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.