Let’s Ban the “Distance Formula”!

One of the things that can be hardest for kids learning algebra is to be able to understand the value of abstraction and of using symbols to help one analyze and think about a problem.  I myself remember learning algebra (way back in 1978)  from Dolciani/Wooton, a textbook that valued formal manipulation above all else; there was almost no motivation given for where the rules came from, just lots of practice in learning how to manipulate symbols correctly.  Indeed, for a number of years afterward I thought that formal manipulation was all there was to algebra.

The idea that there are ideas to be discovered in algebra was completely foreign to me.  I knew in 9th grade that perpendicular lines had slopes that were negative reciprocals, but if you had asked me why, I would never have known, or even thought I should have known.  Dolciani gives a proof, but I would bet a lot of money that very few students read it, as it is, to be frank, inappropriately abstract for a high schooler learning the subject for the first time. 

Our curriculum ( http://parkmath.org/curriculum/ ) approaches the topic by having students first draw a line on graph paper with a slope of 2, and then try to figure out experimentally what slope a line perpendicular to it should have.  Working out concrete examples using lines of different slopes can lead to a much deeper understanding for a 9^th grader than an algebraic proof that they have little chance of following, never mind retaining.  When students are older and have more experience with algebra, of course, a formal proof can make good pedagogical sense.  But for a freshman? 

Which is why I think we should ban(!) teaching the “Distance Formula”, at least for the large majority of 9th and 10th graders.  Why, you might ask, if I am trying to teach them the value of abstraction?

Park School Math in the “Mathematics Teacher”

We’ve been fortunate enough to have an article published in the February 2012 issue of the Mathematics Teacher, “Geometry in Medias Res”.  We teach Geometry in a fairly unusual way, we think, so we decided to write about it and see what people thought about our approach.

One of our main ideas is that we want students to encounter interesting problems on the first day.  So we ask them non-trivial questions right away (e.g. can every triangle be circumscribed?), and in the process of them discussing/arguing with each other, we start to develop with the kids the necessity for a standard of proof other than “it really seems like it to me!”. 

There’s a lot more, but the idea is to have a more natural and intuitive introduction to the axiomatic nature of Geometry than one usually finds.  We are big believers that proof is completely accessible to all levels of students, but that it has to be introduced gradually, as a way of resolving questions students have, not as a forced superstructure like the way it was often taught in the past.

If you have a chance to take a look at the article, we’d love your feedback and thoughts.

The Cruel Irony of Algebra

You would think it would have occurred to me sooner, but it wasn’t until a few years ago that it really hit me what I think is the biggest problem most students have with Algebra:  they don’t actually think the letters represent numbers.   Here’s the kind of question I’ve asked that illustrates what I mean:

Chuck says that (xy)(wz) is always equal to (xyw)(xyz).  Is he right?

Over the years, I have found that if a student is unsure of whether or not they are supposed to “distribute the xy”, they often just guess.  When asked why, they say that they were unsure of the rule they were supposed to use, so they just took their best shot.  For many years, I tried to show them which rule to use in various situations and the principles involved, hoping that over time they would catch on to the logic of algebra.

But invariably, for many of my students, even a slightly changed question presented what seemed like a freshly baffling challenge.  After all, does the question below really seem all that different (other than to a math teacher)?:

Chloe says that (xy)(w+z)  is always equal to (xyw) + (xyz).  Is she right?

So what to do?  While it may seem like taking two steps backwards in the march towards abstraction and generalization, these days I ask my students how they could possibly figure out for themselves if the two expressions are always equal, and the ensuing discussions leads us to the question of what the heck those x’s and y’s and z’s and w’s represent—numbers!  So why not try out these equations with numbers?  The cruel irony of algebra is that what is intended to make generalization easier actually becomes so abstract for many kids that numbers are the last thing on their minds. They end up seeing algebra as a bunch of arbitrary rules that are hard to predict.

Of course, just because two expressions are equal with a given set of numbers doesn’t mean they always will be—and we discuss that eventually as well.  But as an entry point into algebraic identities, and as a gut check to see if my students “get” what algebra is about, I find this works because students believe what they can test and see for themselves.  Having them practice distributing multiplication over addition (and also practicing NOT distributing multiplication over multiplication), while it has its place, I find isn’t a good substitute for the intuition that develops by playing with the raw numbers.

Next, semi-related entry:  The joys and sorrows of “flip and multiply”!

The “Wow!” Factor

Sometimes you see things that are so amazing that it seems almost criminal not to share them with other people.  Being a teacher, I have a ready-made audience to share my enthusiasms, and I long ago concluded that anything that is extraordinary and mathematical, even if it is unrelated to the topic we are studying, is worth sharing.  There is nothing trivial about having students say “Wow!”  So here’s something that did that for me a year or so ago– Hans Rosling’s Ted.com talk about the world’s demographics, using data-rich graphs:

In that talk, he uses a program that makes animations of data in beautiful and intuitive ways and that yield real insight.  Since he made that video, he has started a website where that program is available for use directly on the website using Adobe Flash. The site is   http://www.gapminder.org/   and it really is a wonder.  My son, in the 6th grade, couldn’t stop playing with it, and was immediately drawn in to the questions the graphs you create present.  Why was there a “Bangladesh Miracle”?  Why does China’s life expectancy suddenly drop 20 years around 1960, and then quickly rise again?  Why did the average number of children per women in the U.S. rise sharply between 1940 and 1970?  Why is the connection between average income and mortality so varied in different parts of the world?

I never was that interested in statistics and data analysis when I was in school.  With this program, it takes a herculean effort NOT to be interested.

Tests- The Mathematical Bogeyman

I have always found assessment in the math classroom really quite difficult.  There are so many issues, but the biggest ones I can think of are the problems of mastery and depth.   Traditional math tests, like the ones that I give about every 3 weeks, and have done for 22 years, are basically a test of mastery.  Some of my colleagues strenuously object to this goal, or at least this framing of what a test is for, but in my opinion that is what a test basically is for.  Other teachers in the department would say that by giving a student another chance to solve a similar, analogous problem (sort of retesting on a problem-by-problem basis), or by showing extensive corrections, or by creating their own analogous problem along with an answer key, we allow them the space to revise their understanding and show them that instantaneous mastery is not being demanded of them.

But note the structure that all of us have accepted in the first place:  tests.  My colleagues who disagree with me give them as well, they just don’t want them to feel as punitive or “final” as they did when they were students in school.  They want them to be more of a teaching tool.  And I am sympathetic to that goal, I really am.

But I don’t do that in my classroom.  Basically, I give a test to my kids about every 3-4 weeks.  I give them past tests with answers and/or review problems with answers to study from ahead of time so that they have some idea of what will be expected of them on the test, in terms of content, difficulty, and structure.  They know that the test may look very different, or quite similar, to the kinds of problems I have given them for review, but that if they can’t at least do the review problems, they will likely have a hard time on the test.

So why do I do it this way, which evokes memories of damp English boarding schools?  Competence in mathematics means, to all the math teachers that I know, the ability to solve problems in mathematics, and not only ones that are similar to those you have seen before, but also those that are unfamiliar.  To be able to do that well, you need a good deal of exposure to a range of problems on the material before someone checks to see if you really are able to solve problems on a given topic.  Homework is one way to do that, and our text , being in my opinion much less rote than many traditional texts, has definitely helped in that matter.  But students need more practice, in more contexts, before being tested—thus the review problems and my instructions to make sure to complete a healthy number of them before a test.  What a test positively offers is a chance to know that you know; one good thing about mathematics having answers that are so specific is that a student can see if their chain of reasoning is correct at a glance, because the answer to that review problem was indeed 113.67 and not something else. 

And also for most kids, in my experience, when I did offer revision, they became much less focused before the test, and didn’t complete the review problems as thoroughly, because they knew they could “revise” later.  I don’t blame them—it is human nature.  Now I offer test revisions in unpredictable fits and starts, so that students take tests seriously the first time.

Which is not to say that I am completely happy with tests, not at all.  But they are very effective at helping a student to know what they know, and to know what they do not know.  They also are clear and are perceived by students as being fair; one thing I always tell them is that when grading tests I look to see if there is any question that basically all the students missed, because then I am likely to throw that one out as it clearly was not a fair expectation that they could get it.  So students like the finitude of tests, and they like that have a good sense of what will be on them.  And to be clear, I ask a wider range of questions on tests than I used to with previous texts, because the problems in our texts have greater range and diversity, so students naturally expect that the questions they have to answer are quite varied. 

But, ironically, the reason I’m writing this entry is because I have been thinking about the limitations of tests as assessment. What else do I want out of assessment?  Well, I want to see if students can ASK questions about extending and generalizing a problem, as well as answering them.  I want to see if students can use mathematical habits of mind more systematically and consciously when confronted with a difficult problem.  I want to see how a student handles an open-ended question, rather than one that has a specific algebraic or numerical answer.  I would like students to be able to lead a discussion of a problem, much like they do in an English class or a History class. I would like students to feel, for certain limited topics, that they have gone in more depth than their peers and have attained a real mastery of something.

Currently, the best my tests do is see if my students can handle a range of types of questions about the material at hand, maybe with a little bit of using mathematical habits of mind to make progress on the more unusual problems.    I’m not really addressing much else of what I listed in the last paragraph with tests.

So how am I going to do that?  This blog entry is already too long, so I’ll try to begin to answer that next time, or the time after if the start of school brings up a topic or two that merits an immediate response.

How long should I let kids struggle with a problem?

I’ve been thinking a lot about that question the last few weeks. I teach math to three different “ability groupings” of kids, and yet in many ways they have similar sets of reactions in grappling with a difficult problem.

When they first hear the problem, there generally is interest and excitement, especially if I have chosen a good problem. (By the way, by “problem”, I don’t mean a routine exercise, but rather something that requires them to think, to use what they have previously learned but in a different way.) Students generally start talking among themselves about what they might do, or they raise their hands to ask clarifying questions and to propose a plan of attack. At this point, things have a really good feel– students are engaged, and they are anticipating solving a thorny problem if they just make a sincere effort. And indeed, that is often how things go– students spend a few minutes trying an approach or two, make a mini-breakthrough, and then solve the problem.

But often, after 5-10 minutes of thinking, students find themselves at an apparent dead end.