One of my high-school age learners is working on chemistry with me this year.  We went through the standard high school curriculum up through the point of stoichiometry.  After that point, the curriculum branches off into topics like oxidation-reduction reactions, acids and bases, and so forth.  We found that picking up the topics in order was starting to be tiresome.  And so, as we are wonderfully spoiled with a non-traditional learning environment where we expect learning to be driven by curiosity and exploration, we started to wander in search of a better method of study.

What we hit upon was browsing around various resources, finding interesting topics and questions and doing a kind of guided cooperative research discussion.  I provide some background knowledge, we both contribute curiosity and questions, and we are both enjoying it and learning a lot.

Lately we’ve been investigating some basic organic chemistry, looking at the structure of alkanes (hydrocarbons with only single bonds connecting the carbons).  The textbook gave us a table of the melting points and boiling points of the different alkanes and we noticed something surprising.  The boiling points increase in a smooth logarithmic curve, but the pattern in the melting points is much more erratic and is not even always increasing as the number of carbons increases.  Here’s a graph he made.  Orange dots are boiling points, green dots are melting points.  The x-coordinates are proportional to the number of carbons in each alkane.

What’s up with those first three – methane, ethane, and propane?  And then look how they’re paired off for a while – butane close to pentane, hexane close to heptane, and octane super close to nonane.  So strange!  We’re curious and we’re investigating.

We found this excellent web application, MolView, that allows easy 3D modeling of molecules.  We think the different non-structural isomers of the various alkanes may explain the strange pattern in their melting points.

Anyone know the answer to our mystery?

I’ve mapped out in my mind the different stages I want to help learners move through as they learn a new math concept.  I think sometimes people jump straight to explanation and practice when I think there are a couple key stages that should come before those.

1.  Inspiration.  Before a learner starts in on learning a new concept, I want them to be feeling at least some curiosity about it.  Not only will it make the learning more engaging and more fun, but it will set them up for a better understanding.  It helps so much to have a good feel for what the question or problem really is about, before coming to understand a solution, method or concept that addresses that problem.  Dan Meyer talks about creating mathematical headaches to which mathematical tools are the “aspirin”.  Interest can also be kindled by giving some interesting historical context about how the concept was invented, or by finding an interesting problem that the learner can relate to but finds they are stuck without some additional mathematical tools.
2. Inquiry.  Before learners turn to explanations of the concept presented by others who already have an understanding, I want them to have an opportunity to struggle with the problem themselves.  It can be a brief struggle, if there’s limited time, patience or interest, but even if it’s brief I feel like the struggle is important.   The feeling of being stumped by a problem can make the explanation of the concept seem magical.  Getting stuck gives a feel for what is difficult about the problem, and allows insight into why the solution is so useful.  Once in a while this step can even be stretched into the learners inventing their own complete solution and explanation.
3. Explanation.  After a good struggle, even if learners have invented a complete mathematical tool themselves, it is important to think through some good explanations of a new concept.  It’s good to look at a couple different explanations and perhaps invite learners to reflect on which explanations make the most sense to them.  Since the concept is new I wouldn’t necessarily expect learners to be able to give a full explanation of the concept itself yet, but they should be able to tell when a concept “clicks” for them.
4. Practice. In my enthusiasm for the above three steps it is easy for me to under-emphasize the importance of practice in math learning.  It was again Dan Meyer (infinite font of teaching wisdom that he is!) who joked on the Vrainwaves podcast a while back that he wouldn’t want anyone who discounts the importance of memorization to operate a vehicle.  It’s a good point. Once a concept has been understood in a way that makes sense, the next step is to get it to where it’s easy so that the learner no longer has to think hard to do it.  This allows higher level thinking about the concept, connections with other concepts and makes it easier to apply the concept to challenging problems.  Sometimes a little practice is even needed before an explanation can be completely understood.
5. Mastery. To complete the learning journey a learner needs to make the concept their own.  This can be done by writing down a complete explanation or explaining the concept to someone else.  Learners can apply the concept to difficult problems that require a little creative thinking, or extend or generalize the concept.  At this stage it is great to ask lots of “Why?” questions and challenge learners to prove that the concept always applies (or analyze when it does or does not apply).

What do you think?  How do you view the journey of a learner?

I got negative feedback from a parent.  I ran across it unexpectedly.  The sheet with learner and parent feedback was sitting in a pile of papers and it caught my eye as I was digging around looking for something else.

“We, as parents were disappointed…”

My heart sank and my stomach clinched up.  Why am I so sensitive to one little piece of criticism?  I have been delighted over and over this year by the positive comments about me, my classes and my teaching.  Yet that one little comment rings so much louder in my ears.

I want to accept it with calm, and appreciation.  I want to let it inform how I refine my classes in the future without having to doubt my entire self-concept as a teacher.

The comment was about the structure, or rather, the lack of structure in my Puzzles & Mindbenders class.  There were no “learning objectives”, it said.

My thoughts react with defensive retaliation.  People don’t learn because of learning objectives!  People learn because they are interested and focused, because they are engaged with something that challenges them.  It builds confidence and new ways of thinking.  And different learners need different learning objectives!  Even if I had the time to formalize objectives for each learner, would that really enhance my ability to guide them significantly?   Aren’t learning objectives simply a way to make it appear that learning is happening, regardless of whether it actually is?  Aren’t we in a non-traditional setting precisely so we can avoid this kind of song and dance and focus on what really matters?

But, it may be that my classes can use more structure.  Maybe homeschoolers don’t need a class to have plenty of opportunity to work on puzzles they choose.

Actually, though, I think of the value of a class like my puzzles class as similar to the value of a practice group for swimming or running.  Sometimes there is focused learning around technique.  But most of the value comes from building up your muscles and developing muscle memory.  It is much easier to work hard and practice regularly if you have a practice group.

Puzzles and Mindbenders is exercise for your brain.  It gives learners a chance to work hard at puzzles that are the right level for them.  They are building up the circuits in their brains that allow them to think logically, creatively and carefully.  They are learning persistence and gaining confidence.  Perhaps I can communicate this objective more clearly in the course description next year.

Just a note at the end here – if by some chance the parent who authored the comment happens to be reading – this kind of feedback is essential and welcome.  Please keep it coming!

My nickname as a kid was ‘Monkeyflower’.  I was always up a tree, always looking for “the hard way up” when rock scrambling (usually at Vedawoo, Wyoming), and always up for competition and challenge.  These days I carry forward that spirit of adventure into how I learn, and help others learn math.  Math is best when approached with an adventurous spirit.  There is a deep and mysterious abstract world to uncover, and one beautiful thing about it is that we can all rediscover mathematical truths for ourselves.  We don’t need science labs or expensive equipment.  All we need is our brains, a pencil or a white-board marker, and that sense of adventure.

So join me in my quest to bring Monkeyflower spirit into math learning!