iPad as Mathematical Communication Tool – Part Deux

I have been learning about educational uses of the iPad. My daughter has been learning about repeating patterns at school (Grade 1). Also, she has been asking me to show her how to use iMotion. A win-win situation.

She built four patterns, taking a photo each time she added a piece. Then, she created a video which I dragged into iMovie. Finally, I recorded her as she talked about her patterns. The movie would be better if the audio were synced to the video, but I wanted to see what we could create in ten minutes. Here it is:

In primary classrooms, students could share their videos and have classmates describe or translate the patterns. Similarly, in high school mathematics classrooms, students could build functions and have classmates determine equations. See an interview of UC Berkeley Math Education Professor Dor Abrahamson for the inspiration behind this idea.

These student-created movies could be used by classroom teachers to assess what students are able to do. There are nine mathematics learning outcomes in the BC Kindergarten IRP. One addresses Patterns:

B1 demonstrate an understanding of repeating patterns (two or three elements) by
– identifying
– reproducing
– extending
– creating
patterns, using manipulatives, sounds, and actions [C, CN, PS, V]

In Grade 1, two small changes are made to B1 and a second PLO is added:

B2 translate repeating patterns from one representation to another [C, R, V]

What judgements could you make about my daughter’s performance in relation to the prescribed learning outcomes? A rhetorical question – I’m not expecting or even wanting a reply.

My daughter also told me that sometimes shape and size can be used to describe patterns (e.g., “circle, circle, square, circle, circle, square” or “small, big, small, big”). Our movie doesn’t demonstrate this knowledge. This speaks to the importance of having conversations with our students – from Kintergarten to Calculus.

Running naked again. This time, with scissors.

In an earlier post, I shared a poster that I created using photos of odd numbers taken by local photographer mag3737. He shared this pictorial representation of the Pythagorean Theorem with me. Very cool – both the image itself and the online sharing.

I created another poster from mag3737’s photos – a pictorial representation of exponential growth.

The rate at which exponential functions grow can be a difficult concept to visualize. Starting at 2^4 = 16, the area is doubling but the height is not. By doing this, I’m not sure if I accomplished my goal of illustrating exponential growth (although I did manage to have the numbers fit on the page).

I suggest taking scissors to this poster. The eight columns of eight 64’s that represent 2^6 stacked on top of each other reach a height of over five feet. This is a powerful (and perhaps surprising?) image of exponential growth.

Instead, because of necessary scaling, students often see something like this:

What other mathematical concepts could be represented using these photos of numbers?

PDF’s of the posters: sum of consecutive odd numbers & powers of 2

“Me and Math … are barely on speaking terms.”

“Me and Math were good buds until high school. Then we started to drift and now we are barely on speaking terms.”

“I am naturally drawn to stories. The romance of a and b just doesn’t resonate with me like that of and j (Romeo & Juliet).”

This week, we met with a group of student teachers beginning their first teaching practicum. They were asked to reflect on their experiences learning math. Most of the responses were negative. I listed two of the more creative ones above.

Similarly, whenever we meet with a group of experienced teachers (and/or administrators) and announce that we will begin by solving a math problem, there is always a noticeable groan. You can feel the stress level rise in the room. And remember – this is always a group of educators who have voluntarily signed up to learn about teaching mathematics.

Two themes came out of the student teachers’ comments. First, for many there was a specific grade when math stopped making sense. Often this was grade 8, but grades 7, 11, and 12 were also mentioned. Poor teaching and/or an inability to remember which rule to apply were given as reasons for this. Second, many didn’t see themselves as “math people”.

From this, there are two things to consider. First, it is important that non-specialists be self-aware and not transfer this fear to their students. Second, it is important that we (secondary math teachers) provide our students with opportunities to make sense of mathematics concretely. An emphasis on the procedural may have worked for us math geeks (I might argue this later), but it didn’t work for the majority.

A problem-based approach addresses the above. Students (and teachers!) build conceptual understanding while having fun.

Aside from getting the chance to teach students in many different classrooms, the most rewarding part of my job is hearing teachers say “I used to hate teaching math, but now I love it”. This can’t help but be a great thing for Surrey students.

Multiple Multiples

“Can you show me another way?”

Multiple representations show students there is more than one correct way to do the math. This is an important message in itself.

Multiple representations also allow students to learn new mathematical concepts and procedures.

For example, division can be thought of as sharing or grouping.

8 ÷ 2 = 4 can be thought of as:

  • I have 8 items. I share them equally between 2 people. Each person gets 4 items.
  • I have 8 items. I put them in groups of 2. I can make 4 groups.

I prefer the adjective flexible over multiple. Adaptability of, not number of, is what is important.

To learn how to divide integers and fractions, students must be able to visualize both representations.

For example, -8 ÷ 2 can be thought of as sharing equally between 2 groups. Each group contains four negative counters. However, -8 cannot be put in groups of +2.

Alternatively, having a negative number of groups does not make sense. However, -8 can be put into groups of -2.

Think about why 6 ÷ ½ is 12 (without simply applying the invert and multiply rule). Having a fraction for the number of groups doesn’t make sense. However, students can explore how many halves there are in 6 using pattern blocks or number lines.

3 × 2 is more than simply 3 groups of 2.  An understanding of an area model of multiplication helps students to learn two-digit multiplication.

An understanding of this model will help students make connections between multiplying binomials and multiplying two-digit numbers.

As a secondary department head pointed out a meeting last year, teaching how to multiply binomials may be easier than teaching how to multiply two-digit numbers – in algebra, there isn’t the added complication of place value.

Now if only we would stop using the term FOIL…

A pictorial representation that will have you running naked through the streets

The sum of the first consecutive odd numbers is a square number.

Why? What do perfect squares have to do with odd numbers? At first glance, these are two seemingly unrelated types of numbers.

Some of us (okay, me) may have presented something like this:

1     +     3     +     5 + … + (2n – 1)
(2n – 1) + … + 5     +     3     +     1

The sum of each column is 2n. We have n columns. The total is then × 2= 2n². We added the sum twice so 2n² ÷ 2 = n².

Can you see what perfect squares have to do with odd numbers? Me neither.

Compare that with the following explanation¹ given in Paul Lockhart’s “A Mathematician’s Lament”.

Inspired by this pictorial representation, I created this poster below.

 ¹ Lockhart might say it’s not the fact that perfect squares are made up of odd numbers which can be represented as L-shapes. What matters is the idea of chopping the square into these nested shapes.

The more sides you have, the smarter you are.

“How does shape affect your place in society?”
“The more sides you have, the greater your angles. So, the smarter you are.”

Two years ago, I created a lesson on Angles in a Polygon. The ‘hook’ was the opening minutes of the animated film Flatland: The Movie. In the story, Arthur Square asks his curious granddaughter if she has memorized her ‘laws of inheritance’.

Hex replies “Isosceles triangles have baby equilateral triangles. Equilateral triangles have baby squares. Squares have pentagons. Pentagons have hexagons, like me! And each new generation gets one new side until they get so many sides they look like a circle and become a priest.”

This film interestingly addresses many mathematical concepts, such as points, lines, and shapes in zero, one, and two dimensions as well as larger themes such as critical thinking.

Here it is:

I think it’s a pretty good lesson, but I decided to tinker with it. Here’s the new and improved version:

Yep. That’s it. Blank space.

I learned that from Sandra Ball when planning together for elementary school demonstration or team-teaching lessons. Just one of the many things I have learned from Sandra since joining the team a year ago.

The first activity is overly scaffolded. In the second version of the activity, the scaffolding is removed. Students will ask “How can I solve the problem?” versus “How does Mr. Hunter want me to solve the problem?”. Some students may need scaffolding, but I can better support these students by listening to and observing them. In the first assignment, I assumed all students would need scaffolding. And, really, if my students can’t think of using a table to organize information, what does that say about how numeracy is taught in my classroom?

Here are the documents as well as the three-part lesson plan:

Flatland Assignment
Flatland Assignment 2.0
Flatland Three-Part Lesson Plan

Math Manipulative of the Month – Pattern Blocks

MMM September 2011 Pattern Blocks (colour printer, double-sided)

Last year, a group of Surrey teachers suggested having a “Math Manipulative of the Month” at their school. Instantly, I thought this was a great idea. After this conversation, I created the brochure above. My hope is that this series of brochures can be used to generate conversations between teachers (and students, of course!).

Before trying the problems, I would ask teachers to get to know each MMM and list all they know about them. For example,

  1. “Two reds cover 1 yellow”, “Three triangles make 1 trapezoid”, etc.
  2. “All sides are the same length, except the base of the red trapezoid. It’s twice as long.”
  3. “The orange square and tan rhombus do not cover the other tiles.”

The symmetry problem ended up on the cutting room floor. Here it is: Pattern Blocks Symmetry.

Also, please see how the question “How many ways can you make 360 degrees?” becomes a problem-based lesson in Grade 6. Here’s the three-part lesson plan: Angles (format from Van de Walle).

I attempted to have a balance of primary and intermediate problems. How can each problem be adapted for the grade level that you teach?

Next month… Base Ten Blocks.

Never let them see you smile.

At least ’til November.

Anyone else remember being given this advice by veteran educators at the start of your teaching career? The thinking here was that it would prove too difficult to get students back on track once you loosened the reins. If you must, loosen up at the end of the semester. I could never pull this off. My true self, or at least my true teaching self, would make a special guest appearance by the end of the first class.

I often struggled with planning for the first day of classes. I’m just not able to lecture students for 75 minutes about consequences of unexcused absences, procedures for handing in homework, and lists of food & drink items that are acceptable to have in the classroom. Imagine sitting through this four times on Day 1. Welcome back!

“And one more thing… here’s a review worksheet that covers everything you should know from Math 9. See me or a counsellor if you’re having difficulties with it.”

I was also uncomfortable with the let’s-get-to-know-all-about-each-other approach. No “Find someone who…” searches for me.

When students left my classroom for the first time, I wanted them to believe that

  1. We were going to get to know each other as people, and
  2. We were going to do this while learning mathematics.

Here’s a PMa 10 1st Day Jigsaw activity that, although not perfect, attempts to convey this message.

I cut the squares and placed them in envelopes. In small groups, students pieced the puzzle back together so that questions and answers shared a common edge. An answer key is not provided, but the jigsaw puzzle part of the activity does provide students with some feedback.

These are not rich problems – they are review questions of important concepts & procedures from Math 9. However, I did listen to some interesting conversations. For example, in many groups, there were debates about which power (3^-2, -3^2, or (-3)^2) was equal to -9. One student said he remembered that a negative means flip (his words, not mine) and matched 3^-2 with 1/9. His group members asked him to explain why this works.

Please let me know what you think of this activity. Also, do you have a Day 1 lesson to share?

As a new school year begins, are you looking for posters to decorate your classroom? Learn how to create a gigantic math poster of your own.

Revisiting Pictorial Representations of Functions

The K-7 word walls were developed by my Numeracy Helping Teacher colleagues to help students and teachers communicate mathematically. They were not meant to ‘teach’ concepts but to help make visual and conceptual connections. The cards have been very popular with Surrey teachers. The Math 8 cards have been created and we will be sharing them with Surrey secondary teachers starting in September.

See the sample cards to the right. In an earlier post, I mentioned how concrete and pictorial representations of linear functions can enhance understanding. For example, in the expression 2n + 1, the coefficient of 2 can be interpreted as adding 2 tiles as the pattern continues.

The coefficient can also be visualized in another way. It may be easier to describe by looking at the card for constant. In the first figure, we can see two groups of one (one white square above the red square and one white square to the right of the red square). In the second figure, we can see two groups of two (one group of two white squares above the red square and one group of two white squares to the right of the red square). Similarly, in the third figure, we can see two groups of three. Finally, in the nth figure, there will be n groups of 2, or 2n, white tiles.

This can also be an interesting investigation when teaching quadratic functions (or a challenging extension when teaching linear functions). In the pattern to the right, the red squares in the first figure make a 2-by-3 rectangle. The red squares in the second figure make a 3-by-4 rectangle. We can see a 4-by-5 rectangle in the third figure. In the nth figure, there will be a rectangle with width n and length n + 1 . In each figure, there are also two white squares. Therefore, the expression is n(n + 1) + 2.

This pattern, too, can be be visualized in another way. For example, in each figure, the red tiles can be seen as being made up of a square and a rectangle. In the first figure, we can see 2 squares on top of a 2-by-2 square. In the second figure, we can see 3 squares on top of a 3-by-3 square, and so on. In the nth figure, there will be n squares on top of an n-by-n square. Remembering the 2 white squares, the expression is n^2 + n + 2.

The two expressions are equivalent but reflect different ideas.

How do you know that a relationship is linear? quadratic?
How are the pictorial representations of linear and quadratic functions the same? different?

To see more on this approach, visit I Hope This Old Train Breaks Down.

One more thing… I purposely did not circle the groups and shapes discussed above… I didn’t want to take away the fun of visualizing them for yourself.