Pattern Stories Droujkova Moss 2010

8/8/2019 Pattern Stories Droujkova Moss 2010

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Pattern stories

By Maria Droujkova and Joan Moss, 2009-2010

The Power of Pattern .......................................................................... 1Part 1. Invite a kid to make a pattern out of 10 or fewer counters; whendone, ask to make the pattern grow...................................................... 1Part 2. Ask kids to make the next growth step POSSIBLE to guess. Then,make 100th step EASY to guess. The dip vs. the dead end. ...................... 4Part 3. Out of the dip! ......................................................................... 5

Solution 1. Attenuating the complexity of the example space................. 6Solution 2. Programming .................................................................. 6Solution 3. Amplifying individual knowledge to match the complexity ofthe example space. .........................................................................14

Part 1. Invite a kid to make a pattern out of 10 or fewercounters; when done, ask to make the pattern grow

This sets up one example that, eventually, will grow into an example space (John Mason).

What we see: discrete, countable shapes the idea of growth figural shapes kids name after objects ("tree pattern, star pattern") unpredictable growth

non-orderly complexity no symbolization no analysis of pattern's parts

It's important to start from kid examples. If a grown-up initiates the example space with herown patterns, they will be informed by formalized stages. We need to see what questionsnovices ask that masters don't.

Many early patterns kids make have chaotic and unpredictable growth. We later use theseas non-examples of predictable growth. Non-examples make an important part of theexample space.

Kids like this activity: it's a self-initiated and self-supporting, free-play childhood game,probably "psychologically necessary" (like playing dolls, playing house). Well-lovedcommercial games: peg mosaics, wooden peg boards, fuse beads.


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Growing Yin-Yang, Grace, 12"I want to make it swirl around and move in spirals. But I can't make counters do that."

Growing baby chicken, Abbey, 11

"I will make something cool" Dasha, 11Complexity over predictability


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More complexity examples Michael, 13, Carol, adult (math club)

Part 2. Ask kids to make the next growth step POSSIBLE toguess. Then, make 100th step EASY to guess. The dip vs.the dead end.

Some researchers (H.Bass, Radford, Riviera, Becker) observe patterning activities coming to

the dead end when the question of closed formulas comes up. I found several ways aroundit, and so did Joan Moss. We believe this is the dip, not the dead end.

What we see: Sequences: consistent and predictably growing shapes Kids going back to fix the start of the pattern for predictability Break of predictability/pattern for roleplay or aesthetic reasons; drawing, words, or

other representations supporting consistency Between-reasoning, local reasoning (predictions from step to step) Within-reasoning, global reasoning ("closed formula" predictions within each step) -

PROBLEMATIC ("dead end" of patterning) Describing sequences and predictions in words, tables and formulas Abstract or figural (object-named) patterns

Gestures expressing predictability, e.g. preparing enough counters for the next step Transition from qualitative to quantitative, especially in younger kids

Kids may go through several chaotic, mostly random steps of growth before, during andafter predictable steps. These episodes can be:

Anomalies at the beginning steps as the kids try for predictable growth Disintegration of goal-oriented activity into free play Aesthetic reasons ("shape needs to look right") overriding patterning at the end


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Hypothesis: From kids' point of view, mx+b growth pattern - what Joan Moss's kids called"the bump" - may like "the first growth step is different from the rest." Striving to make thepattern predictable, kids may spontaneously remove that "bump" to simplify the patterninto mx. Picture from Joan:

Kids sometimes see the pattern mx as predictable and the pattern mx+b as broken/unpredictable! They don't generalize is as a valid functional relationship.

This is an activity that supports local reasoning, that is, thinking about pattern "growthsteps" two by two. Inviting kids to skip (to 10th, 100th step) leads them into proportionalreasoning. Kids would comment, for example: "Because 100 is 10 times more, the resultwill be ten times more." However, when functional relations aren't multiplicative (e.g.mx+b), kids get lost, since multiplicative proportion does not work. This is one way patterns

"dead end" - how do you move even to mx+b, let alone to the general functionalrelationship idea?

Part 3. Out of the dip!

There are several ways out of the "patterning dip." Here are some we found.


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Solution 1. Attenuating the complexity of the example space

Follow kids' natural inclination and MAKE their proportional guess work well by infusing rightexamples into the example space.

Example: Joan suggested ant's swimming pool and other designs that have direct

proportions.Maria: "Let's trim antennae off this caterpillar" or "Is the change always the same?"(pointing to the start)."Can you make a pattern where 10th, 100th, any step is easy to make?" Note that this isstill local reasoning: kids read it with the stress on any ONE step vs. the rule

Solution 2. Programming

When children program their steps, it forces formalization, as computers only understandformal languages, and consistency, through automatic repetition. Pointing out features ofthe program is also a very straightforward way to clarify the task or "reveal constraints

quickly" (Dan Meyer).

Maria, clarifying the task: "Program a spreadsheet with row (closed) formulas rather thancolumn (recursive) formulas." This does not resolve the "dead end" feeling about going frommx patterns to others: cognitive leaps are required, rather than gradual learning.

Both these solutions are "similarities over differences" - but not at the level of similarity(concept) that is enough to promote global reasoning (hypothesis).

Examples for Solution 1 and 2

Georgia, 10
http://blog.mrmeyer.com/?p=7950http://blog.mrmeyer.com/?p=7950http://blog.mrmeyer.com/?p=7950http://blog.mrmeyer.com/?p=7950


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S1 is not visually like the rest, so G. discards it, but leaves the numbering from s2, whichcomplicates her formulas. Here is what Georgia did for black counters (Maria wrote down herwords):

Caterpillar pattern, Abbey, 11Abbey programmed recursive formula and dragged it down to fill a column to 200. However,moving toward closed formula did not happen. We contemplated beheading the caterpillar, thenjust clipped the caterpillar's antennae to two counters, producing 2x closed formula for blackcounters, which Abbey could easily do.


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Coleson, 5, planning to make some leaves grow

Maria: "Are these leaves the same?" Questions about sameness produce predictable patterns.


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Delta, 5, joins in making consistent leaves. "The sun makes the plant grow!"

"Now we will add the top and roots" (roleplay over patterning).


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Abbey, 11, expresses predictability by grabbing three counters at once. Preparing enoughcounters for the next step is typical. Use this for questions: if kids can't answer "what is next?"questions from paper representations, often they can "prepare enough counters for the nextstep" with physical counters.

Yellow notes read "-1, 0, 1, 2, 3." Abbey started from "making the rest of the shape predictable- if I make one leg, you can predict the other leg." Then she made people grow, starting from1. She initially named them "family, kid and parents" but then renamed "kid, teen, grownup."Then she added the figure labeled 0 and then -1. Labeling came afterward. This is transitionalfrom figural to abstract, from qualitative to quantitative.


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Growth-within (figural - arm by arm, leg by leg) and growth from figure to figure. Chart, word,formula representations.


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Dasha, 11

The coin shape was somewhat "wild" and unpredictable, and the task of placing coins trampedthe task of pausing after each "growth step" to take a picture, or the task of making sure stepswere predictable/similar. On paper, though, steps grew orderly, and Dasha spent significantattention on counting and re-counting and making sure everything is "just so." Several kidswere more orderly with paper representations.


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Solution 3. Amplifying individual knowledge to match the complexityof the example space.

Hopes and dreams: Coordinating between sequences of values, and functions Generalizing sequences to split "between steps" e.g. f(1/2) - this can potentially

lead to limits; or go into negative steps Coordinating between sequences and series A taxonomy of sequences/series/functions (e.g. linear, quadratic, exponential) Coordinating between recursive and closed formulas From series to infinite products (fractal patterns lead there)

How can we help kids to reason globally, and to build concept structures thatinclude a variety of functional relationships - mx, mx+b, x^2, Gauss's Formula(1+2+3...n - triangular numbers) and so on?

The manipulatives are promoting addition of items one by one. It's difficult to operate with

constructed structures as a whole. Working with a software tool that allows arbitraryselection copy and paste will show more global reasoning.

Hypothesis: Bump patterns mx+b don't provide enough examplehood, because they aretoo close to mx patterns. Kids see mx+b as broken mx patterns, sensing they belong tothe same class. Some kids arrive at the concept of linearity through pattern itself or itsrepresentations.

Joan's perimeter problem is a neat transitional step: thought the formula is linear,reasoning may not be


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Task idea: Make up a closed formula and then make a matching pattern out of counters.What this does: kids who formalized the "step formula" (closed, not recursive) idea with mxpatterns now create more examples at this formal level, and return (fold back) to patternsfrom this perspective. The minus is that pre-literate (or maybe pre-formal) kids can't takethis route. I don't know how to do this with my five year olds. Maybe they can make up apattern that's not "like all those" mx patterns, or "grow very differently." I did squarepatterns with the 4-5yo math club. They sensed fast (non-linear) growth with many excitedexclamations.

Pattern Stories Droujkova Moss 2010

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