Author Archives: Bill McCallum

About Bill McCallum

I was born in Australia and came to the United States to pursue a Ph. D. in mathematics at Harvard University, met my wife, and never went back. I am a professor at the University of Arizona, working in number theory and mathematics education.

Talking about fractions, decimals, and numbers

When students first learn about fractions, we want them to learn that they are just numbers; new numbers, but numbers nonetheless, that fit into the same system as the whole numbers they are familiar wtih. The number line can help … Continue reading

Posted in Curriculum, Illustrative Mathematics, Language | 2 Comments

Ways of thinking and ways of doing

Somewhere back in days of Facebook fury about the Common Core there was a post from an outraged parent whose child had been marked wrong for something like this: $$ 6 \times 3 = 6 + 6 + 6 = … Continue reading

Posted in Standards | 13 Comments

Misconceptions about Multiple Methods

You may have noticed that I am back to publishing regular blog posts! My goal for now is a blog post every second Wednesday. I am now also trying to answer forum questions promptly. I want to thank the readers … Continue reading

Posted in Algorithms | 11 Comments

Curricular Coherence Part 3: Using Deep Structures to Make Connections

In this post I’d like to describe the third aspect of coherent content that Al Cuoco and I have been thinking about. In my fourth and final post on this subject I will talk about coherence of practice. A difficult … Continue reading

Posted in Curriculum | Tagged | Leave a comment

Curricular Coherence Part 2: Evolution from Particulars to Deep Structures

In my previous post on curricular coherence I talked about how the principle of logical sequencing can determine the ordering of a set of topics. Since time is one-dimensional, and curriculum occurs over time, some principle for ordering is necessary. … Continue reading

Posted in Curriculum | 2 Comments

What Does It Mean for a Curriculum to Be Coherent?

Al Cuoco and I have been thinking about this question and have developed some ideas. I want to write about the first and most obvious one today, the principle of logical sequencing. I’ll write about others in the weeks to … Continue reading

Posted in Curriculum | 3 Comments

Quantity progression

The last remaining progression, the quantity progression, is here. Comments in the forums welcome!

Posted in Progressions | 1 Comment

Catching up

The site was down for a few hours today because of a malware attack, but I think we have it fixed now. I took the opportunity to catch up on comments in the forums; I was way behind! Thanks to … Continue reading

Posted in Uncategorized | Leave a comment

Geometry Progression Grades 7 to High School

It’s been a while coming, but here is the draft Geometry Progression for Grades 7 to High School. As usual, please make comments and corrections in the relevant forum.

Posted in Progressions | 1 Comment

Transformations and triangle congruence and similarity criteria

While we are all waiting eagerly for the geometry progression I thought people might be interested in this article by Henri Picciotto and Lew Douglas on a transformational approach to the criteria for triangle congruence and similarity. There is also … Continue reading

Posted in Curriculum | 1 Comment