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Best Practices - 3 Teaching Strategies for Improving Algebra Knowledge

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Best Practices - 3 Teaching Strategies for Improving Algebra Knowledge

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Back Behavior Postsecondary Evidence of effectiveness in the topics selected above. A colored icon with a box indicates positive or potentially positive effects on outcomes for that topic. A grey icon with no box indicates a lack of positive effects. By default, results are sorted by this column, based on the amount of evidence.

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"EdReports.org was spearheaded by Maria M. Klawe, the president of Harvey Mudd College, in Claremont, Calif., and was launched in August. Its nine-member board includes representatives from the National Board for Professional Teaching Standards, the National Council of La Raza, and the Education Trust.
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Meanings and Models For Operations - Tutorial - Annenberg Learner

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Meanings and Models For Operations - Tutorial - Annenberg Learner

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14:30 Integer Chips Introduction

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Previously Introduced:
identity element
If I is an identity element for operation *, then a * I = I * a = a for all elements a in the set. The identity element for addition of real numbers is 0, and the identity element for multiplication of real numbers is 1.
inverse element
If b is the inverse element for a for operation *, then a * b = b * a = I, the identity element for that operation. The inverse for element a for addition is -a, because a + -a = -a + a = 0 for all values of a. The inverse for element a for multiplication is 1/a, because a * (1/a) = 1/a * a = 1 for all values of a except 0. Zero does not have an inverse for multiplication.
whole numbers
Whole numbers are the counting numbers and zero.
New in This Session:
algorithm
An algorithm is a recipe or a description of a mechanical set of steps for performing some task.
asymmetrical multiplication
An asymmetrical multiplication problem is one where the order of the operands is important. Switching the order of operands in this type of problem presents a different situation, even though the product is the same. For example, buying 10 tickets at $5 each is quite different from buying 5 tickets at $10 each, although the total cost (i.e., the product) is identical.
partitive division
A partitive division problem is one where you know the total number of groups, and you are trying to find the number of items in each group. If you have 30 popsicles and want to divide them equally among your 5 best friends, figuring out how many popsicles each person would get is a partitive division problem.
quotative division
A quotative division problem is one where you know the number of items in each group and are trying to find the number of groups. If you have 30 popsicles and want to give 5 popsicles to each person, figuring out the total number of people is a quotative division problem.
symmetrical multiplication

A symmetrical multiplication problem is one where the order of the operands is not important. Finding the area of a field that measures 150 feet by 50 feet, or finding the number of different sandwiches that can be made from 4 types of bread and 6 types of meat, are both symmetrical multiplication problems.

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Meanings and Models for Operations - Annenberg Learner

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Previously Introduced: identity element If I is an identity element for operation *, then a * I = I * a = a for all elements a in the set. The identity element for addition of real numbers is 0, and the identity element for multiplication of real numbers is 1.

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BLOG FOR THE MATHEMATICAL ASSOCIATION OF AMERICA

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Celebrating 40 Years of Collaborative Research

MSRI begins our 40th anniversary celebrations with the announcement of a landmark joint endowment gift of $70 million from Drs. James and Marilyn Simons and Drs. Henry and Marsha Laufer to support the mathematical sciences research community. Starting this summer, MSRI will become the Simons Laufer Mathematical Sciences Institute.

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The Mathematical Sciences Research Institute (MSRI), founded in 1982, is an independent nonprofit mathematical research institution whose funding sources include the National Science Foundation, foundations, corporations, and more than 90 universities and institutions. The Institute is located at 17 Gauss Way, on the University of California, Berkeley campus, close to Grizzly Peak, on the hills overlooking Berkeley.

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The programs of the National Council of Teachers of Mathematics present a variety of viewpoints. The content and views expressed or implied in these presentations should not be interpreted as official positions of the Council. References to particular commercial products by a speaker are not an NCTM endorsement of said product(s) and should not be construed as such. Any use of e-mail addresses beyond personal correspondence is not authorized by NCTM.

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NCTM - Professional Development Webinars

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The programs of the National Council of Teachers of Mathematics present a variety of viewpoints. The content and views expressed or implied in these presentations should not be interpreted as official positions of the Council. References to particular commercial products by a speaker are not an NCTM endorsement of said product(s) and should not be construed as such.

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Shelves > Mathical >

Mathical Books

Showing 1-42 of 42

ONE Very Big Bear (Hardcover)
by
Alice Brière-Haquet
(shelved 1 time as mathical)
avg rating 3.42 — 199 ratings — published 2014

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Round (Hardcover)
by
Jennifer Ward
(shelved 1 time as mathical)
avg rating 3.80 — 90 ratings — published

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Grasping Mysteries: Girls Who Loved Math (Hardcover)
by
Jeannine Atkins (Goodreads Author)
(shelved 1 time as mathical)
avg rating 4.21 — 91 ratings — published 2020

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Can You Crack the Code?: A Fascinating History of Ciphers and Cryptography (Hardcover)
by
Ella Schwartz (Goodreads Author)
(shelved 1 time as mathical)
avg rating 4.34 — 92 ratings — published 2019

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Really Big Numbers (Paperback)
by
Richard Evan Schwartz
(shelved 1 time as mathical)
avg rating 4.15 — 54 ratings — published 2014

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Circle Under Berry (Hardcover)
by
Carter Higgins (Goodreads Author)
(shelved 1 time as mathical)
avg rating 4.17 — 296 ratings — published

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If the World Were 100 People: A Visual Guide to Our Global Village (Hardcover)
by
Jackie McCann
(shelved 1 time as mathical)
avg rating 4.26 — 182 ratings — published

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Shape Up, Construction Trucks! (Board Book)
by
Victoria Allenby
(shelved 1 time as mathical)
avg rating 3.87 — 82 ratings — published

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Lia & Luís: Who Has More? (Hardcover)
by
Ana Crespo (Goodreads Author)
(shelved 1 time as mathical)
avg rating 3.84 — 130 ratings — published

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One Fox: A Counting Book Thriller (Hardcover)
by
Kate Read
(shelved 1 time as mathical)
avg rating 4.13 — 803 ratings — published 2019

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One Is a Piñata: A Book of Numbers (Hardcover)
by
Roseanne Thong
(shelved 1 time as mathical)
avg rating 3.97 — 295 ratings — published

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Which One Doesn't Belong?: Playing with Shapes (Hardcover)
by
Christopher Danielson
(shelved 1 time as mathical)
avg rating 4.19 — 89 ratings — published

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Crash! Boom! (Hardcover)
by
Robie H. Harris
(shelved 1 time as mathical)
avg rating 3.48 — 252 ratings — published

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Sheep Won't Sleep: Counting by 2s, 5s, and 10s (Hardcover)
by
Judy Cox (Goodreads Author)
(shelved 1 time as mathical)
avg rating 3.60 — 137 ratings — published 2019

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Mind-Boggling Numbers (Hardcover)
by
Michael J. Rosen (Goodreads Author)
(shelved 1 time as mathical)
avg rating 4.03 — 30 ratings — published

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Billions of Bricks (Hardcover)
by
Kurt Cyrus (Goodreads Author)
(shelved 1 time as mathical)
avg rating 3.69 — 567 ratings — published 2016

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Genius: The Game (Hardcover)
by
Leopoldo Gout
(shelved 1 time as mathical)
avg rating 3.74 — 2,361 ratings — published 2016

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One Big Pair of Underwear (Hardcover)
by
Laura Gehl (Goodreads Author)
(shelved 1 time as mathical)
avg rating 3.91 — 1,139 ratings — published 2014

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The Great Trouble: A Mystery of London, the Blue Death, and a Boy Called Eel (Hardcover)
by
Deborah Hopkinson (Goodreads Author)
(shelved 1 time as mathical)
avg rating 3.90 — 3,343 ratings — published 2013

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Navigating Early (Hardcover)
by
Clare Vanderpool
(shelved 1 time as mathical)
avg rating 4.08 — 14,033 ratings — published 2013

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Eat Your Math Homework (Hardcover)
by
Ann McCallum Staats (Goodreads Author)
(shelved 1 time as mathical)
avg rating 3.93 — 70 ratings — published 2011

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Hippos Go Berserk! (Board Book)
by
Sandra Boynton
(shelved 1 time as mathical)
avg rating 4.29 — 5,471 ratings — published 1977

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We Are One: How the World Adds Up (Hardcover)
by
Susan Hood (Goodreads Author)
(shelved 1 time as mathical)
avg rating 3.96 — 103 ratings — published

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Uma Wimple Charts Her House (Hardcover)
by
Reif Larsen
(shelved 1 time as mathical)
avg rating 3.74 — 172 ratings — published

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It's a Numbers Game! Baseball: The math behind the perfect pitch, the game-winning grand slam, and so much more! (Hardcover)
by
James Buckley Jr.
(shelved 1 time as mathical)
avg rating 3.50 — 4 ratings — published

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The Animals Would Not Sleep! (Paperback)
by
Sara Levine (Goodreads Author)
(shelved 1 time as mathical)
avg rating 3.92 — 160 ratings — published

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Aftermath (Hardcover)
by
Emily Barth Isler (Goodreads Author)
(shelved 1 time as mathical)
avg rating 4.40 — 269 ratings — published 2021

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Much Ado About Baseball (Hardcover)
by
Rajani LaRocca (Goodreads Author)
(shelved 1 time as mathical)
avg rating 4.03 — 172 ratings — published 2021

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It's a Numberful World: How Math is Hiding Everywhere (Paperback)
by
Woo Eddie
(shelved 1 time as mathical)
avg rating 4.06 — 48 ratings — published

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Seven Golden Rings: A Tale of Music and Math (Hardcover)
by
Rajani LaRocca (Goodreads Author)
(shelved 1 time as mathical)
avg rating 4.32 — 257 ratings — published 2020

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Solving for M (Hardcover)
by
Jennifer Swender
(shelved 1 time as mathical)
avg rating 4.34 — 357 ratings — published 2019

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The Girl with a Mind for Math: The Story of Raye Montague (Hardcover)
by
Julia Finley Mosca
(shelved 1 time as mathical)
avg rating 4.39 — 591 ratings — published 2018

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Nothing Stopped Sophie: The Story of Unshakable Mathematician Sophie Germain (Hardcover)
by
Cheryl Bardoe
(shelved 1 time as mathical)
avg rating 4.24 — 784 ratings — published 2018

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A Hundred Billion Trillion Stars (Hardcover)
by
Seth Fishman (Goodreads Author)
(shelved 1 time as mathical)
avg rating 4.16 — 687 ratings — published 2017

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Animals by the Numbers: A Book of Infographics (Hardcover)
by
Steve Jenkins
(shelved 1 time as mathical)
avg rating 4.37 — 528 ratings — published 2016

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Secret Coders (Secret Coders, #1)
by
Gene Luen Yang (Goodreads Author)
(shelved 1 time as mathical)
avg rating 3.94 — 3,710 ratings — published 2015

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Have You Seen My Dragon? (Hardcover)
by
Steve Light (Goodreads Author)
(shelved 1 time as mathical)
avg rating 3.96 — 2,157 ratings — published 2014

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The Number Devil: A Mathematical Adventure (Paperback)
by
Hans Magnus Enzensberger
(shelved 1 time as mathical)
avg rating 3.92 — 6,048 ratings — published 1997

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All of the Above (Hardcover)
by
Shelley Pearsall (Goodreads Author)
(shelved 1 time as mathical)
avg rating 3.78 — 1,659 ratings — published 2006

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Count with Maisy, Cheep, Cheep, Cheep! (Hardcover)
by
Lucy Cousins
(shelved 1 time as mathical)
avg rating 3.63 — 166 ratings — published 2015

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Shapes, Reshape!: A Minibombo Book (Hardcover)
by
Silvia Borando
(shelved 1 time as mathical)
avg rating 3.94 — 150 ratings — published

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Count on Me /anglais (Hardcover)
by
TANCO MIGUEL
(shelved 1 time as mathical)
avg rating 4.50 — 2 ratings — published

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Welcome back. Just a moment while we sign you in to your Goodreads account.

# data[].attributes.attachment_link.description
Books shelved as mathical: ONE Very Big Bear by Alice Brière-Haquet, Round by Jennifer Ward, Grasping Mysteries: Girls Who Loved Math by Jeannine Atkins,...

# data[].attributes.attachment_link.readable_text
Resources for the titles recognized by this award since its inception in 2015 -- honoring the most inspiring math-related fiction and nonfiction books

Sponsored by the Mathematical Sciences Research Institute (MSRI), NCTE, NCTM, and the CBC

1,327 Resources including

140 Videos

67 Book Readings

177 Lesson Plans

List Analysis Report

Showing 1-48 of 110 book results

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Secret Coders

by Gene Luen Yang and Mike Holmes

(Winner, 2016)

40 Resources 3 Awards

Add to Reading List

Classified: The Secret Career of Mary Golda Ross, Cherokee Aerospace Engineer

by Traci Sorell and Natasha Donovan

(Honor, 2022)

36 Resources 7 Awards

Add to Reading List

How We Got to the Moon: The People, Technology, and Daring Feats of Science Behind Humanity's Greatest Adventure

by John Rocco

(Winner, 2021)

34 Resources 10 Awards

Add to Reading List

Press Here

by Hervé Tullet

(Honor, 2019)

30 Resources 5 Awards

Add to Reading List

Counting on Katherine: How Katherine Johnson Saved Apollo 13

by Helaine Becker and Dow Phumiruk

(Honor, 2019)

30 Resources 9 Awards

Add to Reading List

Unusual Chickens for the Exceptional Poultry Farmer

by Kelly Jones and Katie Kath

(Honor, 2016)

26 Resources 4 Awards

Add to Reading List

Baby Goes to Market

by Atinuke and Angela Brooksbank

(Winner, 2018)

26 Resources 7 Awards

Add to Reading List

Animals by the Numbers: A Book of Infographics

by Steve Jenkins

(Honor, 2019)

25 Resources 11 Awards

Add to Reading List

Nothing Stopped Sophie: The Story of Unshakable Mathematician Sophie Germain

by Cheryl Bardoe and Barbara McClintock

(Winner, 2019)

24 Resources 11 Awards

Add to Reading List

Slay

by Brittney Morris

(Winner, 2020)

24 Resources 4 Awards

Add to Reading List

Bird Count

by Susan Edwards Richmond and Stephanie Fizer Coleman

(Honor, 2021)

24 Resources 2 Awards

Add to Reading List

Alice's Adventures in Wonderland

by Lewis Carroll

(Commended, 2015)

23 Resources 1 Awards

Add to Reading List

The Miscalculations of Lightning Girl

by Stacy McAnulty

(Winner, 2019)

23 Resources 3 Awards

Add to Reading List

Pass Go and Collect $200: The Real Story of How Monopoly Was Invented

by Tanya Lee Stone and Steven Salerno

(Honor, 2021)

23 Resources 6 Awards

Add to Reading List

The Phantom Tollbooth

by Norton Juster and Jules Feiffer

(Commended, 2016)

22 Resources 1 Awards

Add to Reading List

The Boy Who Loved Math: The Improbable Life of Paul Erdos

by Deborah Heiligman and LeUyen Pham

(Honor, 2016)

22 Resources 7 Awards

Add to Reading List

Have You Seen My Dragon?

by Steve Light

(Winner, 2015)

22 Resources 2 Awards

Add to Reading List

Seven Golden Rings: A Tale of Music and Math

by Rajani LaRocca and Archana Sreenivasan

(Winner, 2021)

22 Resources 4 Awards

Add to Reading List

Hedy Lamarr's Double Life: Hollywood Legend and Brilliant Inventor

by Laurie Wallmark and Katy Wu

(Honor, 2022)

21 Resources 5 Awards

Add to Reading List

One Big Pair of Underwear

by Laura Gehl and Tom Lichtenheld

(Winner, 2015)

20 Resources 3 Awards

Add to Reading List

What Will Fit?

by Grace Lin

(Honor, 2022)

19 Resources 3 Awards

Add to Reading List

Count the Monkeys

by Mac Barnett and Kevin Cornell

(Honor, 2015)

18 Resources 2 Awards

Add to Reading List

Hidden Figures: The American Dream and the Untold Story of the Black Women Mathematicians Who Helped Win the Space Race

by Margot Lee Shetterly

(Honor, 2018)

18 Resources 1 Awards

Add to Reading List

Round

by Joyce Sidman and Taeeun Yoo

(Honor, 2020)

18 Resources 6 Awards

Add to Reading List

Zero the Hero

by Joan Holub and Tom Lichtenheld

(Honor, 2015)

17 Resources 3 Awards

Add to Reading List

Edgar Allan Poe's Pie: Math Puzzlers in Classic Poems

by J. Patrick Lewis and Michael Slack

(Honor, 2015)

17 Resources 4 Awards

Add to Reading List

All of the Above

by Shelley Pearsall and Javaka Steptoe

(Honor, 2020)

16 Resources 2 Awards

Add to Reading List

Numbers in Motion: Sophie Kowalevski, Queen of Mathematics

by Laurie Wallmark and Yevgenia Nayberg

(Honor, 2021)

16 Resources 2 Awards

Add to Reading List

Binti

by Nnedi Okorafor

(Honor, 2019)

15 Resources 2 Awards

Add to Reading List

Cao Chong Weighs an Elephant

by Songju Ma Daemicke and Christina Wald

(Honor, 2018)

15 Resources 5 Awards

Add to Reading List

One Fox: A Counting Book Thriller

by Kate Read

(Winner, 2020)

15 Resources 7 Awards

Add to Reading List

Counting the Stars: The Story of Katherine Johnson, NASA Mathematician

by Lesa Cline-Ransome and Raúl Colón

(Honor, 2021)

15 Resources 2 Awards

Add to Reading List

Lia & Luís: Who Has More?

by Ana Crespo and Giovana Medeiros

(Winner, 2021)

15 Resources 3 Awards

Add to Reading List

Feel the Fear

by Lauren Child

(Honor, 2017)

14 Resources 1 Awards

Add to Reading List

Marta! Big & Small

by Jen Arena and Angela Dominguez

(Honor, 2022)

14 Resources 4 Awards

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One is a Piñata: A Book of Numbers

by Roseanne Greenfield Thong and John Parra

(Honor, 2021)

14 Resources 2 Awards

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Meanwhile

by Jason Shiga

(Honor, 2017)

13 Resources 3 Awards

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Navigating Early

by Clare Vanderpool

(Honor, 2017)

13 Resources 5 Awards

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Through the Looking-Glass and What Alice Found There

by Lewis Carroll and John Tenniel

(Commended, 2015)

12 Resources 1 Awards

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Absolutely One Thing

by Lauren Child

(Winner, 2017)

12 Resources 2 Awards

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The Number Devil: A Mathematical Adventure

by Hans Magnus Enzensberger and Rotraut Susanne Berner

(Commended, 2019)

11 Resources 2 Awards

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One Grain of Rice: A Mathematical Folktale

by Demi

(Commended, 2020)

11 Resources 2 Awards

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Just the Right Size: Why Big Animals Are Big and Little Animals Are Little

by Nicola Davies and Neal Layton

(Honor, 2016)

11 Resources 2 Awards

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Eat Your Math Homework: Recipes for Hungry Minds

by Ann McCallum and Leeza Hernandez

(Honor, 2018)

11 Resources 2 Awards

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If You Find This

by Matthew Baker

(Honor, 2019)

11 Resources 2 Awards

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A Hundred Billion Trillion Stars

by Seth Fishman and Isabel Greenberg

(Winner, 2018)

11 Resources 3 Awards

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Crash! Boom!: A Math Tale

by Robie H. Harris and Chris Chatterton

(Winner, 2019)

11 Resources 2 Awards

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100 Bugs!: A Counting Book

by Kate Narita and Suzanne Kaufman

(Honor, 2019)

11 Resources 1 Awards

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Michael Jordan - Twitter (@chicagobulls)

Practice like you've never won and play like you've never lost - Michael Jordan

Who's the best Data Scientist in the world? Cassie Kozyrkov? Yann LeCun? Someone else? Whoever this person is, you can be sure of one thing: This person has worked hard. Because to reach the top, it doesn't only take talent but hard work. Hard work starts with motivation until persistence takes over. Indeed, motivation always wears off over time. And what do you do when it happens?

This is at this precise moment that you're taking an important decision; should I continue or should I stop? Continuing is hard because it is hurtful, and stopping would be the equivalent of giving up. Giving up to who? To yourself. Because you, and only you, is your main competitor.

This article will share 5 tips to become better than your yesterday-you.

The 1% rule consists of practicing at least a little bit, but everyday. Basically, doing even a tiny little bit everyday proves in the long run to be more powerful than being irregular. Someday you'll do great; someday you'll struggle, but everything will eventually pay off.
The key takeaway here is to build a habit, a routine.

Put differently, we can also talk about memorization. We've certainly all spent late nights, the days before our exams trying to fill our brains as much as we could. Chances are that we've already forgotten most of this knowledge. The reason is simply because we've different parts of our brain who retain information differently.

Might come off as common sense; but the more time you give yourself to learn something, the more you will retain. Which can echo what we've just developed previously. Something you'll learn on day 100 will resonate differently than something you learn on day 1.

In the knowledge economy, it is all about compound interest.

Sometimes, it doesn't really matter if you learn again something that you already knew. Since in between you've improved your understanding of the domain, your comprehension of it will have evolved too. And you will remember things that you didn't last time, or understand things differently.

Here some resources to practice your craft everyday:

Get the fundamentals down and the level of everything you do will rise - Michael Jordan.

On the back of what has been just said, going through the basics over and over can be powerful. For instance, someone mastering the ins-and-outs of Pandas can bring a lot of value. Potentially more value than someone playing around with cutting-edge technology.

Pandas and SQL are the bedrock of any analytics project. A deep understanding of these 2 tools enable a great optimization of your work. Someone knowledgeable will know what data type or what function to use that runs 10x to 100x faster and ultimately contributes to ML ops success.

Equally, being unbeatable with the basics will make you faster at delivering results. In industry, being able to deliver proof of concepts, 1st iteration of work, or quick & dirty ad-hoc solution satisfy most of the stakeholders more than we think.

The Lindy effect

The Lindy effect is a theorized phenomenon by which the future life expectancy of some non-perishable things, like a technology or an idea, is proportional to their current age. - Wikipedia

Not only will it optimize your projects, and make you more efficient at your job, but it will also make your role more future-proof.

The basics, by definition, have been in the game for a long time. And statistically, the longer they've been in, the longer they are likely to remain. With this in mind, you'll protect your career significantly if you decide to become the master of something standing the test of time rather than trying to pick all the new trades.

Don't get me wrong. I'm not advising you to not give new tools a go, but use your learning-time strategically instead. 80% on the basics, and 20% on the new for instance?

My best skill was that I was coachable, I was a sponge and aggressive to learn - Michael Jordan

One needs to work strategically. Your learning time is limited; therefore you need to focus on what's really important.

Kaggle comes in handy for that matter. It allows you to look at the best, and identify what separates you from them. Alternatively (or additionally), you can attend webinars or local meet-ups. Identify things you don't know. Then, you can build your learning plan accordingly. Ultimately, it will help define an "As it is" versus a "To be", which will naturally design your learning plan. The key here is to be curious, to google everything that you don't know or are unsure about.

There is "what you know that you know", and "what you know that you don't know".

But also "what you don't know that you know", and more importantly: "What you don't know that you don't know" (The unknown unknowns).

The Dunning-Kruger effect

The Dunning-Kruger effect is a cognitive bias whereby people with low ability, expertise, or experience regarding a certain type of a task or area of knowledge tend to overestimate their ability or knowledge - Wikipedia

To further expand on the previous point, this knowledge map relates to our own bias. Hence the importance of putting effort into mapping out what we don't know. Only then will one realize where they really stand on their learning journey.

One can tell someone's strength by their weak spot

Your strength is only equal to your weakness. On the importance of knowing yourself. If you're great at fine-tuning models, story-telling, data visualization but it actually struggle to clean a dataset this will catch you up.

Fortunately, today resources to learn are abundant. Online courses, Youtube videos, newsletters, podcasts. Name your favorite way of learning, and chances are that a support for it exists. Be specific in what you learn. And work on your main weakness.

Stay on top

Subscribe to newsletters with your professional email address and and read at least one new thing per day. Another idea, start the day with a routine of reading articles, or doing a training exercise as per point #1 of this article. You can also create a list on twitter of data analytics experts.

"Students who spend time teaching what they've learned go on to show better understanding and knowledge retention than students who simply spend the same time re-studying." - The British Psychological Society

When you learn a new language, you'll hear plenty of new words. It's hard to retain them all. So a good way to remind these words is to use them as soon as possible after hearing it for the first time.

Reading and listening are passive, speaking and writing are active. Learning actively has proven more powerful than simply learning passively.

It is the same for data analytics. As much as you can, try to repeat, share, teach what you've learnt. Whether it be a show-and-tell at work, giving a a public talk, or simply writing an online article anonymously, there's no bad way of spreading your knowledge.

In this article I've tried to share five immediately actionable tips that will help you become a better analyst/scientist than yesterday. All in all, it doesn't have to take you hours per day; it just needs to be constant. Also, it goes without saying that you should be kind to yourself. Looking up at other people or what you can be is a great motivation, but don't be disappointed if you don't meet your expectations. Set up a system, trust the process, and enjoy the ride.
If you enjoyed this piece, please make sure to follow my profile, so you don't miss any of my weekly articles. Released previously:

If this post was helpful, please click the clap 👏 button below a few times to show your support for the author 👇

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Practice like you've never won and play like you've never lost - Michael Jordan Who's the best Data Scientist in the world? Cassie Kozyrkov? Yann LeCun? Someone else? Whoever this person is, you can be sure of one thing: This person has worked hard.

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Class handouts from an algebraic geometry class taught by Judy Holdener at CMU in Spring 2013

Handout |
01IntroductionToAlgebraicGeometry.pdf
02UnitCircleExample.pdf
03GroupStructureofUnitCircle.pdf
05IdealofaVariety.pdf
06PolynomialsofOneVariable.pdf
07MonomialOrdersandDivisionAlgorithm.pdf
08ExploringtheIdealMembershipProblem.pdf
09MonomialIdealsDicksonLemma.pdf
10HilbertBasisTheorem.pdf
12ExamIReview.pdf
14PropertiesofGroebnerBases.pdf
15BuchbergersAlgorithm.pdf
16BuchbergersAlgorithm.mw
17FunProblem.pdf
18Elimination.pdf
19EliminationwithGroebnerBases.mw
19GeometryofElimination.pptx
20ImplicitizationProblems.pdf
21Resultants.pdf
22Envelopes.mw
22ProofofExtensionTheorem.pptx
23Nullstellensatz.pdf
24MoreApplications.mw
24RadicalMembership.pptx
27OperationsonIdeals.pdf
28QuotientIdeals.pdf
31ReducibleandIrreducibleVarieties.pdf
32UniqueFactorizationofVarieties.pdf
34CoordinateRingonAffineVariety.pdf
35AlgorithmicComputations.pdf
36ClassifyingAffineVarieties.pdf
37VarietyofMonomialIdeal.pdf
40HilbertFunctionIntro.pdf
41ComplementofMonomialIdeal.pdf
42HillbertPolynomialWrapup.pdf

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Class handouts from an algebraic geometry class taught by Judy Holdener at CMU in Spring 2013

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Q&A for people studying math at any level and professionals in related fields

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This organization in the past would develop wonderful newsletters with ideas on developing math events but has since scaled back.  Just keeping an eye on them to see ifthey return.

In the meantime reread their newsletters in your mailbox.

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With many schools currently closed due to COVID-19, educators and mathematicians worldwide are providing materials for students, teachers, and families to continue engaging with mathematics from their homes. We are highlighting online events that are free and interactive, and that have a primary focus on building community through sharing joyful mathematics.

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Schools are obligated to provide interpretation services for families of English learners, but most don't. Now there's some momentum to change that

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Schools are obligated to provide interpretation services for families of English learners, but most don't. Now there's some momentum to change that

Go beyond test scores and rankings to find out what the culture on campus is like

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We cover inequality and innovation in education with in-depth journalism that uses research, data and stories from classrooms and campuses to show the public how education can be improved and why it matters.

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Medium.com - Writer on Probability and statistics.

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Medium.com - Writer on Probability and statistics.

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Read writing from rita on Medium. learning SWE this quarantine. Every day, rita and thousands of other voices read, write, and share important stories on Medium.

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Press Releases for the NJ Department of Education

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2022 Press Releases for the New Jersey Department of Education

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Research - Free Mathematics Content Research

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Research - Free Mathematics Content Research

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math.MP is an alias for math-ph. Articles in this category focus on areas of research that illustrate the application of mathematics to problems in physics, develop mathematical methods for such applications, or provide mathematically rigorous formulations of existing physical theories.

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Constructivism in Mathematics - Charlotte Mason / Montessori

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Constructivism in Mathematics - Charlotte Mason / Montessori

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Editor's Note. In February of 1943, readers of The Parents' Review were able to read the second part of Elsie Kitching's monumental "Children Up to School Age and Beyond." The first part had been published in the January issue and has also been transcribed and recorded by the Charlotte Mason Poetry team.

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The Mathematical Intelligencer | Volume 43, issue 3

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Volume 43, issue 3 articles listing for The Mathematical Intelligencer

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Infinity is not a number, and trying to treat it as one tends to be a pretty bad idea. At best you're likely to come away with a headache, at worst with the firm belief that 1 = 0.

You may have met various ''paradoxes'' that play on this fact (and our instinct to ignore it) before. An infinitely long line, for instance, is surely infinitely many centimetres long. It's also, equally surely, infinitely many miles long. But each centimetre is a great deal shorter than each mile, so does this mean an infinitely long line is two different lengths at once?

The answer, of course, is to assert confidently that the question is meaningless (and, if you're in the mood to be unkind, just shows how little the asker knows about proper mathematics) and then go back about your business untroubled by such silly quibbles.

Or, as I will hopefully convince you over the course of this article, the answer is a simple ''No.''

Counting the natural numbers

We shouldn't treat infinity as a number. We can't count up to it, and if we have an infinitely large set we will never be able to count all of the objects in it. A set is a collection of objects. They can be any sort of objects, although generally mathematicians use them to talk about mathematical objects, such as numbers. An infinitely large set is, of course, one containing infinitely many objects. It seems strange, therefore, to talk about certain infinitely large sets being ''countably infinite'', but that is indeed what I'm about to do.

Consider the natural numbers (the positive, whole numbers: 1, 2, 3, ...).

There are infinitely many natural numbers (if there were only a finite number of them, there would have to be a largest natural number - what would happen if we added 1 to that number?) so the set containing all the natural numbers must be infinitely large. We'll never be able to count them all. However, we can list them in such a way that if we counted forever, we'd be sure not to miss any out.

The natural numbers are what we use for counting anyway, so we can think of them as coming ready-listed. There is an obvious starting point (1) and a sensible order (1, 2, 3, ...).

To illustrate what I mean by a sensible order, imagine what would happen if we tried to count the natural numbers randomly. However long we counted, we'd never be sure we'd counted them all - we could check for individual numbers in our random list, but we'd never know if they were all there, or when (if ever) we were going to reach a particular one.

Now imagine what would happen if we tried to count the natural numbers by counting the odd numbers first and then the even ones. We'd count forever, and never start counting the even numbers.

However, if we count them in the order given, we'll know once we've reached 100 that we've counted everything between 1 and 100 once and only once. We might never reach the end (in a finite time, at least) but we'll know we've not missed anything on the way and that it's just a matter of time before we reach any given number.

That is, the only thing stopping us from counting them all is that we'll run out of breath before we run out of numbers. The proposed method of counting them all is sound, just impractical. This is what we mean by saying the natural numbers are countably infinite.

In much the same way, any infinite set of numbers that can be put in a sensible, systematic order with a clear beginning such that we're sure to get everything if we count forever is thought of as countable.

When we talk about comparing the sizes of two sets that contain finite numbers of objects (a set, $C$, containing some cats and a set, $M$, containing some baby mice, say) then one way to do so is to match each object in one set with exactly one object in another set and see if we have any left over. For the sets $C$ and $M$, we can instruct each cat to catch exactly one baby mouse - no sharing allowed, nor hogging more than one mouse, nor letting one escape if the cat doesn't already have another one.

If each cat does catch exactly one baby mouse - no more, no fewer - we know there are (or, at least, were) the same number of cats as mice. That is, the sets $C$ and $M$ were the same size. If any cats go hungry, there were more cats than mice ($C$ was larger than $M$) and if any mice go free, there were more baby mice than cats ($M$ was larger than $C$).

This is called putting the objects into one-to-one correspondence, and the same can be done when comparing the sizes of infinite sets.

Another way of thinking about countably infinite sets is that they are those sets whose objects can be put in a one-to-one correspondence with the set containing the natural numbers only. That is, if for each element in the set of the natural numbers there is exactly one - no more, no fewer - element in the set we're trying to count, then the set is the same size as the set containing the natural numbers: countably infinite.

Putting the elements of an infinite set in sensible, systematic order with a clear beginning such that it's just a matter of time before we reach any particular element is, in fact, the same thing as putting those elements in one-to-one correspondence with the natural numbers.

If the natural numbers are the cats from the above illustration and the elements to be counted are the baby mice, then the cats already have an order to line up in. Lining the mice up next to them so none escape and none are eaten together (putting them in a sensible order) is the same as allotting the cats their dinner (putting the elements of the two sets into one-to-one correspondence).

Are the integers (all the whole numbers: ..., -2, -1, 0, 1, 2, 3, ...), therefore, countable?

At first, you might think not. First we need to find a sensible beginning, and that may not be obvious. With the natural numbers, we just started at the smallest and worked up. But the integers don't have a smallest number (just as before, when showing there was no largest natural number, think to yourself: if there were a smallest integer, what would happen when we subtracted 1 from it?), so where can we begin?

Say we start at 1 and work up, as before with the natural numbers. Then we count 1, 2, 3, ..., putting them into one-to-one correspondence as follows:

Here, the problem is however long we count for, we'll never start on the negative numbers. We'll never even get to zero.

It is possible to do. Before reading on, try to think how. It may help to remember the problem we would have had counting the natural numbers if we had tried to count all the odd ones first.

How to count the integers

Consider the following table:

Will each integer be listed once and only once?

It may help (or it may not - don't dwell on this if it just confuses you) to visualise the integers as distinct points on an infinitely long line. Think of 0 as the centre of the line, and imagine a circle, centre 0, which increases in diameter as we count up, covering the numbers we've counted. After we've counted to the seventh integer on the list, say, the circle has a diameter of 6, and covers every number from -3 to 3.

In other words, yes.

Once we reach any integer on the list ($p$, say), any integer with the absolute value (that is, $+n$ if $n$ is positive and $-n$ if $n$ is negative) smaller than the absolute value of $p$ will have been counted exactly once, and any number with a modulus larger than it will be yet to count. Whether $-p$ has been counted obviously depends on whether $p$ is greater or less than 0.

So, while at first glance it might seem that there are more integers than natural numbers, this is not the case.

This is exactly what happens in the so-called paradox I mentioned at the start of this article. You might at first think that for each natural number, $n$, there are two integers, $\pm n$ and so there are twice as many integers as naturals, which is not the case. In the same way, you might think an infinite number of miles takes you further than an infinite number of centimetres. In fact, the centimetres that go to make up the infinitely many miles can be put into one-to-one correspondence with the centimetres that go to make up the infinitely many centimeteres, so both take you the same distance.

How about the rational numbers? (all the numbers that can be written as one integer divided by another: ..., -2/3, 0/1, 1/1, 1/2, ...)

In the case of the natural numbers and the integers, it was easy to check we'd counted every number in a given range. To check we'd counted all the natural numbers less than $n$, we merely needed to have counted up to $n$. To check we'd counted all the integers between $n$ and $m$, we needed to check we'd counted as far as the absolute value of $n$ or $m$, whichever is the largest.

However, we can't count all the rational numbers in a given range, however small the range is.

This is because between any two rationals, there is another rational. (If you're unconvinced, imagine $x$ and $y$ are two rationals between which there isn't another rational. How can $(x + y)/2$ not be rational?) Between -2 and 4 there are seven integers (including -2 and 4) and four natural numbers. There are, however, infinitely many rationals.

How, then, could we possibly count them? We can never get anything in a given range, and there are infinitely many (non-overlapping) ranges we could be given. In fact, if there are infinitely many naturals (which there are) and infinitely many integers (which, again, there are) then surely there must be infinity-squared many rationals, because to get all the rationals you take each integer in turn and divide it by each natural number in turn, as in the following table:

How can infinity-squared possibly be countable, and therefore the same as infinity?

The first thing to do is to take a deep breath and remember that infinity is not a number . In fact, the rationals are countable. To prove this, consider the above table.

Is the table a list of all the positive rationals?

Consider the general term for a positive rational, $p/q$, where both $p$ and $q$ are positive. If it's on the table, then we know we've got all the positive rationals there. (If we hadn't, yet $p/q$ were on the table, there would be a positive rational that can't be represented as $p/q$, which is not the case.) Is it there?

Yes, it's on the $p^{th}$ row in the $q^{th}$ column.

So, we have a complete list. However, trying to count along each row or column in turn gives problems - each row and column is infinitely long, after all, and so we'll never reach the end of the first to start on the second. Is there, then, a way of counting them without missing any out? Yes. Just follow the red line in the image below:

So we know we've counted every rational at least once. Have we counted them at most once?

Obviously, we haven't. 1/1 = 2/2 = 3/3 = ... means 1 is the first, fifth and thirteen number in the list.

Why does this matter? It's a good habit to get into, certainly, but more than that it's useful here to convince ourselves that these sets of numbers are all the same size. If we don't check we're not counting the rationals too many times, what's to stop there being more natural numbers than there are rational ones?

Intuitively, this seems to be a fairly silly fear (after all, the natural numbers are just the first column of the entire table of rationals) but if you're not doubting your intuition by this stage, I haven't explained what we're doing well enough.

We could just check each number we reach against all the previous numbers, making sure it's not equal to any of them. That would work, but ignores the fact that the order in which we're counting these numbers means the first time we meet each one, it's in its simplest possible form.

Consider our trusty $p/q$, in row $p$, column $q$. Assume this is the number in its simplest form - $p$ and $q$ have no common factors greater than 1. The next time we meet the same number, it'll be $2p/2q$, which is in row $2p$, column $2q$, therefore on a diagonal further from our start point. $(n+1)p/(n+1)q$ is always further along than $n p/n q$ (if you need convincing, find a formula to tell you which integer is the leftmost point on the diagonal passing through $y/x$).

So every time we reach a rational not in its simplest form, ignore it.

Hence the positive rationals can be put into one-to-one correspondence with the naturals, and so are countably infinite. Multiply by $-1$ and we get a similar list of the negative rationals. Then, just as we did with the integers, start at 0 and interleave the two lists:

We know we will have every positive and every negative rational, and zero, once and only once.

Thus the natural numbers are countable, the integers are countable and the rationals are countable. It seems as if everything is countable, and therefore all the infinite sets of numbers you can care to mention - even ones our intuition tells contain more objects than there are natural numbers - are the same size.

This is not the case.

Counting the reals: Cantor's Diagonal Proof

Are the real numbers countable? (The real numbers are all the irrationals - those numbers that cannot be written as one integer divided by another: $\pi$, $\sqrt{2}$, $e$, ... - and the rationals together: 1, 4/5, $\pi$, ...)

Every other set of numbers we've met so far has been countable. Each new set of numbers that feels as if it should be larger than the set of the natural numbers has been put into one-to-one correspondence with the natural numbers - all we needed was to work out how to list the numbers sensibly.

The real numbers are made up of the rationals and the irrationals. The rationals are countable, so if the irrationals are countable then the reals must be countable - just interleave our two systematic lists and we'll get another systematic list.

For the same reason, if the reals aren't countable, we'll know that the problem comes with the irrationals.

You've probably met rational numbers in at least two guises - one is that they can be written as one integer divided by another, and the other is that they can be written as decimal expansions that eventually become repeating patterns.

1/3 = 0. 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 ... - the threes repeat forever.

3/8 = 0. 375 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ... - the zeros repeat forever.

22/7 = 3. 142857 142857 142857 142857 ... - 142857 repeats forever.

In fact, all real numbers can be represented as infinitely long decimal expansions. The rationals are the ones that eventually repeat and the irrationals are the ones that don't.

$\pi$ = 3. 14159 26535 89793 23846 ... - at no point does a finitely long string of digits start repeating for ever.

Now, suppose the real numbers were countable. Then we could write a systematic list of all the real numbers. What is more, we could do it as list of decimal expansions.

(In the following list, $a_{(n,m)}$ is the $m^{th}$ digit along in the $n^{th}$ number down the list.)

$a_{(0,0)}. a_{(0,1)}a_{(0,2)}a_{(0,3)}a_{(0,4)}a_{(0,5)}...$

$a_{(1,0)}. a_{(1,1)}a_{(1,2)}a_{(1,3)}a_{(1,4)}a_{(1,5)}...$

$a_{(2,0)}. a_{(2,1)}a_{(2,2)}a_{(2,3)}a_{(2,4)}a_{(2,5)}...$

$a_{(3,0)}. a_{(3,1)}a_{(3,2)}a_{(3,3)}a_{(3,4)}a_{(3,5)}...$

$a_{(4,0)}. a_{(4,1)}a_{(4,2)}a_{(4,3)}a_{(4,4)}a_{(4,5)}...$

We might run into trouble with repeating a number without realising it (0.9 recurring = 1, for instance) but we can just check every number in our list off those that came before. It might be that the system we use to list the numbers lends itself to a neater method, as we found with the rationals, but even if it doesn't, this method will work.

But can we be certain we haven't missed any out?

One way of answering this question is to assume our list is, in fact, infinitely long. Then is it possible for it to contain every possible real number?

The answer is that it isn't. Not all infinities are, we finally see, the same size. The problem, however - one raised and answered by Georg Cantor - is how to show this. How can we write down or describe a number that we know won't be on the list?

Before reading on, take a moment to think about this yourself. It may help to think back to the title of this section: Cantor's Diagonal Proof.

Consider the number $A$, where $A = a_{(0,0)} . a_{(1,1)}a_{(2,2)}...a_{(n,n)}...$ (If the list above had started 1.111..., 2.222..., 4.379..., $A$ would start 1.27..., as 1 is the noughth digit of the noughth number, 2 is the first digit of the first number, and 7 is the second digit of the second number)

Now, letting $m$ take the value of all the natural numbers (and zero) in turn, do the following:

If $a_{(m,m)} \neq 5$, let $b_{(m,m)} = 5$.

If $a_{(m,m)} = 5$, let $b_{(m,m)} = 7$.

Then we have a new number, $B = b_{(0,0)} . b_{(1,1)}b_{(2,2)}...b_{(n,n)}...$

Is $B$ on our list?

Well, it's not the first number, because the first digits don't agree (if the first digit of the first number isn't 5, then the first digit of $B$ is. And if the first digit of the first number is 5, then the first digit of $B$ isn't). And it's not the second number, because the second digits don't agree. And it's not the $n^{th}$ number, because the $n^{th}$ digits don't agree...

So $B$ can't be the first number, and it can't be the second, and it can't be the $n^{th}$, for any value of n, so it isn't on our list.

We could still run into the problem mentioned earlier, that some numbers with recurring digits are the same number, even when the digits are different. However, this is a problem that only occurs when the digits involved are 0's and 9's, and the digits making up $B$ were specially chosen so this would not happen.

Therefore, we know that $B$ is a unique number. This means the list isn't complete, and can never be complete. However cunning our system, even if the list is infinitely long it won't contain every real number.

This means that we can't count all the real numbers - there are uncountably infinitely many.

At the time of writing, Katherine Korner was a second year undergraduate studying Mathematics at Balliol College, Oxford.

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Infinity is not a number, and trying to treat it as one tends to be a pretty bad idea. At best you're likely to come away with a headache, at worse the firm belief that 1 = 0. This article discusses the different types of infinity.

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PZ's Thinking Routines Toolbox | Project Zero

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The Toolbox organizes the Thinking Routines into categories that describe the types of thinking the routines help to facilitate. Some routines appear in more than one category, and some routines have different versions that offer modifications for specific age groups or more specific conceptual challenges.

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December 2013/January 2014 | Volume 71 | Number 4
Getting Students to Mastery

Ceri Dean and Susan M. Brookhart

Ask a random sample of adults (excluding any math teachers) what they dislike most about math, and you'll probably hear two words: word problems. But when we ask adults this question in the future, their responses might be radically different- if their teachers helped them get accustomed to using the Common Core Initiative's recommended standards for mathematical practice while they were in school. Maybe they'll even answer, "I loved word problems!"

These eight mathematical practices are the antidote to teaching mathematics as a series of "plug and chug" procedures ("Do this for this kind of problem"). Students who make sense of word problems don't start by asking, "What kind of problem is this?" They start by trying to figure out what the problem means. What does the problem ask? What information is given, what needs to be found, and which mathematical procedures will lead to that information? Students will approach word problems with questions like these once they master the mathematical practices. They'll not only be able to solve mathematically rich problems, but they'll also appreciate math's usefulness.

Students won't master the standards for mathematical practice overnight, even after schools begin implementing the Common Core State Standards. The practices address habits of mind, thinking processes, and dispositions that help students develop "deep, flexible, and enduring understanding of mathematics" (Briars, Mills, & Mitchell, 2011, p. 20). Teachers will need to both give students problems that require them to use the practices and create environments that support student discourse and risk taking. And students will need a steady diet of feedback on their performance.

Math educators will need to change three things to nurture these practices. Besides changing instructional strategies and materials, teachers will need to change their assessments (so items measure mathematical practices as well as computational skills), and make their feedback focus on students' mathematical reasoning, modeling, and other practices-not just on correct answers. Instruction, assessment, and feedback will all need to focus more on higher-order thinking skills, communication, and collaboration.

The Key: Formative Assessment

Formative assessment can be a powerful tool to integrate cognitively demanding tasks into students' math work. Teachers should share with students clear learning targets that include the eight practices. Because a "target" isn't a target unless students are aiming for it, the first step is finding ways to show students what this kind of mathematical thinking looks like and how they'll know whether they are engaging in it. Students must have clear "look-fors" to monitor the quality of their work and their thinking. Most important, lessons should include ungraded opportunities for students to try out strategies, construct mathematical arguments, and receive feedback on their efforts.

Helping Students Think About Their Thinking

To explore how a teacher might leverage the power of formative assessment, let's look at how 3rd grade teacher Renee Parker modified her instruction to help students develop Mathematical Practices 1 (make sense of problems) and 3 (construct viable arguments). Renee and her colleagues developed a student-friendly rubric to improve their learners' ability to communicate about their thinking while solving problems. They devised five problems, each of which required students to draw diagrams, write number sentences, and explain their reasoning and the steps they used to solve that problem. They then focused on one problem each week (Parker & Breyfogle, 2011).

The rubric provided a clear set of things students should look for in their work (for example, "Was all the important information from the problem used?"). The students discussed weekly problem work with their teacher and classmates, so they got a clearer idea of items on the rubric (for example, "I write what I did and why I did it") looked like. Working through these problems helped all Parker's students learn, but her lower-achieving students improved the most. Students especially improved in writing explanations that included mathematical vocabulary and strategies.

This project included changes to all three areas-instruction, assessment, and feedback. Instruction came to include clear targets, which were clarified by the rubric, student work examples, and discussion around both. Assessment changed: The weekly problems required students to demonstrate that they could use mathematical tools, mathematical models, and viable arguments. These assignments were formative; they allowed students to practice and improve, think intentionally about their own reasoning, and talk with others as they continued to reflect and learn. Feedback changed: It included dialogue about rubric items that helped students make sense of the problems.

Teaching with the Mathematical Practices in View

Zeroing in on one problem shows how a teacher might change these three elements of lessons to help learners simultaneously master both that problem and the mathematical practices. The sample word problem in Figure 1 is aimed at 5th graders and maps to the domain of problem solving at Depth of Knowledge Level 3 on the Common Core framework. The problem asks learners to first compute the total number of crayons in four boxes holding 64 crayons each, and then find the answer to several other questions focused on how to divide these crayons among 32 students. Math problems at this level "require reasoning, planning, or use of evidence to solve problems ... citing evidence and developing logical arguments for concepts" (Webb, 2002, p. 10).

We use this problem because it's a good example of the kind of work students need to do to become skilled with the mathematical practices. Let's look at how teachers might shift their teaching methods to reinforce the practice of constructing viable arguments as students tackle this problem.

Instruction.

Instructional objectives for this lesson should include something about students explaining their reasoning (not just, "Students will solve problems using two-digit multiplication and addition"). You might give students learning targets like, "I can explain what I did and why I did it" or "I used mathematics language."

After sharing the learning targets, show students examples of mathematical explanations former students made for a sample problem. As a group, discuss how one might rate these various explanations, using a rubric that measures how well a student constructs a viable mathematical argument. Rubric items might include "use definitions of mathematics words correctly" or "use drawings to explain steps in solving the problem."

Give students a few minutes to read the problem in Figure 1 and think about how they would approach Part A. Ask each learner to share with a partner what they think this part of the problem is asking and their plan for solving it. Remind students to ask questions like "Why are you choosing to solve it that way?" as they listen to their partner's explanation.

As students solve Part A individually, circulate around the room, noting different approaches students use. Have a few kids explain their solutions to the whole class. Encourage students to refer to the mathematical practices rubric as they listen to these explanations and to question any presenter whose explanation isn't clear. Next, have students respond in their math journals to a prompt like, "Reflect on how you explained 'what you did and why you did it' in solving this problem."

Ask students to solve the rest of the problem-Parts b and C-in pairs and then share their solution with another pair. To wrap up, students might write about what they learned about constructing viable arguments.

Assessment.

Student work should give you lots of evidence about their skills at constructing or explaining mathematical arguments-and what they should do to improve. Evidence comes from three sources: watching students work, talking with students about their process, and observing the final product. Give ongoing, formative feedback, both oral and written, that will move students forward as they take their next steps. Most important, structure classes so that students have immediate opportunities to take those next steps.

Feedback.

Feedback can foster improvement in mathematical practices in four ways:

Feedback should center on the practice. If students are to develop skill at writing an explanation of how they solved the sample problem, your feedback should be about their explanation. For example, if a student wrote for Part A, "I subtracted 4 from 64 and got 60," asking a student "Why did you subtract?" would be more effective than crossing out "subtract" and writing "multiply." A student who multiplies because you told him to won't necessarily learn how to explain when one should multiply.

Feedback should be descriptive. Good feedback comments on what the student did and makes at least one suggestion for a next step. For example, if a student wrote 64 × 4 = 256 for an explanation in Part A, responding, "You've identified the right mathematical expression. Can you tell me why that's the right mathematical expression?" would be better than "Great!" or "Write more."

Feedback takes into account how students learn mathematics (Daro, Mosher, & Corcoran, 2011)-and differs depending on where students are in their learning trajectory with the content. For example, if a student explains his or her work in Part A by writing 64 + 64 + 64 + 64 = 256 or drawing pictures to express the same reasoning, you might point out that the student is correct and ask whether he or she can think of any explanations that involve a process other than addition. Such feedback affirms her current understanding and helps push his or her thinking; as students develop an understanding of multiplication, they generally grasp additive reasoning before multiplicative reasoning.

Feedback helps students see that there are more and less effective arguments, as well as correct and incorrect ones. Imagine one student wrote for Part C: "Students have 8 crayons from 4 boxes. To get 12, each student needs 4 more crayons, that's half as many as they already have. So half as many more boxes is 2. 4 boxes + 2 boxes is 6 boxes." Say another student wrote: "12 crayons × 32 students = 384, so we need 384 crayons. 384 ÷ 64 in a box = 6, so we need 6 boxes." As feedback to each student, you could say, "You've both written clear explanations that get you to the same number of total boxes. Work together to compare your explanations. Are they both right? If so, how is that possible? Explain to one another the next step you'll take to answer the question."

Applying These Principles

We've zeroed in on one of the eight mathematical practices and one word problem, but the principles illustrated in this example apply to helping students develop skills in all eight practices and in a wide range of mathematical content.

Note that the changes to instruction, assessment, and feedback we've portrayed here all involve formative assessment strategies: communicating clear learning targets and criteria for success, designing performances of understanding that match the learning targets, providing feedback that feeds students forward, and asking questions that make student thinking visible. Teachers will need a thorough understanding of what each practice looks like and how students typically learn mathematics content to use these strategies effectively.

All these strategies are interdependent. Focusing instruction on the quality of students' mathematical arguments won't work unless you've communicated to students what a mathematical argument is-and made constructing one their learning target. Having a target won't help unless students have multiple opportunities to solve word problems in various ways and talk with others about that problem solving. The formative learning process is a cycle. At its best, it's an upward spiral, getting students to mastery.

Use accurate vocabulary and appropriate symbols and labeling to explain their thinking and justify their solutions.

Represent problems in multiple ways and reason with models and pictorial representations.

Use technology tools to demonstrate a solution to a problem, and justify their tool selection.

See and use structure to solve problems, notice patterns in calculations, and look for ways to generalize what they learn to broader mathematical strategies.

References

Briars, D., Mills, V., & Mitchell, S. (2011, February 23). Deeper dive into the Common Core State Standards: Leading with the mathematical practices [Webinar]. Denver, CO: National Council of Supervisors of Mathematics. Retrieved from www.mathedleadership.org/docs/events/webinars/NCSMCCSSWebinar2011-02-23Presentation.pdf

Daro, P., Mosher, F., & Corcoran, T. (2011). Learning progressions in mathematics: A foundation for standards, curriculum, assessment, and instruction (Report #RR-68). Philadelphia, PA: Consortium for Policy Research in Education Research Report. Retrieved from www.cpre.org/learning-trajectories-mathematics-foundation-standards-curriculum-assessment-and-instruction

Parker, R., & Breyfogle, M. L. (2011). Learning to write about mathematics. Teaching Children Mathematics, 18(2), 90-99.

Webb, N. L. (2002). Alignment study in language arts, mathematics, science and social studies of state standards and assessments for four states. Washington, DC: Council of Chief State School Officers.

Endnote

The eight mathematical practices are make sense of problems and persevere in solving them, reason abstractly and quantitatively, construct viable arguments and critique the reasoning of others, model with mathematics, use appropriate tools strategically, attend to precision, look for and make use of structure, and look for and express regularity in repeated reasoning. For a detailed description of each practice, see www.corestandards.org/Math/Practice.

Ceri Dean is a senior fellow at McREL in Denver, Colorado. Susan M. Brookhart is an independent education consultant based in Helena, Montana, and a research associate at the Center for Advancing the Study of Teaching and Learning at Duquesne University in Pittsburgh, Pennsylvania. Her latest book is How to Create and Use Rubrics for Formative Assessment and Grading (ASCD, 2013).

KEYWORDS

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Mathematical Practices for Deep Understanding

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December 2013/January 2014 | Volume 71 | Number 4 Getting Students to Mastery Ceri Dean and Susan M. Brookhart Ask a random sample of adults (excluding any math teachers) what they dislike most about math, and you'll probably hear two words: word problems. But when we ask adults this question in the future, their responses might be radically different- if their teachers helped them get accustomed to using the Common Core Initiative's recommended standards for mathematical practice while they were in school.

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With the traditional method of math instruction -- the way many of us were taught -- the goal was to get the right answer. Our instructional model, however, requires students to explain and show their thinking. Today we'd like to share some of the cognitive and learning science research behind this approach, and why it's so important that students explain their reasoning.
1. Teachers need to know how students arrive at their answers.
Right and wrong answers don't reveal much about student thinking. We also need to be able to connect what students do with what they know. For example, let's look at a real piece of student work:

If the only information the teacher had was this answer, s/he might think the student doesn't know anything about fractions. Click here to hear the student explain his thinking.

He said, "I have to find a multiple of 10, so half would go to 5/10 and 1/5 would go to 2/10, and multiply that to make one whole."

Now we know that he worked to find the common denominator and knew right away that it was 10. He also quickly found that 1/2 = 5/10 and 1/5 = 2/10.

So we know that this student knows how to find a common denominator (but not when to find a common denominator). We also know that the student understands equivalent fractions. His mistake was in applying the process for adding fractions to multiplication.

Why would a student do this? The answer is the next advantage of having students explain their reasoning:
2. Explanations encourage students to explain the why and when, not just the how.
This student likely learned to add fractions by following a rote procedure: find common denominators, add (or subtract) the numerators, and simplify. If you learn that procedure without knowing why to find common denominators in addition or subtraction, you might, like this student, assume that all operations with fractions require common denominators.

Asking students to explain their reasoning can make a connection between the procedure and the underlying conceptual knowledge, and that connection helps students know when to apply procedures like common denominators.

In addition, students solve problems in different ways, and if you don't know the way they're solving it, you don't know what the student is capable of doing. For example, one student might use a formula to solve a problem while another uses context clues from the word problem. If a test question only asks for the answer, all you know is that they got it right, and you might assume that they all have the same knowledge when they don't.
3. Students learn better when they self-explain.
Several research studies have shown that self-explaining can have a positive impact on student learning. In one study, students who were prompted to self-explain demonstrated a more robust mental model of what they learned than students who were not.*

Self-explaining establishes connections between conceptual and procedural knowledge. These connections both contextualize the knowledge (providing the why and when) and make it easier to remember. Self-explaining can be a powerful tool for students both when they learn new concepts and when they access that knowledge while solving a problem, so giving them opportunities to do so can have an impact on their success.

Useful Strategies to Get Students to Explain Their Thinking

The simplest way to start having your students explain their thinking is to ask them questions like:

Why?

How do you know?

Are you sure?

Will that always work?

You can also build a student's ability to explain their thinking by providing opportunities for students to analyze the work of their peers and explain what's correct or incorrect about their thinking. You can, for example, provide several samples of student work and ask students to figure out which is correct and explain why.

We want to improve not just test scores, but real understanding of mathematics, which is why our textbook provides countless opportunities for students to demonstrate their thinking and MATHia analyzes and adapts to how students solve problems, not just the answer they give.

*"You Got Some Explaining to Do! Self-Explaining," Dr. Bob's Cog Blog

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Why Students Need to Explain Their Reasoning

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With the traditional method of math instruction -- the way many of us were taught -- the goal was to get the right answer. Our instructional model, however, requires students to explain and show their thinking. Today we'd like to share some of the cognitive and learning science research behind this approach, and why it's so important that students explain their reasoning.

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School sector leaders are calling on the chancellor for extra funds to deal with spiralling SEND costs, the crumbling school estate and low minimum per-pupil funding

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A selection of number crunching programmes from the Radio 4 Archive.

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Nils Ahbel of Deerfield Academy gave a thought provoking presentation at the 2011 Anja S. Greer Conference on Secondary School Mathematics (held at Phillips Exeter Academy in Exeter, NH) on the history and potential future of the American High School mathematics curriculum. The Prezi that he used to illustrate his talk can be found here.

As I recall, his core points about the state of things today were that:

- our curriculum has remained largely unchanged for 119 years (witness the content of the textbook whose pages fill the number 8 in his prezi).

- the current goal of most high school curricula is to prepare students to take calculus, yet only around 8% of the student population ever seems to (eventually) pass a calculus course.

- as a society, we consistently depict mathematics and other quantitative topics as being un-intelligible, obscure, and/or arcane - yet every student will need to be able to decide for themselves on questions such as: are they on track with their retirement fund? Does data support or refute global warming? Is driving to the airport more or less risky than flying?

He then turned to ideas on how we might change this situtation:

- our syllabi are full. If we wish to add a topic, we almost always have to find one or more topics to drop in exchange.

- Instead of pursuing incremental change, it is more sensible to re-think our curriculum from the ground up. Begin by adding the topics that are most important to the futures of the students we are teaching.

- Calculus is still a useful and needed course of study, particularly for students going into the sciences and engineering, but why prepare 92% of students for a course they are not likely to complete or make use of?

- Exponential equations and statistics are currently and seem likely to continue to be the most relevant topics our students should master .

- The problems that most students will need to understand in this day and age involve: compound interest, present value, population growth, resource depletion, metabolic assimilation rates, descriptive statistics, understanding standard deviation, and supporting arguments with statistics.

Update 7/16/11: Some of these points are not new, as R. Wright pointed out in a comment below. Sheldon P. Gordon wrote about these and related issues in a 2005 paper titled " What's Wrong With College Algebra? ", which is interesting to read.

Published by

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Re-thinking Our High School Math Curriculum

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Nils Ahbel of Deerfield Academy gave a thought provoking presentation at the 2011 Anja S. Greer Conference on Secondary School Mathematics (held at Phillips Exeter Academy in Exeter, NH) on the history and potential future of the American High School mathematics curriculum. The Prezi that he used to illustrate his talk can be found here.

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Achievethecore.org :: Instructional Content Nav - Mathematics: Focus by Grade Level

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The Common Core and other college- and career-ready (CCR) standards call for a greater focus in mathematics. Rather than racing to cover topics in a mile-wide, inch-deep curriculum, CCR standards require us to significantly narrow and deepen the way time and energy are spent in the math classroom. We focus deeply on the major work of each grade so that students can gain strong foundations: solid conceptual understanding, a high degree of procedural skill and fluency, and the ability to apply the math they know to solve problems inside and outside the math classroom. The following documents illustrate these concepts using the Common Core State Standards, but Focus, Coherence, and Rigor are integral to all college- and career-ready standards.

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math mathematics search sites resources
collection ~steves dump

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The Math Forum Internet Mathematics Library

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The Internet Math Library is our best resource for teachers, students and parents looking for web sites across the Internet on math topics. Sites are gathered from our own explorations and submissions from users like yourself.

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Many rules taught in mathematics classrooms "expire" when students develop knowledge that is more sophisticated, such as using new number systems. For example, in elementary grades, students are sometimes taught that "addition makes bigger" or "subtraction makes smaller" when learning to compute with whole numbers, only to find that these rules expire when they begin computing with integers (Karp, Bush, and Dougherty 2014). However, middle-grades students, especially those who are struggling, often try to force-fit the rules that they remember from the elementary grades to new concepts or skills.

In this article, we present 12 persistent rules that expire. These are "rules" that we have found prevalent in our many years of working with students, from mathematics education literature, or in some cases, rules that we ourselves have taught and later regretted. In each case, we offer mathematically correct and more helpful alternatives.

The Common Core's Standards for Mathematical Practice (SMP) encourage precision, including the appropriate use of mathematics vocabulary and notation and the reasoned application of "rules" (CCSSI 2010). This leads to instruction focused on sense making and reasoning-the very experience described in NCTM's Principles to Actions: Ensuring Mathematical Success for All (2014).
GETTING STARTED
Imagine this scenario in your mathematics classroom, in which you present the following set of one-variable equations:

Which of these equations would your students choose to solve first or find the easiest? In our work, we found that students were comfortable solving equations 1 and 3 because they "looked normal" with the "operation" first (on the left side), followed by the "answer" (on the right side). Students often hesitated at equations that were similar to 2 and 4 because the perceived "operation" and "answer" were arranged in a seemingly reversed order (see table 1). This inflexibility can be linked to the idea that middle school students often have yet to interpret and understand the equal sign as a symbol indicating a relationship between two quantities (Mann 2004). Additionally, students may think that the solution to an equation always goes on the right side of the equal sign. These overgeneralizations are not helpful and can have a negative impact on students' conceptual understanding. We suggest that these students are experiencing rules that expire (Karp, Bush, and Dougherty 2014).

We highlight rules sometimes used with middle school students that seem to hold true at the moment, given the content the student is learning at that time. However, students will later find that these rules expire. Sometimes taught as shortcuts with content that students learned in the previous grades, these rules expire when students use them inappropriately with more advanced problems and find that they are incorrect. Such experiences can be frustrating and can promote the belief that mathematics is a mysterious set of tricks and tips to memorize rather than concepts that relate to one another. For each rule that expires, we do the following (similar to Karp, Bush, and Dougherty 2014):

1. State the rule.

2. Discuss how students overgeneralize it.

3. Provide counterexamples.

4. State the expiration date or the point when the rule begins to fall apart.
12 RULES THAT EXPIRE
1. KFC: Keep-Flip-Change.

When learning to divide fractions, students are sometimes taught to KFC (Keep-Flip-Change) or told "Yours is not to reason why, just invert and multiply." Although both versions align with the standard algorithm, students might overgeneralize this rule to other operations with fractions. Additionally, these mnemonics and sayings do not promote conceptual understanding, making it challenging for students to apply them in a problem-solving context. Instead, division of fractions can be linked to whole-number division by asking how many groups of the divisor make up the dividend. Although students will eventually use the algorithm, they should gain a conceptual understanding of dividing fractions through the use of physical models (Cramer et al. 2010) or other methods, such as the common denominator strategy. Expiration date: Grade 6 (6.NS.1)

2. Use the factor rainbow to factor.

Sometimes students are taught to create a "factor rainbow" to list all factors of a number. For example, if finding the factors of 20, students would write 1 and 20, then 2 and 10, and 4 and 5 (see fig. 1). The rule is taught so that once you identify factors that are consecutive numbers (e.g., 4 and 5), you have identified all factors. However,factors can be missed with this approach. Some numbers do not have consecutive factor pairs, as in when finding the factors of 40: 5 and 8 are the closest factors. Moreover, this rule expires when factoring a square number (its factors are not all paired) or when working with non-whole-number factors, such as in algebra when students factor out fractional parts as well as variables. Expiration date: Grade 6 (6.NS.4)

3. The absolute value is just the number.

Students are sometimes told that the absolute value of a number is that number, with a positive sign. For example, |-4| = 4 because you drop the negative sign. Confusion sets in when students are presented with -|-4| because they are unsure what this represents. How can absolute value be negative? Without making sense of the meaning of absolute value (that is, its distance from zero on a number line), students may not interpret it correctly within particular contexts.

Expiration date: Grade 6 (6.NS.7)

4. Multiplication is repeated addition.

Considering multiplication as only repeated addition can result in students thinking that the expression 33 is equivalent to 3 + 3+ 3. This thinking leads to overgeneralizations because students come to believe that 3 raised to the third power means that 3 is used as an addend 3 times. Writing such expressions in correct expanded form can help with this misunderstanding.

Expiration date: Grade 6 (6.EE.1)

5. PEMDAS: Please Excuse My Dear Aunt Sally.

This mnemonic phrase is sometimes taught when students solve numerical expressions involving multiple operations. At least three overgeneralizations commonly occur with this rule:

* Students incorrectly believe that they should always do multiplication before division, and addition before subtraction, because of their order in the mnemonic (Linchevski and Livneh 1999), instead of performing them in the order in which they appear in the expression.

* Students perceive that the order of PEMDAS is rigid. For example, in the expression

30 - 4(3 + 8) + 9 ÷ 3,

there are options as to where to begin. Students actually have a choice and may first simplify the 3 + 8 in the parentheses, distribute the 4 to the 3 and to the 8, or perform 9 ÷ 3 before doing any other computation-all without affecting an accurate outcome.

* The P in PEMDAS suggests that parentheses are first, but this should also represent other grouping symbols, including brackets, braces, square root symbols, and the horizontal fraction bar. We suggest making sense of a problem. However, if using a hierarchical model, consider this order: (a) Grouping symbols or exponents; (b) multiplication or division; and (c) addition or subtraction. Expiration date: Grade 6 (6.EE.2)

6. A solution to an equation must be in the form × = □.

Students are often taught that the variable and/or operation comes first, followed by the answer (e.g., the constant) in an algebraic equation (Dougherty and Foegen 2011). However, this rule has no mathematical necessity because the equal sign indicates that two quantities are equivalent. Therefore, variables, operations, and constants can be located on either or both sides of the equal sign. Instead of overgeneralizing that an equation should "look" a certain way, we as teachers should promote flexibility in students' thinking. When the teacher uses a specified set of steps and the placement of the solution in that format, students lose sight of the conceptual aspects of equations and instead focus on implementing algorithmic steps. Expiration date: Grade 6 (6.EE.4)

7. The "Butterfly Method" for comparing fractions.

Students are frequently taught the "Butterfly Method," which refers to cross multiplying two fractions to determine which fraction is greater. For example, in

students may draw a loop around the 3 and 8 and around the 4 and 7 (which looks like a butterfly) and multiply 3 × 8 = 24 and 4 × 7 = 28 to determine that 24 is less than 28, which tells them that

However, this rule is problematic for several reasons. First, it does not foster conceptual understanding of the numerical value of fractions because it removes the need to understand the relationship between the two fractions or consider the quantities they represent. Second, students begin to overgeneralize and incorrectly apply this rule to other situations whenever they see two fractions, such as when they add, subtract, multiply, or divide fractions.

Expiration date: Grade 7 (7.RP.2)

8. The most you can have is 100 percent of something.

Students are sometimes taught that because 100 percent is equivalent to 1 whole, that is the most they can have. However, increases and decreases can be of any size, including more than 100 percent. This rule expires as students work with ratios and proportional relationships involving markups, discounts, commissions, and so on. Expiration date: Grade 7 (7.RP.3)

9. Two negatives make a positive.

This rule may be taught when students learn about multiplication and division of integers and is used to help students quickly determine the sign of the product or quotient. However, this rule does not always hold true for addition and subtraction of integers, such as in -5 + (-3) = -8. Additionally, this rule does not foster the understanding of why the product or quotient of two or more integers is negative or positive. Instead of focusing on the rule, consider using patterns of products to develop generalizations about the relationship between factors and products.

Expiration date: Grade 7 (7.NS.2)

10. Use keywords to solve word problems.

A keyword approach is frequently introduced in the elementary grades and extends throughout a student's school career as a way to simplify the process of solving word problems. However, using keywords encourages students to overgeneralize by stripping numbers from the problem and using them to perform a computation outside the problem context (Clement and Bernhard 2005). This removes the act of making sense of the actual problem from the process of solving word problems. Many keywords are common English words that can be used in many different ways. Often a list of words and corresponding operations are given so that word problems can be translated into a symbolic, computational form. For example, students are told that if they see the word of in a problem, they should multiply all the numbers given in the problem. Likewise, although the keyword quantity sometimes signifies the need for the distributive property, at other times it does not. Keywords are especially troublesome in the middle grades as students explore multistep word problems and must decide which keywords work with which component of the problem. Although keywords can be informative, using them in conjunction with all other words in the problem is critical to grasping the full meaning.

Expiration date: Grade 7 (7.NS.3)

11. A variable represents a specific unknown.

When students work with one-variable equations, the solutions to the equations are almost always one specific value (e.g., x = 5 or x = -3). However, students overgeneralize this as being true for all situations involving variables, yet this rule quickly expires as variables take on other meanings, such as varying quantities or parameters (e.g., y = mx + b), labels (A = bh), or generalized unknowns. Additionally, students may not accept equations that represent identities (such that the variable can take on any value) or equations that have no solution (such as 3x + 4 = 3x - 4). This rule expires when students begin to work with linear functions.

Expiration date: Grade 8 (8.EE.7)

12. FOIL: First, Outer, Inner, Last.

When learning to multiply two binomial expressions, students might be taught to FOIL, that is, to multiply the first term in the first binomial by the first term in the second, then multiply the outer terms of each binomial, then the inner terms of each binomial, and then the second (last) terms of each binomial. Although this rule works for binomials, it soon expires as students begin multiplying other polynomials, such as a binomial and a trinomial, or two trinomials. Instead, have students explore how they are really using the distributive property multiple times, to multiply each term in one polynomial by each term in the other polynomial.

Expiration date: High school (A.APR.1)
EXPIRED LANGUAGE AND NOTATION
We must also consider the mathematical language and notation that we use and that we allow our students to use. The ways in which we communicate about mathematics may bring with them connotations that result in students' misconceptions or misuses, many of which relate to the previously discussed Rules That Expire. Using terminology and notation that are accurate and precise (SMP 6) develops student understanding that withstands the growing complexity of the secondary grades. Table 2 includes commonly used expired language and notation, gathered from our years of experience in the classroom, paired with alternatives that are more appropriate.
"BUT MS. JONES SAID SO"
Coherence is one of the major emphases in CCSSM. By having a series of rigorous standards at each grade, with less overlap and structured alignment, students can progress more purposefully through the content. By building a schoolwide plan for the consistent and precise presentation of rules, terminology, and notation used by all teachers, students will never get caught in the "But Ms. Jones said so" mode of finding something in their past instruction that is no longer accurate. Through such intentional consistency, students are able to focus on the new ideas presented as the language and tenets continue to be the foundation for lessons. Because the middle-grades years are pivotal in cementing the ideas from elementary school and building the concepts needed for high school, this explicit, systemic consistency is critical. As we avoid these 12 Rules That Expire, we instead find ways to present a seamless and logical world of mathematical ideas.
Are There Other Rules That Expire?
We invite MTMS readers to submit additional instances of "rules that expire" or "expired language" that this article does not address. Join us as we continue this conversation on MTMS's blog at http://www.nctm.org/12rules, or send your suggestions and thoughts to mtms@nctm.org. We look forward to your input.
REFERENCES
Clement, Lisa, and Jamal Bernhard. 2005. "A Problem-Solving Alternative to Using Key Words." Mathematics Teaching in the Middle School 10 (March): 360-65.

Common Core State Standards Initiative (CCSSI). 2010. Common Core State Standards for Mathematics (CCSSM). Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers. http://www.corestandards .org/wp-content/uploads/Math_Standards.pdf

Cramer, Kathleen, Debra Monson, Stephanie Whitney, Seth Leavitt, and Terry Wyberg. 2010. "Dividing Fractions and Problem Solving." Mathematics Teaching in the Middle School 15 (February): 338-46.

Dougherty, Barbara J., and Anne Foegen. 2011. "Evaluation of the Mathematics Project for Special Education and General Education Teachers." Report Prepared for the Iowa Department of Education. Ames, IA: Iowa State University.

Karp, Karen S., Sarah B. Bush, and Barbara J. Dougherty. 2014. "13 Rules That Expire." Teaching Children Mathematics 21: (August): 18-25.

Linchevski, Liora, and Drora Livneh. 1999. "Structure Sense: The Relationship between Algebraic and Numerical Contexts." Educational Studies in Mathematics 40 (November): 173-96.

Mann, Rebecca. 2004. "Balancing Act: The Truth behind the Equals Sign." Teaching Children Mathematics 11 (September): 65-69.

National Council of Teachers of Mathematics (NCTM). 2014. Principles to Action: Ensuring Mathematical Success for All. Reston, VA: NCTM.

Any thoughts on this article? Send an email to mtms@nctm.org. -Ed.

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12 Math Rules That Expire in the Middle Grades

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Many rules taught in mathematics classrooms "expire" when students develop knowledge that is more sophisticated, such as using new number systems. For example, in elementary grades, students are sometimes taught that "addition makes bigger" or "subtraction makes smaller" when learning to compute with whole numbers, only to find that these rules expire when they begin computing with integers (Karp, Bush, and Dougherty 2014).

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New Jersey Student Learning Standards

Mathematics

Deidre Richardson, Coordinator

New: The New Jersey Student Learning Standards (NJSLS) are reviewed and revised every five years. The 2020 NJSLS were adopted by the State Board of Education on June 3, 2020. The 2020 New Jersey Student Learning Standards webpage provides links to the 2020 NJSLS and information regarding curriculum implementation dates. The NJSLS in English Language Arts and Mathematics were adopted by the New Jersey State Board of Education in May 2016 and are not currently under review.

The principles for school mathematics address the overarching themes of equity, curriculum, teaching, learning, assessment and technology. (NCTM, 2000)

Equity: Excellence in mathematics education requires equity - high expectations, worthwhile opportunities, accommodation for differences, resources, and strong support for all students.

Curriculum: A coherent curriculum effectively organizes standards and mathematical ideas, focuses on important mathematics, and is well articulated within and across grades.

Teaching: Effective standards-aligned mathematics instruction is a complex endeavor that requires understanding what students know and need to learn, and then challenging and supporting them to learn it well. Effective teaching requires continually seeking improvement.

Learning: Conceptual understanding is an important component of proficiency. Students must learn mathematics with understanding, actively building new knowledge from experience and prior knowledge. Learning with understanding is essential to enable students to solve the new kinds of problems they will inevitably face in the future.

Assessment: Standards-aligned assessment, a routine part of ongoing classroom activity, should enhance students' learning and inform instructional decisions.

Technology: Technology, not to be used as a replacement for basic understandings and intuitions, is an essential tool in teaching and learning mathematics; it influences the mathematics that is taught, supports visualization, facilitates organizing and analyzing data, and offers efficient computation.

Intent and Spirit of the Mathematics Standards
For more than a decade, research studies of mathematics education in high-performing countries have concluded that mathematics education in the United States must become substantially more focused and coherent in order to improve mathematics achievement in this country. To deliver on this promise, the mathematics standards are designed to address the problem of a curriculum that is "a mile wide and an inch deep."

The math standards provide clarity and specificity rather than broad general statements. The standards draw on the most important international models for mathematical practice, as well as research. They endeavor to follow the design envisioned by William Schmidt and Richard Houang (2002), by not only stressing conceptual understanding of key ideas, but also by continually returning to organizing principles (coherence) such as place value and the laws of arithmetic to structure those ideas.

In addition, the "sequence of topics and performances" that is outlined in a body of math standards must respect what is already known about how students learn. As Confrey (2007) points out, developing "sequenced obstacles and challenges for students...absent the insights about meaning that derive from careful study of learning, would be unfortunate and unwise." Therefore, the development of the standards began with research-based learning progressions detailing what is known today about how students' mathematical knowledge, skill, and understanding develop over time. The knowledge and skills students need to be prepared for mathematics in college, career, and life are woven throughout the mathematics standards.
New Jersey Student Learning Standards for Mathematics NJSLS Mathematics Crosswalk

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New Jersey Student Learning Standards: Mathematics

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New Jersey Student Learning Standards Mathematics Deidre Richardson, Coordinator New: The New Jersey Student Learning Standards (NJSLS) are reviewed and revised every five years. The 2020 NJSLS were adopted by the State Board of Education on June 3, 2020. The 2020 New Jersey Student Learning Standards webpage provides links to the 2020 NJSLS and information regarding curriculum implementation dates.

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Teachers' Circle Online Workshop - mathcommunities.org

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Description: This free workshop offered stand-alone sessions throughout the week of July 13 to 17. The daily program included math sessions, "Teachers' Lounge" seminars, and social time to connect with math friends near and far. All MTC members were invited to join the workshop for as many sessions as they liked!

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Journal of Mathematics Education at Teachers College

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The JMETC is intended to provide dissemination opportunities for writers of practice-based or research contributions to the general field of Mathematics Education.

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The Brown Center released a report today called "The Algebra Imperative." The report shows that current national and international math assessments do not adequately measure how well American students are learning algebra. In this chalkboard post, I elaborate upon a point made in the report: that taking and successfully completing an Algebra II course, which once certified high school students' mastery of advanced topics in algebra and solid preparation for college-level mathematics, no longer means what it once did. The credentialing integrity of Algebra II has weakened.

The declining significance of successfully completing Algebra II highlights a dilemma. Pushing students to take more advanced coursework has been a mainstay of American school reform for several decades. That prescription has worked in boosting enrollments. In 1986, less than half of all 17 year-olds (44%) had completed Algebra II, and for Black and Hispanic students, the rate was less than a third. Completing Algebra II is now commonplace. In 2012, about three-fourths of students completed Algebra II, and the race/ethnicity gaps associated with taking the course have narrowed significantly. (All NAEP data below are from the NAEP data explorer.)

Table 1. Percentage of 17 Year-Olds who Completed Second Year Algebra (1986-2012)

Getting more students to take higher level math courses may be a hollow victory. It has not coincided with students learning more math. Figure I shows the average NAEP scores for Algebra II completers (in the following discussion, "completers" include students enrolled in the course when they took NAEP). As enrollments boomed, test scores went down. Caution must be exercised in infering causality. We cannot be certain of a causal connection between rising enrollments and falling scores, but it is certainly plausible-even likely-that enrollment gains have been achieved by drawing in students who are not prepared to take the course. Research does not indicate a decline in the quality of Algebra II teachers or Algebra II textbooks or some other deterioration in course components from 1986-2012 that can explain the decline in NAEP scores.

Figure 1. NAEP Math, 17 Year-Olds who have Completed Second Year Algebra (1986-2012)

A Pipeline Problem

Previous research has documented serious problems in the earliest stages of the advanced math pipeline, in which students traditionally progressed from Algebra I to a final course (Pre-Calculus or Calculus) over four or five years. More and more unprepared students are being pushed into advanced math in middle school. In 2008, I published a study analyzing eighth grade NAEP data. The data were from 2005, and even then a large percentage of low achievers (about 29% of those performing at the 10 th percentile or lower in math) were enrolled in advanced math courses. I called these students "Misplaced Math Students" because they did not function at a level commensurate with their math courses. They functioned at about the second to third grade level. They were students who had trouble answering NAEP items involving fractions, decimals, and percentages, concepts that should have been learned several years earlier, and yet they were enrolled as middle schoolers in courses that until recently were considered high school courses.

Several subsequent studies have raised questions about policies that make low performing students take Algebra I in 8 th grade.[i] A study out of California found that marginal math students who spent one more year before tackling Algebra I were 69% more likely to pass the algebra end of course exam in 9 th grade than ninth grade peers who were taking the course for the second time after failing the algebra test in 8 th grade. The researchers referred to a "leaky pipeline" put under increased pressure by California's aggressive placement of low achieving 8 th graders in algebra. The policy succeeded in boosting middle school enrollments in Algebra I, but a large number of students quit taking math just as soon as they could in high school. Even low achievers who matriculate into higher level math courses may not benefit. A study of Charlotte-Mecklenburg students by Clotfelter, Ladd, and Vigdor found that low achievers who took 8 th grade algebra experienced negative long term effects, including lower pass rates in Geometry and Algebra II.

Consequences

The misplaced students are not the only ones who pay a price. Recent studies suggest that a systematic erosion of the legitimacy of advanced courses is taking place. By that I mean: students are taking advanced, college-prep courses, passing them with good grades, and yet do not know the advanced subject matter signified by the titles of the courses they have taken. Courses have lost their credentialing power. An National Center for Education Statistics (NCES) study of high school transcripts analyzed the textbooks students used in high school Algebra I, Geometry, and Integrated Math courses. NCES conducted the study to solve a riddle.

The 2005 National Assessment of Educational Progress (NAEP) High School Transcript Study (HSTS) found that high school graduates in 2005 earned more mathematics credits, took higher level mathematics courses, and obtained higher grades in mathematics courses than in 1990. The report also noted that these improvements in students' academic records were not reflected in twelfth-grade NAEP mathematics and science scores. Why are improvements in student coursetaking not reflected in academic performance, such as higher NAEP scores?[ii]

The researchers found that course titles often don't mean what they say. NCES Commissioner Jack Buckley summarized the study's main finding in an interview with Education Week, "We found that there is very little truth in labeling for high school Algebra I and Geometry courses."[iii]

As unprepared students flow through a series of counterfeit courses, the entire curricular system is corrupted. Algebra II teachers are expected to teach mathematics to students who passed Algebra I with good grades but who, in reality, have not mastered elementary grade concepts that are fundamental to understanding algebra. Parents get false signals about how well their sons and daughters are prepared for college. Schools misallocate resources dedicated to remedial programs by assuming that students know material that they, in fact, do not know.

In the end, the transition to college unmasks the charade. In California, the California State University System draws students from the top one-third of graduating seniors. In 2012, about 30% of entering freshmen taking the Entry Level Math test failed the exam and were placed in remedial math classes, despite earning a mean GPA of 3.15 in college prep high school programs. That doesn't make sense. Good grades in tough courses, yet remediation was needed.[iv]

The problem is not confined to California. Nor is it limited to mathematics. A report from the National Center for Public Policy and Higher Education (NCPPHE) estimates that about half of all four-year college freshmen must take remedial classes. The NCPPHE pinpoints a flawed assumption:

Over the last 15 years, many states have emphasized mastery of specific content and performance standards, as shown through grades and statewide assessments; however, this shift to standards-based performance in the schools generally has not been extended to higher levels of achievement associated with college readiness, whose indicators still focus on courses taken. The flawed assumption has been that if students take the right courses and earn the right grades, they will be ready for college.[v]

We need to restore the legitimacy of college prep courses. Let's ensure that all students who study and pass advanced subject matter have learned what that their parents, teachers and colleges assume they have learned.

To read more about the declining significance of coursetaking-and my thoughts on potential solutions to the problem in terms of algebra courses-please read the full paper, "The Algebra Imperative: Assessing Algebra in a National and International Context."

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Algebra II and The Declining Significance of Coursetaking

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The Brown Center report "The Algebra Imperative" shows that current state, national, and international math assessments do not adequately measure how well American students are learning algebra. Here, Tom Loveless elaborates upon a point made in the report: that taking and successfully completing an Algebra II course, which once certified high school students' mastery of advanced topics in algebra and solid preparation for college-level mathematics, no longer means what it once did.

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Browse all issues of Gifted Child Quarterly

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Click an issue image to see its table of contents.

Spring 2020
Table of Contents

COMMENTS FROM THE EDITORS
Carolyn Briles, Loudoun County Public Schools
Brian Buckhalter, Buck Wild About Math, LLC

1

USING ROLE PLAY TO REHEARSE PROBLEMS OF PRACTICE
Shannon Larsen, University of Maine at Farmington
Cheryl Tobey, Maine Mathematics and Science Alliance

3

PURPOSEFULLY COACHING MIDDLE SCHOOL TEACHERS: THE CASE OF AN EXEMPLARY MATHEMATICS SPECIALIST
Joy W. Whitenack, Virginia Commonwealth University
Aimee J. Ellington, Virginia Commonwealth University

13

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Click an issue image to see its table of contents. Spring 2020Table of Contents COMMENTS FROM THE EDITORS Carolyn Briles, Loudoun County Public SchoolsBrian Buckhalter, Buck Wild About Math, LLC 1 USING ROLE PLAY TO REHEARSE PROBLEMS OF PRACTICE Shannon Larsen, University of Maine at FarmingtonCheryl Tobey, Maine Mathematics and Science Alliance 3 PURPOSEFULLY COACHING MIDDLE SCHOOL TEACHERS: THE CASE OF AN EXEMPLARY MATHEMATICS SPECIALIST Joy W.

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Mathematics Teaching in the Middle School

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“Mathematics Teaching in the Middle School” published on by National Council of Teachers of Mathematics.

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Issues in 2021

53(1) Mathematics Teacher Noticing: Exploring and Expanding the Terrain
Eds.: Jaguthsing Dindyal, Edna O. Schack, Ban Heng Choy, Miriam G. Sherin

53(2) Designing and enacting teaching that supports language in mathematics classrooms
Eds.: Susanne Prediger, Kirstin Erath, Jenni Ingram and Judit Moschkovich

53(3) Calculus in High School and College Around the World
Eds.: Patrick Thompson and Guershon Harel

53(4) Empirical Research on Mathematical Problem Solving and Problem Posing around the World
Eds.: Peter Liljedahl, Jinfa Cai

53(5) Implementation and implementability of mathematics educations research
Eds.: Boris Koichu, Mario Sánchez Aguilar, Morten Misfeldt

53(6) Mathematics Textbooks as Instruments for Change
Eds.: Sebastian Rezat, Lianghuo Fan and Birgit Pepin

54 (2) Understanding complexity in the mathematics classroom: In memoriam David Clarke
Eds: Yoshinori Shimizu, Berinderjeet Kaur, Carmel Mesiti and Man Ching Esther Chan

54 (3) Mathematics Learning in Chinese Contexts
Eds.: Jinfa Cai, Yingkang Wu, Qimeng Liu, and Meixia Ding

54 (4) Educational Research in Discrete Mathematics: Current Work and Future Directions
Eds: James Sandefur, Elise Lockwood and Eric Hart

54 (5) Mathematical Competitions in Mathematics Education
Eds: María Falk de Losada and Peter Taylor

54 (6) Early Algebraic Thinking: A Multi-Dimensional and Ever-Expanding Global Vision
Ed: Carolyn Kieran

54 (7) Exploring the significance of the history of mathematics in mathematics education
Eds.: Renaud Chorlay, Kathleen M. Clark and Constantinos Tzanakis

Virtual special issue - COVID 19 pandemic – its mathematical background, its social and educational consequences
Eds.: Johann Engelbrecht, Marcelo Borba and Gabriele Kaiser

Issues in 2023

55(1)

Emotions and Motivation in Mathematics Education and Educational Psychology
Eds.: Stanislaw Schukajlow, Katrin Rakoczy and Reinhard Pekrun

55(2) Conceptualizing teachers’ interactions with resources in crossing languages and cultures
Eds: Luc Trouche, Jill Adler and Janine Remillard

55(3) Connecting citizenship education and mathematics education
Eds: Katja Maass, Vince Geiger, Iddo Gal and Mellony Graven

55(4) Exploring and Strengthening University Mathematics Courses for Secondary Teacher Preparation
Eds.: Orly Buchbinder, Nicholas H. Wasserman, Nils Buchholtz

55(5) Reconceptualizing the relationship between language, mathematics, and the learner in mathematics education research
Eds.: Higinio Dominguez, Miwa Takeuchi, Marta Civil

55(6) Mathematics and Interdisciplinary STEM education
Eds.: Merrilyn Goos, Susana Carreira, Immaculate Kizito Namukasa

55(7) Reviews on important themes in mathematics education
Eds.: Gabriele Kaiser, Stanislaw Schukajlow

Issues in 2024

56(1) tba

56(2) Reviews on important themes in mathematics education
Eds.: Gabriele Kaiser, Stanislaw Schukajlow

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ZDM - Mathematics Education is one of the oldest mathematics education research journals in publication. The journal surveys, discusses, and builds upon ...

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Abstract

What shapes do you get when you spin a cube on one of its vertices? This article explores the algebraic aspects of a spinning cube and establishes equations for the two cones and the hyperboloid using secondary school mathematics. Both the analysis and the verification take advantage of dynamic mathematics learning technologies.

Recommended Citation

Bu, Lingguo (2021) "The Spinning Cube: An Algebraic Excursion," The Mathematics Enthusiast: Vol. 18 : No. 1 , Article 5.
Available at: https://scholarworks.umt.edu/tme/vol18/iss1/5

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The Spinning Cube: An Algebraic Excursion

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What shapes do you get when you spin a cube on one of its vertices? This article explores the algebraic aspects of a spinning cube and establishes equations for the two cones and the hyperboloid using secondary school mathematics. Both the analysis and the verification take advantage of dynamic mathematics learning technologies.

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JUME

2151-2612 (Online)

​

Homepage
Publisher: Aggie STEM

Society/Institution: Texas A&M

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Platform/Host/Aggregator: OJS

Date added to DOAJ: 9 Jun 2011

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LCC Subject Category: Education: Special aspects of education | Science: Mathematics

Publisher's keywords: mathematics education, urban education, black/african american education, latino/a education, critical inquiry

Language of fulltext: English

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JUME 2151-2612 (Online) ​ Homepage Publisher: Aggie STEM Society/Institution: Texas A&M Country of publisher: United States Platform/Host/Aggregator: OJS Date added to DOAJ: 9 Jun 2011 Record Last Updated: 13 Jan 2020 LCC Subject Category: Education: Special aspects of education | Science: Mathematics Publisher's keywords: mathematics education, urban education, black/african american education, latino/a education, critical inquiry Language of fulltext: English Full-text formats available: PDF OPEN ACCESS & LICENSING Type of License: CC BY-NC-SA License terms Open Access Statement Year Open Access Content Began: 2008 Does the author retain unrestricted copyright?

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The most downloaded articles from The Journal of Mathematical Behavior in the last 90 days.

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Journal for Research in Mathematics Education - National Council of Teachers of Mathematics

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An official journal of the National Council of Teachers of Mathematics (NCTM), JRME is the premier research journal in mathematics education and is devoted to the interests of teachers and researchers at all levels--preschool through college.

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Educational Studies in Mathematics presents new ideas and developments of major importance to those working in the field of mathematics education. It seeks to…

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Full Description

Forewords by Paul Killen

Mathematics in lockdown by Tony Cotton

Education on the move: A project in Belize by Shanine Campbell and Michelle Turto

Sharing learning from Shanghai on school reopening under Covid-19 by Linda Wang

Coping with Covid by Heidi Kirkland

Plotting coordinates with Barley's Maths by Jenny Forsythe

Being mathematical 1 by Ashley Compton, Gerry McNally and Mary Pardoe

Spirals of activity: Examining a pattern block sequence by Simon Gregg

Mathematical white lies by Colin Foster and Mike Ollerton

Centre pages: Resources from Median

Being mathematical 2 by Livia Mitson

Intelligent answers by Anne Watson

The fraction question by Jonny Griffiths

Using a mathematics project to connect with GCSE resit students in an FE College by Valerie Sampson

Insight by Kasia Tatys

The Gresham lectures

The Jo Walters Trust grants 2019-2020

Remembering Don Steward

From the archive 1: Attack by Don Steward

From the archive 2: Pattern blocks by Tom Barclay

Book review by Ian Benson

General Council report

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Full Description Forewords by Paul Killen Mathematics in lockdown by Tony Cotton Education on the move: A project in Belize by Shanine Campbell and Michelle Turto Sharing learning from Shanghai on school reopening under Covid-19 by Linda Wang Coping with Covid by Heidi Kirkland Plotting coordinates with Barley's Maths by

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The Mathematics Educator strives to provide a forum for collaboration of mathematics educators at varying levels of professional experience. Its purpose is to promote the interchange of ideas among the mathematics education community, locally, nationally, and internationally and to present a variety of viewpoints on a broad spectrum of issues related to mathematics education.

The Mathematics Educator got its start in 1989 at the University of Georgia. Since its first publication in 1990, TME has been a student-produced journal supported by the Mathematics Education Student Association and the Department of Mathematics and Science Education. We are grateful for our readers, reviewers, and authors who allow us to continue to produce TME and openly share peer-reviewed knowledge with the field.

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The Mathematics Educator strives to provide a forum for collaboration of mathematics educators at varying levels of professional experience. Its purpose is to promote the interchange of ideas among the mathematics education community, locally, nationally, and internationally and to present a variety of viewpoints on a broad spectrum of issues related to mathematics education.

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Frequency: Yearly ISSN: 1947-7503 eISSN: 2472-7466 https://www.tandfonline.com/doi/abs/10.1080/19477503.2018.1548222 ABSTRACT This study focuses on proportional division, a specific problem type of proportions and proportional relationships, which embeds a part-part-whole relationship.

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This special issue will detail how to judge, select, and communicate research questions that are researchable. However, papers for this special issue should not merely list researchable questions in mathematics and science.

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international journal of science and mathematics education

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International Journal of Science and Mathematics Education

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This special issue will detail how to judge, select, and communicate research questions that are researchable. However, papers for this special issue should not merely list researchable questions in mathematics and science.

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The Hong Kong Academy for Gifted Education

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We are a non-governmental organisation, providing research-based information and support to all gifted students aged 10-18 (through our programmes), teachers (through our training opportunities) and their parents (through our programmes and hotline) acros

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Fall 2022 Current Issue Where We Stand: Reclaiming Our Future Randi Weingarten

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Articles by: Kevin Hogan | Tech & Learning

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The launch of Tech & Learning University, coupled with the relaunch of TechLearning.com, ensures Future's Education Group will remain the go-to resource for education technology news, reviews, and insights.

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Despite the man's awkward gestures, unkempt hair, and ill-fitting suit, it was one of the most extraordinary speeches that Reverend John Gulliver had ever heard. It was March 1860, and the venue was Norwich, Connecticut. The following morning Gulliver struck up conversation with the speaker, a politician by the name of Abraham Lincoln, as he caught a train down to Bridgeport.

As the pair took their seats in the carriage, Gulliver asked Lincoln about his remarkable oratory skill: "I want very much to know how you got this unusual power of 'putting things.' " According to Gulliver, Lincoln said it wasn't a matter of formal education. "I never went to school more than six months in my life." But he did find training elsewhere. "In the course of my law-reading I constantly came upon the word demonstrate," Lincoln said. "I thought, at first, that I understood its meaning, but soon became satisfied that I did not." Resolving to understand it better, he went to his father's house and "staid there till I could give any propositions in the six books of Euclid at sight." 1

Elements proposed that definitions were at the foundation of knowledge, and led to self-evident axioms that needed no proof. From these definitions and axioms, Euclid showed how to prove dozens of mathematical propositions, producing knowledge that was objective and undeniable. A person of reason would have to accept a proven fact, no matter what their personal beliefs or convictions were.

Elements would become a best-selling work, second only to the bible in printed editions, and used until recently as the standard text for mathematics classes. It profoundly influenced Western thought, and shaped Western science and art. What's less recognized is its role in the creation of modern politics: The distance from proofs about equilateral triangles to the foundations of democracy in Europe and the United States turned out to be just about two millennia.

John Locke was an early pioneer of Euclidean thinking in politics. Born in 1632, the Englishman grew up in a time of turbulence, with a nine-year civil war beginning in 1642. Locke, whose father had fought against the Royalists in the war, would go on to develop a great interest in the concept of morality. In forming his ideology, Locke took guidance from the logical structure of Elements. He believed that by following the logical consequences of self-evident statements "as incontestable as those in mathematics," it would be possible to demonstrate what was right and wrong.

Historically, absolute monarchs and the church had dictated laws and justice. Enlightenment thinkers such as Locke sought to challenge this tradition. Rather than defining equality from above, he wanted to root it in natural, objective laws. Locke believed that the "natural rights" of a society could be established in a similar manner to geometric theorems, and would therefore be "as certain as any demonstration of Euclid."

Like Euclid, Locke defined terms, then used these definitions to prove moral claims. For example, Locke defined property as "a right to any thing," and injustice as "the invasion or violation of that right." The statement "where there is no property there is no injustice" resulted naturally from these definitions. In his Two Treatises of Government, published in 1689, Locke noted that "creatures of the same species and rank, promiscuously born to all the same advantages of nature, and the use of the same faculties, should also be equal one amongst another." It was therefore self-evident that, in the absence of rules or laws, humanity's natural state was "free, equal, and independent." He concluded that the aim of a government should be to preserve the natural rights of life, liberty, and property.

Locke's thinking would flow across the Atlantic, into the pens of the American Founding Fathers. The final version of the United States Declaration of Independence, drafted by Thomas Jefferson in 1776, opens with "We hold these truths to be self-evident, that all men are created equal." A keen student of mathematics himself, Jefferson had adopted Euclid and Locke's logical style, from the "self-evident" nature of the truths to the subsequent right to "life, liberty, and the pursuit of happiness."

Nearly a century later, Lincoln had found his own way to Euclid. While working as a lawyer on the Illinois circuit during the early 1850s, he carried Euclid's texts in his saddlebag. One time, his colleague William Herndon entered their shared office in Springfield to find Lincoln was surveying a vast canvas of diagrams. Lincoln was so taken with the shapes on the thick paper sheets that he barely acknowledged Herndon as he walked in. Bottles of ink lay scattered on the table, alongside pencils, a compass, a ruler, and a large pile of fresh paper-the results of a fruitless effort to "square the circle," a classic Euclidean puzzle which involves drawing a square and a circle of equal area.

While Lincoln had to admit defeat after three days of drawing and measuring, Euclid's mathematical principles and system of logic would go on to become a crucial tool throughout his political career. Euclidean logic would shape his arguments against slavery, and his debates against its proponents. It would convince him of what was just, and what was unacceptable. As he rose to the presidency, and as the nation descended into civil war, Euclid would be there to guide him.

A person of reason would have to accept a proven fact, no matter what their personal beliefs or convictions were.

"If A can prove, however conclusively, that he may, of right, enslave B," Lincoln conjectured in a private 1854 essay, "why may not B snatch the same argument, and prove equally, that he may enslave A?" The argument was clear and irrefutable: Regardless of whether the right to own slaves was defined by color, intellect, or money, an enslaver could always use the same reasoning to enslave their supposed superior.

It was a textbook example of a kind of proof often used in mathematics, called proof by contradiction. In Elements, Euclid used this proof method to show that if two angles in a triangle are equal, the sides opposite them will be equal too. First, he assumed that one side was longer than the other. If he cut off the excess, however, so both sides were the same length, the smaller triangle would be in every way identical to the original. This contradicted his axiom that "the whole is greater than the part." Hence both sides must always be equal. Lincoln's proof had followed this same structure.

Lincoln's main rival on the Illinois political circuit was Democrat Judge Stephen Douglas, who did not share Lincoln's opposition to slavery. When the Founding Fathers talked of equality, Douglas claimed in 1857, they were not really referring to all men, "they were speaking of British subjects on this continent being equal to British subjects born and residing in Great Britain." Lincoln countered by once again following the logic to its absurd conclusion. If the Declaration only referred British subjects, Lincoln argued, "the French, Germans, and other white people of the world are all gone to pot along with the Judge's inferior races."

During 1858, Lincoln and Douglas-who were running against each other for the Senate-engaged in a series of debates. Once again, Lincoln turned to Euclid. When Douglas criticized a legal point made by Senator Lyman Trumbull by calling him dishonest, Lincoln told Douglas that logical arguments could not be defeated with personal insults. "If you have ever studied geometry, you remember that by a course of reasoning, Euclid proves that all the angles in a triangle are equal to two right angles," Lincoln said. "Euclid has shown you how to work it out. Now, if you undertake to disprove that proposition, and to show that it is erroneous, would you prove it to be false by calling Euclid a liar?"

Douglas argued that slavery was a sovereign right, to be decided by the people of each state. Lincoln disagreed: The notion of popular sovereignty should not include slavery. The following year, Lincoln challenged Douglas to construct a logically sound argument to support his case. "If Judge Douglas will demonstrate somehow that this is popular sovereignty-the right of one man to make a slave of another, without any right in that other, or anyone else to object-demonstrate it as Euclid demonstrated propositions, there is no objection."

Euclid would have been proud.

As powerful as they were, there was also a limit to using Euclid's axioms to argue the issues of the day. Euclidean logic relied on the concept of common axioms, agreed upon by all. But what if there wasn't general agreement on those axioms? Progress could be difficult to come by. Speaking once in the city of Peoria, Illinois in October 1854, Lincoln became frustrated with Douglas' view that Illinois was not founded as a free state. "To deny these things is to deny our national axioms," Lincoln said. Which, in his view, made argument hopeless. "If a man will stand up and assert, and repeat, and re-assert, that two and two do not make four, I know nothing in the power of argument that can stop him." By the end of the 1850s, when slavery had drawn a dividing line through the United States, Lincoln would write "One would start with great confidence that he could convince any sane child that the simpler propositions of Euclid are true, but, nevertheless, he would fail, utterly, with one who should deny the definitions and axioms. The principles of Jefferson are the definitions and axioms of free society. And yet they are denied and evaded, with no small show of success."

Not only did Lincoln begin to reach the limits of axiomatic persuasion, the very idea of a fixed set of Euclidian axioms began to dissolve during this period, even in its native field of mathematics. In the 19th century mathematicians revisited classical questions in Euclidean geometry, such as whether it was possible to square the circle as Lincoln had attempted (it wasn't). Others challenged the nature of geometry itself. Researchers like Carl Gauss explored non-Euclidean realms, in which Euclid's theorems no longer held: Draw a triangle on the surface of a sphere, and its three angles no longer add up to two right angles.

"Would you prove it to be false by calling Euclid a liar?"

By the end of the century, mathematicians such as Georg Cantor were going further, and finding paradoxes and inconsistencies at the very heart of pure mathematics. Their predecessors had developed theorems based on the "intuitively evident" laws of nature, such as Euclid's axiom that "the whole is greater than the part." Real life intuition suggests this statement is always true, but Cantor found that it failed when applied to infinite collections of numbers. For example, Cantor demonstrated that the set of positive integers (1, 2, 3 ...) had an equivalent "size" to the set of square numbers (1, 4, 9 ...). The whole, in other words, was not always larger than the part.

Original photo by ROBYN BECK / Getty Images

In the early 20th century, researchers tried to tackle the growing number of paradoxes by building a set of complete, consistent axioms, from which all mathematical theories could be derived. Unfortunately, Kurt Gödel's 1931 incompleteness theorems showed this aim to be impossible. He proved that no matter how detailed the axioms, there were always some situations that they would not cover. Gödel's work demonstrated the limitations of axiomatic systems: Mathematics could not be both complete and consistent. Axioms were simply not enough.

So how could they be for politics? Abolishing slavery eventually required a new rule-the Thirteenth Amendment-implicitly implying that slavery was not so self-evidently prohibited by the constitution as it stood. Other new interpretations of American national axioms also made themselves felt. In 1896, the U.S. Supreme Court had ruled that "separate but equal" facilities for different races did not contradict the constitution. But attitudes gradually changed, and in 1954 they reversed the decision, noting that "separate educational facilities are inherently unequal." Denial of women's suffrage in the early 20th century and internment of Japanese Americans during World War II-both deemed constitutional at the time but subsequently condemned-further illustrate the shifting the notion of equality.

All of this amendment and re-evaluation mirrored Lincoln's realization that we cannot rely on untouchable and self-evident truths. In his 1863 Gettysburg Address, he described the U.S. as a nation "dedicated to the proposition that all men are created equal." The axiomatic project had to give way to a more modern project of ongoing interpretation-of propositions-whether it was in math or in politics. Barack Obama put it this way in a speech last year, following a U.S. Supreme Court ruling that same-sex marriage was a constitutional right. "Our nation was founded on a bedrock principle that we are all created equal. The project of each generation is to bridge the meaning of those founding words with the realities of changing times." Equality, over time, has changed from a self-evident conclusion to a proposition. The proof of that proposition, Lincoln hoped, would be the United States itself.

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Adam Kucharski is an assistant professor at the London School of Hygiene & Tropical Medicine, and the author of The Perfect Bet: How Science and Math Are Taking the Luck Out of Gambling.
References
1. Carpenter, F.B. The Inner Life of Abraham Lincoln: Six Months at the White House Hurd & Houghton, New York, NY (1874).

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londonschool of hygiene & tropical medicine

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Euclid as Founding Father - Issue 41: Selection - Nautilus

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Despite the man's awkward gestures, unkempt hair, and ill-fitting suit, it was one of the most extraordinary speeches that Reverend John Gulliver had ever heard. It was March 1860, and the venue was Norwich, Connecticut. The following morning Gulliver struck up conversation with the speaker, a politician by the name of Abraham Lincoln, as he caught a train down to Bridgeport.

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Welcome to OpenMind, the knowledge community of BBVA, an information portal with the best free and accessible knowledge both in Spanish and English

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Edutopia is a free source of information, inspiration, and practical strategies for learning and teaching in preK-12 education. We are published by the George Lucas Educational Foundation, a nonprofit, nonpartisan organization.

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2:05

Ask a dozen educators to define personalized learning and you're bound to get 12 different answers about why this approach to learning has strong benefits and real risks for students. (November 6, 2018)

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2:05 Ask a dozen educators to define personalized learning and you're bound to get 12 different answers about why this approach to learning has strong benefits and real risks for students. (November 6, 2018)

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Pre-service teachers designing serious games for 21st century classrooms

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Serious games are fun and powerful vehicles for learning, yet initial teacher education courses rarely require pre-service teachers to design serious games they can be used for future teaching and learning. Wanting to disrupt this reality, we

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Education Week American Education News Site of Record - Education Week

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Find breaking news, analysis, and opinion on K-12 education issues affecting school and district leaders, teachers, policymakers, and ed-tech professionals.

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Colm Mulcahy, of Spelman College in Atlanta, joins us to share his ice cream trick from his CardColm mathematical card trick column on the MAA website! You're invited to explain how this works in the comments below.

Colm also shares a quick puzzle, tweeted on his What Would Martin Gardner Tweet feed @WWMGT. And finally we touch on the Gathering For Gardner and the Celebration of Mind, held all over the world around the time of Martin Gardner's birthday, October 21.

And at last we get around to answering our quiz from a few weeks ago. There are indeed two solutions for correctly filling in the blanks in:

The number of 1's in this paragraph is ___; the number of 2's is ___; the number of 3's is ____; and the number of 4's is ___.

[spoiler] namely (3,1,3,1) and (2,3,2,1) [/spoiler]

We can vary this puzzle at will, asking

The number of 1's in this paragraph is ___; the number of 2's is ___; ..... and the number of N's is ___.

For N=2 or 3, there are no solutions (Asking that all the numbers we fill in are between 1 and N); for N=4 there are two. For N=5 there is just one, for N=6 there are none and beyond that there is just one. I think we'll let the commenters explain that.

But here's the cool thing.

One way to approach the problem is to try filling in any answer at all, and then counting up what we have, filling that in, and repeating. Let's illustrate, but first stipulate that we'll stick with answers that are at least plausible- you can see that the total of all the numbers we fill in the blanks has to be 2N (since there are 2N total numbers in the paragraph).

So here's how this works. Suppose our puzzle is:

There are ___ 1's;___ 2's; ___ 3's; ___ 4's; ___ 5's

Let's pick a (bad) solution that totals 10, say, (2,4,1,2,1). So we fill in:

There are __2_ 1's; __4_ 2's; _1__ 3's; __2_ 4's; _1__ 5's

That's pretty wrong! There are actually three 1's in that paragraph, three 2's; at least there is just one 3, and two 4's and one 5. In any case this gives us another purported solution to try: (3,3,1,2,1). Let's fill that in:

There are __3_ 1's; __3_ 2's; _1__ 3's; __2_ 4's; _1__ 5's

That attempt actually does have three 1's; but has only two 2's; it does have three 3's but only one 4 and one 5. So let's try (3,2,3,1,1):

There are __3_ 1's; __2_ 2's; _3__ 3's; __1_ 4's; _1__ 5's

Lo and behold that works! We do in fact have three 1's; two 2's; three 3's and yes, one 4 and one 5.

So we can think of it this way: filling in a purported solution and reading off what we actually have gives another purported solution.

In this case (2,4,1,2,1) -> (3,3,1,2,1) -> (3,2,3,1,1) -> (3,2,3,1,1) etc,

We can keep following this process around, and if we ever reach a solution that gives back itself, we have a genuine answer, as we did here.

So here's an interesting thing to think about.

First, find, for N>=7, a correct solution; and a pair of purported solutions A,B that cycle back and forth A->B->A->B etc.

Second, find a proof that this is all that can happen (unless I'm mistaken)- any other purported solution eventually leads into the correct one or that cycle.

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Colm Mulcahy, of Spelman College in Atlanta, joins us to share his ice cream trick from his CardColm mathematical card trick column on the MAA website! You're invited to explain how this works in the comments below. Colm also shares a quick puzzle, tweeted on his What Would Martin Gardner Tweet feed @WWMGT.

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EdSurge - Education Journalism and Research

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A nonprofit newsroom reporting on the powerful forces, fascinating people and innovative practices shaping teaching and learning.

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