From Numberphile: “Some stuff about Pi, the ‘celebrity number’. This video features maths-loving author Alex Bellos and Professor Roger Bowley from the University of Nottingham.”
Did you notice the error? It was supposed to be “a”…
The Math Student’s Manifesto
[Feature photo above by Texas A&M University (CC BY 2.0) via Flickr.]
Note to Readers: Please help me improve this list! Add your suggestions or additions in the comment section below…
What does it mean to think like a mathematician? From the very beginning of my education, I can do these things to some degree. And I am always learning to do them better.
(1) I can make sense of problems, and I never give up.
- I always think about what a math problem means. I consider how the numbers are related, and I imagine what the answer might look like.
- I remember similar problems I’ve done before. Or I make up similar problems with smaller numbers or simpler shapes, to see how they work.
- I often use a drawing or sketch to help me think about a problem. Sometimes I even build a physical model of the situation.
- I like to compare my approach to the problem with other people and hear how they did it differently.
Teaching the Standard Algorithms
[Feature photo above by Samuel Mann, Analytical Engine photo below by Roͬͬ͠͠͡͠͠͠͠͠͠͠͠sͬͬ͠͠͠͠͠͠͠͠͠aͬͬ͠͠͠͠͠͠͠ Menkman, both (CC BY 2.0) via Flickr.]
An algorithm is a set of steps to follow that produce a certain result. Follow the rules carefully, and you will automatically get the correct answer. No thinking required — even a machine can do it.
This photo shows one section of the first true computer, Charles Babbage’s Analytical Engine. Using a clever arrangement of gears, levers, and switches, the machine could crank out the answer to almost any arithmetic problem. Rather, it would have been able to do so, if Babbage had ever finished building the monster.
One of the biggest arguments surrounding the Common Core State Standards in math is when and how to teach the standard algorithms. But this argument is not new. It goes back at least to the late 19th century.
Here is a passage from a book that helped shape my teaching style, way back when I began homeschooling in the 1980s…
Ruth Beechick on Teaching Abstract Notation
“Understanding this item is the key to choosing your strategy for the early years of arithmetic teaching. The question is: Should you teach abstract notation as early as the child can learn it, or should you use the time, instead, to teach in greater depth in the mental image mode?
“Abstract notation includes writing out a column of numbers to add, and writing one number under another before subtracting it. The digits and signs used are symbols. The position of the numbers is an arbitrary decision of society. They are conventions that adult, abstract thinkers use as a kind of shorthand to speed up our thinking.
“When we teach these to children, we must realize that we simply are introducing them to our abstract tools. We are not suddenly turning children into abstract thinkers. And the danger of starting too early and pushing this kind of work is that we will spend an inordinate amount of time with it. We will be teaching the importance of making straight columns, writing numbers in certain places, and other trivial matters. By calling them trivial, we don’t mean that they are unnecessary. But they are small matters compared to real arithmetic thinking.
“If you stay with meaningful mental arithmetic longer, you will find that your child, if she is average, can do problems much more advanced than the level listed for her grade. You will find that she likes arithmetic more. And when she does get to abstractions, she will understand them better. She will not need two or three years of work in primary grades to learn how to write out something like a subtraction problem with two-digit numbers. She can learn that in a few moments of time, if you just wait.”
— Ruth Beechick
An Easy Start in Arithmetic (Grades K-3)
(emphasis mine)
2015 Mathematics Game
[Feature photo above by Scott Lewis and title background (right) by Carol VanHook, both via Flickr (CC BY 2.0, text added).]
Did you know that playing games is one of the Top 10 Ways To Improve Your Brain Fitness? So slip into your workout clothes and pump up those mental muscles with the Annual Mathematics Year Game Extravaganza!
For many years mathematicians, scientists, engineers and others interested in math have played “year games” via e-mail. We don’t always know whether it’s possible to write all the numbers from 1 to 100 using only the digits in the current year, but it’s fun to see how many you can find.
Rules of the Game
Use the digits in the year 2015 to write mathematical expressions for the counting numbers 1 through 100. The goal is adjustable: Young children can start with looking for 1-10, middle grades with 1-25.
- You must use all four digits. You may not use any other numbers.
- Solutions that keep the year digits in 2-0-1-5 order are preferred, but not required.
- You may use +, -, x, ÷, sqrt (square root), ^ (raise to a power), ! (factorial), and parentheses, brackets, or other grouping symbols.
- You may use a decimal point to create numbers such as .2, .02, etc., but you cannot write 0.02 because we only have one zero in this year’s number.
- You may create multi-digit numbers such as 10 or 201 or .01, but we prefer solutions that avoid them.
My Special Variations on the Rules
- You MAY use the overhead-bar (vinculum), dots, or brackets to mark a repeating decimal. But students and teachers beware: you can’t submit answers with repeating decimals to Math Forum.
- You MAY NOT use a double factorial, n!! = the product of all integers from 1 to n that have the same parity (odd or even) as n. Math Forum allows these, but I’ve decided I prefer my arithmetic straight.
December Advent Math from Nrich
[Feature photo (above) by Austin Kirk via Flickr (CC BY 2.0).]
Click on the pictures below to explore a mathy Advent Calendar with a new game, activity, or challenge puzzle for each day during the run-up to Christmas. Enjoy!
Advent Calendar 2014 – Primary
Advent Calendar 2014 – Secondary
Math Debates with a Hundred Chart
Wow! My all-time most popular post continues to grow. Thanks to an entry from this week’s blog carnival, there are now more than thirty great ideas for mathematical play:
The latest tips:
(31) Have a math debate: Should the hundred chart count 1-100 or 0-99? Give evidence for your opinion and critique each other’s reasoning.
[Hat tip: Tricia Stohr-Hunt, Instructional Conundrum: 100 Board or 0-99 Chart?]
(32) Rearrange the chart (either 0-99 or 1-100) so that as you count to greater numbers, you climb higher on the board. Have another math debate: Which way makes more intuitive sense?
[Hat tip: Graham Fletcher, Bottoms Up to Conceptually Understanding Numbers.]
(33) Cut the chart into rows and paste them into a long number line. Try a counting pattern, or Race to 100 game, or the Sieve of Eratosthenes on the number line. Have a new math debate: Grid chart or number line — which do you prefer?
[Hat tip: Joe Schwartz, Number Grids and Number Lines: Can They Live Together in Peace? ]
Roadmap to Mathematics: 3rd Grade
A frequently-asked question on homeschooling forums is, “Are my children working at grade level? What do they need to know?”
The Council of the Great City Schools has published a handy 6-page pdf summary of third grade math concepts, with suggestions for how parents can support their children’s learning:
Whether you are a radical unschooler or passionately devoted to your textbook — or, like me, somewhere in between — you can help your children toward these grade-level goals by encouraging them to view mathematics as mental play. Don’t think of the standards as a “to do” list, but as your guide to an adventure of exploration. The key to learning math is to see it the mathematician’s way, as a game of playing with ideas.
The following are excerpts from the roadmap document (along with a few extra tips) and links to related posts from the past eight years of playing with math on this blog…
Roadmap to Mathematics: 2nd Grade
A frequently-asked question on homeschooling forums is, “Are my children working at grade level? What do they need to know?”
The Council of the Great City Schools has published a handy 6-page pdf summary of second grade math concepts, with suggestions for how parents can support their children’s learning:
Whether you are a radical unschooler or passionately devoted to your textbook — or, like me, somewhere in between — you can help your children toward these grade-level goals by encouraging them to view mathematics as mental play. Don’t think of the standards as a “to do” list, but as your guide to an adventure of exploration. The key to learning math is to see it the mathematician’s way, as a game of playing with ideas.
The following are excerpts from the roadmap document (along with a couple of extra tips) and links to related posts from the past eight years of playing with math on this blog…
Roadmap to Mathematics: 1st Grade
[Feature photo (above) by woodleywonderworks. (CC BY 2.0 via Flickr)]
A frequently-asked question on homeschooling forums is, “Are my children working at grade level? What do they need to know?”
The Council of the Great City Schools has published a handy 6-page pdf summary of first grade math concepts, with suggestions for how parents can support their children’s learning:
Whether you are a radical unschooler or passionately devoted to your textbook — or, like me, somewhere in between — you can help your children toward these grade-level goals by encouraging them to view mathematics as mental play. Don’t think of the standards as a “to do” list, but as your guide to an adventure of exploration. The key to learning math is to see it the mathematician’s way, as a game of playing with ideas.
The following are excerpts from the roadmap document, along with links to related posts from the past eight years of playing with math on this blog…
Roadmap to Mathematics: Kindergarten
[Feature photo (above) by MIKI Yoshihito. (CC BY 2.0 via Flickr)]
A frequently-asked question on homeschooling forums is, “Are my children working at grade level? What do they need to know?”
The Council of the Great City Schools has published a handy 6-page pdf summary of kindergarten math concepts, with suggestions for how parents can support their children’s learning:
Whether you are a radical unschooler or passionately devoted to your textbook — or, like me, somewhere in between — you can help your children toward these grade-level goals by encouraging them to view mathematics as mental play. Don’t think of the standards as a “to do” list, but as your guide to an adventure of exploration. The key to learning math is to see it the mathematician’s way, as a game of playing with ideas.
The following are excerpts from the roadmap document, along with links to related posts from the past eight years of playing with math on this blog…
Denise Gaskins' Let's Play Math 