# Author Archives: Darrell Blanks

### The CASTE System in Math

Teaching middle school I often had students who still used their fingers to find the difference between 12 and 7. They would start with 7 and then count 8, 9, 10, 11, 12 while putting up fingers each time they would say a number. When we worked with integers and subtraction the idea that positive 6 minus negative 3 related to how far apart the numbers were was hard for them to understand even though, in their procedure, they were finding out that very thing. Something about the way they thought about subtraction and their procedure did not allow them access to thinking of subtraction in a way that made sense beyond finding the answer. I think because of that the answer didn’t make sense beyond “I did the procedure correctly”.

This way of understanding math could be referred to as the Correct Answer for Some Tasks Expression or CASTE system. Yes, I totally made this up for this blog, but indulge me. The CASTE system of knowing mathematics limits the ability to move into new area of mathematics and expand the contexts. We see this system in math in many forms. It might be the FOIL (first, outside, inside, last) method of multiplying polynomials or Keep, Flip, Change (KFC) for dividing fractions. While these work for the immediate context of math they limit math down the road because the CASTE system focuses on the math of the moment. You are dividing fractions today and you will divide fractions tomorrow. Don’t be concerned if the answer makes sense or the applicability of your method outside of today’s problems.

Fighting the CASTE system in math is a goal of the AIMS Center although it has not been expressed in precisely those terms. The work we have been doing at the center in the units coordinating team in subtraction, composite units and fractions is all aimed at developing ways of attending to student thinking and responding in such a way as to not promote math understanding that will limit students. It is an area of service that I have enjoyed and for which I have a passion. Thanks for indulging me.

### Cotton Balls and Place Value

It is important to remember that when we engage students in experiences meant to help them build meaning behind math concepts, what might appear to be happening may or may not be actually happening. Let me explain. Place value is one of the times in math when understanding what the concepts represent requires more than… Continue Reading

### Giant Business Cards and the Art and Science of Communication

Recently, I was listening to an episode of the Hidden Brain podcast entitled, “Alan Alda Wants Us to Have Better Conversations.” The episode details Alan Alda’s work with scientists and health care professionals to help improve their communication. During the interview, he talked about an experience working with the TV show Scientific American Frontiers, during… Continue Reading

### Games That Promote Mathematical Thinking, Part 2

Before Christmas our team developed a game to work with students on building the concept of equal size groups. We named the game “The Great Wall.” Students are given the task of building the Great Wall of China for the emperor. They are directed to build sections of the wall, each a certain number of… Continue Reading

### Top Lessons Learned in 2017

I would like to take a small detour for this entry and use the start of the new year reflect on the previous year. Reflecting on the past as a teacher can help us to think about what we might consider for the future. So what were the top lessons learned at AIMS in 2017?… Continue Reading

### Games That Promote Mathematical Thinking

A few years ago, I went to a conference where I was able to listen to Keith Devlin, a noted mathematician from Stanford, talk about using technology, particularly computer games, to help students think mathematically. He made the case that the symbols we use to represent mathematics, like numerals, operation signs, etc. create a barrier… Continue Reading

### What Every Student Needs to Know for Multiplication (Part 4)

***This is part 4 of a series. Click the links to go back and read part 1, part 2, and part 3*** In this series, we have been discussing a progression of tasks that give students the opportunity to construct meaning for working with two types of units, the towers and the cubes that make… Continue Reading

### Same Task, Different Situation

When I taught middle school, I always found it interesting that my students could do this task: Johnny rode 34 miles on Tuesday and on Wednesday he rode 27 miles. How far did he ride over the two days? Yet they often had no idea what to do when I gave them this task: Johnny… Continue Reading

### How much knowledge is enough knowledge?

The semester has started and I am confronted with the same question the AIMS Center is currently wrestling with: How much knowledge is enough for a teacher to know to make effective decisions with students? We read research by researchers who have been spending years if not decades studying how students come to learn to… Continue Reading

### Lesson From Mexico: Pan De Mujer

Every day on the way to our work site in Mexico, we passed a small sign on the side of a similarly small building that said, “Pan de Mujer, 5 pesos.” I speak Spanish but I was still a little unsure what this meant. I knew the words, but “woman’s bread” didn’t make sense, so… Continue Reading