# Category: BlogPosts

## Test-taking Time Pressure

Why do we have timed tests? asks Adam Grant (University of Pennsylvania) in this New York Times article. Because many educators believe that speed is a sign of students’ aptitude and mastery. In fact, says Grant, completing a test more quickly isn’t an accurate measure of knowledge or intelligence; it assesses the much narrower skill […]

## Why we need a new cultural script for math instruction

The other day I was working with the teachers at a school who are investigating how they might improve their math instruction. We began by brainstorming characteristics of the “ideal” math lesson. I gave teachers this prompt: What does an effective mathematics classroom and lesson look like and sound like to meet the needs of […]

## Mathing vs Studenting

TL;DR Mathing is the process of students understanding math concepts. Studenting is the process of students applying that understanding to a worksheet. Both need to happen. Mathing is more important than studenting. — Let me briefly share a recent 1st grade lesson I taught to explain what are mathing and studenting and why we need […]

## Planning a Bansho Lesson

In my last blog post, I shared my mental model for how to teach a lesson that incorporates both student-centered inquiry and direct instruction. There are eight steps in this instructional model, but it really is three main phases: introduction, inquiry, and direct instruction. This lesson structure is very similar to the 5 Practices for […]

## A formula for inquiry AND direct instruction

In earlier blog posts (this one and this one) I’ve talked about two – seemingly opposing – views of math instruction: direct instruction and student-centered instruction. There is plenty of evidence that students benefit tremendously when given an opportunity to invent/investigate their own understanding prior to formal instruction by the teacher. There is also plenty of evidence that […]

## Part 2: Three Aspects of Rigor (Why is it necessary?)

The California Mathematics Framework calls on teachers to teach conceptual understanding, procedural fluency, and application with “equal intensity”. The way I think of “equal intensity” is with a Venn diagram. I first saw a Venn used to describe rigor here. The ambitious instruction we are seeking – rigor – is the sweet spot in the […]

## Part 1: Three Aspects of Rigor (Ambitious Instruction)

When it comes to mathematics instruction, what are we aiming for? Exactly what do we want students to know and understand and be able to do? Do we want students to have a big-picture understanding of the mathematics they are learning? Do we want students to be skillful in using procedures and algorithms? Do we […]

## Discovering Pick’s Formula

Finding the area of polygons is pretty easy if the shapes are familiar to you like rectangles, triangles, circles, etc. Those shapes have nice formulas we can learn that will give us the area of the figure without much hassle. But what about unusual shapes like this? Sure we could find the area of this […]

## The JOY of becoming a mathematician

Children should be led to make their own investigations and to draw their own inferences. They should be told as little as possible, and induced to discover as much as possible. -David Eugene Smith (1904)   I will never forget the day I finally became a mathematician. Was it when I was placed in the […]

## Desmos: Effective lesson planning

A teacher recently asked me to co-plan a lesson with her and then co-teach it. She wanted to teacher scatterplots and found this activity, but she wanted my support in planning and teaching the lesson.: https://teacher.desmos.com/activitybuilder/custom/5ae8aa5436ae340a1d96ad4c We began by printing the Teacher Guide to see if the activity’s author provided any suggestions for pacing and […]