Here’s a novel idea …
Bring back math memorization … at the Algebra 1 level!
No — I’ not suggesting that we ask students to memorize the times tables from the 12s to the 20s.
Nix on that because it is NOT critical information for algebra students to have. So it would not serve the greater purpose of these students.
But I am suggesting that we require students to memorize a handful of facts that will make
their algebra experience considerably less painful.
For starters, let’s require students to memorize a limited set of exponential powers for small integers.
Initial proposal: students memorize all powers of 2 from 2^0 through 2^10.
On the benefit side. Consider how much easier it would be for students to simplify √ (32) if they knew that 2^5 = 32.
They just figure like this:
= √ (2^5) [this step done in a flash, instead of the typical half-minute]
= √ (2^4) (2^1)
= 2^2 √ 2
= 4 √ 2
Or how much easier it would be to simplify this:
Instead of agonizing over this, first being puzzled, then realizing that they have to “figure out” their powers of 2, students would instantly recall the relevant fact:
log2 (32) = 5, since 2^5 = 32
And there are other situations in Algebra 1 and 2, Trig, PreCalculus and Calculus, which would be easier if students simply memorized the basic powers of 2.
If you’re thinking that the students would object, saying they “are not able” to memorize these facts – or that it will take huge amounts of time — I have news. They will. But you as the teacher can respond that it is actually quick and easy to memorize these facts. Most students can do this in less than a day; some in as short as 10 minutes.
As a tutor I require my late Algebra 1 and all Algebra 2 students to memorize these facts. And they all get it done in 24 hours.
Here are a few mnemonic tips, serving as the proverbial “spoonful of medicine” …
2^4 = 16 [Remember the reverse — 4^2 is also equal 16. This math “coincidence” helps students remember this one.]
2^5 = 32 [5 = 3 + 2]
2^6 = 64 [6 is the first digit of 64; next digit 2 less]
2^7 = 128 [Ignore both 2’s, and you get: 7 = 1 8. Think of this as backwards reading: 8 – 1 = 7. A bit complicated but it works for me, and I’ve seen it work for students, too.]
2^8 = 256 [8 = 2 + 6]
2^10 = 1024 [1024 starts with 10, then just add 2 x 12]
If your students don’t like these mnemonic devices, suggest that they develop their own. But most importantly, take the mission seriously. Give your students graded quizzes on these facts till they have them down cold. Or play “Fact Baseball” with them, choosing teams and having them compete at beating the other team.
A few years later, when you see them after high school, they’ll tell you that they’re the only one in their Calc 2 class who knows these powers without a calculator. And if you look closely, you’ll see a little “thank you” half-hidden in their expression.
Comments on: "Memorization in Algebra — They’ll LOVE you for it (later)" (4)
I’m going to disagree with this approach. Memorizing lists of numbers in an effort to make other calculations easier is not doing mathematics. Doing mathematics is problem solving, exploring the world through the lens of mathematics, and pattern finding. If a student happens to memorize some calculation facts as a result of this exploration fine, but I would never force students to memorize information that they can look up.
Perhaps having students create their own booklets of “important facts to know” would be more productive?
I suppose one’s definition of “memorization” will determine how this is viewed. Personally, I’m not a big fan of memorization in Math; I much prefer developing concepts using smaller numbers and from there connecting the algebraic representation to its geometric counterpart. Once this understanding is owned by the student, more complex scenarios can be put forth and solved using technology as an aid. I’m not sure there is any benefit in using large numbers when teaching any concepts.
Memorizing multiplication facts 1 through 12 does make Algebra easier, especially for those students who don’t automatically realize that multiplying by 12 is the same as 2 times 6 or 3 times 4. Try teaching a kid to factor when he doesn’t know what two numbers multiply to -39 and add to 10!
Thanks for all of your comments.
I am familiar with the arguments made by David and Earl — the idea that memorization is generally a bad thing in math education, and we should do all we can to avoid having students merely memorize.
I do think, though, that this argument can be taken to an extreme point where it stops being useful.
Certainly math educators need to focus on helping students understand math facts, skills and concepts. I am a big advocate of trying to help students understand math skills and procedures as much as possible. Anyone who reads my books would see that I am very much a concept-driven math educator. I also believe in helping students discover mathematical understandings as much as possible, the heuristic approach. My PreAlgebra Blastoff book, for example, is all about helping students grasp and wrestle with the idea of positive and negative numbers in an experiential way.
But I really cannot see taking this perspective as far as some people do. If we go so far as saying that students should never be asked to “memorize information that they can look up,” that would be tantamount to saying that students should never memorize the times tables, that they should not even memorize addition facts. Certainly we should try to teach the times tables in such a way that we encourage students to see and appreciate the patterns inherent in our number system: using 100 boards, ten-frames, and so on. But there also comes a time when it behooves students to commit mathematical information to memory, so they can go on and master higher mathematical tasks, as Zero-Sum so aptly pointed out. Without the times tables mastered, students cannot factor. And as I argue further, without the powers of basic integers mastered, students will endlessly struggle when reducing radicals, as I have seen occur in my daily tutoring work.
There simply is a time to “move on” and help students go to the next level, even if the prior level is not learned with the ultimate degree of pedagogical perfection.
As the saying goes, the perfect is the enemy of the good. And here the call to never memorize but always understand from scratch is the “perfect ideal” that could, if carried out to its idealistic end, stop students from advancing in the somewhat imperfect way that learning does actually occur in the real world.
In fact, I have seen something interesting happen with regard to the times tables. I have seen students who did not really grasp the concept of multiplication but were required to learn the times tables. In watching these students work on their memorization, I have seen these very students grow to understand the concept of multiplication BECAUSE they had memorized them. With all of these number facts in their mind, these students started to play with the numbers and gradually — through a lot of independently sparked “aha” moments — they start to figure out what multiplication is. So learning can occur in surprising ways. And therefore, to cut off one form of learning by saying that all memorization is bad is, in my opinion, doing a disservice to students. Memorization, I believe, is part of how we learn.