Neat idea!

Are the points travelling together or on separate journeys? Who is leading the journey — the blue point or the red point or they take turns?

Who is travelling farther — blue or red or it depends on the journey?

Will the points ever be in the same location at the same time? Will the points ever be in different quadrants?

Where would you place the red point if you wanted the points to be close together? Far apart?

I am sort of seeing two points on a line segment rotating around the origin in the current form — maybe a little bit more random movement and places where you pause. Did you experiment with hiding one point, then moving the other point, and then un-hiding to show the new location?

Before algebraic—-> ask for arithmetic

Students can often solve problems arithmetically that teachers want algebraic representations for. Show students the connection.

“How do you see these dots moving” as am entry point before looking at mapping one on the other.

The hiding/un-hiding was a suggestion for disguising the relationship a little bit more at the initial stage. The initial movement you show kind of has a rigid/mechanical feel. Maybe points sort of flickering on and off in more random locations would be a little better.

Before I harness math I harness intuition.

Before I ask for a micro perspective I seek a macro perspective.

Before I ask for the general rule, I ask for a specific example.
Or, before I ask for the abstract, I ask for the concrete.

In this case, I’d like to see students freezing the picture to see if they can see what’s happening in a specific (hopefully easy) case, like red dot on (5,5), or (5,0). (And, following the link to the original lesson, it looks like that’s already built in.)

This is a little like Jamie’s, but also:
Before I ask for the literal, I ask for the analogy.

In other words, what is this behavior like? Have we seen it before?

“Before I ask for a solution, I want to ask students to guess and check.”

I’ve no problem with the others, but I have strong reservations about this one.

When I teach logs or solving linear equations, for example, I really, really do not want them to guess and check. So much so that I often write the answers up as I write the question.

I explain that what they are learning is a step upon which they will build later. But only if they can do it properly. Many students stop once they get an answer, so not only do they waste time guessing and checking, but it allows exactly the most resistant ones to think they can then stop as they have the answer.

Quite early on we need to get away from thinking algebraic maths is about “the answer”. The important skill is being able to write the equation for a problem. Solving is usually the easy part.

Personally I would happily remove the concept of “guess and check” out of school Maths entirely. The concept of estimating is hugely useful, but wasting time on getting a closer answer appeals not at all. What are they learning as they plug repeated numbers in until they get to 3 s.f.?

The arithmetic before algebra is also dangerous at high school levels, pretty much for the same reasons. It leads to the idea that you can “solve” x^2 + 4x = 12 by writing 2×2 + 4×2 = 12, so x = 2. At least where I teach x = 2 is not the solution to that problem because it omits x = -6.

This is really interesting and I don’t know if it is because I am immersed in the elementary world that the informal feels more accessible than the formal in most cases. Even in that informal language, however, I find it tricky to be mathematically accurate so the move to formal language doesn’t seem contradictory or muddied for those who don’t need that extra obstacle. I am not saying I do this well, but it is something I am becoming more aware of as I work with middle school curriculum. For example, before students have the word congruent…is it the same figure? is is a copy? are they identical copies? is it a figure that can be placed on top of the other one and fit exactly? It then becomes about the shortest, informal way of saying something without having to state the entire definition.

Anyway, you asked about the graph above, so here are a few I thought about…

Before I ask for quadrants, I ask for directional language.
Before I ask for coordinates, I ask for locations.
Before I ask for absolute value, i ask for distance.
Before I ask for vertices, I ask for points at the corners.

Would be nice to add a prompt as a third step (before finding the algebraic rule) asking students for ONE aspect the two points have in common. Or, as suggested by l hodge, have them figure out whether they can be in the same location at the same time.

Hi Dan,

I assume you are familiar with the Van Hiele model for learning geometry (see https://en.wikipedia.org/wiki/Van_Hiele_model).

Based on that list, if it is correct, then there is a step before conjectures.

Before I ask for conjectures, I ask what do you notice?

Also, based on this list there is a distinction made between deductive reasoning made at the high school level and the formalization required by mathematicians. One could argue that at the high school level, assumptions being made are not always being made explicitly, and that this is a key difference with mathematicians who have designed different sets of axioms on which everything else can be built up.

Before I ask for rigorous proofs, I ask what assumptions are we making here?

My geometry students were often confused when we got to proof-not by formal proof itself, but rather because they saw it as no big deal, in contrast from what they had heard from older siblings, cousins, etc. They expressed it as having gotten so used to being asked how they knew what they knew, and how they knew what they didn’t yet know that might be useful (and why), that proof was more a matter of formatting than anything else. So perhaps before I ask for proof, I ask for whys/hows… what Lani refers to as “say your becauses”

RE: “What’s my rule?” [Comments 15, 17]

I sometimes play a variation on this called “Guess my rule & Guess my mistake!”

A couple of preservice teachers in one of my classes were doing a standard presentation on “Guess my rule” as a way to broach linear functions, but inadvertently miscalculated.

I thought it was a great way to explore mistake making; for example, when you want to guess linear functions, any 2 input/output combos uniquely define the rule (function). I would consider this game “pretty well understood” if folks realize that any two points uniquely determine a line, but play using 0 as the first input to find the y-intercept, and 1 as the second input to compute the slope. (What about 0 and -1? And how is the equation for a line being written? And…)

Okay: Suppose I may make exactly one error. How many input/output combos do you now need to be sure that you’ve guessed the right linear function?

(It’s no longer 2. Is it 3? 4? More?)

And yes, for those who favor generalization: One may ask analogously about an nth degree polynomial for which up to k errors are made. But I think the line game alone (with 1 or 2 errors) is already a pretty rewarding activity!

Dan — that EDC use of “guess and check” isn’t what I would call “guess and check” at all.

It’s putting numbers in to see how something works. In some cases I might even do a similar thing with a class. (The bit at the bottom about students making mistakes with variables that they wouldn’t with numbers is certainly spot on the money.)

But the statement posed was “Before I ask for a solution, I ask students to guess and check”. That’s a whole different kettle of fish.

At least where I come from, guess and check means going through a cycle of numerical calculations to get closer and closer to the actual answer. That process, to me, is inimical to getting students to move past arithmetic solutions and on to algebraic ones.