She uses highlighters which are the same colors as her sticky notes.  She goes through her lecture notes highlighting certain concepts, then adds the appropriate sticky note to the page.  She shows how to combine the two for a set of well done final notes.

Cornell note taking is often recommended to students because it divides the paper into three sections.  The right hand side is for key points or topic headings while the left side is for the notes and the bottom part can be used to clarify and identify the main points or summarize the notes.  In math, we find that harder to visualize but it is possible to do it.  If you check this site , there are great examples of both good and bad note taking using this system in math.

This pdf shows how to set the page up with notes on how to use the three column note taking method which is very similar to Cornell.  To follow up, this 20 minute video shows how to use the Cornell format in Math done by a teacher who has her students use this method.  In addition, she includes information on how to use the notes later.

As for the actual notes, it is recommended students:
1. Ask questions to clarify the material as needed.

2. Identify important elements of the lecture - could be done by reading the material before coming to class.

3.  Review notes after class.

4. Consider using a tablet or computer to take notes if you feel this would allow you to take notes better. (my students often snap pictures of notes for later.)

5. Record the lecture for use later.

6. Skip words, not numbers.

7. Use color for emphasis.

8. Use a form of shorthand.

9.  Use a three column paper for notes.

These suggestions are strongly recommended as a way of helping students take better notes in math.

Check these out or have your students check these out if they need help learning to take better notes in math.  Let me know what you think.

## Friday, November 25, 2016

### Black Friday and Football

Today is quite well known due to the extreme sales offered by some stores while others are excited due to the extra number of football games being offered.  The question becomes, how do you integrate mathematical activities with this theme to get the attention of all students.

Yummy Math has some lovely activities for the whole thanksgiving weekend.

1.  Macy's Thanksgiving Day Parade.  This activity has students look at the route of this famous parade to determine how far people must march, beginning and ending times, volume of a couple balloons, and a few other things.  Although this is geared for upper elementary, it could easily be adjusted for upper grades.

2. Black Friday Sales. This activity makes students look at various "sale items" to determine the discount and the actual amount saved on popular items like televisions and game boxes.

3. Consumer spending - this activity has students examine graphs on our spending to see if they can spot historical patterns to get a better idea of what might keep spiking.  Almost the same thing that retailers use to determine their best sales.

4. Home Team Advantage for the NFL - helps students learn to read and interpret infographics.  This infographic shows the wins and losses for games played at home and away.  Students are asked to prove whether the team will win more at home or away by analyzing the data.  Perfect for the sports enthusiast.

Yummy Math offers 6 more activities dealing with topics associated with this weekend.  Topics from cooking Turkey and mashed potatoes to football to food banks and shelters.  All very appropriate and very relevant.

Check it out and have fun.  Monday, I'll be talking about ways to take notes.  Have a good weekend.

## Wednesday, November 23, 2016

### Perfect Time of Year.

We are heading into a six week period that is perfect for creating and using infographics.  Infographics are a great way to present information to the world in an easily understood way.

Students need to find and present information in a variety of ways.  The first thought is using graphs, or posters but infographics are the newest way to share information.

The reason I say this is the perfect time of year for creating infographics is that there is a ton of information being collected by companies which students can use.

For instance, one student could easily research the most popular varieties of candy bought on Halloween while another student looks at the most popular costumes bought by adults for themselves or for children.

At thanksgiving time, students can research information on black Friday to see what are the most popular categories of items sold.  They can check into the top brands sold in each category or even what items are best purchased during black Friday sales.

In addition, they can check out the number of people who travel the day before thanksgiving, the day of and the day after by air, train, or car.  There is also the types of meat such as turkey, duck, or geese as the main meat.  What food items are purchased the day before thanksgiving and the day of thanksgiving?

Furthermore, they could check out the projected sales numbers between black Friday and Christmas day. They could check various categories such as electronics, cars, etc over the past 10 to 15 years to look for trends that could be covered in an infographic.

This is the perfect project for just before winter holidays where you need something to keep the students interested but you can't start anything new.  Let me know what you think.

## Tuesday, November 22, 2016

### Differentiating Textbooks

As you know, I work with ELL students who have problems reading the textbook fully.  They have not been taught the skills needed to read a textbook so I have to take time to do it.  I recently came across a book titled “Differentiating Textbooks” by Char Forsten, Jim Grant, and Betty Hollis.

Until I’d run across this book, I’d never heard of differentiating textbooks.  The only method I know to use when reading a textbook, especially a math textbook is what I learned back in high school.  We just read it.  In college, I learned to go over examples but not to do much more.

This book has so much information from grouping students to creating smaller books to techniques and graphic organizers.

I like a couple suggestions made by the authors in chapter two on selecting and adapting textbooks.  I do not have time to create new textbooks but I like the suggestion of substituting headings and subheadings in the form of questions.  This simple move requires students to find the answer to the questions.

They also suggest that students circle and box their math problems because this helps reduce on careless errors.  They suggest circling one type of problem while boxing another type of problem so as to distinguish between the two.  Students work all the circle ones first and the other ones second.

The rest of the book is divided into pre-reading, reading, and post reading strategies with examples.  One of the pre-reading strategies is the Clear Up Math Visuals one.  It is suggested that students had the word problem and then together decide on the visuals one should use to represent the problem.  This is one I need to use with my students.  Its perfect and has them taking more ownership of their work.

A during reading strategy which resonates with me is called Power Thinking.  Students use powers to indicate main idea down to details all on the same idea.  So Power 1 is the big thought such as sports.  Power 2 might be Wrestling while Power 3 names some of the wrestlers they see on television.  Another Power 2 could be Basket Ball while the Power 3 could be the teams.

In math it might look like Power 1 is exponents.  Power 2 might be positive exponents, Power 3 could be examples. So over all it could appear like this:

Power 1  Exponents.
Power 2 - Positive
Power 3  - Makes the result bigger.
Power 2 - Rational
Power 3 - Seen as Fractions
Power 3 - Represents roots.
Power 2 - Negative
Power 3 - represents fractions
Power 3 - numbers get smaller.

This is a nice way to summarize the material.

For the after reading strategy, you might try three facts and a fib where people create groups of four facts but only three are true.  The other people have to determine which is wrong.  This one could easily be used in math.  For instance:

A pentagon is made up of 3 triangles.
A hexagon is made up of 5 triangles.
A decagon is made up of 8 triangles.
A octagon is made up of 6 triangles.

A student has to decide which one is wrong.

Although not all of the suggestions can easily be used in math, there are enough suggestions that I can use in the classroom to make this worth it. I would also say that this book is not so much about differentiating the textbooks as giving students additional reading strategies they can apply.

## Monday, November 21, 2016

### Perfection

There is beauty and perfection in certain numbers in math.  Look at the 3-4-5 triangle which is one of our standard examples in math to explain the Pythagorean Theorem but its also considered the standard in industry.

Its one of our perfect triplets of all time.  I love it and so do carpenters or concrete forms.  In order to create the perfect 90 degree angle, carpenters and people who make concrete forms know to use the 3-4-5 triangle to create the perfect right angle.

This particular triangle has recorded uses back in Ancient Egypt.  They are not sure if the Pythagorean Theorem was known back then but they do know surveyors used the concept for building.

The way to apply this is to take a corner.  You measure 3 feet from the corner and mark it. Then you measure 4 feet from the corner, in the other direction and make a mark.  If you get a measurement of 5 feet between the two marks, you have a 90 degree angle.  If its less, the angle is less than 90 degrees and if its more, the angle is over 90.

I know how important this concept is in building because I helped build a small cabin when I was in college.  They laid the flooring and then build each side so when hefted into place, it was supposed to fit together perfectly.  When we went to put it together we discovered no one used this particular mathematical concept when building.

The floor was not square and each side listed slightly so they were not square.  After a lot of swearing, shoving, and pushing we got it to fit but I will not guarantee how long it stood since the right angles were put in under pressure.  I don't think our supervisors had ever built anything and I didn't know you used the 3-4-5 triangle under this circumstance.

It could also be used to make sure tile or carpet is perfectly square so that it fits in the corner.  So this lovely triangle can be used in any situation where you want a perfect 90 degree angle.  Let me now what you think.

## Saturday, November 19, 2016

I am off working a local wrestling tournament. Be back Monday!

## Friday, November 18, 2016

### Calculating volume

The other day in class, during warm-ups, we discussed the words much vs many.  Since I work with ELL students, they are always mixing the two words up.

I explained that much is usually used with quantities that cannot be counted while many indicated countable quantities.

During the conversation, the topic of calculating the amount of fuel remaining in a tank came up.  This is something that has to be calculated when you buy or sell a house.  A nice real world application.

This skill is also great to know if you want to know how many gallons your fuel tank takes.  Up here in Alaska, most people heat with fuel oil so we buy several gallons at once and its stored in cylinders connected to the house.  In addition, there are water tanks which are cylindrical although I know someone who bought a rectangular shaped one for collected rain water.

So how does one calculate the volume of a tank sitting in your yard? If its cylindrical you measure the radius of the tank and the height(length)  Next you square the radius, multiply the result by the height or length. The final step is to multiply the result by 3.1415 to get the cubic volume but you still have to divide this figure by 231 to find the number of gallons.

To find the amount used, you could use a stick, dip it and based on the depth you could easily do an approximation of the volume left.  For instance if the diameter is 30 inches and the stick is covered up to 12 inches, then 12/30 or 2/5 is left so you can multiply 2/5 by the volume such as 200 gallons so you'd have 80 gallons left.

This could be extended to certain types of travel cups, cylindrical water troughs, columns to calculate the amount of cement needed, etc.

On the other hand if you tank is rectangular, volume is simply length times width times height.  So to find the number of gallons by dividing by 231.  This would be great for calculating the amount of water for a pool, aquariums, holding tanks, etc.

So many real world applications and possibilities for students to work through.  Where are some places students might need to know this?  Let me know what you think.