In my last post, I showed the powers of 3 as a series of fractal drawings. I had lots of other doodles in my math notebook, so I spent the morning dotting little pieces of paper to share them with you.

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These could serve as exercises for a student wanting to practice using exponent notation, but they could also serve as decent stand alone problems. Something as simple as “How many dots?” maybe. Perhaps I’ll post some on 101qs.com and see what kind of questions they get. (I did.)

Problems and exercises can both be valuable, but they aren’t the same thing. Math students should know the difference between doing math in its most free and open form, and completing exercises to build familiarity, skill, and speed. I need to do a better job of including this in my teaching.

What do you make of these?

Perhaps you’re wondering “How much time did he spend drawing all those dots?

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Through school and work, I’ve been able to download two copies of Mathematica, and I LOVE it! It’s especially good for all of my mathematical art and imaging needs. It also has an exceptional documentation system with hyperlinks and examples that make it very easy to teach yourself and play. I’m amazed how much I’ve learned in two weeks.

Mathematica also handles recursive programming nicely, which is perfect for fractals. I wrote some code this afternoon and generated these images. Enjoy.

Now how long would THAT have taken me to draw?

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Max and I did some, “How many dots?” questions with our 3rd graders at the beginning of the year. It got them thinking about counting in groups and good ways of grouping for counting. They found it fun to eventually be able to count enormous bunches of dots using their grouping methods.

I find that I have the most trouble giving problems, not just exercises, in Algebra 2 class. So many topics there are: Learn this complicated procedure that you’d never figure out on your own, and now practice it. I often try to back exercises with a problem – mostly a “What’s the trend or pattern here?” sort of problem. But this kind of “problem,” if it counts as one, just doesn’t have the apparent richness or accessibility of, say, the Envelope-Flipping Machine problem we did in Modern Algebra, or even, “Count the dots!” There’s value in exercise – it hones the mental muscles – but exercise for exercise’s sake loses its pull quickly. I really need better Algebra 2 problems.

You bring up a good point Anna, and it got me thinking. The envelope-flipping problem was great for our modern algebra elective, but for you or I, or maybe even an undergraduate math major, it’s an exercise.

These fractal counting questions could be exercises for an algebra 1 class, but for a young enough mathematician, these could serve as very interesting stand alone problems that bring forth the entire notion of exponentiation!

“Problem” is in the mind of the beholder?

Next year we should do more talking about Algebra 2…

Pretty. Snowflakes. My geometry kids do some fractals on GSP. Really cool what you did.

I tell the kids doing exercises in textbooks/worksheets is like doing sit-ups and pull ups, which gets boring and tiring after awhile. But having a well conditioned body or a set of fine tuned skills will help on game day or in solving problems.

Glad you like them, Fawn. Somehow fractals are some sort of core component to our curriculum at Saint Ann’s. Bizarre.

You know, I’ve said the very same thing about practice and “game day,” but I ultimately decided that test day is not a good enough goal, especially at our school where students don’t get grades! It was a calculus student’s comment, actually, that made me rethink that, I’m proud to say.

@Fawn I wonder if “game day” actually ever comes for our students outside of school? What do you think? If it doesn’t, then why make them practice?

I think this may have to do with what we see as the goal/benefit of a mathematical education. When mathematical thinking has found its way into your natural sensibilities, “game day” is every day, perhaps. You look around the world and analyze. You are keenly interested, and wonder how things work. When you think mathematically, your mind is engaged in mathematical activities that require a richness of tools and experience.

I often feel my goal is just to amplify and develop my students’ mathematical sensibilities, and encourage mathematical thinking. Math students (and programs) should simply focus on becoming increasingly mathematical.

What do you think?

All of what you write I completely agree with. I think my issue comes with the whole practice/game day analogy that I often hear. Usually, this takes some variation of the form, “students must practice skills so that they can apply them or problem solve with them (game day).” This view implies that “mathematics” and mathematical thinking are separate entities. Not only is one separate from the other, but there is also the implication that one must precede the other. Or, stated another way, you must learn “mathematics” before you can be “mathematical.” This I just can’t vibe with.

My suggestion above was that much of the middle/high school math curriculum is a set of skills that are never applied outside of a classroom. Students are valid in their complaint of having to learn this. But, as you suggest, we all reason and make sense of things on a daily basis. For this reason, I can’t see why “mathematics” and “mathematical thinking” are (or should ever be) separate in our approach to teaching and learning mathematics. They are one. I would also suggest that this type of “mathematical thinking” is not necessarily something we need to cultivate in students but is, rather, something that the human mind does automatically in reasoning, sense making, and problem solving. (My post)

Unfortunately, many students have “unlearned” how to trust their natural intuitions because of the skill-based nature of math education.

stated another way, you must learn “mathematics” before you can be “mathematical.” This I just can’t vibe with.Me neither! Yuck.

I couldn’t agree with your second paragraph more.

Mathematics IS mathematical thinking. (period)

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@Bryan I meant “game day” literally for many of the kids who play in a sport and must practice during the week or coach does not let them play in the upcoming game. Athletes go through all sorts of drills to play football although no specific drill shows up on the football field during a game.

I wrote “But having a well conditioned body or a set of fine tuned skills will help on game day or in solving problems.”

Sorry if I wasn’t clear, but “a well conditioned body” goes with “on game day,” while “a set of fine tuned skills” go with “in solving problems.”

You’re right, from the way I said it, it might sound like one math entity preceeds another, but that can’t be further from the truth. I agree with you wholeheartedly that mathematics and mathematical thinking are one of the same, heck, it’s all math!

The day that my students complain as to why they must do the exercises, or “when are we ever going to use this?” is the day I need to leave teaching.