Teaching math without memorization March 2, 2011
Posted by mareserinitatis in education, gifted, homeschooling, math, older son, teaching.Tags: abacus, arithmetic, bead frame, gifted education, homeschooling, math, multiplication, tables
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If there is one thing I learned in my junior level electromagnetics class that will always stick with me, it’s this:
The permittivity of free space (ε0) is 8.854 * 10 -12 F/m.
And realizing that will always stick with me gave me a lot of insight into what is wrong with the typical approach to teaching math, especially in elementary school.
I know it’s a stretch, but bear with me.
(Totally gratuitous picture of a cute baby bear.)
One of the things I’ve encountered with both my kids is that their teachers are very set on them memorizing math facts. My older boy spent 3rd and 6th grade enrolled full-time in regular public school programs, and his third grade teacher was constantly railing on about how he was ‘bad in math’. In fact, she blamed it on the fact that he’d been homeschooling. (We got this every time we talked to her.) She would go on and on about how he didn’t have his tables memorized. Why, he had to stop and think every time she asked him a basic addition problem!
OMG…thinking in school?! Can’t have that.
I therefore found it very amusing when, prior to the the next school year, his principle pulled out the results of his spring MAPS testing and commented on how good his math scores were because, in her words, “He must know his tables really well.”
By these two comments alone, you can tell what is important to elementary school teachers: memorization of arithmetic tables.
Aside from having a BS in physics, I minored in math in college. Despite the fact that I had enough credits for a major, the credits were in overwhelmingly applied math classes, and there was no applied math major at my school. Suffice it to say that I do have at least a basic knowledge of math.
I also have homeschooled my older child for most of his educational career, and as a freshman in high school, he’s finishing a course in college algebra and trigonometry.
During the older child’s homeschooling years, I never once made an attempt to have him memorize tables of any kind. I did not practice a lot of repetition of basic facts, either. This was because of my experience in my electromagnetics class: I didn’t memorize the value of the permittivity of free space due to repetition and drill; I memorized it because I used it in nearly every problem I did for four months in that class. Yeah, I had to look it up the first dozen times I used it, but after that, it was lodged in my brain. And look…it’s still there a decade later!
I came to the conclusion that if you really need to know something, you’ll learn it through frequent use. But how do you use something that you don’t know?
Addition and subtraction are fairly simple: you give a kid a bead frame, abacus, or even a ruler (the original slide rule!) and show them how to perform addition and subtraction operations using beads or moving up and down a number line. Then you can move them quite quickly through addition and subtraction of infinitely (okay…not infinite) finitely large numbers. You can let them go through increasingly complex topics without ever making them memorize a table. In fact, after a short time, you’ll find that they are pointing at beads or rulers in the air, counting out the solution to their problem. And after that, the invisible ruler or beads will be sitting in their head, being manipulated by mental fingers. Finally, they won’t even have to think about it…they’ll just know.
The image on the left is an abacus, while the image on the right is a bead frame. Bead frames are easier to find and manipulate, in my experience.
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Multiplication should be taught as addition of groups of objects, and division as ‘counting’ of the number of groups in the whole. Once kids have mastered the process of multiplication and division, you can then simply print out a multiplication table. I had my older son paste it on the inside cover of a notebook or the front of a folder so that he could always find it. You may find that some kids prefer to go back to the bead frames. (And if you are really lucky, you have an abacus and know how to use it for multiplication…which I don’t.) Any method is fine as long as it works for your child. But the point is that you can then let them progress through more and more complicated arithmetic involving those operations (such as multiplying large numbers or long division) using the table or other device to look up values. Again, as you progress through these concepts, they will slowly begin to memorize them.
As a homeschooler, the curriculum that I chose for math was fairly important as well. I liked Singapore math for it’s focus on simplicity and conceptual explanations. Everyone raves about the ‘mental math’ tricks that are taught in the series. And they’re right: mental math is awesome. However, the only reason the series does it so effectively is because it teaches the concepts in a way that you can then make logical simplifications in process that result in ‘mental math’ and good estimation skills.
(My only complaint is that they don’t teach the lattice method of multiplication. I think the ‘traditional’ method of multiplying large numbers is much better suited for estimation methods, whereas the lattice method is definitely superior for calculating with precision.)
To me, the important part of all of this is to make sure that they understand what the process of addition, subtraction, multiplication, and division. Kids can memorize facts for quick recall, but if that is the emphasis and they can’t recall a fact, they’re going to be stuck. If the emphasis is instead on teaching arithmetic as a process, they can always figure it out should they forget.
And really, arithmetic is a process. I came across an article on Hoagies’ Gifted site called Why Memorize? I have to take big exception to the article because it says that math is a lot of dry facts. If you teach it as memorization of facts instead of a process of manipulating numbers (or objects or motion in space), it sure is! But I can tell you that it’s not, and as you advance to higher level classes in mathematics, reliance on the notion that math is memorization will cause you problems and impede your progression.
Finally, I have to wonder if this is why so many elementary school educators fear math: it’s boring memorization of facts. They are never taught how it’s actually a really cool process. If it were taught properly, preferably with a lot of enthusiasm instead of dread, I wonder if a lot of teachers would lose their ‘math-phobia’. And that would, of course, mean their students might start to like math, too.
I also blog at Engineer Blogs, home away from home to some of the best engineering blogs.
Great post!
One of the things I’ve recently begun to bemoan is how much of math I’ve “lost.” I haven’t used many of the cool “processing” tools in well over a decade and have forgotten them. For example: There was this cool factoring thing I learned once. But have I needed to factor recently? Nope.
The psychology adage is true: Use it or lose it.
~Luke
My regret is my foreign language skills. I studied both Spanish and German. I can remember some of both, but it’s really spotty.
On the other hand, I’ve been considering trying to pick up German again.
Fanstisch! Wir konnen zusammen in Deutsch bloggen!
I wish I had read this post years ago. So does my daughter. She is good at math – she really is. But, she thinks she’s bad at it because memorizing the facts has been such a challenge for her. In recent months I have made a deal with her – show me that you can process the math, and I won’t make you do the memorization drills. She’s stopped dreading math – has actually started liking it – and has been completing her lessons in a fraction of her usual time. Meanwhile, I’ve felt guilty. All I’ve ever learned is that math is memorization. But, I’m not going to feel guilty anymore. Thanks!
This is one area I haven’t felt guilty…but there have been many others where I’ve wondered!
If only they would teach this method in school, so many more students would fall in love with math.
I tend to agree, but a couple counter thoughts.
1. Its generally accepted that higher levels of learning take more time and teaching effort. Blooms taxonomy right or wrong supports such a view.
2. Teaching to understand rather than rote often results in greater disparity from student to student for a given amount of time rather than effort.
Public schools by and large place a huge emphasis on equality. Those calling for accountability put a huge emphasis on efficiency.
[...] Is it merely the ability to do well on a standardized test? Is it the ability to parrot back canned answers via memorization? Is it the ability to understand, derive and create? All of these issues must be considered should [...]
[...] with practice, I ended up memorizing my math facts. I’ve written before about how math facts are better memorized through practice than rote. I think this would be another great method for teaching facts, as it obviously worked for me. [...]
I hope what you say is true. I was just thinking about this today when I was working some simple multiplication in my head. I thought it was a good thing I memorized my multiplication tables, or I wouldn’t be able to do this. I would get lost trying to calculate each simple step and would lose track of carrying and adding the decades up. But maybe that’s not true.
You also mentioned Spanish. I am very proficient in Spanish. I know all the verb forms for regular and irregular verbs and when to use them. I remember doing tedious memorization and some rules about the morphology of verbs with certain letter patterns. Now I just I know which one and often cannot recall the rules at all or where I learned it. I normally credit the tedious exercises I did.
So my belief has always been that tedium is the price you have to pay at various stages of learning. I still tend to believe that even though I’m not good at memorization.
This will come up as my kids get older. Do I try to sell them on tolerating the pain for a higher purpose of being able to use these painfully-memorized bits of information in larger interesting problems? Or do I teach them the cool stuff and let the excitement of those problems encourage them to learn the mundane stuff without realizing they’re learning it.
I don’t know that the first 11 significant digits of π (pi) are 3.1415926536 because I had to memorize them. I know them because I used π so many times and was interested in the concept and the importance of the number. I had seen the number repeatedly to varying degrees of significance, and at some point it became fun to remember several of the digits. But it certainly wasn’t because one of my teachers insisted that I know 11 digits. In fact, you can’t memorize all of the digits of π because they never end.
If a teacher has a class memorize π, the teacher would have to have them memorize a few digits, not all. 3.14 is a fairly good approximation depending on the purpose. But if my child came home and told me that π is equal to 3.14 I would tell him or her that 3.14 is all the teacher knows because they want to memorize math, not learn it.
My point is not that students should not memorize math facts; in their own time and in their own way, they will each probably memorize a plethora of math facts. But, they should not be taught that the way to be good in math (and to get good grades) is to memorize math facts.
Remember, you can lead a horse to water, but you can’t make him memorize seven times eight.
[...] talked before about simply giving kids multiplication tables to work from. My younger son, however, seems like he’s a little better with memorization, [...]