Math Library

Have a question about elementary mathematics content? Pedagogy? Language? Explore our library of posts answering questions teachers most frequently ask about elementary mathematics or ideas that they often want to investigate. These informational pages give insight and clarify concepts about elementary mathematics for teachers.

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Found 47 Results

December 3, 2019
five and one

Addition and Subtraction

Addition and subtraction of whole numbers can be developed together, systematically, starting with recognition of quantities and combinations of quantities, the 10-based structure of the language […]
November 20, 2019

Algebraic Thinking

Number tricks are fun for children. The fun, all by itself, is valuable, but is not mathematics. But understanding how the trick works is good mathematical, often algebraic, learning.
November 20, 2019
Base 10 Blocks

Base Ten Blocks

This is a school term, sometimes used in teaching math. The word has no formal mathematical definition. A model for base 10 numeration. Ten copies of […]
February 7, 2020

Classroom Stories: Building Mathematical Structure

Paul shares a story about a surprising discovery he made while observing Emma and challenges us with a question: “How can we run a school in a way that gives teachers the time and support to watch and listen actively and to recognize, understand, and help children build on the underlying mathematical ideas?”
February 7, 2020

Classroom Stories: Building on Strengths

Teachers are often encouraged, sometimes compelled, to pre-assess students and then focus on their weaknesses. After all, what’s teaching if not helping a child move from weakness to strength? But this story is about a different approach: recognizing and using strengths that are camouflaged by apparent weakness or hidden behind real weakness.
February 7, 2020
weather graph

Classroom Stories: Correct Answer, Wrong Question

Very often—in fact, it seems to me, most of the time—hesitations or wrong answers in math class that are attributed to weakness in mathematics are the fallout from some confusion of communication. The source is not in the mathematics. In fact, “wrong” answers are often correct answers to an entirely reasonable alternative interpretation of the question.
February 7, 2020

Classroom Stories: Oral and Written Mathematics

Paul describes the connection between oral and written mathematics through the eyes of six-year-old Aaron.
November 21, 2019

Developing Mathematical Vocabulary

How to build mathematical vocabulary: Understand it well yourself (see glossary), and use it naturally (and correctly) in communication when you need it, in a rich enough context so that students can “figure out” the meaning from context, just as you might expect them to figure out the meaning of a new word that occurs in the middle of a story.
December 3, 2019

Difference of Squares

This page describes one silent teaching activity that serves, at the superficial level, to give students useful practice with multiplication facts, but that also builds deeper […]
December 3, 2019

Dimension

The word dimension is related to the word measure. It is used in two ways in geometry. It is used to specify a measurement: “What are […]
November 20, 2019

Factor

The terms factor and multiple are sometimes confused with each other. Factors of 15 include 3 and 5; multiples of 15 include 30, 45, 60 (and […]
November 21, 2019

Guess My Rule

How to Play Guess My Rule games are games in which one person thinks up and gives examples of some “rule,” and the player(s) try to […]
November 21, 2019
A hundreds chart in which "higher numbers" are higher, and "the 30s" are all on the same row.

Hundreds Chart and the Number Line Hotel

Hundreds chart A hundreds chart in which “higher numbers” are higher, and “the 30s” are all on the same row. Unlike the number line, which is […]
January 9, 2020

Language and Mathematics

Children are extraordinary linguists! They acquire half their adult vocabulary and nearly all their adult grammar by the time they are 5 years old, mostly informally […]
November 21, 2019

Measurement: Discovering formulas for area

Area formulas Students who have the informal notion that area is the “amount of 2-D ‘stuff'” contained inside a region can invent for themselves most of […]
November 4, 2019

Measurement: Length, width, height, depth

Length, width, height, depth Outside of the mathematics class, context usually guides our choice of vocabulary: the length of a string, the width of a doorway, […]
January 9, 2020

Measurement: Volume

Informally, volume is the amount of (three-dimensional) “stuff.” The volume of a body of water is how much water there is. The volume of a sculpture […]
December 3, 2019

Mental Math

Mental Math exercises are lively, brief, highly focused, and daily. They are strategically selected to build students’ mental math skills quickly increase their capacity to hold […]
January 9, 2020

MP: Why do we need standards for Mathematical Practice?

Standards in mathematics suitable for 21st century learners must address two kinds of knowledge: mathematical content and mathematical practice. The content standards are adjusted to reflect […]
November 21, 2019

MP1 Make sense of problems and persevere in solving them

Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution.