Here’s one more quote from homeschooling guru Ruth Beechick. It applies to classroom teachers, too!
Everyone thinks it goes smoothly in everyone else’s house, and theirs is the only place that has problems.
I’ll let you in on a secret about teaching: there is no place in the world where it rolls along smoothly without problems. Only in articles and books can that happen.
— Ruth Beechick
You Can Teach Your Child Successfully (Grades 4-8)
[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…
“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)
[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.
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.
[Featured Image (above) by Math Giraffe, and Math Goggles image (right) by Moebius Noodles — two great posts from this month’s carnival.]
Number sense, measurement, place value, functions, calculus for kids, Christmas math activities, art, and much more — check out the December math education blog carnival:
Math Teachers at Play (MTaP) Blog Carnival #81
Welcome to the 81st edition of Math Teachers at Play (MTaP) Blog Carnival. I am extremely exited to host this post in my favorite month of the year, December…
Understanding 81: An interesting fact is that 81 is a tribonacci number (sounds a lot like Fibonacci) – the sequence of tribonacci numbers start with 3 predetermined terms (0,0,1) and each term afterwards is the sum of the preceding 3 terms. Thus the sequence starts like this: 0,0,1,1,2,4,7,13,24,44,81,… (you can go further if you want to see how fast the numbers go).
Now the maths posts…
[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!
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? ]
The new Math Teachers at Play math education blog carnival is up for your browsing pleasure. Each month, we feature activities, lessons, and games about math topics from preschool through high school. Check it out!
[Photo by Paul Downey. (CC BY 2.0 via Flickr)]
Welcome to the 80th Edition of the Math Teachers at Play (MTaP) blog carnival.
Before we dive into some math posts from around the web, let’s see what is special about the number eighty.
80 is…
- how long it took Phileas Fogg to travel around the world in Jules Verne’s novel Around the World in Eighty Days.
- 4 scores.
- commonly used in the “80:20 rule,” which originated from Vilfredo Pareto, an Italian economist…
…And more! Click here to read the whole carnival, featuring 17 posts of mathy awesomeness!
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…
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…
[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…
Denise Gaskins' Let's Play Math 