## Virtual Manipulatives Revisited

Occasionally, I am asked if I know of any virtual math manipulatives. “I do. Why?” I reply.

I have a tough time with this type of app. Wanna know what make excellent pattern blocks? These:

I am not an “ever optomistic techno-cheerleader.” Asking questions such as “What are the benefits of replacing a tactile experience with a simulation of a tactile experience?” make it difficult not to be seen as a cynic

or a grump.

Geoboard has softened my position on virtual manipulatives. Last week, as part of an investigation (Pick’s theorem), we asked teachers to figure out the area of the shape below (from Marian Small).

Teachers calculated the area in a variety of ways. Filling shapes with colour in Geoboard helps illustrate each strategy.Â Most groups divided this shape into a rectangles and two triangles, used the formulas for the area of a rectangle and a triangle, and calculated the sum.

Some groups emphasized the relationship between rectangles and triangles.

Other groups subtracted the area of three triangles from the area of a square.

Many groups counted squares and visualized pieces being rearranged to create new squares.

The use of geoboards (real ones) led to answers of “approximately eleven.” With the elastic bands having to wrap around the pegs, the relationships between partial squares were more difficult to see. Similarly, in the investigation of Pick’s theorem, it was sometimes difficult to tell whether a lattice point was a border point. It’s not an issue within the iPad app; the virtual elastic bands connect rather than wrap around the virtual pegs. Plus, working with virtual bands was easier than working with real bands. This encourages even more “what if?” thinking. I don’t think this is true of all virtual manipulatives.

More importantly, learners can share their solutions through an AppleTV. This can also be accomplished with real manipulatives and the iPad’s built-in camera. True, students can push, pull, or drag their real geoboards to the front of the class to show and share their solutions, but technology just makes this seamless.

While I may have warmed up to virtual manipulatives, don’t expect me to warm up to virtual flashcards any time soon. Some teaching practices are harmful to students. Retina display doesn’t change that.

## iPad as Mathematical Communication Tool â€“ 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, sheÂ 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 anÂ interview of UC Berkeley Math Education Professor Dor AbrahamsonÂ 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 actionsÂ [C, CN, PS, V]

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 â€“ 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 â€“ from Kintergarten to Calculus.

## 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.Â (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 gameÂ 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.

*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 KhanÂ on his Mathalicious blog.