Friday, August 26, 2011

Math problems of the week: 1900s Algebra vs. Interactive Mathematics Program

I. The 21st homework assignment in Wentworth's New School Algebra (published in 1898) [click to enlarge]:

II. The 21st homework assignment in Interactive Mathemetics Program: Integraded High School Mathematics Year 1 [click to enlarge]:

III. Extra Credit:
Estimate the ratio of effort to learning in each problem set. Draw a diagram that indicates how well each one works.


Brian Rude said...

These two examples are certainly dramatically different, and if the point is to suggest that a century ago we had pretty good math textbooks, whereas today we have mush, they serve pretty well. But I think there is a lot more to be said about them.

I have long argued that a problem, or an assignment, cannot be judged outside of a knowledge of what the problem or assignment is supposed to do. Context is everything. Since context is not given for either of these assignments, we'll have to do a little imagining. You can't prove anything on the basis of imagined premises or situations, but we might get things to think about.

The context of the 1900s algebra page is pretty easy to imagine. I imagine that the text book presents ideas about algebra, gives a little explanation for each topic in turn, a few examples, and then a page of problems that provide something for the students to apply the ideas about algebra to. In other words that algebra book is probably pretty much like the algebra books I was teaching out of until I retired a year ago. I didn't have the best algebra books in the world, but they provided what was needed, a presentation of ideas about algebra, a little explanation, a few examples, and then a page of problems for each section, problems that provide something for the students to apply the ideas about algebra to. However there was a difference. I imagine the 1900s algebra book was a slim volume of at most a couple of hundred pages. That's about what it takes to provide what's needed for a course in algebra The algebra books I used would be a minimum of 500 pages, and would provide pages and pages of irrelevant something or other that just got in the way. I didn't care for all this fluff, but one learns quickly enough what to ignore. The basics were there, a reasonably coherent arrangement and presentation of algebraic ideas, some explanation, some examples, and problems to assign, so the textbooks I had were quite usable.

That, I presume, would be the context thinking just of the textbook. Just as important, or considerably more important, is the context of the classroom, or in the context of the course, or in the context of what the teacher is trying to do. Here again we must imagine. I imagine the teacher will rather closely follow the text. He or she will probably go straight through the textbook, each day explaining a new algebraic idea (or explaining more on an idea that is not so new, but perhaps difficult), each day working through a few problems, each day answering questions about homework problems, each day assigning homework that will give students practice in applying the ideas that were explained and discussed. In other words things in the early 1900's were much the same as they were for me a year ago before I retired.

Brian Rude said...

(continuation of my previous comment)

Is this a good assignment, this example from the 1900's algebra text? That's the wrong question, in my humble opinion. A better question is, is this assignment useful for what the teacher is trying to do? All a textbook can be is a tool. It's a good tool if it has a reasonably coherent arrangement and presentation of algebraic ideas, some explanation, some examples, and problems to assign. Having taught algebra for a few years I know how these problems fit into the context of the course. I know what topics came before this assignment, and what topics come after. I recognize this topic as important. It's the culmination of some algebraic ideas, and the foundation of other ideas.

Now what about the other example, the one from the Interactive Mathematics Program? Is it a good assignment? On what basis would we judge it? It's unwarranted, in my humble opinion, to say that it is either a good or a bad assignment as it stands. One would have to ask good or bad for what? If the students need to understand how to add signed numbers, it's not a good assignment. If the students need to practice solving simple equations, it's not a good assignment. If the teacher wants to explain the quadratic formula, it's not a good assignment.

What would be the context in which this is a good assignment?

I draw a blank here. This assignment could bring in some algebra, I suppose, but it is not at all clear what. And it could bring in some arithmetic, I suppose. Or it might have something to do with geometry. To say this is a good assignment I would like to fill in the blanks in something like this. This is a good assignment because the learning task of the moment is to ________________________, and this assignment gives practice in __________________________, or this assignment demonstrates that _________________, or this assignment leads to the conclusion that ________________________, or this assignment reinforces the idea that ____________________.

This is high school math? I don't get it.

A possibility comes to mind, but it is not a pleasant thought. The possibility that comes to mind is that in the world of fuzzy math it is accepted that if students are engaged in a thinking activity, if the students appear to be using their brains, and appear to be goal directed, then the activity qualifies as an "educational activity", and no further justification is required.

Does doing puzzles qualify as learning mathematics? Is the mind a muscle that will get stronger by exercise? Can we let students do puzzles for four years of high school and expect them to come out as critical thinkers? The idea of the mind as a muscle lost popularity in the early 1900's. As it understand it, it was considered thoroughly debunked at that time. (Actually I don't think it should be totally dismissed, but maybe 98% dismissed.) The thinking at that time, if I understand it right, was that subjects, such as math, grammar, history, science, and languages were considered by old fashioned educators to exercise the mind and make it stronger, but modern educators knew better. Modern educators of the early 1900s dismissed this idea of the mind getting stronger by exercise, and therefore wanted to dismiss the subjects, the math, grammar, history, science, languages, etc. The idea that these subjects were not just a collection of puzzles to entertain students, but were bodies of knowledge of great value, apparently didn't enter the thinking of that time.

Do supporters of the newest math think there is no body of knowledge of mathematics to be taught? Do they think that doing a series of disconnected puzzles for years in school is somehow going to make mathematically capable adults? I hope not, but if not, then what is the explanation for this garden planning problem?

Katharine Beals said...


Thanks for your thoughts. I agree that a full comparison of Reform Math and earlier math programs requires much more than a problem-by-problem comparison. Teaching methods, prior instruction, and, of course, what else is in the textbooks all play a role. My purpose in publishing these weekly comparisons is simply to add to the debate. Much of the debate is focused on general issues of instruction and curriculum. In the same way that we don't get a full picture if we merely discuss specific problems, we don't get a full picture if we don't look at specific problems. Indeed, I've long felt that any debate over Reform Math absolutely must include a close examination of the specific tasks that children are given--something that seems to me that supporters of Reform Math often try to avoid.

Crimson Wife said...

Around the turn of the last century, schools did not attempt to teach algebra to every teenager. Many students attended grammar school but did not continue on to high school. In 1900, fewer than 2% of young people in the U.S. graduated from high school. Today, that figure is in the 75-80% range depending on the precise methodology.

The 21st century textbook has to deal with the reality of the "algebra for all" movement and that's why it has been "dumbed down" significantly.

Katharine Beals said...

Crimson Wife, Good point, though we have no idea how many of the many who didn't attend high school wouldn't have been able to handle this kind of algebra if they had.

If our goal is to teach algebra to all, programs like IMP aren't the answer. Not only is IMP dumbed down, but it also is diluted with non-algebra, and it doesn't provide nearly enough practice with algebraic algorithms. The weaker students, in particular, need much, much more practice than they are getting here.

A better strategy would be to teach everyone via a straight-up, problem-intensive algebra text (like Wentworth, or any number of other pre-Reform texts), but to buck today's educational orthodoxy and teach different people at different rates.

Lsquared said...

Have I ever mentioned that IMP is my least favorite of all of the high school texts? I think it's all summed up in the fact that the table of contents lists the names of the stories, but not the math they are supposed to teach (and there is no index at all).