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 RoÍ¬Í¬Í Í Í¡Í Í Í Í Í Í Í Í sÍ¬Í¬Í Í Í Í Í Í Í Í Í aÍ¬Í¬Í Í Í Í Í Í Í 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)
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…
[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…
[Feature photo above by Jimmie, and “79” image (right) by Steve Bowbrick via flickr (CC BY 2.0).]
Do you enjoy math? I hope so! If not, browsing this post just may change your mind.
Welcome to the 79th edition of the Math Teachers At Play (MTaP) math education blog carnival — a smorgasbord of links to bloggers all around the internet who have great ideas for learning, teaching, and playing around with math from preschool to pre-college.
Let the mathematical fun begin!
By tradition, we start the carnival with a puzzle, game, or trivia tidbits. If you would like to jump straight to our featured blog posts, click here to see the Table of Contents.
Since I’ve been spending all my free time working on my upcoming Math You Can Play book series, I’m in the mood for games. So I found a few games featuring prime and nonprime numbers [which category is #79 — do you know?], and I’ll sprinkle some of my best-loved math game books throughout the carnival.
Continue reading Math Teachers at Play #79: Puzzles and Games
[Feature photo is a screen shot from the video “the sausages sharing episode,” see below.]
How in the world can 1/5 be the same as 1/10? Or 1/80 be the same as one whole thing? Such nonsense!
No, not nonsense. This is real-world common sense from a couple of boys faced with a problem just inside the edge of their ability — a problem that stretches them, but that they successfully solve, with a bit of gentle help on vocabulary.
Here’s the problem:
Think for a moment about how you (or your child) might solve this puzzle, and then watch the video below.
[Feature photo (above) by U.S. Army RDECOM. Photo (right) by Stephan Mosel. (CC BY 2.0)]
On your mark… Get set… Go play some math!
Welcome to the 76th edition of the Math Teachers At Play math education blog carnival — a smorgasbord of links to bloggers all around the internet who have great ideas for learning, teaching, and playing around with math from preschool to pre-college.
By tradition, we start the carnival with a puzzle in honor of our 76th edition. But if you would like to jump straight to our featured blog posts, click here to see the Table of Contents.
[Dragon photo above by monkeywingand treasure chest by Tom Praison via flickr.]
Over the years, some of my favorite blog posts have been the Word Problems from Literature, where I make up a story problem set in the world of one of our family’s favorite books and then show how to solve it with bar model diagrams. The following was my first bar diagram post, and I spent an inordinate amount of time trying to decide whether “one fourth was” or “one fourth were.” I’m still not sure I chose right.
I hope you enjoy this “Throw-back Thursday” blast from the Let’s Play Math! blog archives:
Cimorene spent an afternoon cleaning and organizing the dragon’s treasure. One fourth of the items she sorted was jewelry. 60% of the remainder were potions, and the rest were magic swords. If there were 48 magic swords, how many pieces of treasure did she sort in all?
[Problem set in the world of Patricia Wrede’s Enchanted Forest Chronicles. Modified from a story problem in Singapore Primary Math 6B. Think about how you would solve it before reading further.]
How can we teach our students to solve complex, multi-step story problems? Depending on how one counts, the above problem would take four or five steps to solve, and it is relatively easy for a Singapore math word problem. One might approach it with algebra, writing an equation like:
… or something of that sort. But this problem is for students who have not learned algebra yet. Instead, Singapore math teaches students to draw pictures (called bar models or math models or bar diagrams) that make the solution appear almost like magic. It is a trick well worth learning, no matter what math program you use …
[Click here to go read Solving Complex Story Problems.]
You can help prevent math anxiety by giving your children the mental tools they need to conquer the toughest story problems.
Read Cimorene’s story and many more in Word Problems from Literature: An Introduction to Bar Model Diagrams—now available at all your favorite online bookstores!
And there’s a Student Workbook, too.
![x - \left[\frac{1}{4}x + 0.6\left(\frac{3}{4} \right)x \right] = 48](https://s0.wp.com/latex.php?latex=x+-+%5Cleft%5B%5Cfrac%7B1%7D%7B4%7Dx+%2B+0.6%5Cleft%28%5Cfrac%7B3%7D%7B4%7D+%5Cright%29x++%5Cright%5D++%3D+48&bg=ffffff&fg=303030&s=0&c=20201002)
Denise Gaskins' Let's Play Math 