Tag Archives: developmental mathematics

Systems of inequalities: Really?

I wrote a while ago on the topic of word problems,

I will consider my career a success when my students no longer tell me that they are bad at (or good at) word problems.

And then I had the opportunity to meet Dan Meyer recently. He has written extensively on pseudocontexts, his word for the word problems I was railing against.

In that spirit, I need to get the following example off my chest, from a College Algebra text I no longer use.

We are supposed to set up a system of inequalities (“at most an additional 10 gallons of gas”) and solve with techniques of linear optimization.

But how could the answer be anything but, “Put as much as possible into the moped; put the rest into the car”?

And wouldn’t the problem be more interesting if Omar spent a few bucks on a gas can? Then some of his previous money goes to something other than gas, but buys more miles because he can put it in the moped.

And why must he spend all of his gas money at once?

And if he does have to do so, how is he getting both car and moped to the gas station at the same time?

And does that little yellow car really only get 20 mpg?


Landscape of learning

I wrote quite some time ago about a student of mine in the Math Center (the site of our developmental math courses) and her struggles to learn our textbook’s algorithm for finding least common multiple.

I have brought that post up in personal conversations with students and colleagues several times and I now realize that not everyone sees in the example what I do. So now I will be more explicit.

I see three important themes playing out in my post:

  1. Algorithms (Karen and I developed a new one),
  2. Standard topics in algebra (of which finding LCM of algebraic expressions is one, and which deserve critical reexamination on a regular basis), and
  3. A plea for compassionate practice in mathematics teaching.

What I really had in mind was the third.

Karen was struggling. She was frustrated. I knew she had been working hard and that the textbook explanation simply made no sense to her. Her ideas and the textbook techniques were too far apart.

In their book series Young Mathematicians at Work, Catherine Fosnot and Maarten Dolk write about metaphors for learning mathematics. In particular they contrast a linear metaphor with a richer one.

learning math as a linear process

We have many, many ways of talking about learning mathematics as a linear process. We speak of students being ahead or behind. We talk about accelerating students through material. We speak of filling gaps in student knowledge. We work hard on sequencing material for students.

In this linear metaphor, we presume that one student’s learning process is much like that of each other student. The main difference between students is how quickly they acquire each skill and move on to the next one.

It is as if each student is traveling a highway. The teacher’s job is to keep students moving down this highway at an appropriate speed. When a student, such as Karen in my LCM example, strays from the highway, it is the teacher’s job to get her back on the highway.

Learning math as moving through a landscape

Fosnot and Dolk challenge us to think about learning in a richer way. Their enriched metaphor supposes that there are many ways to know a mathematical topic. They want us to consider the learning process as navigation through a landscape. We want students to head in a particular direction, but we do not assume that there is a single, linear path.

Instead, different students will follow their own paths. It is the teacher’s job to know this landscape very, very well so that when a student is lost, the teacher can help her find a way forward.

A thought experiment

Imagine you are visiting a dear friend in an unfamiliar city. You go out on your own for the day and get lost on your way home. You call your friend. Which of the following two responses will lessen your anxiety and make you trust that you will end up getting where you are going?

  1. Get on I-94…I know you’re lost, that’s why you need to be on I-94; it will take  you right to my house.
  2. Where are you and what do you see? What landmarks do you remember passing recently? Can you see any street signs?

In the linear metaphor, we constantly tell students to get back on the highway (even if they have no idea where the entrance ramp is). In the landscape metaphor, we begin with where they are and help them to get where they need to go.

Karen was way, way off the highway when she was canceling factors. She saw a procedural connection to canceling common factors in rational expressions and she was following that path. I could have tried forcing her back onto the highway (citing the union of the sets of factors). But the more compassionate route was to help her develop an algorithm that was connected to her thinking; one that would always work.


I am pleased to report that “Karen”, after four semesters of beating her head against the Math Center wall, finally passed with a “C”. She then took my College Algebra course where she earned an “A”. I attribute this success to her hard work, and to the College Algebra course being about ideas (the landscape) more than about an arbitrarily chosen set of algorithms (the linear highway).

And she is in someone else’s section of Precalculus where she recently earned an A on her first exam.

The end of word problems

I will consider my career a success when my students no longer tell me that they are bad at (or good at) word problems.

The direction of K-12 mathematics curriculum in the United States in recent years has been towards giving students a sense of how people actually use mathematics. There are many people who use abstract mathematical ideas on a regular basis outside the mathematics classroom-physicists, statisticians, mathematicians and the like. There are many more people who regularly use mathematics to solve practical problems that matter. (See for instance, this recent New York Times article on airline bumping). But no one solves word problems such as the following unless they are in math class:

The apartments in Vincent’s apartment house are numbered consecutively on each floor. The sum of his number and his next-door neighbor’s number is 2409. What are the two numbers? (from Introductory and intermediate algebra: full citation below)

This is a classic word problem of the sort that I hope to eradicate from mathematics instruction (at least from my own).

Why eradicate them? I have already presented my first argument-that they do not represent the ways anybody actually uses mathematics. At their very very best, word problems are intriguing puzzles and perhaps like a clever lyric or a piece of a melody they provide a bit of satisfaction to the aesthetic soul. But few word problems meet this standard. Instead, they represent to many students roughly half of what mathematics is (most the other half is abstract symbolic manipulation). Students who find these problems silly come to believe that mathematics has nothing to offer them.

I believe students are right in asking When am I ever going to use this? I can honestly tell students that they will not ever be in a situation where they know the sum of two addresses, and where they know that the addresses are consecutive, but where they do not know either address.

Occasionally I let myself believe that American math teaching has made substantial strides since the release of the 1989 NCTM Standards for School Mathematics and the revised Principles and Standards 2000.

I recently had an experience that set me straight. I was in my College’s Math Center, where the developmental math courses are housed (i.e. those courses not bearing college credit because they cover remedial topics). The apartments problem cited above comes from the textbook we use for these developmental courses.

Things were slow so I browsed a small selection of books on a shelf. At least one had been culled from the college library recently. It was titled, How to solve word problems in algebra: A solved problem approach by Mildred Johnson (full citation below). I became intrigued by the Table of Contents, which closely matches the kinds of problems in our text: Numbers; Time, Rate and Distance; Mixtures; Coins; etc. Of course the problems in this book match those in our text as well. Consider:

The sum of three consecutive integers is 54. Find the integers. (p. 13)

This problem, at least, is more honestly posed as a puzzle. The apartment problem puts the same idea into an unrealistic setting. But How to solve word problems commits the same crime. Consider:

Mrs. Mahoney went shopping for some canned goods which were on sale. She bought three times as many cans of tomatoes as cans of peaches. The number of cans of tuna was twice the number of cans of peaches. If Mrs. Mahoney purchased a total of 24 cans, how many of each did she buy? (p. 14)

Seeing the similarities between the perspectives of our textbook and this supplementary book, I noticed the Preface.

There is no area in algebra which causes students as much trouble as word problems…Emphasis [in this book] is on the mechanics of word-problem solving because it has been my experience that students having difficulty can learn basic procedures even if they are unable to reason out a problem.

And here is the crux of the matter. I have already argued that the very nature of word problems is such that people’s actual experience has no bearing on solving them. But in this preface is the rarely stated truism that we can train students to work these problems even when we cannot teach them to think mathematically. Entire sections of textbooks are devoted to the translation of word problems into algebraic symbols and Ms. Johnson has written the book on it.

While I appreciate Ms. Johnson’s efforts to help students through the arcane world of word problems, I am saddened by the uncritical approach. There is no discussion of why students should be forced to learn to solve artificial word problems, nor a questioning of whether there might be better uses of their (and their teachers’) time.

I began to wonder how recently the book had been written, given the similar perspectives of our text with this supplement. The copyright is 1976. In thirty-four years, we have truly made no progress.

Back to work.


Bittinger, M. & Beecher, J. (2007). Introductory and intermediate algebra: Third edition. Boston: Pearson.

Johnson, M. (1976). How to solve word problems in algebra: A solved problem approach: Updated first edition. New York: MacGraw Hill.

A new algorithm for finding Least Common Multiple?

A student in a developmental algebra course was struggling with problems involving least common multiples. The lesson she was working on involved finding the least common multiple of numbers first, and then using that process as an analogy for finding least common multiples for variable expressions. Surprisingly, she felt confident with the variable expressions and was struggling with the numbers.

When I sat down with her, Karen (not her real name) could not understand why the technique she was using for variables was not working with numbers. She was able to correctly find the prime factors of numbers, so she wrote:

24=2*2*2*3 and


Then, drawing on her experience with fractions, she cancelled the common factors and used what was left:

24=2*2*2*3 and


She ended up with only a 2 and a 3 remaining, which is six and she knew this was not correct. Six is not even a multiple of these numbers, never mind the least common multiple.

I restated our textbook’s approach, to wit: We want to use each factor the same number of times as it appears in the number in which it appears most often. There really is no simple way to state this and I worked a couple of examples for her.

But I was intrigued by her ‘cancelling’ approach. In addition I had offered a strategy that, while supported by our textbook, bore very little relation to her way of thinking about the problem. This is not a recipe for success. We need to help our students refine their ways of thinking, not give them yet another rule to remember. So I explored her idea of cancelling and suggested this:

When we use each factor the greatest number of times as it appears most often, we can think of this as gathering all of the prime factors, then getting rid of the ones that come from numbers where they appear less often. When we have:

24=2*2*2*3 and


We want to “cancel” the 2’s in 36 and the 3 in 24-we don’t need those. So Karen’s Cancelling Algorithm-perhaps new to the world, and perhaps new only to her and to me is this:

Cancel the common factors only in one of the two numbers:

24=2*2*2*3 and


We cancel two of the 2’s in 24 because they match up with the two 2’s in 36. And we cancel the one 3 in 24 because it matches up with one of the 3’s in 36. We could have cancelled the two 2’s in 36 instead-that’s not important. What is important is that we cancel them only once.

Karen loved this algorithm and was very, very pleased to have her thinking changed into an algorithm that works.

The interaction was partly satisfying for me and partly disturbing. Karen began very frustrated and ended feeling successful and bright. That is always satisfying. But what had she really learned? She is not studying least common multiples for their interesting mathematical properties. Instead she is studying them in order to be able to add, subtract, simplify and solve rational expressions. Without questioning the larger goal of the whole enterprise of developmental college mathematics, it is still reasonable to ask how important least common multiple is for operating on rational expressions.

The only argument for finding least common multiple in this context is that it gives us a simpler form of the resulting rational expression than any other multiple will. If I am working with numbers, I can use the least common multiple to add:


But I can use any common multiple:


The least common multiple results in a simpler fraction, but it’s the same answer either way. Indeed, I can always use the common multiple that results from multiplying the two denominators:

1/24+1/36=36/864+ 24/864=60/864

And the same is true of the rational expressions Karen will be working with shortly. So why do we induce this stress in our students? If the only reason to find least common multiple is to work with rational expressions, and if at the same time any common multiple will do, why do put this artificial barrier in front of our students? And why do we, as teachers, allow ourselves to work as though these barriers were real?