## Blue Point Rule

My biggest professional breakthrough this last year was to understand that every idea in mathematics can be appreciated, understood, and practiced both formally and also informally.

In this activity, students first use their informal home language to describe how the red point turns into the blue point. Then, more formally, I ask them to predict where I’ll find the blue point given an arbitrary red point. Finally, and most formally, I ask them to describe the rule in algebraic notation. Answer: (a, b) -> (a/2, b/2).

It’s always harder for me to locate the informal expression of a idea than the formal. That’s for a number of reasons. It’s because I learned the formal most recently. It’s because the formal is often easier to assess, and easier for machines to assess especially. It’s because the formal is often more powerful than the informal. Write the algebraic rule and a computer can instantly locate the blue point for any red point. Your home language can’t do that.

But the informal expressions of an idea are often more interesting to students, if for no other reason than because they diversify the work students do in math and, consequently, diversify the ways students can be good at math.

The informal expressions aren’t just interesting work but they also make the formal expressions easier to learn. I suspect the evidence will be domain specific, but I look to Moschkovich’s work on the effect of home language on the development of mathematical language and Kasmer’s work on the effect of estimation on the development of mathematical models.

Therefore:

• Before I ask for a formal algebraic rule, I ask for an informal verbal rule.
• Before I ask for a graph, I ask for a sketch.
• Before I ask for a proof, I ask for a conjecture.
• David Wees: Before I ask for conjectures, I ask for noticings.
• Before I ask for a calculation, I ask for an estimate.
• Before I ask for a solution, I ask students to guess and check.
• Bridget Dunbar: Before I ask for algebra, I ask for arithmetic.
• Jamie Duncan: Before I ask for formal definitions, I ask for informal descriptions.
• Abe Hughes: Before I ask for explanations, I ask for observations.
• Maria Reverso: Before I ask for standard algorithms, I ask for student-generated algorithms.
• Maria Reverso: Before I ask for standard units, I ask for non-standard units.
• Kent Haines: Before I ask for definitions, I ask for characteristics.
• Avery Pickford: Before I ask for complete mathematical propositions, I ask for incomplete propositions.
• Dan Finkel: Before I ask for the general rule, I ask for a specific instance of the rule.
• Dan Finkel: Before I ask for the literal, I ask for an analogy.
• Jim Murray: Before I ask for algorithms, I ask for patterns.
• Nicola Vitale: Before I ask for proofs, I ask for conjectures, questions, wonderings, and noticings.
• Natalie Cogan: Before I ask for an estimation, I ask for a really big and really small estimation.
• Eileen Quinn Knight: Before I ask for algorithms, I ask for shorthand.
• Bill Thill: Before I ask for definitions, I ask for examples and non-examples.
• Larry Peterson: Before I ask for symbols, I ask for words.
• Andrew Gael: Before I ask for “regrouping” and “borrowing,” I ask for grouping by tens and place value.

At this point, I could use your help in three ways:

• Offer more shades between informal and formal for the blue dot task. (I offered three.)
• Offer more SAT-style analogies. sketch : graph :: estimate : calculation :: [your turn]. That work has begun on Twitter.
• Or just do your usual thing where you talk amongst yourselves and let me eavesdrop on the best conversation on the Internet.

BTW. I’m grateful to Jennifer Wilson and her post which lodged the idea of a secret algebraic rule in my head.

Featured Comment

Allison Krasnow points us to Steve Phelp’s Guess My Rule activities.

David Wees reminds us that the van Hiele’s covered some of this ground already.

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. More here.

1. #### Avery

May 17, 2016 - 3:28 pm -

I really like how your “therefore” list, what educated guesses and/or estimates might look like in different contexts. One potential additional “shade” might include descriptions that are incorrect and/or are interpreted (because of imprecise language) as incorrect. I’m thinking something along the lines of “If you draw a line from the center to the red dot, the blue dot is always on the line.”

2. #### l hodge

May 17, 2016 - 3:34 pm -

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?

3. #### Mike Davidson

May 17, 2016 - 3:42 pm -

Before I ask one person to share their ideas with the class, I want everyone to take turns sharing their ideas in small groups.

4. #### Bridget Dunbar

May 17, 2016 - 3:52 pm -

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

5. #### Dan Meyer

May 17, 2016 - 3:55 pm -

@Avery, not really following here. You’re saying that student quote might be seen as incorrect because the language is ambiguous?

l hodge:

Did you experiment with hiding one point, then moving the other point, and then un-hiding to show the new location?

That’s effectively the second round where I show the red point, ask where the blue point is, and then reveal on the next screen.

Mike, I’m curious how you see the small group-shared ideas as less formal than the whole class-shared ideas. They both seem informal to me.

Bridget, thanks. Added to the post.

6. #### Elizabeth

May 17, 2016 - 3:56 pm -

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

7. #### l hodge

May 17, 2016 - 4:30 pm -

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.

8. #### Andrew

May 17, 2016 - 5:33 pm -

I’m a big fan of gestures: before it’s on paper, and potentially before there is a verbal description, show me with your finger/hand/arm what’s going on.

I’m tempted to say that most of the authentic (there’s a loaded word…) mathematical reasoning our brains do is necessarily in the informal register. At least within my own head, the formalization of “identify the two coordinates and divide each by two” is much too slow and involved for me to quickly spot the blue point, but splitting the distance between the point and the origin is a breeze. A good chunk of my internal reasonings are nonlinguistic, and it is only through the formalization process that I’ll introduce the words (or symbols) needed to communicate them. Incidentally, that is one of the things I love about Desmos: I get to take the formal interpretation of what I “see” or intuit about a problem and translate it back into a more informal view. And I can ask my students to do the same, which simultaneously assesses whether they can work in the formal and informal domains.

9. #### Sara B. Vaughn

May 17, 2016 - 5:40 pm -

Before I harness math I harness intuition.

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

10. #### Avery

May 17, 2016 - 5:49 pm -

Sorry I wasn’t clear. The statement “If you draw a line from the center to the red dot, the blue dot is always on the line” is true (the blue dot IS always on the line), but it is not telling me where on the line the blue dot lies.

Or, more formally, the statement is necessary but not sufficient.

11. #### Dan Finkel

May 17, 2016 - 7:10 pm -

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:

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

12. #### David Condon

May 17, 2016 - 9:24 pm -

Well, I’d say learning the informal expression is a key step in learning to use the solution fluently. If I have to stop to write down the problem in order to solve it, then that’s going to slow me down whenever I have to use the problem in the real world. The formal expression is more powerful in the sense that it’s more specific. The informal expression is more powerful in the sense that it can be arrived at more quickly.

13. #### Chester Draws

May 17, 2016 - 9:45 pm -

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

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.

14. #### Allison Krasnow

May 17, 2016 - 10:52 pm -

This isn’t exactly what you’re asking, but you should look at Steve Phelps’ work on GeoGebra Tube on this. He has a whole series which isn’t all on 1 nice link for me to share. Some of them are called “Guess my rule” and others are “What’s my rule.” If you haven’t seen his work on this, you may get more ideas. He has a few using complex numbers as well. I’ll try to circle back with thoughts on what you’re actually asking about tomorrow when it’s not midnight.

15. #### Kristin Gray

May 18, 2016 - 2:34 am -

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.

Before I ask for vertices, I ask for points at the corners.

16. #### Steve Phelps

May 18, 2016 - 4:17 am -

Thanks, Allison.

I have been playing around with these “What’s my rule?” ideas on the Nspire and in GeoGebra for a while now

http://ggbm.at/CMNJVztR

I have also explored “Color rules:” http://ggbm.at/Nmq9UHkX

However, I realized the problems when I had a color-blind student. Instead of color rules, you can change the point style.

17. #### Björn Beling

May 18, 2016 - 4:21 am -

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.

18. #### Ruth

May 18, 2016 - 4:40 am -

This post makes a lot of sense, especially as it is written for its audience: teachers. It is not written for an audience of postdoctoral researchers, for instance, in which case the language would be less accessible and more formal.

Similarly, math should make sense to the audience in our classrooms. Allowing and encouraging informal language and methods not only makes learning easier, as you state in your post, but it allows students to make sense of the concepts. Formalizing then becomes part of the learning process.

19. #### Dan Meyer

May 18, 2016 - 4:56 am -

Many, many thanks for all the contributions. I’ve added a number of them to the body of the post. I’ve also added Steve Phelp’s work to the post, as he seems to have covered this specific kind of task in a lot of depth. (Thanks for the link, Allison.)

Chester Draws has reservations about “Before I ask for a solution, I want to ask students to guess and check,” reservations which I’d buy if the point of guessing and checking were only to get a more precise answer. I encourage Chester and everyone else to look into EDC’s guess-check-generalize pattern. In their work, guessing and checking is used to reveal the structure of a problem, making its variable representation clearer and easier to use.

20. #### Mike Davidson

May 18, 2016 - 6:22 am -

Dan,

Sharing ideas within small groups lets them refine their ideas, more informally and in a more relaxed atmosphere. When someone is then selected to share, they have had a chance to refine their wording.

21. #### David Wees

May 18, 2016 - 7:21 am -

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.

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?

22. #### Dan Meyer

May 18, 2016 - 7:44 am -

David, I’m familiar with the van Hiele’s work, but my brain hadn’t yet looped their hierarchy into this post. Thanks for making the connection. Added to the post.

23. #### Peg Cagle

May 18, 2016 - 8:00 am -

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”

24. #### Andrew Sommer

May 18, 2016 - 12:38 pm -

25. #### Benjamin D.

May 18, 2016 - 6:03 pm -

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!

26. #### Travis

May 19, 2016 - 1:03 pm -

This got me thinking about estimating…
How about a Goldilocks Corral? Think N.E.S.W. layout and x-y grid.
Estimate ln(50)=? => 2.7^?=50

3^4=80 too high

2^5=32 < 50 < 2^6=64

3^3=27 too low

Goldilocks Diamond might also be apropos.

27. #### Chester Draws

May 20, 2016 - 8:08 pm -

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.

28. #### Roberto Catanuto

May 28, 2016 - 1:41 am -

Great ideas.

What do you do when you have to handle the unavoidable passage from informal to formal?

I mean, there is a time when a student “has” to make this leap. You can’t keep going in high school doing arithmetic all the time, or can you?

So how do you deal with the reluctant student who refuses to abandon the soft spot of informal for the more formal?

Great ideas again.

• #### Björn Beling

May 28, 2016 - 2:28 am -

As you said, it’s a passage, not a leap. Having students look for similarities between the two points might help smooth that transition. (e.g. “They’re always opposite each other”, “They’re on the same height”, “They’ve got the same y-coordinate”, etc). If students still struggle, why not have students write down the coordinates of some red-blue pairs, then see how they differ. Or you ask students to write down a rule that MIGHT be right (no pressure) and one rule they know is wrong. Then put them up and discuss why some of them don’t work. I haven’t tried it yet, but I’m fairly confident students will arrive at the formal end eventually.