iPad as Mathematical Communication Tool

When children think, respond, discuss, elaborate, write, read, listen, and inquire about mathematical concepts, they reap dual benefits: they communicate to learn mathematics and they learn to communicate mathematically(NCTM)

In general, I’ve been disappointed with many of the iPad apps categorized under Education. With new apps being added (270/day in June 2011), I’ve got to admit it’s getting better. A little better all the time.

As Orwell Kowalyshyn and/or Kevin Amboe mentioned last spring, apps from other categories such as Games or Photography may provide richer educational opportunities for students.

My daughter (6) is currently enjoying the game Slice It!. The goal is to slice shapes as evenly as possible. The number of slices you are allowed and the number of pieces the shape is to be sliced into is given. The challenges get increasing difficult. I can imagine using this app to explore mathematical concepts such as area, fractions, percents, and line symmetry. Perhaps students could take screenshots and explain their strategies to their classmates. Maybe they could explain how they know the pieces have approximately the same area. (The FAILED text that appears when not sliced into the correct number of pieces may turn off some educators. No noticeable signs of this affecting my daughter, at least so far.)

Students will benefit from iPads in the classroom not because there’s an app for practicing number operations, but because there’s an app for communicating their thinking. ShowMe, ScreenChomp, and Explain Everything have been listed/discussed in many math ed blogs. Students, not their teachers or Sal Khan, can create video explanations using these interactive whiteboard apps.

Meeting with Surrey & Vancouver secondary math teachers this summer, one teacher showed us a picture of two containers each filled with chocolate eggs. The number of eggs in the smaller container was given and we were asked to guess the number of eggs in the larger container (see Dan Meyer’s blog). Using her iPad 2, the teacher filmed us giving and justifying our estimates. In a classroom, teachers or, better yet, students could interview peers, administrators, parents, members of the community, etc. and then share and discuss these guesses and strategies.

One app that I had fun with this summer is iMotion HD. This app allows you to create and share stop motion movies from pictures you have taken. In the video below, I show why 1/2 + 1/3 = 5/6 using pattern blocks.

Using iMovie, I could have added narration but I chose not to. Why? Because I have no plans to share this with students*. I chose not to narrate my movie because students, not the teacher, should be doing the math. In this way, students communicate to learn and learn to communicate.

photo by pong0814

*Also, I have only one nephew. He is 18 months old and so far has been able to complete his algebra homework without asking his uncle to tutor him. The Khan Academy has already been widely and deservedly criticized by others. Please check out Karim Ani’s An Open Letter to Sal Khan on his Mathalicious blog.

How come we’re playing games today? I thought this was math class.

Inspired by reading the tweets & blogs of Surrey teachers over the summer, I thought I’d resurrect my blog.

In his blog, Richard deMerchant writes about how games, in addition to being fun, can help develop conceptual understanding of mathematics (http://rvdemerchant.wordpress.com/2011/08/31/count-down-part-two-games/). He also writes about the impact that debriefing strategies (“Why did you make that move?” etc.) has had on his son’s thinking.

I have seen this in my daughters as well. My 6-year-old loves the game/puzzle Camouflage. The challenge is to place polar bears on ice and fish in water while also having the game pieces fit on the board (see http://www.smartgamesandpuzzles.com/inventor/Camouflage.html for a better description). As she was playing, she went to place a piece down and then stopped herself saying “That can’t go there. It’ll make a square”. I asked her to explain this to me. She had figured out that if a move created a blank one-by-one square, then she would not be able to fit all the pieces on the board. (The game pieces are one-by-two dominoes or L-shaped triominoes). She developed this strategy on her own. As she completed the increasingly more difficult challenges, I could see her develop problem solving and reasoning skills (as well as spacial sense).

This year, I’m excited by the inclusion of the games learning outcomes in the Foundations and AWM pathways. This one comes from FoM 11:

But games/puzzles can also be used to address/enhance other learning outcomes in the math curriculum. For example,
  • rotations in Pentago
  • translations in Rush Hour
  • combinatorics in Mastermind
  • area in TopThis!
  • isometric drawings & volume in Block by Block
Each secondary school in Surrey will be receiving a games kit in the fall. Here’s the list: Secondary Games Kit
At a workshop in June, I asked teachers to play Blokus. Immediately, one teacher asked “What’s the point? Why are we doing this?”. It didn’t feel like math for him and it probably won’t feel like math for our students. However, aside from the strategy aspect, think of the possible connections to traditional math topics. For example,
  • transformations (when determining the number of game pieces, or ‘free polyominoes’)
  • area/ratios/percent (when determining the winner)
  • square roots (If the 4-player game board is 20-by-20, what should the dimensions of the 2-player game board be?)
In defining math, most of us math teachers will probably use words like ‘problem-solving’, ‘reasoning’, ‘patterns’, ‘estimation’, etc. (Would our students use these words or would they use words like ‘memorize’, ‘rules’, ‘formulas’…?) Compare a lesson in which students play (and discuss!) Blokus to one in which the teacher shows students how to divide rational expressions (1. factor numerators/denominators, 2. invert and multiply, 3. cancel) and students practice questions similar to the examples. In which lesson might you see the words listed above? In which classroom are students doing math?

Linear Functions – Concretely, Pictorially, Symbolically

Welcome to my blog!

I really enjoyed Marc’s Patterning the Blues activity (taken from Marian Small’s Big Ideas book that department heads received).

Teachers often talk about how manipulatives can help the struggling learner. I’m suggesting that having students solve problems concretely can assist all learners.

When I experienced this problem using the blue and yellow tiles, I gained a deeper understanding of the problem. The equation y = 3x + 2 now had meaning. I was able to find the pattern in the table to determine the number 3. By modelling the problem using tiles, I was able to see this as adding an extra 3 blue tiles every time the figure grew.

In the past, I had difficulty explaining to students where the 2 came from. I could convince them that it had to be there. For example, take the point (2, 8). Multiplying the 2 by 3 gives  6, so we need to add 2 more. Looking at this concretely & pictorially, the 2 now has meaning. For me, it is how many blue tiles there were before we start adding yellow & blue tiles. (See the photo below.)

Your students who used to get it symbolically will still get it if they approach it concretely. However, what it means to “get it” in your classroom will start to change.

Patterning the Blues
Patterning the Blues Concretely

I’d appreciate your comments. Maybe you have some thoughts on how this activity addresses the 7 processes?