I was intrigued by this post on Anthony Cody's blog in so many ways. It's about science, not math, but I wonder if we can apply these ideas to math instruction?

Channon Jackson, a fourth grade teacher in Oakland, writes:

In my class [the students] do a science conclusion at the end of every investigation. It is a paragraph where they introduce the topic, give a definition of the key term, make a claim that goes with the focus question, give evidence and then give the scientific reasoning behind the claim. I ... found out that a lot of students were not writing.Eventually, she allowed them to skip the vocabulary words in their writing.

...

I decided to have this new structure, where they have these free writes. They don't have to use complete sentences. They just can write whatever they want to write about the topic we were learning about. So if we were learning about rocks, they could write whatever they learned about rocks. If we were learning about magnets, it's just whatever you learned about magnets, but they still had to use vocabulary words. They would get five minutes free write, then sit with their table groups and share out to their table about what they learned.

Still, I had some kids that weren't writing.

And all of a sudden, ALL my students had something to say. They clearly understood the content. They now were competing with each other to see who could have the most things to share out at their table group.She didn't want to give up on the vocabulary...

So the third thing I changed in my teaching was they would do the free-write, they would still do the table talk, we would have a class discussion, and then I would have them pull out their vocabulary words (we keep them on these cards.) I would tell them "Go back and look at the sentences you wrote, and see if there are any words that you could replace with one of your vocabulary words?" I found out that most of them could do that. Most of them could figure out "Oh, I used 'non-living' here, and I could say 'abiotic' instead."

I love that she found ways to open up the process until it worked for her students. They knew so much more than she realized at first. And I believe the ones who weren't writing previously are learning so much more now that they're writing. I think they are internalizing the concepts more fully, and learning how to incorporate strange new words into their own personal vocabularies.

One of the things I loved about this post was that, without ever mentioning standardized tests, it gave a beautiful critique of a system built around testing. Standardized tests make no room for this organic process between a teacher and her students. They take time away that could be used in these delightful community-building ways, and they frame learning in such shallow ways.

The only example she gave of vocabulary - abiotic, which means non-living - confirms me in my bias against teaching vocabulary. The kids may need to know abiotic for the testing they must endure, but how does knowing a word like that help them in their understanding?

When I explain vocabulary in my math classes, I often tell students I would never test them on it, I just want us to be better able to communicate the ideas to each other. I think my position may be a bit extreme. I'm curious.

**Do you teach vocabulary, and can you help me understand how it helps students to learn math?**

**I know that certain concepts take a long time to internalize. We don't get very far along that path by teaching the definition of a term explicitly, but it is still part of the process. Here's a bit from a great post about how definitions are handled in math:**

Aspects of definitions that people new to abstract math don't always understand [include]: The definition gives a small amount of structural information and properties that are enough to determine the concept, [but] the information in the definition may not be the most important things to know about the concept. (from SixWingedSeraph, who blogs at Gyre and Gimble)

**Definitions in Math**

I can give an example of the necessity of precise understanding of vocabulary from my studies last night from W.W. Sawyer's book,

*A Concrete Approach to Abstract Algebra*. I believe I took courses in both group theory and field theory about 25 years ago, so I know I've worked with these ideas before. But either the examples used were different, or I've completely forgotten the experience. I am loving my discoveries as I work through the exercises.

Sawyer definitely doesn't focus on vocabulary, and clearly avoids the typical definitions given at the beginning of most advanced math courses. Mostly, that's great. But then in an exercise, he asks whether two structures he has described are

*isomorphic*. And, although I thought I knew what isomorphic means (having the same structure), I'm not sure. He's given me no definition of isomorphic to use, and I'm puzzled whether the difference I see makes the two non-isomorphic. Mathematics depends on careful definitions of words used, and I need a precise definition of isomorphic to work from to answer the delightful question he posed.

Here are the two structures he asks the reader to compare. (At this point, it may get a bit hairy if you're not used to thinking abstractly.)

One is the addition and multiplication tables for modulo 5 (also called mod 5) arithmetic, where we are only interested in the remainder after division by 5. This looks like

+ | 0 | 1 | 2 | 3 | 4 | * | 0 | 1 | 2 | 3 | 4 | ||

0 | 0 | 1 | 2 | 3 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | ||

1 | 1 | 2 | 3 | 4 | 0 | 1 | 0 | 1 | 2 | 3 | 4 | ||

2 | 2 | 3 | 4 | 0 | 1 | 2 | 0 | 2 | 4 | 1 | 3 | ||

3 | 3 | 4 | 0 | 1 | 2 | 3 | 0 | 3 | 1 | 4 | 2 | ||

4 | 4 | 0 | 1 | 2 | 3 | 4 | 0 | 4 | 3 | 2 | 1 |

Can you see why? 2*4 = 8, which is 3 more than 5. In modulo 5 arithmetic, 8 is equivalent to 3, and we forget about the 8 and just write the 3. It turns out that this new arithmetic still has all the properties we would want: commutativity (

*a+b = b+a*), an additive identity (0 works as it should,

*a+*0

*= a*), a multiplicative identity (1 works as it should,

*a**1

*= a*), and so on.

Now we're going to compare this to another, slightly stranger structure. This time we'll use modulo 10 (so we get to just pay attention to the ones digit), and we'll only use the even numbers: 0, 2, 4, 6, and 8.

+ | 0 | 2 | 4 | 6 | 8 | * | 0 | 2 | 4 | 6 | 8 | ||

0 | 0 | 2 | 4 | 6 | 8 | 0 | 0 | 0 | 0 | 0 | 0 | ||

2 | 2 | 4 | 6 | 8 | 0 | 2 | 0 | 4 | 8 | 2 | 6 | ||

4 | 4 | 6 | 8 | 0 | 2 | 4 | 0 | 8 | 6 | 4 | 2 | ||

6 | 6 | 8 | 0 | 2 | 4 | 6 | 0 | 2 | 4 | 6 | 8 | ||

8 | 8 | 0 | 2 | 4 | 6 | 8 | 0 | 6 | 2 | 8 | 4 |

I had predicted that they would be isomorphic, and felt so sure that this was a simple problem that I almost skipped it. Imagine my surprise when I saw that the multiplicative identity wasn't in the position I had expected (the 2, since it's the first non-zero element) but was the number 6!

So here's my dilemma. The addition table is really the same, so if we were just considering addition, we would say that the two systems were isomorphic. And actually, the multiplication tables can be thought of as being the same in a strange way. We saw that 6 is "our new 1", since it's the multiplicative identity. And we could match up the other rows: in both tables 4 gives the numbers in reverse order (I think of it was being like -1), in both tables 2 goes through the numbers in the same order, and the 8 row in the mod 10 table seems to give the numbers in the same order as the 3 row in the mod 5 table. So considering multiplication alone, we could say the two systems were isomorphic. But. We need to consider both operations at once, I think. And then things are tangled up.

I just read the definitions over at wikipedia, and I know my next step, but it's pretty technical. (It took some vocabulary to help me wade through the articles and find the one I wanted. I had thought I was working with 'groups', but realized as I read wikipedia that having two operations makes these structures 'rings' or 'fields'.) After seeing how the numbers were tangled up in their multiplications, and not tangled in the same way in their additions, I wanted to believe the two structures were not isomorphic. But now I'm not so sure.

You're welcome to discuss this question in the comments too. (

**Are these two structures isomorphic?**) I'll be working it out before I peek, and will enjoy any discussion of this issue. My main question in this post, though, is how vocabulary is related to the learning of deeper subject matter, whether it be science, math, or something else.

I've shown that they are in fact isomorphic, but I don't think what I've done is elegant yet. I'd like better explanations than what I've given myself.

ReplyDelete[Order the numbers: 0,6,2,8,4. (This is in order by their remainder mod 5, actually.) The function used for mapping from the mod 5 structure to the mod 10 structure is f(a)={a if a is even, a+5 otherwise}. I used Excel to check that f(a)+f(b) = f(a+b) and f(a)*f(b) = f(a*b), for all a and b. Messy.]

My inclination would be to bypass any explicit construction, and use the fact that any two finite fields with the same number of elements are isomorphic. Of course you might consider that cheating without a full proof of that fact. Usually that is proven in the context of Galois theory but there probably is a fairly short self-contained proof.

ReplyDeleteWell, I haven't studied Galois Theory so far. Perhaps Sawyer will get to that proof in this book. If not, you've given me reason to keep going when I finish this book. :^)

ReplyDelete