# Category Archives: Language

## Words to avoid in the middle school classroom (continued)

I have this to add to the collection so far:

Long and hard.

In trying to put the Common Core mathematical practices into kid-friendly language, a colleague transformed this:

1. Make sense of problems and persevere in solving them.

Into this:

1. Think long and hard to solve problems.

## Words and images to avoid (addendum)

We had a little fun back in April with words and images to avoid in the middle school classroom.

Those who do not learn history are doomed to repeat it, no?

Consider the following from a recent draft of a project that will remain nameless, but which is intended for sixth grade:

Two boys who live near a golf course search for lost golf balls and package them for resale.

How many packs of 12 golf balls can be made
from a supply of 6,324 balls?

or

If a supply of 6,324 golf balls is packed in 12 boxes,
how many balls will be in each box?

## Words (and images) to avoid in the middle school classroom

When I taught middle school, I sometimes worried that they taught me more than I taught them. Middle schoolers are masters of the double entendre and they love nothing more than twisting their teachers’ innocent words and drawings in perverse ways.

I leave the following list to your imagination. How might a middle school teacher have meant to use the following words and images, and how might a student take deviant pleasure in hearing and seeing them used?

### Final warning

The following is completely immature and inappropriate. Yet each example comes directly from my own middle school classroom. Middle school teachers, back me up on this, please!

### words

Balls

Period

Score

Pull out (especially if what we are pulling out is balls)

In a Connected Math probability unit, there is a problem involving blue and orange blocks. Students are trying to list out all the ways two orange blocks and two blue blocks can be put into two containers. Students typically abbreviate orange with O and blue with B. This all goes well until they put all four blocks in one container and list them out: blue, orange, orange, blue.

The formula for area, A, of a square as a function of its side length, s, is A=s*s

## 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.

### 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.

## Let’s agree not to use these words

Obvious or obviously.

Clearly

It has to be

It’s the only way

Trivial

Easy

These words and phrases are pervasive in mathematics classrooms, and they are increasingly common the further we go in mathematics.

But each has a toxic effect on student engagement. Consider the following two scenarios.

### Scenario 1: I don’t get it

If I don’t understand what others describe as obvious, I feel stupid and I disengage. I feel like I can’t do this.

### scenario 2: I do get it

If I am feeling smart because I understand something, and then others describe it as obvious, I don’t get to feel smart anymore. The thing I worked so hard to understand is obvious to everyone else. I am likely to disengage and to feel like I can’t do this.

### So what?

As a teacher, I have two roles related to this issue.

The first is to eliminate this language from my own teaching vocabulary. There is no upside to these terms, so I have to stop using them.

The second is to challenge the “obviousness” of what others claim to be obvious. Is it really that obvious? Is that really the only way to do it or to think about it? As the one who sets the tone for discussion in the classroom, I have a responsibility to include everyone. Sometimes that means pretending I don’t understand something I do.