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:

- Algorithms (Karen and I developed a new one),
- Standard topics in algebra (of which finding LCM of algebraic expressions is one, and which deserve critical reexamination on a regular basis), and
- 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?

- 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.
- 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.

### postscript

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.

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Final update here: Karen took Calculus 2 with me last spring. She struggled. She worked her tail off and was not satisfied with her grade, but she passed.