New Math, New New Math....WTF????

Cecil's Columns/Staff Reports
101

Another example:

Today we were working on word problems. One of them was something like, “The teacher had 22 pencils. She shared 9 of them with her students. How many did she still have?”

I can easily teach students to do 22-9. But it’s incredibly rare in life that you encounter a problem that starkly; more often you encounter a math problem coached in terms of physical objects, or money, or something else similar. Numbers are, in the real word, adjectives, not nouns, so to speak.

And before a student can solve that problem, they need to be able to understand that it’s asking for subtraction. I had at least two students who couldn’t tell me, given the problem as stated, whether the answer would be greater than or less than 22.

What good is arithmetic in such a circumstance?

Kids need to understand the underlying mathematical concepts–that subtraction involves breaking up a set, that the difference in a subtraction equation (with whole numbers) will be less than the minuend–before they can make heads or tails of such a problem.

So how did I deal with this difficulty with abstract principles? By reverting to the concrete, of course: I provided them with 22 cubes, asked them to pretend they were pencils, told them I was about to give away 9 of those cubes, and asked them to predict whether I’d end up with more or less than 22 cubes after I gave 9 away. Given that concrete example, both of them correctly predicted I’d end up with fewer than 22 cubes.

And that’s what it’s all about: seeing how math works, moving from concrete examples into abstract principles.

At the same time, I was working with a high-performing student. She’d solved a word problem with 16-9=7, and another with 26-9=17, and I was pushing her and pushing her to find the relationship between the problems. Her first answer to the question, “Can you find a relationship between the two problems?” was “No way hosay, I can’t.” I didn’t accept that, and because she’s pretty adroit at math, I didn’t let up on her until she noticed what was going on in the tens place. I don’t think she entirely got it this lesson, but I made a note to hit this principle again with her, so that she can build her understanding of place value and use it to her advantage in problem-solving.

A counterexample:
I recently went to a training on differentiating math instruction for struggling students and adroit students. One piece of advice for struggling students was to review key words in story problems and teach them to underline these key words: “altogether” and “More” mean addition, for example, while “Have left” and “gave” mean subtraction.

I immediately wrote up some story problems:

-John had 10 marbles. Sarah gave him some more marbles, and now he has 17 altogether. How many did Sarah give him?
-Larry had 15 iguanas. Sarah has 8 iguanas. How many more iguanas does Larry have than Sarah?
-20 people are at lunch. 3 get up to throw away their trash, and 2 go to the restrooms. How many people have left?
-I have 18 angry monkeys. John gave me 5 angry monkeys. How many do I have?

Of course, each of these problems breaks the “key word” principle; each problem can easily be solved by a student who’s learned to visualize the problem and work it out from scratch, rather than by using some memorized list of key words.

Which is exactly the point: the “key words” method is a highly traditional method taught to struggling students. On a test designed to measure the use of traditional methods, all the story problems will adhere to these key words. Real life, however, won’t. It’s important for kids to be able to reason through problems, rather than simply memorizing a magical formula for solving them.

102

I like this story a lot. It compactly illustrates quite a lot about the different mindsets we’re discussing, I think.

(Actually, I first misread the beginning and thought you were teaching the the key words business yourself, immediately responding “Oh no…”. Then, as I read on, I experienced such relief at your actual response.)

103

I absolutely did not find this to be the case, in any way. Your “of course” is totally presumptuous.

104

You don’t think math and science faculty are good judges of their students’ skill in math?

105

Well, never mind actually. My point was that a letter signed by 400 math and science faculty doesn’t have much worth when it comes to specific educational reform; they are math and science experts, not pedagogy experts.

Thanks for sharing some of your experiences, LHoD. I find it very interesting to read accounts of a teacher incorporating constructivist ideas into their practice; as a novice teacher, I wouldn’t feel comfortable with it. Mainly because I haven’t seen it done, I think. You give off the impression that you gave a good notion of what you’re doing. :slight_smile: It’s reassuring.

106

Indistinguishable

Okay, thanks for the clarification. I agree that the ultimate goal is for students to understand when to use multiplication, etc, rather than just be able to do computations. I guess my point is the comment you made at the end - students may be perfectly able to identify that they need to do multiplication, but then unable to actually compute the correct answer because they don’t have enough practice actually doing computations.

I speak from my own experience here, admittedly at college level, not elementary. My chem II class was a very fun class. The instructor was knowledgeable and gave interesting lectures. He was also laid back - lectures weren’t required, and he didn’t assign homework. I did a few problems in labs and stuff. Then when it came time for final exams, I was woefully unprepared to actually do the computations on the test. I knew the principles involved and what the intent was, but couldn’t actually set up the computations because I had not practiced them.

Left Hand of Dorkness said:

Seems you’re talking about two different skill sets. One skill set is doing the arithmetic - getting the computation and the result. The other skill set is recognizing what computation is required. Both skill sets are necessary.

What the critics seem to be complaining about is the lack of developing the computation skill set. Students may be able to recognize what they need to do, but they aren’t able to do it. Seems that’s a serious failure of education.

I’ll give you a counterexample: it doesn’t matter if I know 27 ways to compute 27 - 9 if I can’t get the correct result with any method I choose.

I certainly don’t claim to be an expert on pedagogy, and this really isn’t an issue that I’ve been following. I recall my experience was first we learned how to do the computations, and had to practice them “to mastery”. Then we were exposed to word problems, and had to learn how to interpret them. I don’t remember exactly when we started word problems - somewhere in elementary school. I just recall that at some point we had assignments that involved straight computations for practice, then associated word problems that used those types of computations. You had to do the method, and you had to be able to decipher the problems.

I certainly understand the need to be able to take a situation and determine what computation is required. That is the essence of my physics and engineering problems.

107

The majority of universities give only lip service to evaluation of professors for teaching effectiveness–if even that. It would be extremely surprising if there were a lot of professors who were good teachers.

I say this as a university prof myself btw…

108

Is there any good reason to think that there is any such thing as an effective teaching method as opposed to an effective teacher?

I often suspect that the unit of effectiveness is the teacher rather than the method. But I don’t really have anything to back this up–except my personal experience (anecdotal) and a vague notion that I read something like this once somewhere.

I did go look at the website of something called the “What Works Clearinghouse” or something like that, part of the Department of Education. I only looked at what they have to say about math. There wasn’t much, but what little I did see all seemed to explicitly conclude, in each case, that the evidence of the effectiveness of the method was only inconclusive…

LHoD is telling very good and inspiring stories (literally, in that I went and did the dots and rectangles thing with my kid the very morning I read it) but I’m afraid what we’re seeing is that LHoD has just the right kind of empathic, “mind-reading” (non-supernaturalistically speaking) and explanatory abilities that make for good teaching. If all LHoD knew were traditional methods, I suspect s/he still might have been just as effective a teacher.

But what evidence is there against this suspicion of mine?

109

I don’t know. But measuring the effectiveness of teaching is caught up in the question which came up before of along what axes to assess students in the first place. To a large extent, I think the debate in this thread is really not so much about “What makes for effective teaching of the specific material X? Which teaching styles are better than others for conveying X?”, but more about the even prior questions, “What is the X we should be trying to teach? What is the purpose of teaching X? What material should be the goal of our curriculum and what material is less necessary to worry about?”.

Which is to say, there are a number of posters in here who feel certain teaching styles are less effective than possible when it comes to developing the ability to perform certain calculations by hand and that this is a big problem, and there are also a number of posters in here who feel that the ability to perform those calculations is not actually of much significance and that teaching styles which focus on such are themselves problematic in their misguided neglect of other, potentially more important aspects of mathematics education. This difference of viewpoints is, I think, the actual main bone of contention in this thread.

110

I was assuming everyone here agrees that understanding how and when to calculate (and with what instrument–the head or the calculator for example) is the ultimate goal, and that the disagreement was over whether the best way to get there is to learn calculation as an exercise abstracted from any particular application, (then learning how and when to apply calculations,) or whether instead all learning of calculation should be inextricably tied to understanding how to apply these calculations to problems that exist somewhere other than in the ink-and-paper fusion sitting on the desk in front of the student…

That’s what I’m on the fence about anyway. Myself, I kind of like the idea of learning the methods abstractly, as kind of a game divorced from reality, then coming to see how this game can be made useful in other domains. But that’s just me. It may well be that that way of doing things works for some people and not others. Or it may even be that I’m completely wrong to move from “I like the idea of doing it this way” to “It is best for me that I learn things this way.”