Attention

I'm sure we've all had the issue where we make sure every student has their eyes on the teacher before explaining something and then have a few students who ask "What are we supposed to do?".  It seems to happen every time.

Have you ever wondered about why they didn't hear you? We’ve  all heard that if you do not get people’s attention within the first few minutes minutes, you won’t get it.


 It turns out, it is a bit more complicated than that.  Yesterday, I listened to some TED radio show as I headed into town.  They interviewed a neuroscientist who has spent her whole life studying attention.  During the first part of the interview
she stated that most people would only hear about 4 out of 8 minutes of her talk.  In certain professions, hearing only 50 percent of what goes on could be detrimental such as a judge or perhaps a pilot.

According to Amishi Jha, we take our mind away from what is happening to let it wander.  We often allow our minds to travel back to events that have already happened.  We want to revisit them or we head to the future as we plan what we want to do.  We do this about 50 percent of the time because thoughts spontaneously pop in and our mind goes off.  This time wandering is one of two types of mind wandering. 

The other type is  often referred to as “daydreaming” where people all their mind to drift.  This type of mind wandering is what causes us to be creative.  This is also the type that occurs without missing other information.  Unfortunately, the down side is when our brains are occupied by other things, we have fewer chances for spontaneous thought to happen.

She said we are caught in this circle of when we experience external interruptions, we begin interrupting ourselves such as when we take a minute to check out the latest book by our favorite author or check our e-mails or texts.  These self imposed interruptions often become more and more frequent especially using mobile devices.  Many of our students stay up late checking out the latest text or responding before waiting for a reply.

Unfortunately, due to the exposure to more and more data, our brains do not have a chance to reboot.  Consequently, the term mindfulness has come into popularity in the last 20 years as the amount of data we subject our brains to has increased.  Mindfulness is where people take time away from life and meditate or enjoy some peace and quiet.  It gives our brains a chance to reboot and level out before we expose ourselves to the gigabytes of data.

So now you know why students often ask you "What am I supposed to do?"  Its because their minds wandered off.  Let me know what you think.  I'd love to hear.

Bingo

Every time I see suggestions for bingo, they are for use in the lower grades but I have been thinking of ways to use in my math classes.  So I am going to share some of the ideas I have come up with. Most of the possibilities require students to choose the answers from a list of possibilities. It is also expected, the list of possibilities will be two or three times the number of blank squares on a bingo card.

1. Common denominators - prepare a list of factors to produce a list of common denominators.  Have students write down several denominators on a blank bingo sheet.  Once students have filled out their cards, it is time to draw two numbers at a time. Students have to figure out the LCM and see if they have it.  See who gets the first Bingo.

2. Trinomials - prepare a list of factors which students choose two for each square. The teacher draws a trinomial for the students to factor. If they have the pair on their card, they can mark it off.

3. Dimensions for area or perimeter - students choose the measurements for a shape. The teacher draws the area or perimeter and students figure out the measurements. This could also be used for volume on certain shapes.

4. Words and equations - students choose from a list of equations and the teacher selects a verbal description and students see if they have it.

5. Algebraic fractions - provide a list of solutions for an assortment of problems. When ready start pulling problems out for students to solve. Once they find the answer, they see if they have it.

Other topics that would work in this same way include ratios, proportions, markups or mark downs, percent increase or decrease, nteger math, or just about any type of math you can imagine. It’s a matter of providing answers for their bingo cards.  It is just a matter of taking a little time to work things  out in preparation.

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

Writing Open Ended Questions

Ask students

We hear so much about integrating open ended questions in class but have you ever wondered
why?  What advantages are there to using open ended questions?  What is the best way to use them in class?

One great thing about open ended questions is there is not one correct answer.  Most textbooks are filled with problems designed to have only one possible solution. These problems are designed to be solved using a specific process.  Unfortunately, real life problems are not always as neatly solved as those in the textbooks.

In addition, these types of problems can be solved by students of different abilities with a differing level of knowledge.  These are perfect for differentiating the lessons to meet the needs of all students. These problems encourage student thinking and decision making. Textbook problems do not encourage students to develope their mathematical thinking.

The last thing these types of problems do is to help students develop their reasoning and communication abilities. The justification for the method chosen when working textbook problems is they use the method taught in the section. The reason for knowing the answer is it was in the back of the book.

The great thing about textbooks is they can provide problems which can be rewritten into open ended questions with a bit of adjustment.  First step is to start with the answer while throwing out the actual problem.  Instead of asking what 1/2 of 12 is, begin with the answer of 6.


Then create a question asking students to design a problem that gives produces that answer. So you could say 6 is a fraction of a number - what is that number and what is the fraction?

In addition there are a few other ways to create open ended questions for students.  One way is to ask students to compare and contrast numbers, shapes, graphs, probability, or measurements.  This requires more than something like 95 and 100 are different because one has 2 digits and the other has 3 digits.

Another way is to have students explain how something such as 5 is a factor of two numbers, what might be true for the two numbers? Or you could write a statement using soft words like almost or close such as two numbers multiplied together is almost 600?

Finally, have students make up sentences using numbers and mathematical words such as 8, 5, less, and. This allows for more than one answer.

If you have students who are not good at expressing their thoughts or using a more in depth explanation, these are some good ways to start students on their paths.  Let me know what you think, I would love to hear.

Happy Memorial Day

Warm-up

Warm-up

Data Mining in the Math Classroom.

  I've been hearing more and more about data mining.  I've decided it is time to look at what it is and how math is used because it is a real life application.   In many ways, it something to know more about as companies use data mining.

Data mining is simply the automated analysis of data looking for patterns and relationships among the data.  Sounds a lot like the definition of mathematics. 

Data mining uses aspects from linear algebra, multivariate statistics and optimization and the calculations can be quite sophisticated.   It is used in real life to find the best traffic route for the morning and evening traffic reports provided by both radio and television or places the tagging on our selfies, or prices of sale items or which movie to see.  We've all used the results of data mining.

This topic is important for students to be introduced to because  it is a prevalent part of our world.  There are a variety of lessons out there on data mining.  Texas CTE Resource Center has a nice introduction to the topic which requires students to do research to find the answers to several questions. 

The Kenan Fellows Program has a lesson on understanding data mining.  This one requires the school to download R statistical software to the servers along with the necessary data sets to analyze.  The lesson includes the lesson plan, student worksheets, and appropriate assessments.  This is geared more for AP statistics but could be done with any higher level math class.

We know that sites such as Facebook have gotten in trouble for using data mining but where does some of the data come from?  The site Teaching Privacy has a full lesson plan focused on your digital footprint and what information you leave with sites as you surf the internet. 

This lesson describes big data which is data that is too big to analyze by normal methods which in a sense describes data mining.  I love that it includes videos, questions, and a discussion of the exponential growth of data. 

Once students understand the basics of data mining, the University of Edinburgh offers data sets appropriate for students to practice data mining.  The sets include those the university deems as interesting and those that are not as good so students can look at both to determine which is better.  The data sets cover topics like particle physics, psychological data, brain-computer interface data, molecular bioactivity for drug design, etc.

So if you want to address the topic of data mining in your math class, this will give you the ability to teach it.  I am trying to figure out where I can integrate it in my lesson plans for next year. 

Let me know what you think, I'd love to hear.  Personally, I believe this is an important topic to introduce and it provides the basis of an authentic task using real world information and provides an opportunity for students to learn more about who uses data mining and for what.

Authentic Tasks

  We hear the words "Authentic Tasks" bandied around but how can we tell if it is indeed authentic or just something labeled that way.  Many of the "Authentic tasks" I find have a feel of not quite being real.  Perhaps its the way they are written or perhaps they are something my students have little experience of.

But I was interested in what makes a task "Authentic".   So I looked up the definition for authentic tasks.

1.  The task must have a clear connection to the real world which means the one I found concerning the ratio of cans of red paint to blue paint was not as authentic as it could have been due to a lack of context such as the ratio needed to produce a certain shade of purple paint for the house.  I have seen houses painted a lovely shade of purple but I'd never do it.

I've seen comments about the real may not be totally real but it might have a connection to literature or to history.  One year I had students determine the number of boats in the village, the average number of passengers per boat so they could figure out how long it would take the people in our village to move the number of soldiers they moved at Dunkirk.

2. It must make students think about it and it must take some time.  If students are finished in five minutes, it is probably not as authentic as it could be. 

3.  It should require students to use different approaches and different ways of representing the problem.  In other words, students should look at it with more than one perspective.  In addition, they need to know and understand a variety of mathematical concepts in order to solve the task.

4. It must allow for collaboration  and discussion which is something my students need to work on.  Right now, collaboration usually means one student does it and everyone else copies while discussion consists of "Let so and so do it".

5. The task must have students find a solution and interpret the solution in context of the situation. The reality is that too many tasks do not allow students to interpret the solution.

6.  A good task will challenge and motivate all levels of learners in the classroom.  Even the struggling students will want to try rather than give up while the more advanced students are not bored.

The next thing in regard to authentic tasks is to consider writing them yourself.  For many situations writing your own authentic tasks is much better because students do not always relate to what we consider the usual.  Out in the village, students are more likely to tell you how much gas the boats will use if you are going over to another village to hunt.   They can't even pop down to the hardware store to buy paint to a specific color specification which is what most people are able to do.

If you want to make your won authentic tasks consider the following:

1.  Make sure it has a real-world connection because it provides a relevant frame for student work.  Look at topics such as immigration, buying a house, entertainment, jobs, throwing a party, etc.  This can provide a bridge between that learned in the classroom and why it should be learned.  It makes classroom learning relevant.

2. The task requires students to create some sort of product.  The product could be an argument, creating a policy, etc. Just keep in mind what type of product can students make to address the problem or situation, who is their audience, and what value does the final product have.

3. The task should be interdisciplinary so students see connections between the disciplines, math, and life.

4. In order to solve the task, students must collaborate because through collaboration and discussion, they increase their knowledge.  In addition, collaboration and discussion are skills needed in the modern work place.

5.  It helps if the task requires students to look for information to help solve the task.

I will probably try writing a few authentic tasks over the summer that meet these criteria.  I've come to the conclusion that not all found on the internet meet the criteria for authentic tasks and if they do, many of my students do not relate to them.

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

Legos and High School Math

Yesterday, I wandered through Barnes and Nobel to check out the latest technology and stem kits they offer.  Right across the aisle, I saw row after row of Legos but they were kits to build things, not the plain blocks to use one's imagination.

I thought about purchasing a few to play with because I'd like to use them in class.  I already know how to use them to teach fractions but I wondered if they could be used in Algebra.  It was easier to find information on fractions, addition, subtraction, multiplication and division but I did find activities for using Legos in high school.

Colin Graham's blog has a list of 20 ways to use Legos as manipulatives in math showing the final result as a picture so you would have to figure out how to do it.  There are no instructions.  I found two that are applicable to the math I teach but most are for maths I do not do such as box fractals.

Some of the idea's I've come up with on my own:

1.  Squares and square roots for perfect squares such as 2 x 2 or 8 x 8.  Most of my students need the activities.

2.  Area - students come across problems which ask them to create as many ways as they can to develop a shape with a specific area such as 24 square feet. 

3.  Volume - students could build three dimensional figures to either figure out the best arrangement for say 108 cubic inches. 

4.  Edges, vertices, faces of cubes.  They could build their own shapes to identify the edges, vertices, and faces for certain shapes.  These could be used as they explain the concept in a video.

5. multiplication of binomials can be shown using legos.

6. Factoring of numbers for common denominators.

7. Adding, subtracting, multiplication and division of variables or exponents.  Many of my students do not visualize what these mean without some sort of manipulative.

8. Cubes and Cube roots - again students can build these using Legos as a way of visualizing this material.

9.  Scale Models - it isn't that hard to take one item and build another based on a scale model.

These are just a few ideas I have but I still have to find some Legos in town I can get to explore these on my own.  I see how to do them in my head but I have to play with them so I can write up the activity for them with the appropriate questions.

I'd love to hear if you have any ideas on this.  Let me know what you think.

Assessment and Reality.

Last year, our school finally invested in and began using MAPs to give us data on our students and their learning.   After about a year, we are starting to get some real results.

The first two times we gave the test, students did not take it seriously at all.  Most students just clicked through it but over time they are beginning to take things seriously.

I got a look at the results for all three tests given over the year.  Yes, most of them increased their scores but I discovered one reason my students have struggled so much when they reach high school.

The majority of them arrive with the skills of a fourth grader.  Unfortunately, this is not just in math, its the same in reading and language arts.  Imagine, trying to teach high school math to students who still struggle with their multiplication and division tables.

I do tend to focus on the students who attend class most days and who actually do the work.  I have a certain number of students who are regularly absent and if you talk to their teachers from earlier grades, they were not in school all that much which means they did not get a chance to learn the foundational material needed to succeed in he higher level maths.

In addition, the same applies to students who choose not to work in class.  I spoke with their earlier teachers and they didn't work there either.  Unfortunately, many of these students are testing in at a second grade level.  I'm glad to have these assessment results because I now have a better idea of where they are at so i can plan accordingly. 

This is the first time in all the years I've worked here that I've gotten an overall picture of my student's ability.  There is a place I can go to get information on what they are supposed work on but its not as detailed as I'd like so I'm just going to have to do my best. 

This is not the only assessment I use but it gives me an overall picture so I can adjust my assessments to focus on learning those specific skills.  Many of my students are so low the skills they need to work on are things such as multiplication or division, foundation skills required for higher level math.

So over the summer, I'll be checking skills, figuring out how to sneak these into my math classes while teaching to their level and meeting the standards the state has determined students must learn. I will admit one last thing. It is frustrating for students to arrive in high school with only a fourth grade ability. 

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

Mental Math Strings.

  I tend to buy books, briefly read them before I put them on a shelf for later.  I've started going through several so I can plan for next year.  One of the activities that facinated me when I read about them are mental math strings.

In mental math strings, you give students a series of mathematical statements such as "The number of inches in two feet" or "The number of vertices in a pentagon" instead of the actual number itself. 

As you give each statement, students need to work the math out in their head. They are not allowed to use pencils, paper, or calculator during the exercise because the activity is for developing understanding of relationships.

Before giving the mental math, the teacher needs to preview the content, procedures and skills needed for the mental math but it does not have to be done immediately before the math string.  It could be done the day before.  When doing the math string, give one step at a time with a break so students have enough time to think of the number and do the math.  This engages the students and allows them the opportunity to do keep the answer in their head through the activity.

Once the students have the answer, do a quick check to see if they got the correct answer.  If any students made a mistake, they can make corrections and explain why the answer is incorrect.  Redo the math string using the same vocabulary, mathematical concepts and procedures but with different numbers.

This is a way for students to recall important vocabulary, concepts and procedures so the information moves from short term to long term memory and they develop a deeper understanding.  In addition, it can be done anywhere in the lesson depending on where you need it.  It can be used as a bell ringer, exit ticket, or just a quick assessment in the middle.

Mental math strings have advantages such as they improve student flexibility in thinking, builds their fluency and confidence, and exposes students to multiple ways to solve these problems. 

There are some things that can be done to help students be less intimidated by those who finish first.  Rather than raising their hands, students can put the fist on the table with a thumbs up or if you use the red/yellow/green cards, the student can choose the green card indicating they have an answer.  Once everyone has an answer, ask students for theirs and to explain the steps they went through to get the answer. 

There are plenty of examples on the internet for having students practice number strings to strengthen their calculation abilities or the more complex ones as explained earlier in the entry.  Give it a try because it will help students improve their over understanding of math.

Warm-up

If I weigh 5.2 pounds with clothing and 5 pounds without, what percent of my weight is my clothing?

Warm-up

Each jelly bean measures .5 inch by .25 inch.  How many can fill a cylinder with a diameter of 4 inches and a height of  6 inches?

Study Skills in Math.

Over the past few years, the number of students who know how to apply study skills has declined.  Very few who arrive in high school know anything about study skills and I think I'm going to have to begin integrating it into my classes.

Some of these skills will include how to apply them to technology because we are using it more and more in class.

Mathematics is one of those subjects which require active involvement to learn the material.  Students need to be active participants because very few become proficient by just listening and reading the material.  Working the problems is an important part of active involvement.

When students do not do the assigned problems, they have less opportunity to understand the formulas, internalize their learning, and transfer their knowledge.  In addition, math builds on previously learned material.  You can't solve algebraic fractions if you never learned to work with fractions earlier on.  Its hard to isolate a variable if you can't work with integers.

So to help students do better, I'll be having students learn to use Cornell Notes for note taking in class.  This is the same system the science teacher has them use.  This will make it more consistent across the curriculum.  I hope to teach them to apply the same system to taking notes off of videos because I'll be showing some but I recommend they check You Tube for help.  Furthermore, I hope to help them learn to review their notes on a regular basis by asking questions from their notes.

The next thing I want to do is take time to help students learn how to read a math textbook.  They have not figured out how to read a math textbook because up until 6th grade, they just have to work their way through consumable books which are not set up in the same way as my textbooks.  I don't believe the middle school math teacher requires them to read the textbook.

In addition, it is important for them to have a place where all the formulas are summarized, kind of like the reference sheets they get on tests.  If they can't read a reference sheet, they have more difficulty doing well on the test.  I hope to have students work on vocabulary words, both words for math and words they are likely to find on tests such as justify, or simplify.

I realize there are others but I think these are the skills that are the most important for my students to learn. These skills will help in college or if they go for additional training. 

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

More Multiplication Apps.

  As commented earlier, its harder for students to figure out common denominators for fractions if they are not fluent in their facts but there is another aspect of multiplication they need to be good at and that is when you multiply the numerator and denominator to change the fraction.

Many of my students struggle when multiplying two digit numbers by two digit numbers.  So I found two apps which can help a student with that type of procedure.  First is the Math edge multiplication app.  It is different because it has two different choices.  The first is step by step which has students practice their multiplying two digit by one digit or two digit by two digit number a step at a time.

It bolds the two digits it wants you to multiply and you type in the answer using the number pad.  As you type in the answer, it shows up on the screen. If a digit has to be carried, it will float up to the area for carrying.  If you make  mistake, the answer is in red and fades away.  Once the multiplication is finished, you have to do the adding to complete the process.  The flash cards limit you to selecting multiplication tables for zero to five.  If you want to go to the 12's you have to upgrade.

The other is Multiplication!! which is free and designed to have students practice multidigit multiplication.  This differs from the other one in that it has a square to type in the digit.  You are required to type in everything here including the numeral you are carrying.  If you place the incorrect digit in the space, it is red and will not move on until you place the correct digit in there.

Every problem involves a different number of digits but it is not timed and there is a tutorial to show students how to do each problem.

The last two are games to practice simple multiplication.  One is M: Mission: Multiplication which has students controlling a rocket to fly through the ring with the correct answer.  It has an old fashioned feel to it but it was created by a high school student a couple years ago.  It requires a bit of finesse to get the rocket where you want it.  The number in the ring turns red if it is incorrect and green if it is correct. I don't last long because I"m always crashing into an asteroid belt.  Best of all, it is free.

The other is Marble Math Lite, a nice little game where you guide a marble through a obstacle course.  Sometimes you hit a flashlight so you have to complete it in the light provided by the flashlight, while other times you hit the wrong answer, or get a free ride card.  It asks the questions in three different ways such as 2 times what is 10 or what factors give you 48, or 3 times 4 is.  You have a chance to look at the problem before you move the marble so you know where to aim. 

 Both games give you a chance to practice your multiplication tables but the marble one is easier for me personally to play but I think students would enjoy both.  I hope these reviews help everyone.  Let me know what you think.

Optical Illusions

It is the final couple days of school so we are basically done with teaching and we are providing fun things for students to do.  I'm teaching one art class while the English teacher is teaching the other.  She does the usual drawing and I'm working with students to create optical illusions, mostly because I can't draw worth anything.

The best thing about creating optical illusions such as the one in this entry, is that it can be done with straight lines.

We started with something simple that didn't require much. we used a hand, straight lines, and a few curved lines to make it look as if the lines were painted across the hand.  I did this to introduce them to the concept.

The next thing we drew was a triangle similar to the one in the picture.  They started with three triangles, extended a couple of lines, erase the corners and voila, you have a wonderful triangle whose sides run into another.  They are often referred to as "Impossible Triangles".  A couple of students had a great time with it.

The third piece we worked on was a set of stairs going down into the paper.  Three students got to this one and had an absolute blast.  The hardest part was extending the stairs up so it looked like it went into the ground.  One person used red instead of black which gave it a really interesting look.  Another got so excited that she wanted to do more so I have to find some for her to do tomorrow.

I already plan to use an optical illusion where it looks like a square hole pushes down into the paper.  It is such a cool illusion.  I also have to figure out how to do a double triangle that looks like its made of the letter Z and the reversed Z.  I have an idea of how to do it but I will have to play with it tonight.

Students often find this really interesting and fun.  Yesterday, I took over a fifth grade class for an hour so the teacher could give make up tests.  They did fairly well but I discovered many of them are not that precise when they follow directions.  Its like they are in too much of a hurry but most of them managed to finish two different ones and they were happy with the results.

I can see where we can use this when I next teach two dimensional shapes in Geometry.  Imagine spending a day or two playing with optical illusions and relating them to the two dimensional shapes.  yeah.  Let me know what you think, I'd love to hear.

multiplication and fractions

Right now, I juggle the balance between meeting IEP goals and helping all students learn.  I have students who require a calculator for doing math but if you've ever tried to do fractions on a calculator, it can be extremely frustrating.

The problem with most calculators and fractions is you have to find the button for fractions and it usually takes two to three repeated motions.  Other calculators do not have a choice of providing fractional answers in fraction form, only in decimals which is not great when you try to explain that 1/3 and .33 are not exactly the same.

I learned to work with fractions growing up but I also knew my multiplication tables.  Unfortunately too many students, even the best ones, struggle with this.  Many of them do not have a solid grasp of multiplication.  Unfortunately there are still places in this world where fractions are used such as if you purchase lumber, fabric, cook, discounts, etc.

One of the best ways I've found to have students practice their multiplication facts is to let them play games designed to let them learn their facts while having fun. Today's apps can really help with this.  I'm not talking about the ones designed to do a electronic drill and kill but the ones that give the students challenges, rewards, everything found in their beloved video games.

The other thing is that the games have to be designed for older students, not first or second graders.  I've explored a few free multiplication apps and reviewed them here for you.  A couple are games while others are flash cards.

1. Times Table Game offers both a free version and a paid version.  The free version offers a chance to learn your factors from 1 x 1 to 6 x10 but anything higher, you need to actually purchase the paid version to get everything.  The format is you have a problem with four choices.  You choose the answer.  If it is correct you get points and if not, it flashes the problem with your chosen answer and a question mark all in red.  There are two formats, one is timed and the other is not.

2. Multiplication Table Game has three modes, learn, practice, and test.  Learn is exactly what it says.  It shows the table for the number you want and you learn from times one to times 10.  Practice has you choose the one you want to practice such as 8's.  A problem appears on the blackboard with 8 choices at the bottom for answers.  If you incorrectly identify the answer, it has a red x and it has you choose another answer.

The test gives you 20 problems to complete and if you miss a problem, you get automatic feedback.  There is even a 2 player option where they share a tablet and players are given the same problem to play against each other.  Each works on answering the problems but it records the first person with the correct answer and it keeps giving problems as they are answered.  When one person hits 15 correct answers, they are proclaimed the winner. 

3. Multiplication or Division Flash card game is another one which you have to buy the app to receive full access to everything. Although it is geared for second and third grade,  I believe it would work in high school because of the way it is designed.  When you click on multiplication you have a choice of practice, quiz games, or match games. 

Practice only allows you to practice 1 to 4 .  You have flash cards which can be used in the normal way regular flash cards are used.   You are given a choice of two quiz games.  One is the flash card quiz which has you selecting the correct answer out of three or the speed challenge where answer as many as you can in a certain time.

The last is the match game which gives the player one of two choices.  The first is to match the equation to the answer while the other has players match the equation with another equation that has the same answer.  This is the part of the app I love because it helps students connect equations which have the same answer rather than looking at them separately.

Tomorrow I'm going to review a few more including two which help students learn to do multidigit multiplication.  Let me know what you think, I'd love to hear.

Apps, Free apps, Paid Apps, Which One Should I Get.

  I love free apps.  I love them personally and I love using them at school.  When given a chance I go for the free apps.

Over the years I've come to the conclusion that free apps fall into one of three or four categories.

First, there are the free apps for companies whose material you have to have a school subscription to in order to use such as Power School or certain reading, writing, or math programs.

Next are the apps you can subscribe to privately for so much per month in order to access their services.  The only thing that is free is the app and possibly a trial. Some of these apps are designed by private citizens who are interested in monetizing their product.

The third type of app is the free app which shows people what the app can do but to unlock most of the features or the premium ones, you have to upgrade to the paid version.  I like these because they give you an idea of what the product can do before you buy it.

The last type are those which are totally free and do everything you want them to do.  The only problem with these apps is the developers often move on to their next app and these are not updated so they cannot be used after a while.

As a school teacher, I hate purchasing an app I have not tried.  I am afraid of buying the app only to find out it is not what I wanted or it didn't do what I thought it did.  For me, I love the light or free versions of the app.  It gives me a chance to taste it and discover if it is something that will work for class. 

I know an app that requires some payment is signaled by having the phrase "In-App purchases"  but I can never seem to find the place in the adds which tells what you have to purchase.  Are you purchasing access to the materials?  The server?  What? 

Still, I'd like to know what I am going to have to pay for.  Its nice to know before downloading an app if you have to pay for the service, for additional levels, or for premium items.  Often, I can get by without the premium features.  I'd also like to know if the app needs the internet to work or if you have an offline option.  I often end up in places without reliable internet and cannot use apps that require internet connections.

The bottom line is I prefer the lite or free versions so I can test the app first before investing in it.  This is important for school teachers because we hate buying apps only to find out they are not what we wanted. 

Let me know what you think.  Tomorrow I'm looking at multiplication apps for high school.  Let me know what you think.

Warm-up

Warm-up

JIgsaw and Math

I've had people teach the Jigsaw method in professional development sessions but I've never really figured out how to use it in class other than for reading the textbook.  I don't mind using it for that but I'd like to extend its usage to other areas of math.  Let's start with how the jigsaw technique works in general terms.

The first thing is to divide the reading into four segments so each group of four students can have one segment.  The idea is that each group reads and learns the material well enough to teach it to other groups because each group is an expert on that part.  This is great for reading but I want to apply it to other mathematical learning.

One place the Jigsaw technique can be used is for factoring polynomials with the leading coefficient of one.  You could break the factoring into difference of squares,  binomial squared, or the regular ones.  Or you could have students look at the situations where two factors are positive, the two factors are negative, the larger of the two factors is positive or the larger of the two factors is negative.  Let each group take one situation, become an expert in it before teaching it to the other groups.

Another situation which Jigsaw could be used when teaching the four or five different methods of proving congruent triangles, or the ways for proving similar triangles.  Both are important in Geometry and it seems to me that the Jigsaw technique would be perfect for either one of these.  In addition, it could be used for medians, altitudes and bisectors.

Think about using it for rotation, translation, dilation, or reflection in Geometry or with families of graphs showing parent graphs and their transformations.  So a group could become an expert on the transformations or with families of graphs or both.  It wouldn't be hard to create a jigsaw activity for either topic.

These are just a few ideas of topics in addition to end behaviors, continuity, trig ratios, and so many more possibilities.  I know jigsaw could be easily used for reading but what about having students become an expert is the actual process of solving processes, analyzing graphs for end behavior, continuity, etc.

They could also find real world applications for the concept to share or teach each other how to do it.  There are so many possibilities.  If they are uncomfortable in trying to do it in person, they could create a video for others to watch or they could create a slide show.  The best thing about having them create a technologically based artifact is that students can review the material time again and again.

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

Two Truths and A Lie

I saw something on Twitter which made me want to explore the idea more.  I thought it sounded great and would add something to my math class.  The technique is called "Two Truths and a Lie".

The idea behind it is the teacher provides either a math problem or graphic and they are responsible for creating two truths and a lie about it.

The students then share their truths and lie with the other students to see if the others can determine which are which.  Once the lie is identified, it has to be corrected so it is now true.

The cool thing about this activity is that it encourages deeper thinking especially if you require a bit more than the obvious such as the line is straight, there are two dots, its pink.  You'd have to model what is expected such as the linear equation has a slope of -1/2, the values are decreasing, or it does not have a constant slope.  The third would be a lie.  When they identify the lie, they have to explain what it should be so it is now a truthful statement. 

In addition, it requires students to use mathematical language when they are writing so they are able to develop and extend their base vocabulary.  We are always looking for ways to increase writing in math and this provides it beautifully.  There are at least four ways this activity can be used in the classroom.

1.  It can be used during the warmup by showing students a picture or problem with the three statements already on them so the students just need to identify the lie and explain why its a lie or you can put a picture up and let the students create their own two truths and lie.

2. Create a gallery walk out of the student creations.  Post them around the classroom and have students visit each one to determine which are truth and which are lies.  Students can work individually or in small groups.  If they work in small groups, student can create one note with an explanation they worked out together.

3. Take several pictures and place them around the room with one statement that is either true or false.  You can let students know ahead of time how many of each there are but they have to determine if they are true or false and justify their answer.

4.  Guess who in which one student at a time gets up to share their two true and one false statement with the class, even going so far as to identify which are true and false.  The job of the students are to guess what is being described. This requires analytical thought.

5.  This activity can also be used with vocabulary words to help student develop better usage of words especially words which have different levels of meanings such as product, sum, etc. 

6.  Have students work three problems so two answers are correct and one is incorrect.  Divide students into groups of two and they exchange papers.  Each one has to figure out which problem is incorrect before correcting it and explaining why it was incorrect.

I think these are some great ways to use this ice breaker activity in class.  Let me know what you think, I'd love to hear.    Remember, many of thee activities can be done via Padlet, on white boards or on paper.

VR and Mathematics.

  Personally, I own a couple of Virtual Reality viewers. One is a google cardboard viewer while the other is more like the one in the picture which I got on sale at a Barnes and Noble.  I've seen so many movies and other applications for social studies but not so many for Math but the other day, I found a something that sparked my interest.

I downloaded several VR math apps from the iTunes store but honestly, I wasn't really impressed with them.  I'll go through them and tell you a bit about them.

1. VR Game Math created by Turker Guney.  In the game, the app needs access to your camera.  It flashes problems such as 1/1 with 3 dimensional numbers floating around the viewer.  The idea is to touch the answer a couple of times so the program recognizes you have grabbed the correct answer.  This appears to be for early elementary but honestly I didn't get past level 2 because the problems were so simple.  The good thing is the user does not need the internet to use it.

2. Math VR app by ACE - Learning. I didn't even try it really because the creator charges about $10.00 per month  Singaporean to use the product.  I wasn't willing to do a free trial so I do not know how it works however according to their web site they have information for simple or composite solids, volume, standard forms, inequalities, Venn diagrams, graphs of height against time, trigonometry ratios for acute angles, bearings, and 3D senarios.

3. VR Math by VR-AR Education.  This is a free app but there are indications it is still being built.  The application has some great three dimensional shots for use in Geometry that one can move around to explore the whole shape but it is missing some information to do the work.

It covers on vertices, edges, faces, shapes, volume, calculating volume, calculating area, and distance between vertices.  One big problem I see, is when you choose the activity to find the area of a cube, there is no indication of length. I have a question but nothing to tell me how long a side is.  The other problem is if you choose the wrong answer, it only tells you that you are wrong, not why.

A few topics have videos with explanations which helps.  This app doesn't seem to require internet but it does have an option to get an account.  I didn't bother  as I was able to access certain things without it.  As I stated earlier, it appears to be a work in progress but out of all three apps, this seems the best developed.  With a bit more work, the app should be great. 

I used one of the activities to show a student how the surface area of a cube was calculated.  I was able to move it around to show the faces.  I see being able to use it in class right now to supplement my teaching.  One other thing.  You do not need any fancy VR glasses to see the material. 

Let me know what you think, I'm interested in hearing.

Think Pair Share in Math

  I always hear from the English and Reading teachers, students should do a think - pair - share every time they read something new but I have no idea how to do it properly in math.  Its not like my students discuss the plot or the setting in math so I am not sure how it works.

When I ask them how to apply it in math, I get those funny looks of "I don't know, but you should, you teach math" looks. 

The idea behind Think Pair Share is for students to think and clarify their understanding about a reading, a problem, or a question by talking to a partner.   This process can help deepen their understanding, and develop skills such as listening, rephrasing material for clarity, and learning to disagree politely.

It turns out it really isn't that difficult to incorporate it but it does take a bit of skill because you can't focus on only reading the textbook.  It can be used :

1. To read word problems and identify the most important versus the least important information.

2. Discuss methods of solving a problem and settling on one strategy after discussing the merits of each method.

3. Think Pair Share can be used to help students activate prior knowledge, consolidate knowledge from both partners, or understand a problem.

4. Use it so students review each step of the process so they understand what needs to be done.

5. Have students review use this method to review the material.

The next question is "When do I use this in the lesson?"
Well, it can be used in the beginning, middle or end depending on what you want to accomplish but there are three parts to doing the Think Pair Share. First, you have to decide when it will happen in the lesson.  Choose a time when students need to reflect on the material to increase deeper learning and understanding.  It might consist of reading the text, or doing over the material the teacher just taught.

The second step is to have students take several minutes to think about the material and write down ideas.  Be sure to set specific expectations and the focus of the thinking and sharing.  Students need guidance when they are learning the process.  Remind students to use paraphrasing and clarification.  The last step is to call upon pairs to share their thinking with the whole class.  Include a follow up assignment to write in their journal.  Consider switching  the pairs so the same two people are not always working it together.

You can also use questions to initiate thoughts to write down before they discuss the topic with their partner. Some of the starters might be:
1. List four things you know about _______?
2. Write a definition for ________?
3. What is the difference between ________ and _______?
4. Think about different ways you can ___________?
5. Look at this diagram.  In three minutes, I'm going to cover it and you need to write down what you remember about the diagram?
6.  Read the directions and rewrite them for Grade _______.
7.  Summarize the mathematical concepts we covered over the past week.  What do you still have questions on?

I've never used it in class because I didn't know how to use it.  Now I know so its something I can use in class next year and I hope it helps my students learn the material better.  Let me know what you think, I'd love to hear.

Computational Thinking

I forgot where I read that schools should be teaching computational thinking across the curriculum rather than saving it for computer science. 

Of course being me, I needed to determine exactly what is meant by computational thinking.  After all, computational thinking is usually associated with computer science because it is the way programmers break down a problem in order to create a solutions.

Computational thinking is broken down into four sections.  It is defined as being able to processes, data, or problems into smaller, more manageable pieces.  It requires a person to find patterns and trends in that data along with identifying what created these patterns.  The final step is to create step by step instructions for solving the original and similar problems.  In other words, decompose the problem, recognize the pattern, focus on the important information only or abstraction, and creating the solution.

We can see how to use it in Math but what about other subjects.  In literature, students can break a poem down to analyze its meter, rhyme, structure, tone, diction, etc.  In Economics, it is possible to find the cycles of rises and falls of a country's economy.  In cooking, people create recipes which are step by step instructions.  In chemistry, it is possible to explain the rules for chemical bonding.  Each of these matches a step of computational thinking.

Other examples found in education in areas other than math can be:

1.  For civics, government, or history look at the American justice system system to identify problems and propose solutions so it is more equitable and fair.

2. In P.E. decompose LeBron James's dunk using motion tracking data to understand his throws better.

3.  Back to History, students can create their own civilizations, selecting natural resources, and technologies to see how they grow over the years based only on those two factors.

4. What about creating 3 dimensional amusement parks based on the human body with blood cells represented by bumper cars, and neural networks based on zip lines.

5.  Analyze real world earthquake data in science to determine where the most seismic activity occurs so as to determine how much the plates contribute to them.

6. In Math, have students prepare a list of draft choices for a fantasy football or basketball league by looking analyzing performance data.

If you'd like to learn more about it, google offers a nice class on computational thinking while you can find lesson plans here for math, science, social studies and ELA.  I took a look at some of the lesson plans and they look nice. Although, they are geared for upper elementary, they could easily be adjusted and used in middle school and high school.

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

Warm-up

Warm-up

Robotic Movement and Math.

Yesterday, I finished a class I signed up for on Future Learn.  Future Learn is a site filled with free classes that range from two weeks to eight weeks long.  I tried taking two other classes but life interfered and I was unable to finish it before the time limit hit.  I could have paid a bit of money to get unlimited access to the class as long as it exists on this site but I chose not to.

I will try those classes again in the future but maybe in August of next year.  I just finished a two week short class on teaching students with complex trauma which is a great introduction to the topic.  Today as soon as I finished it, I found another class I want to work on over the next three weeks.

It is a short class - Introducing Robotics - Making Robots Move a class designed to look at the geometry and vectors involved in making robots move along with describing where the robot's position in using position, rotation, translation and orientation.  It looks at forward and backward kinamatics, movement in two dimensions and a touch of movement in three dimensions.

In the process of robotic movement, you have to be aware mathematically of the objects that robots move, along with their position and orientation.  Furthermore, describe the mathematical relationship between the robotic joints and the tool position.

The classes are set up so you can just follow along but if you want to complete all the exercises you are given access to MATLAB via Future Learn.  I've never used MATLAB so I will be learning a new technology.  In addition, I'll learn more about applying mathematics in the form of geometry and vectors in a real world situation. 

This class does require you to be up on analytic geometry and linear algebra including points, vectors, matrices, matrix-vector, matrix-matrix-multiplication, and linear transformations.  I'm up to date on the first but a bit rusty on the second half as I don't usually teach matrices unless I've got students who are that far along. 

This will give me the carrot to wave in front of my more capable students who need a bit of a challenge and need to see the math in a real life situation so they can see the connection between the theoretical and the application of the concept.  I'll keep you posted on this class as I work my way though it.  There are a couple more robotics classes I've seen but I'll wait to share those with you as I enroll in them. 

If this goes well, I might even set up a class within my regular class to have students sign up and work their way through the material rather than have me try to teach it because it will prepare them better for taking distance classes in real life.  So many possibilities.

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

Supporting Mathematical Vocabulary

I've been wondering if there is a time in the lesson when introducing vocabulary is better than others.  Is there a way to introduce it that is better than giving a list of words for the students to copy definitions from the glossary.

Is there a way to help students learn to define mathematical terms in their own words rather than than relying on the textbook?

The thing about mathematical vocabulary is a word can often describe a concept so trying to preteach the vocabulary is much more difficult.  It has been suggested the teacher take time to divide vocabulary into one of three categories.  The first category is words which could be pretaught such as reduce or simplify because these words often mean the same thing in math as they do in general usage.  The second category are those words that should be taught at the same time the concept is being taught such as numerator and denominator.  The final category are words which should be taught after students have had the chance to explore the math.

In addition to this are the fact words can be classified as general everyday words such as hello, good bye, etc while the second group are descriptive or words that could be either mathematical or general such as product, or increasing at a slower rate.  The third group are those words which are quite specific such as vertex or hypotenuse.

There are three places within a lesson where vocabulary can be introduced.  It can be introduced at the beginning of the lesson, the middle or the end.  The choice of where depends on what the vocabulary support provides the student.  In other words, they need the vocabulary to engage in the lesson but not to ruin the challenge of the lesson itself.

When preteaching vocabulary at the beginning of the lesson it can give students what they need to participate in the lesson because they understand the means as they read, write, speak or listen to the mathematics. In addition, it helps them interpret the math so they can explore things better but if done at the wrong time, it can interfere with learning.

If the vocabulary is associated with the new mathematical concept, you do not want to preteach the vocabulary.  You want  to introduce the vocabulary in the middle of the lesson when teaching the concept.  There are several ways to do it such as pausing class to have a student share what they've learned about a new word, or perhaps have all the students can participate in a think, pair, share activity.

When vocabulary is shared at the end of the lesson, it is used to formalize key concepts taught during the lesson.  At this point, students are engaged in using the new mathematical terminology so they have connections between the vocabulary and the concept.  It is important to have this occur so students become more fluent in mathematics.

Once the student has learned the vocabulary and the concepts, it is time to have the student create their own word wall using one or more apps, a poster showing their understanding, or other visualization. 

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

Checking Work? How?

I discovered the other day, my students think checking their work means one of two things.  They either glance at it and make sure there is an answer for each and every question, or they want the answer sheet to check against their answers.

It is really my fault that I did not identify the problem before now but I honestly thought they came into high school knowing how to put the solution back into the original problem.

I admit, I assumed they got through elementary and middle school being told to check their work in the normal manner but they haven't, just as if they are not required to show all their work.  The would rather put answers only.  I do not know why.

Since this school year has about two weeks left, its too late to really start working on having them check their work, so next year I plan to tackle it. In the meantime, I am reading up on ways to help my students learn to check their work be it on a test, a quiz, or on the regular assignment.

I ran across a suggestion of breaking the idea of checking work into three levels. The first level is for students to check to make sure they have answered all questions.   This is a good tactic when you run out of time.  The second level requires the student to read the question before looking at the answer to see if it makes sense.  If it doesn't make sense, you check your work.  This helps eliminate simple mistakes.  The final or third level is when students reread and rework the questions. 

I don't usually do any of those levels myself.  I am more into having students take the answer and put it back into the original equation to see if it works out.  I have no problem with them using a calculator to double check the math in this way.  This is the way I was taught to find a mistake.  Once I found an incorrect answer, I then checked the math in each step to so if I did something like 3 x 2 = 5. 

One thing I have done to help them begin the process of checking their work is to have them solve problems on worksheets with answers already there. If the student does the problem correctly, it will yield a letter to write in a blank.  If they do the work correctly, the end saying answers the question at the top.  The down side of doing this is when my students are about half way through a problem, they look at the answer to "find" the proper one.  Unfortunately, they often write the problem down, show some of the work and jump to the answer.  Sometimes they are right, sometimes not.

Next year, I am going to use fewer of those.  I am thinking for the first few weeks, I'll put the answers up on the wall but the answers will be shown in the form with the answer in the original problem so they get used to seeing it in that form.  I suspect its going to result in "Do we have to check our answer?" because I already get the "Do I have to show my work?" and "Do I have to write all that down?".

If you have any suggestions, I'd love to hear from you.  This is one problem I'd love to hear suggestion on. Thank you.  Have a great day.

8 Informal Assessments.

  Yesterday, I spoke about assessing students who cannot express themselves in written form.  This lead me to wondering if there are informal assessments I can use in class to tell me how they are doing in learning the material other than standard quizzes or tests.

The other day while searching for additional information on assessing students who do not write well, I discovered a lovely eight page pdf filled with informal assessments.  Some are based on writing but some can easily be done using alternative ways so all students can participate.

I am not sharing every suggestion, just a few I like and ways I see using them. Yes I already downloaded the pdf because I need more informal assessments to use.

1. Application cards on which students write real life applications which use the math being studied.  These could be shared on a Padlet or similar type wall so everyone can see.  In addition to the application, students can include a bit more on it such as how the application is used.  Students could include a photo with the description.

2. Blogs replace the paper as math journals.  The good thing about blogs is they can be done in both written and video form.  In the past, Mac OS has had a blog part that could be set up to be viewed for the whole school.

3. Chain notes in which a teacher writes a question on the outside of the envelope.  The envelope is passed around.  Each student reads the question, writes an answer on a piece of paper before slipping the the paper in the envelope.  Then the envelope has had a chance to make it all the way around the classroom, it is returned to the teacher. 

4. Directed paraphrasing where students are asked to explain something in everyday terms with a specific audience in mind.

5. Gallery walk where the teacher places 3 to 5 questions on large sheets of paper or on an iPad.  Small groups of students move around the gallery, looking at questions, discuss the questions before recording an answer.

6. Minute questions  where students are given to answer a question in one minute.  The teacher  collects the answers, makes a quick response, and returns the papers.

7.  Traffic cards - Each student ecieves three cards, ed, yellow, nand green.  The student can flash the cards during the lesson to let the teacher know if they need help.  For instance, if they flash a yellow card while the teacher is talking, it might mean, please slow down., you are going too fast.

8. Umpires is where the teacher glistens to a student answer before calling upon another student to determine if the answer given is correct.

These are just a few ideas given but many of the suggestions can be done quickly and easily.  With a bit of adjustment, technology can be used instead of paper.  I look forward to trying use several of these next year.  Let me know what you think, I'd love to hear.