Alison Blank makes good points in her interesting presentation: "Math is not linear", where she encourages us to make connections, go on tangents, preview future topics and review past ones. In short, we should not be trapped in the inflexible sequence suggested by textbooks and school culture. In a recent blog post, Jim Tanton makes… Continue reading Mind Maps
In my previous post, I listed questions to use in class discussions, or in conversation with a student or a group of students. Today, I'll discuss how to handle wrong answers. This is complicated and there is no single correct answer for all situations. I'll start by clarifying my goals:broad participation by students in the… Continue reading Handling Wrong Answers
On the first weekend of December, the California Math Council held its annual meeting in Asilomar for the 60th time. (I attended for the 33rd time, and presented roughly the same talk I had presented in 1984.) Over the decades, the "must-attend" presenters have changed. Two of my favorites back in the day were Harold… Continue reading Any Questions?
In my last post, I shared some generalities about puzzle creation. Today, I will zero in on the specifics of creating puzzles for the mathematics classroom. I will do this by way of analyzing some examples. Multiple PathsA characteristic of all classrooms is that they are constituted of students whose backgrounds and talents vary widely. … Continue reading Puzzles for the Classroom
John Golden asked whether I had written about my approach to puzzle creation. I've only written a brief post on the subject, five years ago. Yet I believe that my work as a curriculum developer is largely based on my involvement with puzzles: solving them, constructing them, editing them. Of course, puzzling is not the… Continue reading Puzzle Creation
(Previous posts on this topic.)I suspect that by far the most common introduction to geometric construction in US classrooms is a presentation by the teacher (or textbook) on various compass and straightedge construction techniques. "This is how you construct a perpendicular bisector. This is how you construct an equilateral triangle." And so on. "Now memorize… Continue reading More on Geometric Construction
My early forays as a curriculum developer date back to my days as a K-5 math specialist in the 1970's. A key insight of my young self was that activities intended for students were that much more worthwhile if they were also interesting to me. I learned to view with suspicion activities that were boring… Continue reading Polyarcs
Many students have weak arithmetic skills. Many teachers blame this on calculator use, but it is just as likely that the real reason lies elsewhere. For one thing, the teaching of arithmetic traditionally does not involve developing any understanding, so the learning is shallow and fragile. For another, students correctly feel that mindless arithmetic is… Continue reading Calculation
I will offer two workshops this summer (2017), at the Head-Royce School in Oakland, CA. Sign up for either or both!June 26-27: Hands-On Geometry (grades 6-10)June 28-30: Transformational Geometry (grades 8-11)If the times or locations don't work for you, I can offer a workshop for your school or district. Contact me directly. I recommend attending… Continue reading Geometry Boot Camp!
Much can be said in defense of practice exercises, but when all is said and done, very few students develop deep understanding from routine practice. For example, compare these two approaches to the area of a trapezoid. Approach 1The teacher says: ”The area of a trapezoid is given by the formula h(b1+b2)/2, where h is… Continue reading Comparing two approaches