Posts tagged flow channels
Taxophilia
Sep 16th
I went and looked up math words for all the richness of creation from the Tuesday math club. Precise and deep language is a math value for a reason. Once we name something, it becomes a placeholder for a collection, a class, a group of similar objects. “This is a star.” This stage is not math yet, actually. It’s a dream of math to come.
Math happens when we notice similarities and differences. When Cindy made a 5-point star, then we observed it’s one continuous line (making it a knot) and then she made a 7-point star that was similar. And I made a 7-point star that was puzzlingly different. Wolfram claims they are 7:2 and 7:3 stars. Makes sense, now that we made them. But I hesitate to tell the kids yet! Maybe they will discover other things?! This is math proper. You purposefully create differences, keeping similarities, and observe what happens. Delta varied the size of her straws and made smaller and smaller stars. Carter built a smaller pyramid within a larger one. Ben and his family made pyramids with different polygons for bases: triangle, square, pentagon. Caedmon and Nannette kept a 3d pattern of attaching tetrahedrons to one another in a certain way, until a shape emerged – a deltahedron, we called it Caedmon’s shape.
There are layers and layers of noticing to be had. We need to return to activities again and again to reach more layers. That’s why geeks are often told, “You have too much time on your hands!” when an outsider realizes how much time is spent with a single activity. There are riches to be had ONLY if you spend the time, though.
Should we tell kids standard names, taxonomies, theorems? Yes, but! The dialogue with the past mathematics is a delicate matter. Kids are still finding their own voices. The sheer bulk of the past can easily overpower and silence them. My goal is for kids, by the time they dig up something like Wolfram’s star polygon taxonomy, to have made up some of their own names, taxonomies and patterns. I want kids to feel that theories they look up are sisters of theories they make up.
To accomplish this, we develop a certain rhythm of activities. Free exploration comes first, including “the naming of cats” by kids, and it’s a relatively slow stage. Then we need to spend enough time with the topic for kids to notice and to formalize some properties within it. This is tricky to accomplish, and many supposedly “math activities” never bother, either stopping at explorations or skipping to formalization. Notation, photographs, drawing, stories, videos, spreadsheets and other representations come into play, because they promote noticing. Kids can look up other people’s work when they first name objects, or when they notice and name patterns. At this formal stage, when kids compare and contrast different patterns, they are open to looking at quite a few of patterns created by others (existing math), without being overwhelmed. That’s where we can pass on some content from the past generations.
Problems and their camp followers
Sep 7th
What is a problem for one person can be a puzzle or an exercise for another. A smart teacher can turn an exercise into a problem by being “less helpful.” Many curricula and educators also work hard on turning all problems and puzzles into exercises by attaching a step-by-step guide to every one, and by formalizing math before solving as such begins. Lockhart comments on the results: “I’m not complaining about the presence of facts and formulas in our mathematics classes, I’m complaining about the lack of mathematics in our mathematics classes.”
Video: Dan Meyer on being less helpful.
- Problems are mathematical questions for which the solver has no readily available methods of solutions, but has ways and the intrinsic intellectual need to figure them out. In particular, solvers do not know all mathematical concepts or formulas they will use.
- Open-ended problems have multiple correct solutions, often infinitely many of them. The correctness of open-ended problem answers is determined by the mathematical qualities of the solution, such as rigor, logic, definitions, and aesthetics.
- Exercises are mathematical questions for which the solver knows what methods, concepts and formulas will lead to the solution.
- Closed-ended problems and exercises have answers known exactly ahead of time. For example, multiple-choice questions are always closed-ended.
- Puzzles in mathematics are similar to problems in that the solver does not know which methods, concepts and formulas to use. The difference is that problem solving involves eventually developing a mathematical method for solution, whereas puzzles require intuition, finding a trick, or guessing, and may not involve any methods at all. There is no clear line between puzzles and problems, and most collections have a mix of both.
Relationships between problems and puzzles are complicated. Consider the nine-dot puzzle, a classic. Join all nine dots, using four straight lines, without lifting your pencil from the paper.
The difficulty of this puzzle is psychological: most people assume that lines have to start and end inside the assumed square. Once solvers start to experiment with lines that go beyond the square, the solution usually presents itself soon. There is no general math method involved. Unlike another classic, “the wolf, the goat and the cabbage,” or the nine-coin puzzle that open the door to the whole class of interesting math problems and investigations, the nine-dot puzzle stands by itself.
The nine-dot puzzle helps to make a powerful problem-solving point about assumptions. Puzzles may have a powerful role in mathematical problem-solving, similar to “the Mozart effect.” Puzzles support mathematical values and develop the mathematical sophistication, on a meta-level. I do not consider puzzle-solving the same activity as problem-solving. Evidence: quite a few people like one and hate the other, or are good at one and not the other.
What to make of pastimes like Sudoku and KenKen? Their combinatorial complexity quickly overwhelms human processing capacities, such as memory and attention. Human solvers are forced to develop and use intuition-based strategies, which is a puzzle trait. Yet there are also consistent, formulated strategies and rules solvers develop and use, which is a problem-solving trait. Finally, individual steps are routine exercises. The balance between puzzle, problem-solving and exercise traits in Sudoku, in particular, hits the sweet spot for millions of people. Chess and several other abstract games are similar in their balance.
I have a gut feeling, not yet supported by readings or experiments, that there is some sort of 3d flow channel among problems, puzzles and exercises. Each person has particular needs for the balance between the three to develop mathematically. The balance shifts with time. I have no idea how to visualize 3d flow channels.
Its own context?
Aug 31st
The following is a part of the Flow Channels brainstorm in the Natural Math wiki.
Paul Lockhart, in his “Lament” (the book version) talks about the importance of math as its own context, without trying to motivate it through any applications. I think it’s not about applications – it’s about metaphors! You can use “math as its own metaphor” so to speak, or approach it non-metaphorically, as autists supposedly do, which works great – for those who are willing to, and can, follow that path. In my observation, a tiny minority of people self-initiate or choose this self-contained approach to math, given the choice among multiple contexts as metaphor sources.
Here is how metaphors work. At first, there is a single entity, which only retrospectively can be named a metaphor. For example, a kid can think that division IS fair sharing. People frequently feel uneasy and even offended if you call their metaphors “metaphors” in this stage. It’s a defense mechanism, allowing the metaphor to support enough “roleplay” for the person to develop rich images that can later sustain formal structures. Forcing early formalization breaks the play, disrupts the natural learning rhythm, and leads to despondent feelings average math classes currently invoke.
If we manage to sustain rich image making in the all-important early stage, and gently help students move on to noticing some patterns in what they do, the patterns become math, and the metaphor turns into a simile. Patterns at this stage can just as easily become science, philosophy, or any other pattern-based discipline, but let’s focus on math patterns for now. In our example, once the kid starts noticing, say, that you can’t share certain quantities fairly without splitting, she is doing math, namely division, and sharing becomes LIKE division. When the context of sharing becomes unimportant (though the vocabulary may remain), and the focus fully shifts on quantities and their properties, the metaphor “dies” (Lakoff) and the new math structure, now self-sustained, is born.
To recap the life of metaphor in the context of the model of mathematical learning created by Pirie and Kieren:
- Metaphor promotes Image Making and supports Image Having
- Metaphor turns into a simile during Property Noticing, when its source and its target visibly separate
- The source of the metaphor dies, and the newly born math structure stands alone, in Formalizing
If you happen to love a context other than math – dragons, marine biology, car racing, fashion design – using it as a source of your math metaphors can be as powerful as using math as its own context. However, using math as its own context allows for mathematical elegance and depth not available otherwise. It has to happen, as well.
Bonus: Madison’s poem
know what i figured out?
the meaning of words isn’t a fixed thing!
any word can mean anything!
by giving words new meanings, ordinary english can become an exclusionary code!
two generations can be divided by the same language!
to end that,i’ll be inventing new definitions for common words, so we’ll be unable to communicate.
don’t you think that’s totally spam?
it’s lubricated!
well, I’m phasing.
marvy.
fab.
far out.
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