When I was pregnant with Linda, I read an article in Scientific American about how math manipulatives actually make it more difficult for students to learn the concepts. I remember manipulatives from back in elementary school; I also remember thinking they were stupid.
Seems that they've found something more idiotic than manipulatives. Strategies.
Now, to the rest of us, strategies are plans of action. To third-graders learning multiplication, they're pictures.
Let that sink in a second. Third grade students are learning multiplication, but they're having to draw pictures to do so.
Bobbie's homework today included a page of word problems. Something along the lines of Jimmy has three boxes. Each box holds eight cars. How many cars can he store altogether?
Now, were you or I to come across that problem, we'd write out 3 x 8 = 24 and be done with it. But no. Bobbie can't do that. Bobbie has to use her strategies. Bobbie has to draw three boxes, and then eight cars in each box, and then number every freaking "car."
Actually, because Bobbie has me for a mother, she first wrote out the equation and solved it, and then went back and did the drawing. She also got a rant on how idiotic it is that they're being made to do these things.
Why are they doing this with mathematics? Exactly what is difficult about isolating the numbers and then the words which tell you what to do with them and then doing it? I could almost understand if this was kindergarten, but it isn't. It's third grade.
We learned a saying in Russian...повторение мать учения. Repetition is the mother of learning. Drills don't sound fun, but you know what? They work. San Antonio College actually has the best math program in the state. Know how they teach? The professor shows how to do a problem, and then you practice the same type of problem again and again and again until you get it. It works.
Math is incredibly simple. It really is. Just numbers. They go together in the same way every time. Mucking it up with anything else simply does not make sense.
16 comments:
It does seem silly to have to draw out pictures for homework, and I'd be annoyed, too.
however! I disagree that it's useless in general. There's a BIG difference between knowing that 8x3=24 and understanding WHY 8x3=24. Memorizing multiplication tables without knowing what multiplication even is... that's not so good.
So, I think you need both. When first taught multiplication, the child should be shown that it's 3 sets of 8, or 8 sets of 3, and how you could add it all up, but it takes longer.
Then, you move on to learning facts. And that's how it's done in our school, and the school we were at last year. So I wouldn't assume your school is never going to emphasize multiplication facts. Just maybe that the kids need to understand the process.
(it was so fascinating to me when I learned with my 8 year old--he was 6 at the time--WHY we do multiple-digit multiplication like we do. I learned the process, but never knew why we wrote it that way. Well, there's a reason! It's not just a formula! How fun to understand. The same with long division.)
It's a deep dark plot that allows 1 in 100 people to speak the secret "math" language within the ear shot of otherwise smart people.
Strategies work when you are having problem learning the concept or applying it.
You are absolutely right in saying they need repetition. They weren't called skill and drill for no reason.
There's a BIG difference between knowing that 8x3=24 and understanding WHY 8x3=24.
The WHY is contained in the equation. Eight, three times, is twenty-four. I've explained this to both school-aged children. When Bobbie was doing 3+3+3=9 last year, I told her it was the same as 3x3, because it is three, three times. And you know what? She understood that (as did her then-five-year-old sister), without having to draw it. Kids get things much more readily than they tend to be given credit for. What's gained by adding in another step?
Memorizing multiplication tables without knowing what multiplication even is... that's not so good.
Who's to say you can't figure the basic principle out on your own by looking at the tables and how the numbers increase? I don't think that's beyond most kids' intellectual capacity by the time they get to that level. And it's certainly less complicated than putting labels on things it's unnecessary to be putting labels on.
It is teh stupid. I took a class several years ago where the instructor would repeat his points several times throughout the day. He'd preface each repetition by stating that "Repetition is not only the mother, but the father, of learning."
You and he are both right, studying tables is boring, but it does drill the knowledge into you.
Ask any paramedic student and they'll tell you that memorizing drug dosages, actions, side effects, countraindications, is painfully boring. It's also incredibly important to be able to remember all of that in the middle of the night when you have a really sick patient in front of you.
I don't think drawing a picture of a vial of medication would have helped me learn that.
Our 23 yr old son learned pretty traditional math, they were just beginning to introduce "new math" . Our 13 year old has had exactly what you are describing and it is stupid! If a student is struggling to understand multiplication this might be a nice, simple illustration but do we seriously need to dummy down our kids. We made our kids memorize their multiplications and we taught them to use tables and it worked, simple!
I've heard the argument before, that we should be teaching "what" mathematics is, rather than "how" to do it. Only the person who was making the claim, was also making the claim that multiplication isn't repeated addition, and that it will only cause confusion in children to continue to do so.
And do you know what? That person was right, at least about the first part. As an example, how is 1.4 * 3.7 "repeated addition"?
The problem with this reasoning, though, is that when we're talking about integers, repeated addition is multiplication; when we generalize things to get fractions, we simply expand multiplication beyond that. When we get to complex numbers, it becomes "scaling" and "rotations".
But we don't need all that abstraction to teach our children! We just have them learn their times tables, and then we can build from there. Just as a proper Algebraist would! (I suspect that the person advocating "real multiplication" wasn't an Algebraist, but I'm not sure.)
Have you been exposed to matrix multiplication? It's horrid.
My 5th grader learned 4 different methods of long dividion in one week. All his teacher said to me was "We have to show them many ways, and hope one of them works for them."
Everyday Mathmatics is casuing more problems than I can keep up with.
"As an example, how is 1.4 * 3.7 "repeated addition"?"
Seriously?
1.4 + 1.4 + 1.4 + .98 = 5.18.
1.4 * .7 = .98
1.4 * 3.7 = 5.18
That answer your question?
Rick, how do you get "1.4 * .7" while viewing multiplication as repeated addition? That is, how do you get .98 as 1.4 added .7 times?
If you think multiplication as a "dilation", and .7 "shrinks" 1.4, it would work. Similarly, you could create fractions, and then expand fractions to real numbers via sequences...and then show that "addition" and "multiplication" generalize nicely on these new number systems...but we leave "repeated addition" at this point.
When you get into multiplication of complex numbers, or quaternions, or polynomials, or matrices, it becomes more and more difficult to justify these systems as "repeated addition".
But none of this matters, because when we're talking about integers, multiplication is repeated addition, and it's relatively easy to expand yourunderstanding of multiplication to other number systems when they need to be introduced.
14 * 37 = 518. Then figure out where the decimal point goes. (Which you have to do if you're setting up the problem as a long-form multiplication anyway).
1.4
3.7 x
------
098 (14*7)
420 + (14*3)
------
518 = 5.18 when you put the decimal point back in.
That's how I learned it lo these many years ago - as addition and moving the decimal point.
@Ian Argent: That still isn't repeated addition, and it also works only for finite decimals.
It's also a darn fine way to calculate the result--one I'll be teaching my kids!
Well, everything but placing the decimal point can be done by addition - you're multiplying integers, which is iterated addition. Placing the decimal point is, sort of, addition as well - take the number of digits that are to the right of the decimal point in each of the number to be multiplied (1 and 1 in this case), add that count together (2) and that's how many numbers to the right of the decimal point there are in the product.
For bonus points, that's how you figure out where the decimal goes once you're done multiplying on a slipstick as well...
You are correct that it only works for finite decimals, but if the numbers are rational but infinite decimals, convert to fractions and add that way. If they're not rational, what are they doing in a third-grade math problem?
I've tried to explain to my children "number creation" from the bottom up, starting at a very young age (my oldest is 5). When they were younger, it would put them to sleep; the older they, they tend to lose interest quickly. I would do it something like this:
Start with counting and adding (create Natural Numbers); devise subtraction (introduce 0, negative numbers); discuss repeated addition (multiplication); introduce fractions (division); go over infinite sequences and real numbers; discuss powers and polynomials; discover imaginary numbers; then go on to quaternions.
Before going on to quaternions, I should probably discuss vectors and matrices; I've always felt a little awkward transitioning straight from complex numbers to quaternions.
The concept of "repeated addition" definitely breaks down with fractions; it sort-of breaks down with negative numbers. (The best explanation I've seen of negative-number multiplication is that it's a 1-dimensional reflection over 0; the algebraic explanation has never stuck with me.) With complex numbers, in particular, multiplication is a dilation and a rotation--not "repeated addition", because addition is vector addition.
In any case, none of this matters to third graders, because "repeated addition" makes sense when we're talking about natural numbers; when we're talking about fractions, the important concepts are related to "cutting things up".
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