# Author Archives: Darrell Blanks

### How much of a difference makes a difference?

How much of a difference makes a difference? When my dad taught us to play golf he showed us how being out of alignment by a few inches at the ball would result in being off by 20 yards at the place the ball was going. In teaching, we make decisions about materials we use, questions we ask, the order of assignments we give and, in these and the multitude of other decisions we make, we are always faced with this question: how much of a difference makes a difference?

In working with students in developing fractional understanding, we made a shift in our task to have students take pre-constructed fifths of a whole and use one piece to trace out five pieces to make a new bar. We then asked, “How do the bars compare?” What we were hoping was that they would say, “It’s the same!!” Of course, that didn’t happen.

Instead, they traced using different pieces and when comparing the lengths of the bars they made they noticed they were not exactly the same and so they didn’t have that golden “aha” that we had imagined in our planning. Why? Because there was enough of a difference to make a difference. There were a few differences we needed to make in the way we presented the task.

The first was that the pieces we cut were not perfect squares. When students used the pieces they rotated them and then used a short side. We realized this when one student compared rotated and unrotated pieces and determined they were not the same size. This was important because in previous tasks the focus was on the length of the bars the students had made into fifths, and whether their fifths were too small, too big, or just right. The purpose was to get them to reflect on the size of the fifth they had made. Now they were being asked to recreate the whole with a fifth we had made. There were small differences in the length depending on how they had turned their fifth piece. They began to focus on the small differences in comparison to the original whole and could not see that any piece of the whole could be copied five times to recreate the whole. With our pieces not being exact enough, it caused a problem because students pulled in their thinking from the previous task.

Second, the pieces we used were made of paper and the tracing was not exact enough. The paper bent and was slow to trace and therefore the bars did not come out the same size. See the explanation in the previous paragraph as to the source of this problem.

So how much of a difference makes a difference? In this case, what we saw as “not a difference” made a big difference. The biggest difference is that it failed to meet the fundamental goal of the task.

### 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… Continue Reading

### 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