Structure of Solving Visual Pattern Problems, II

Let’s give this a first shot.

Following Lesh, let’s think about what needs to happen with a constant rate of change visual pattern in terms of the modeling cycle.

  • Description: Articulate a “way of seeing”/model that allows one to describe the growth through the given steps.
  • Prediction: Turn that “way of seeing” into a mode of calculation
  • Verification and Modification: See if the model works. Tinker.

There is important thinking, variation in strategy, and levels of sophistication at each of these stages. (Another yiddish word that we lost the chance to incorporate into the broader culture is chochmah. There is a chochma involved in each of these components.)

Description:

One way or another, the point is to identify what’s changing in each step. Here are some levels of this thinking, roughly inspired by CGI.

1. Imagining the action – At an early level of experience with these problems, it is important for students to imagine being able to “add on” the new elements of the step at each stage. Young students can be given manipulatives to act this out. This sort of thinking is, by definition, recursive.

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2. Relational model – Students begin to see growth that relates the step number to the shape of the pattern elements.

There is a ladder of sophistication that students show in their representations of a relational model. This is where, for many students on their way to algebraic models, the magic happens. At an early stage, students will fail to draw a picture that exposes their way of seeing. They might be thinking about the given stages of the pattern with an “acting it out” model and not making the jump into a relational perspective.

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Other students will draw out fully specified sketches of the “jump ahead” step.

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Other students make good use of a “block” model that doesn’t draw out each individual object, but instead exposes the groupings that they see in the pattern.

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Even students with fairly sophisticated representations are not sensitive to under- and over-counting in their models, counting in full groups of objects that “naturally” fall together in rows, columns, lines or other familiar shapes.

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With time, though, students become sensitive to compensating or overcounting and develop ways of adjusting their models.

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3. Algebraic relation – Students, with experience, will begin to selectively drop their visual models when determining “jump ahead” steps. They’ll sometimes very quickly abstract an algebraic model.

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At times, students will attempt this and result in error.

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There’s more to do, of course, since we haven’t talked about prediction or verification. But let’s identify in a nice list all the models above.

  • Acting it out
  • Relational Model – Detailed drawing of the step
  • Relational Model – Schematic drawing of the step
  • Relational Model – Unsensitive to over- or under-compensating
  • Relational Model – Sensitive to over- or under-compensating
  • Arithmetic Model 
  • Algebraic Model

I’ll end this drafty post by trying to draw out some of the feedback/hints/goals that each of these models suggests.

Goal come quickly: use of each of these models would serve as a perfectly sensible mathematical goal for a group of students that hasn’t attained them or their more sophisticated descendants. If my class has never worked with patterns before, maybe my goal is for them to solve visual patterns using, at least, an acting it out model. If I’m working with students that use relational models pretty well, maybe my goal for the day is for them to be able to use arithmetic models of the pattern.

Feedback and hints can help with this, and our articulation of the models can help us come up with sensible hints and feedback.

  • Acting it out

“Can you act this out with some bricks?”

“How would you build this?”

“Can you imagine building this next step with your hands?”

“Which blocks were added on?”

“If you’re not sure how a pattern is growing, imagine yourself building each step. Maybe you’ll realize something new.”

  • Relational Model – Detailed drawing of the step

“Show me a picture of this step.”

“Why does it make sense that this row is 5 long and 4 up?”

“What instructions could you write someone for drawing out this step?”

“Drawing a picture sometimes helps us get clearer about how we’re seeing the pattern.”

  • Relational Model – Schematic drawing of the step

“Do you have to draw each box?”

“Why does it make sense that you counted 10 in this row?”

“If you’ve got a big step, it can take a long time and be hard to count all those pieces. That’s when you might try making a drawing that doesn’t count each little thing.”

  • Relational Model – Sensitive to over- or under-compensating

This gets sort of close to verification and that’s a whole other set of thinking tools. I want to focus just on the model, though, and getting to this way of seeing a pattern.

“Why doesn’t it make sense to count this square twice?”

“When we’re working with patterns, we sometimes have to worry about counting things too many times. Can you find the part of your diagram that you counted twice?”

  • Arithmetic Model 

“What about the 89th step? Or the 103rd? Could you figure those out too?”

“Why does it make sense that this turned out to be 10+10 – 1?”

“If you feel that you understand a pattern really well, it’s often helpful to look for ways of calculating that only use numbers and not pictures, because that gets us closer to finding a rule that always works. Can you try writing this in terms of only numbers?” (I feel somewhat weird about this hint, but I think that it would be entirely appropriate, depending on our mathematical goals for the student.)

  • Algebraic Model

“What about the millionth step?”

“Can you write some instructions that anybody could use to calculate the number of squares in a step?”

“When you’re looking for an algebraic rule for a pattern, it’s often helpful to try calculating a few “jump ahead” steps, because we can often just tinker our arithmetic and end up with a rule. Try to find a rule that always work using this strategy.”

Like I said, first draft. I feel this is promising, but has issues that will become easier to notice on a reread.

The Structure of Thinking About Visual Patterns

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I’m planning a lesson for 3rd Grade right now. We’ve been working on making representations of patterns using graphs and models, and as part of that work I’ve been asking them to find “jump-ahead” steps, e.g. the 43rd or 21st step or whatever.

Their work on Friday was great to see. In particular, there was a controversy over the number of total squares in the 21st step. Some students drew attempts at accurate diagrams that they miscounted or misdrew, or that somehow didn’t capture the structure of the pattern. Other students reasoned without trying to draw the 21st step, and they often made misgeneralizations.

The most accurate students tended to do a particularly sophisticated thing: they drew schematic diagrams of the highest step that accurately captured the structure of these patterns.

Coming in to today, I’m thinking about making explicit two important features that distinguish the most accurate and efficient work on this problem in my 3rd Grade class from the least.

  • Seeing The Structure: The strongest work articulated a productive way of seeing the pattern and then applied that to a higher step. That productive way of seeing it involves thinking about how the number of WHATEVER is related to the step, e.g. “There are 2 columns in the first step, then there are 3, then 4” or “There are two rows on top of each other and at first the grey are just 1 long, then 2 long, etc.”
  • Drawing a Schematic, Rather than Accurate, Picture: The strongest work also used a schematic representation of the jump-ahead step rather than trying to draw an accurate picture and count it.

If I knew what productive thinking looked like for “jump-ahead” work with visual patterns over a wide-span of grades, I could have a bank of mathematical goals, feedback, and hints to give students. Further, I could tightly integrate goals, feedback and hints and really focus my instruction in productive ways.

I’m about to head into class. I wonder what other productive mathematical moves I can find in their work today. I also think that explicitly articulating these important features of the strongest work will help close the gap in the student work. I’m OK modeling it, though I think I could also draw their attention to these features by analyzing some excellent student work.

Maybe articulating the structure of student thinking about visual patterns would be a helpful first mini-project to take on for me?

The First Time I Thought About Problems On Twitter

At this point of things I was (apparently) thinking of “engaging” as a stable property of a problem. In practice, that meant that I spent a lot of time searching for and trying to create great problems.

These days I don’t think of “engaging” as a stable property. The degree to which a problem is engaging depends on everything else — the preceding lesson, whether I launch a problem in a way that draws connections to things that kids already know, if I have hints/feedback ready in advance of the lesson for some common issues, if I selected a task that has a clear goal and that relates to math that my students are working on. This means that I spend more time worrying about what my kids know and what math my students might learn from an activity than I used to.

How did this development happen? I’m wondering whether my twitter archives might have part of the answer.

Minimalistic Hints Are Still Hints (Or: If You’re Helping, You’re Doing The Work)

This post is an argument that there isn’t much of a difference between questions and non-questions when it comes to hints.

Well, wait, hold on, that’s dumb. There are lots of important differences between questions and non-questions. I should focus up my claim.

Claim: Whether you ask a question, make a request or just say something to a student during a problem-solving session, you’re doing mathematical work for that student.

Argument:

Back on my old blog I wrote a post about the mechanism through which feedback can impact learning. I took my cue from Kluger and DeNisi and identified two mechanisms:

  • Locus of Attention – Feedback can draw the student’s attention towards something that matters for their learning.
  • Motivation – You can ask a student to do something that they otherwise wouldn’t do.

I think that this is true of teaching more broadly. I think that everything that a teacher does for learning can probably be dropped into either the canister marked “redirect attention” or the one marked “motivation.”

What we pay attention to matters so much. There is a mathematical way of seeing and noticing.  Our ability to focus our attention in mathematically productive ways is a sign of problem-solving expertise (Schoenfeld 1992).

When a teacher asks a question of a student, they are (inevitably) redirecting that student’s attention to something that they aren’t currently thinking about. To take an example from Max, if you ask “What do you notice about this?” you are redirecting a student’s attention back towards the scenario, asking them to see things with fresh eyes. A more experienced problem solver probably wouldn’t need this question. Per Schoenfeld, that student would know that it’s time to go back and redirect her attention to noticing things about the scenario.

Annie wants the student to do all the mathematical work, but she also wants to redirect their attention in productive ways:

As a sometimes coach in a wide range of schools/districts/populations, I don’t think that I work with students that are so unusual, and I find that when a kid asks me a question or tells me they’re stuck, probably 19 times out of 20 (or maybe more like 49 out of 50, though my colleagues would probably guess at 99 out of 100), what I say is, “Tell me something about the problem.” That’s it. There are small modifications – if they asked about question 3, I’ll say, “Tell me something about question 3.” Or, if I’m feeling radical and they’ve actually done some math, “Tell me about what you’ve tried so far.”

I added that emphasis there, because this seems like a great example of a bit of important mathematical work that experts often do but that our students often don’t. Going back to Schoenfeld, this isn’t just a confidence or a motivational issue. It’s a marker of a certain kind of mathematical expertise, perhaps teachable (though perhaps not so easily at all).

The world of math education is full of strong distinctions and lines in the sand. There are GOODWORDS and there are BADWORDS, and also GOODTHINGS and BADTHINGS. A GOODTHING is a question that asks students to think about their own thinking. A BADTHING is a hint, even one that amounts to asking kids to think about their own thinking. (e.g. “It’s often helpful in moments like these to just ask yourself what you notice about the scenario. Make a list, even.”)

I take from Dewey (and the other pragmatists) a “rejection of sharp dichotomies.” Everything that we say to a kid alters the locus of their attention, hopefully in a good way, and we need to own that if we are to understand it and wield that power productively. Minimialistic hints, if they help learning, work for the same reason that any sort of hint could conceivably work: you’re doing mathematical work for the student that sets them up to think about something valuable.

And this is pretty much all we can do.

How To See It

“Our interpretation of Polya’s heuristics is that the strategies are intended to help problem solvers think about, reflect on, and interpret problem situations, more than they are intended to help them decide what to do when “stuck” during a problem attempt.” (From “Problem Solving and Modeling”)

This interpretation makes sense of Polya’s stated lack of interest in providing helpful, specific advice. I could buy it.

EduSpeak and GOODWORD/BADWORD Dichotomies

I’m interested in the language we use to talk about teaching and what that says about the job and the profession. One of the paradigms that I’m most sure exists is the GOODWORD/BADWORD dichotomy that shows up throughout education. Tonight I was chatting with a teacher who made a strong distinction between EXPLORATION and PRACTICE. Fair enough. But the way she described exploration was as a chance to try out the ideas that have just been directly instructed. Wait, that sounds a ton like practice! What’s the difference? “Practice is repetition for fluency. Like shooting free throws. Exploration involves making connections and testing theories.”

It’s hard for me to pin down why exactly I find this interesting. Maybe it’s that, to me, the dichotomy seems absolute but the distinction doesn’t deserve it. As defined above, the difference between exploration and practice seems to be that exploration is just good practice (or practice is just bad exploration). Anything that makes a classroom exercise dumb, boring, rote, routine, meaningless…YES that’s exactly what we mean by “practice.” All the good stuff is what we mean by “exploration.”

(Note that, as defined, we can’t say that “exploration” is a type of practice, since practice is defined to have all this negative baggage attached to it.)

Some more GOODWORD/BADWORD dichotomies:

  • HINT/STRATEGY
  • PROJECT/ACTIVITY
  • PROBLEM/DRILL

There is well-known ambiguity surrounding use of “problems” and “problem-solving.” Schoenfeld seyz, “problem solving has been used with multiple meanings that range from ‘working rote exercises’ to ‘doing mathematics as a professional.'”

I wish that I could piece all of this together, but I can’t.

Questions:

  • Why do so many technical terms in education end up fitting the GOODWORD/BADWORD paradigm?
  • In writing about teaching, is use of the GOODWORD/BADWORD paradigm productive or confusing?
  • What are some of the terminological sensitive points that surround problem and problem solving (besides for the ones that I’ve already found)? What do they say about how we think about problem solving?

Links to GOODWORD/BADWORDS examples:

Specifying Exercise: Chop a Shape Into Simpler Ones

Parent Strategy: Solve a Simpler Problem

Geometry Strategy: Chop a Shape Into Simpler Ones

Slightly More Specific Geometry Strategy – Keep An Eye Out For Simpler Shapes, Especially Right Triangles

Area Version #1: If you’re trying to find the area of a complex figure, it’s often helpful to chop it into shapes whose areas you can more easily find.

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Area Version #2: If you’re trying to find (or remember) the formula for the area of a type of shape, try cutting and rearranging that shape’s pieces into a more familiar one. In fact, there are two cuts that often help: cutting off a right triangle and making a midline cut.

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This has a natural extension to limits, as we pass to the infinite.

Trigonometry Version: When you’re trying to find missing sides or angles of a polygon, it’s often helpful to chop that shape into triangles and to chop that triangle into right triangles.

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Quadrilateral Proof Version: If you’re trying to prove (or remember) something about the sides or angles of a quadrilateral, it’s often helpful to draw one or both diagonals and divide the quadrilateral into two (or four) triangles.

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Volume Version: Much like with area. Version 1 says, try to chop up the volume of a complex solid into simpler solids you can tackle individually. Version 2 says, if you’re trying to derive or remember the formula for the volume of a solid, try to chop and rearrange its pieces into a box. There is a natural extension of volume to limits, as we pass to the infinite. (Cavalieri’s Principle)

Auxiliary Lines Version: If you’re stuck on a geometry problem, try adding lines that create right triangles, equilateral triangles or isosceles triangles, because you can often use their properties to illuminate other aspects of the problem.

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Questions:

  • How does proof by dissection fit into all of this?
  • Is this a helpful analysis?
  • How would I use this during the year in the course of my planning? (I think this could help me focus on better mathematical goals, give better hints and identify better things to talk about in whole-group discussions. that’s my prediction, anyway.)

Chop a Shape Into Simpler Ones

I think this needs to be the big practical project that I try to tackle. If I think that this habit/strategy/hint/whatever is too big to help (unless it’s somehow been specified for you) then it’s time to show exactly how complex and under-specified it is.

In doing this I’ll, for now, limit myself to a high school geometry class. And I’ll limit myself to studying the CME Geometry text, which I’m pretty familiar with. I’ll look towards my own problem-solving in geometry to support the search. After all is said and done, we’ll start looking to extend this throughout other grades to show how this idea develops.

The question is, what are some of the more specific versions of “Chop a Shape Into Simpler Shapes” that help students?

Some quick thoughts:

  • Trig – Dropping an altitude in trig problems to create two right triangles is a crucial move that needs to be made explicit.
  • Proof – It’s often helpful to add the diagonals to a quadrilateral and chop it up into a series of triangles. This often helps by letting us find some similar or congruent triangles. To prove the Isosceles Triangle Theorem it helps to drop an altitude.
  • Area – This is an important subcategory. It has a few subcategories of its own. Make Right Triangles: Making right triangles in an equilateral triangle or regular polygon to find its area; making right triangles by dropping altitudes in triangles or trapezoids or parallelograms in order to cut and rearrange their pieces into rectangles/triangles/parallelograms and find their areas; cutting circles into triangle slivers. Chopping a complex shape into pieces whose area formulas you know is another. This is helpful for proof by dissection.
  • Volume – Cavalieri’s principle, and natural extensions of lots of the Area stuff
  • Advanced Geometry – A great deal of auxiliary lines problems are chopping problems. It’s often a good idea to chop a shape into an isosceles triangle or a right triangle if you can. (This was a crucial move in GoGeometry #8).

When else?

Experience, of course

There is a pattern here, the pattern of auxiliary figures, which has some promise and which we can describe as follows: Try to discover some part of the figure or some closely RELATED FIGURE which you can construct and which you can use as a stepping stone in constructing the original figure. 

This pattern is very general. In fact, the pattern of similar figures formulated in sect. 1.5 is just a particular case: a figure similar to the required figure is related to it in a particular manner and can serve as a particularly handy auxiliary figure.

Unavoidably, its greater generality renders the pattern of auxiliary figures less concrete, less tangible: it gives no specific advice about what kind of figure we should seek. Experience, of course, can give us some directives (although no hard and fast rules): we should look for figures which are easy to “carve out” from the desired figure, for “simple” figures (as triangles), for “extreme cases,” and so on. We may learn procedures, such as the variation of the data, or the use of analogy, which, in certain cases, may indicate an appropriate auxiliary figure. (Polya, Mathematical Discovery p.15)

Good for Polya for understanding that advice such as “discovery a related figure” is too general to serve as specific advice. As such, it would seem unlikely to be helpful advice for a student struggling on a problem.

What fascinates me is how little interest Polya seems to have in helping the reader grasp the intangible. “Experience, of course, can give us some directives” but he seems uninterested in truly specifying them beyond the bare tantalizing crumbs that he offers here.

What, then, is Polya’s interest? The book is titled “On Understanding, Learning and Teaching Problem Solving.” Why does he think that specifying things such as his “auxiliary figure” pattern is helpful for the reader, whereas “find simple figures” or “check extreme cases” can only be grasped via messy, pick-it-up-as-you-go-along experience?

The Plan: May-June

To Read:

Polya, George. “Mathematical discovery: On understanding, learning, and teaching problem solving.” (1981).

Krulik, Stephen, and Jesse A. Rudnick. Problem solving: A handbook for teachers. Allyn and Bacon, Inc., 7 Wells Avenue, Newton, Massachusetts 02159, 1987.

Schoenfeld, Alan H. “Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics.” Handbook of research on mathematics teaching and learning (1992): 334-370.

To Solve:

Work on more problems, write about the experience (what I learned, what I tried, how I failed, what I figured out, etc.)

Work on problems from Polya, write about the experience.

To Make:

Make a list of specific heuristics for a portion of a high school geometry class?

To Write:

Write a series of posts trying to make sense of Polya, Rudnick/Krulik and Schoenfeld’s ideas.

To Decide:

Whether to continue with this project.