Reflections on Teaching

_{Pi Pie on Flickr!}

Nope, not something else about Web 2.0, technology in education, etc. just a simple observation about mathematics instruction in elementary schools in this country. One training I went to where the trainer described a Chinese educator’s consternation with America’s over-reliance on pies for fraction education (“too much round-food” was the quote).

I’ve run across this both in teaching fifth graders and in helping my fourth grade son with homework. Here is a classic example of how this goes wrong:

See the problems always ask the kids to “draw” a pie chart to compare the two fractions. As you can see, my son has poor drafting skills, so the pies are not equal, and the sections aren’t either. He’s comparing apples and oranges. I think the mistake we’re making is by first acting like the “wholes” of fractions being compared are equal, and then assuming when kids draw them, they are equal. I believe this is one of the antecedents of some of the crappy pie charts that get created in PowerPoint, where the context of whole is skewed or missing (I’m thinking of one that I made-check out slide 3).

What are your thoughts?

8 Comments to

“What’s the “whole” point?”

February 13th, 2008 at 3:59 am Reply Mark Says:
I agree that we do use food analogies an awful lot when we teach fractional parts… it was all part of making it “relevant” and “real-world” like we’ve been told is so essential.

I guess in that 2/3 vs. 3/4 example, I can see why your son went awry, but I’m trying to figure out what a better alternative would be for helping kids compare the two amounts, and nothing is immediately coming to mind. Make them mathematically convert them to common denominators so they can see that 8/12 < 9/12? Make them learn the decimal or percent conversions?

I actually asked a question just like this on one of my Academic Leagues contests because it is so beguiling for kids… I asked which was the greatest: 2/3, 3/4, 4/5, 5/6, or 6/7? The brightest students realize they’re getting all but one piece, so they want the whole to be broken into as many pieces as possible, so the one piece they’re not getting is as small as possible. But that’s just for that example… what if we were comparing 3/7 and 2/5 instead? Then you have to get to the mathematical approaches I suggested above.
February 13th, 2008 at 5:12 am Reply Jenorr Says:
I think one of the problems we face teaching math, especially in elementary school is that we, as teachers, don’t understand the concepts as deeply as we need to in order to teach it well. I’m amazed at what I’ve learned in the past couple of years as I’ve focused more on math. It’s changed my teaching dramatically and I’m sure that I’ve barely begun.
February 13th, 2008 at 2:43 pm Reply Dan Says:
I find that students in 4th, 5th, and even 6th grade have a difficult time building the conceptual understanding of fractions, decimals, and percents. I have found a some success this year, using a constructivist approach. I work with AIS students in 5th and 6th grade, some who are classified. During a particular lesson, a student realized that a 1/2 of a 1/5 was a 1/10 and had never work with fractions prior to being in the class. It was quite the rewarding experience. We did use food though as part of the setting of the problem, except it wasn’t pies but submarine sandwiches. It allowed them to “cut” the subs either on paper or using cubes.
February 13th, 2008 at 8:10 pm Reply alicemercer Says:
Here is what I’ve been doing. I used stacked bar manipulatives to get away from “round” analogues. Also, since they are “machine made” not hand made, they are uniform. I just remembered another resource that I ran across last year:
http://www.philtulga.com/

This guy is a music teacher who does school assemblies in Northern California and has a bunch of online activities using fractions in music. I like his Fraction Pie activity: http://www.philtulga.com/pie.html

I don’t think it’s the “answer” but rather an answer?
February 13th, 2008 at 8:22 pm Reply alicemercer Says:
Oops, went out without saying thank you all for sharing! I really appreciate your thoughts. I liked Jenn’s point. I found I didn’t have great math sense, until I worked with numbers at a bank, then I went into teaching, but I found/find it hard to translate what I learned to kids? I just wonder if it’s a maturity thing, and we’re pushing kids? But, they are getting smarter, so who knows?

Mark and Dan, I did something with Legos, showing how in comparing two legos, they could be 1/2 of another lego, but depending on what each was being compared to, they could be different sizes?
February 14th, 2008 at 8:59 am Reply Dan Says:
The Lego idea sounds good. I have used pattern blocks as well. 1 hexagon equals 2 trapezoids, 3 blue rhombii, 6 triangles, etc. You may want to consider checking out Cathy Fosnot’s materials at contextsforlearning.com. It has some real interesting ideas on the development of math K-6.
February 16th, 2008 at 4:41 pm Reply msdhorn Says:
I too have noted the difficulties that students have when comparing fractions. Their ability to draw figures is so vastly different between students that I have looked for websites to help them see the relationships between fractions. Here is a website that I have found useful in this endeavor.
http://arcytech.org/java/fractions/fractions.html
February 16th, 2008 at 5:09 pm Reply alicemercer Says:
Nice app msdhorn! I like that. Also, it’s in bars. Since they have so darn many pie/round analogues I like giving them bars sometimes to break it up. I’ve been using stacked bar fractions (sorta like unifix cubes) with my son.

Thanks for all your ideas!