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
Tag: Arithmetic
More Notes from NCTM Phoenix
See Part 1 of my notes from Phoenix: A Brief History of Math Education (NCTM President Matt Larson's presentation.)Here is Part 2.Growth Mindset: telling is not teachingIn his short session, Dylan Kane pointed out that talking about growth mindset may be helpful to students "in the middle". But there are students in our classes who… Continue reading More Notes from NCTM Phoenix
Partitions
In this post, I will outline my approach to this partition problem:How many ways can you write a positive integer n as a sum of three or fewer positive integers? Partitions are a standard topic in number theory, but I will limit myself to this specific question. I started trying to figure it out after… Continue reading Partitions
Fractions
I have a new Fractions home page, with links to three pages on my site. In this post, I'll use it as an excuse to discuss some general ideas about teaching. Visual Representations In my Fraction Arithmetic page, I present a visual strategy for figuring out how to add, subtract, and multiply fractions. (There is… Continue reading Fractions
Animated Demonstrations
New on my Web site:→ Animated slides on the Lab Gear model for signed number arithmetic.Note that for each operation, the model is based on what students already know. For addition, you put down the first number, then the second number, and finally count. For subtraction, you put down the first number, take away the… Continue reading Animated Demonstrations
Integer Sequences
Last winter, I attended an interesting meeting of mathematicians and math educators in Banff, Canada. Our charge was to compile a list of integer sequences that would offer suitable problems for students (and teachers) at each level from Kindergarten to 12th grade. It was a sequel to 2014's Unsolved K-12 meeting, and once again was… Continue reading Integer Sequences
Geoboard Triangles
One way to discover or apply the formula for the area of a triangle is to explore area on the geoboard. The initial activities should be along the lines suggested in my Geometry Labs Lab 8.4. (Free download.)After that, one can zero in on triangles, by asking a question like: "Find triangles of area 15."… Continue reading Geoboard Triangles
About hints
A few days ago, I saw a raging debate on Twitter about hint-giving in math class. It was triggered by a short talk by Michael Pershan, a teacher in NYC. Michael argues that high school teachers need to share good hints with each other, and he proposes some guidelines as to what makes a good… Continue reading About hints
Getting Help
In my last post, I described a problem I encountered more than twenty years ago, and my recent attempt at solving it. The problem: Partition the numbers from 1 to 2n into pairs, so that the sum of the numbers in each pair is a perfect square. For what numbers is this possible? I decided… Continue reading Getting Help
I’ve Got a Problem!
Many, many years ago, I saw this problem somewhere: Arrange the whole numbers from 1 to 18 into nine pairs, so that the sum of the numbers in each pair is a perfect square. I liked the problem, and included it in a book I co-authored (Algebra: Themes, Tools, Concepts, following lesson 5.5). In the… Continue reading I’ve Got a Problem!
