"How is a teacher, who uses formative assessments, supposed to know what a child truly understands or is grappling to understand, if they don’t have a window into their thinking?"

"By giving them truly challenging problems and seeing if they consistently get the right answers!"

*From a recent exchange I had on an Education Next thread on whether students should be required to explain their answers.*

It strikes me that there are two things about Reform Math assignments that cause today’s educators to be so insistent that kids explain their answers:

1. Reform Math problems, in comparison with traditional math (and overseas math, including Singapore Math) are often so easy that there’s no work to show. The calculations are relatively simple, and the word problems usually don’t involve more than one or two steps. Many students, unless you instruct them otherwise, have no reason to write down anything beyond the bare answers. Teachers, meanwhile, remembering how important it was to show their work back when they were in school, may conflate showing your work with explaining your answers.

2. Reform math assignments often involve no more than a half dozen problems, again with relatively easy calculations, where traditional (and overseas) assignments involve dozens of problems with much more complex calculations. It’s therefore conceivable for a student to get every problem on a Reform Math sheet correct by chance rather than by understanding. Getting all correct answers for several dozen complex calculations without understanding what you’re doing, in contrast, is so unlikely that, when students accomplish this, then it’s reasonable to assume that (unless they were cheating) they know what they are doing.

Two additional factors underlie Reform math’s insistence on explained answers. One is the notion that, regardless of how much your math skills exceed your verbal skills, if you can’t communicate your mathematical thinking then that thinking must be deficient. The other is that “mere calculation” has little to do with “deep understanding.”

These notions, of course, also help justify the educational malpractice of restricting students to problem sets in which the calculations are a fraction of the frequency and difficulty they once were and still are nearly everywhere else in the developed world.

**Postscript:**

I recently showed the Singapore Math curriculum to a student visiting from Mainland China, and the first thing she observed was that the 4th grade Singapore Math problems were problems that she and her classmates routinely did in

*first grade*.

"You must have gone to one of the top elementary schools," I said.

"Not one of the top schools; just a pretty good school" was her reply.

## 8 comments:

Thank you for putting this so succinctly.

This morning I realized that there IS a place for 'discovery math'-- But it's individual, not group based, and you need a good underlying curriculum to support it.

I use Saxon with my daughter. But I'm also really lazy and don't always check to see if they're introducing a new concept--so I just give her the worksheets, let her go to town, and then step in a 'teach' if it turns out there's something she can't handle.

On her own, she 'discovered' that 4*10 is just like 4 dimes, and that multiplying is really just adding over and over, really quickly. No tables, no group work, no guesstimating....

The same sort of thing has happened with the fractions in second grade Saxon, and with almost every concept that has come up. (I had to step in to explain the meanings of 'symmetry, oblique, horizontal and vertical' Everything else, she just figures out when she hits it.

In a sense, this is what 'discovery' math is supposed to be--kids teaching themselves and wrestling with new concepts until they understand them--- The problem is that Discovery Math in the classroom is often about 'fun!!!' instead of learning. However, I'm seeing that 'guide on the side' DOES work..

If:

1. You have a very small number of students

2. The students are fairly bright

and

3. Your curriculum has a good, orderly structure.

Discovery can be done right and wrong. And as you've "discovered", it can also happen during homework, when students are working problems by themselves. Some books structure the problems so that some discovery does occur in the course of doing them.

HI Katharine,

I have rad your book and now found your blog as well. I live in a community that uses Everyday Math and they are finally realizing that it is not a good fit. Recently the superintendent announced that there has actually been a committee studying whether it works or not in our community. (You would think that 5 years of declining scores would be insightful enough!) At the same time he talked about moving to a "balanced math" curriculum. Have you heard of such a term?

Thanks.

Gillian

"Balanced math" is the same idea as "balanced literacy" in reading, where in theory they are blending more traditional math assignments with discovery based assignments. In practice, the problem is usually that the traditional assignments become homework because the poorly designed discovery assignments take up all the class time. Be suspicious.

"Balanced math" is exactly as ChemProf described it. My first grade son's classroom instruction consists of 90 minutes of group activities using manipulatives, playing "math games" and having calendar time. At the last conference, I asked his teacher how the math instructional time is allocated (group activities, teacher instruction, independent work.) She told me that she spends about 10 minutes showing students how to play the game or use the manipulatives, about 45 minutes in groups of 4 (collaborative learning)playing the games & using the manipulatives, and the rest of the time is spent working with pairs of students if they need extra assistance. She actually admitted that sometimes she isn't aware that a student hasn't understood the concept until long after they've moved on. Ridiculous!

My son has struggled all year with math -- his weekly math homework consists of a packet of worksheets, which we realized is where any and all direct instruction is occurring, in the form of me teaching him. He has also decided now in first grade that he "isn't good at math" and that "math is too hard."

One is the notion that, regardless of how much your math skills exceed your verbal skills, if you can’t communicate your mathematical thinking then that thinking must be deficient.I've just recently discovered, via Barak Rosenshine's article in American Educator, the idea of 'elaboration' in learning, and it makes perfect sense to me. In fact, it's more or less the 'missing piece' in my own thinking about all this.

Elaboration means (as I understand it) that we learn best when we .... manipulate or develop or grapple with the content we're trying to learn (remember) in some way.

That's what writing does for me: I learn via writing because it is through writing that I 'elaborate' on the content I'm trying to master.

That's also got to be what a good class discussion does; a good class discussion allows everyone to elaborate on the novel or poem or history text etc. they are studying.

"Elaboration" also goes a good long ways towards explaining why watching a video of a lecture doesn't work. In a real lecture students interrupt and ask questions or make remarks, or the lecturer self-interrupts and asks questions, or the lecturer simply sees from the looks on student's faces that he/she needs to take another tack, etc.

I'm thinking that progressive educators may have confused "explain your reasoning in words" with "elaborate upon the content you are trying to learn."

Because we all speak English, for most of us 'elaboration' is probably verbal -- or, at least, language is the medium in which we engage in shared elaboration (via class discussion, writing papers, etc).

But offhand, I don't see any reason why elaboration in math wouldn't take place IN MATH.

I don't know what the relationship of math to language is for math specialists; a lot of the 'math people' at ktm seem to be pretty verbal. I'm guessing that 'math people' can engage in useful elaboration via language --- BUT elaboration per se does not mean 'put into words' as far as I can tell.

One is the notion that, regardless of how much your math skills exceed your verbal skills, if you can’t communicate your mathematical thinking then that thinking must be deficient.I've just recently discovered, via Barak Rosenshine's article in American Educator, the idea of 'elaboration' in learning, and it makes perfect sense to me. In fact, it's more or less the 'missing piece' in my own thinking about all this.

Elaboration means (as I understand it) that we learn best when we .... manipulate or develop or grapple with the content we're trying to learn (remember) in some way.

That's what writing does for me: I learn via writing because it is through writing that I 'elaborate' on the content I'm trying to master.

That's also got to be what a good class discussion does; a good class discussion allows everyone to elaborate on the novel or poem or history text etc. they are studying.

"Elaboration" also goes a good long ways towards explaining why watching a video of a lecture doesn't work. In a real lecture students interrupt and ask questions or make remarks, or the lecturer self-interrupts and asks questions, or the lecturer simply sees from the looks on student's faces that he/she needs to take another tack, etc.

I'm thinking that progressive educators may have confused "explain your reasoning in words" with "elaborate upon the content you are trying to learn."

Because we all speak English, for most of us 'elaboration' is probably verbal -- or, at least, language is the medium in which we engage in shared elaboration (via class discussion, writing papers, etc).

But offhand, I don't see any reason why elaboration in math wouldn't take place IN MATH.

I don't know what the relationship of math to language is for math specialists; a lot of the 'math people' at ktm seem to be pretty verbal. I'm guessing that 'math people' can engage in useful elaboration via language --- BUT elaboration per se does not mean 'put into words' as far as I can tell.

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