Nonlinear lesson design: on definitions

Last year I wrote about nonlinear lesson design as a way of fixing issues I had getting students to understand a difficult problem.

I’ve been thinking lately about applying a nonlinear thought process to lesson plans in general. The process is more intense and at times impractical, but I believe it possibly worthwhile in making a lesson plan designed to be read by other teachers.

The general argument is this: if you’re planning the students to explore and discover things on their own, how do you know they will follow in your pre-planned linear format? Would it not be more helpful to anticipate the different directions a lesson could go?

Rather than talk about everything, let me focus on one line common on mathematics lesson plan templates —

Vocabulary and definitions:

Now, one can deposit vocabulary in front of the students like a decree. Sometimes this is useful. However, it is also useful (especially for English language learners) to have them write (or rewrite) in their own words.

Rewording is not a bad thing: math definitions are more mutable than commonly perceived.

Here’s an experiment, before reading any farther: define the word variable.

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Go ahead. Try it!

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How close is your definition to these?

Wolfram Mathworld:

A variable is a symbol on whose value a function, polynomial, etc., depends.

Wikipedia:

In mathematics, a variable is a value that may change within the scope of a given problem or set of operations.

Merriam-Webster:

a quantity that may assume any one of a set of value

Planet Math:

The word variable as used in mathematics (and in other scientific fields that use mathematics) is somewhat vague and may have different meanings depending on the context . . . (i) As “mathematical” variables: These stand for a concrete object, for example, an element of the real numbers That is, when we write the symbol x , it is a stand-in for various numbers: e.g. 2, 3, , , 578.24. But we do not name these numbers specifically, because we may want to talk about all these numbers at once, in a general statement, theorem, or proof about numbers.

Purplemath:

What is a variable? It is a box, and it exists to contain a value. Sometimes the value is already inside the box, and you have to figure out what that value is. Other times, the box is empty, and you get to pick the value to put inside.

MathIsFun:

A symbol for a number we don’t know yet. It is usually a letter like x or y.

How useful would any of the first four be to a student seeing a variable for the first time? And are the last two definitions too imprecise or dangerous?

Definitions are like a boundary rather a discrete point. I’ve found if I’m letting students phrase a definition in their own words, I have trouble telling if they’ve gone wrong.

I also asked people on Twitter to define “variable”. Here are some the responses:

A variable is an entity, in general not known and (usu.) represented by an arbitrary symbol, in a given context of mathematical relations. (zarfeblong)

[a placeholder/template for] an object that can have multiple values [usually typed] (comath29)

a symbol which represents an unknown quantity. JackieB

the idea that reminds us we can think of a “set of related objects” as a single object which varies in some way we care about brianwfrank

An object that is used in expressions to represent an quantity that is allowed to vary calcdave

I really like Wu’s view of a “variable” as a pronoun dcox21 I describe it that way with my kids, too. ‘pronoun’ seems to make it make sense to them as a placeholder for an unknown value Mythagon

a variable is sometimes a symbol representing a fixed, but at present unknown quantity
further, as a symbol, a variable may not simply represent a fixed but unknown quantity, rather an unfixed set of possible values
but a variable can also be changing quantity, that I could elect to fix at any time blaw0013

How do we convey all this information to the beginning (or even veteran) teacher in a way that’s reasonable?

I don’t think the entire calvacade I produced above is quite readable, but I’d start by having an “official” definition and a “kid-friendly” definition. Drawing from the above list let’s say (although one could quibble):

Official definition: A variable is a value that may change within the scope of a given problem or set of operations.

Kid-friendly definition: A symbol which represents an unknown quantity.

(Already there’s trouble here: calcdave calls this a parameter.)

Even the kid-friendly definition would give my 8th and 9th graders blank stares without context, so it’d also be helpful to include a

Metaphorical definition: Think of the variable as a ‘pronoun’ — a placeholder for an unknown value.

In addition to the positive space, to handle true nonlinearity the teacher needs to know about the negative space: what is truly a blunder? Let’s make a new category:

Problematic definition #1: A symbol for a number we don’t know yet. It is usually a letter like x or y. [This implies there will always be a fixed value somewhere along the line. It is possible for a variable to never attain a specific value; for example the line y = 2x + 1 represents a set of values.]

Problematic definition #2: It is a box, and it exists to contain a value. Sometimes the value is already inside the box, and you have to figure out what that value is. Other times, the box is empty, and you get to pick the value to put inside. [The box metaphor is a good idea and connects with early algebraic concepts from elementary school, but “you get to pick the value to put inside” is too open-ended, and muddles the situation of y = 2x + 1 where one could plug in for y and solve for x or plug in for x and solve for y.]

Maybe that’s too much? Perhaps on a standard paper form, but I am thinking of lesson plans on computer with boxes that can be expanded or contracted with information rather than the printed paper; the teacher can click on Official definition, Kid-friendly definition, Metaphorical definition, and Problematic definitions to read them in a non-linear way.

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