According to this article by Jo Boaler — professor of mathematics education at Stanford and co-founder of www.youcubed.org — math memorizers scored poorly on the international PISA test, and the U.S. has more memorizers than most other countries in the world. The highest achieving students internationally were those who thought of math as a set of connected, big ideas.
Here’s what we see:
1. A visual approach to fractions gives students better number sense, and better access to word problems.
When we require drawing, every problem becomes a word problem. In the problem below, all students recognized that 1/2 is 6 out of 12, visually. This is a “12-peak Toblerone”, so a total of 17 twelfths (by simply counting!) . Then this student imagined moving one 12th from the top row to make the 2nd row equal to one, leaving 5/12 on top. This shows number sense! Our students can do fraction addition and subtraction mentally. More importantly, visualization helps facilitate the transfer to word problems, as below.
Egyptian fractions: We spent a few days answering word problems by building fractions with Cuisenaire rods. Here, for example, is a TWELVE-WIDE wall:
One fourth — the light green rod — is called one fourth because four of them fit in a whole. The purple rod is called one third because 3 of them fit, the red is 1/6, etc.
This student had no trouble finding a way to make 11/12 with Egyptian fractions:
After long exposure to physical representations, word problems become easier. This problem, for example, would be difficult to do with algorithms.
How about this problem: Erin and Kana went shopping for groceries. Each of them had an equal amount of money at first. Then Erin spent $80 and Kana spent $128. After that Kana had 4/7 of what Erin had left. How much money did Erin have left after shopping? Solve by drawing a fraction model.
This is very difficult to do without algebra. Try it yourself before looking at the answer here. Once you see the solution, it’ll make sense, and all of this will transfer to stronger algebra students in 3 years.
2. A visual approach to math is the ONLY approach that works for some students.
In the past, visual learners struggled with the algorithmic manner in which math was taught. (Challenge: randomly survey a couple dozen adults – we predict almost 1/3 of them will say they were ‘never very good at math’)
However, in the past, there were good middle class jobs available to high school graduates – jobs that are now disappearing. It is our duty to make math accessible to ALL students.
The good news is that requiring visualization of math also benefits the innately abstract math learners. Visualization skills helps students in Chemistry, Physics, Trigonometry, and other STEM subjects these students gravitate towards. Here’s an article about visualization in physics.