Author Archives: Atrytone

Mathematics: A Human Endeavor (Textbook review)

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This book is amazing.  Written by Harold R. Jacobs almost 50 years ago, it is a blessing that the content will not likely become out-dated.  It presumes nothing more than fluency with arithmetic from the learner, but covers a range of topics that span pre-algebra to advanced algebra.  The subtitle is “A Textbook For Those Who Think They Don’t Like The Subject”, but in my opinion it is simply a textbook for everyone (including those who think they don’t like the subject!)

First and foremost, this book goes beyond interesting into downright fascinating.  It takes the learner on a safari of mysterious patterns, both in abstract and in nature.  It unveils the beauty of geometric solids and mathematical curves.  It inspires awe in large numbers, and delight in mathematical tricks.

Despite all this depth, it is accessible.  The chapters are short and the prose is comfortable and inviting to read.  Much of the instruction happens in the well-crafted exercises.  The learner is allowed to discover many concepts for themselves, so that the learning feels more like inquiry than instruction.

The treatment of logarithms in the fourth chapter is particularly impressive.  In my years as a tutor I have often needed to provide mathematical triage to learners who become hopelessly confused by this or the other explanation of logarithms.  This book puts to rest any question of whether logs can be explained in a clear and intuitive manner.  The author draws on the learners comprehension of arithmetic and geometric series and gradually leads the learner through the use of logarithmic patterns and functions without introducing the obscuring notation until the concept is solidly established.

The book is also filled with “experiments” which range from abstract inquiries to mathematical arts and crafts.  I opened the book expecting to build a class around it, only to find that the class is already built.  All we need is some graph paper, compasses, rulers and a room full of minds ready to learn!

 

Can ethics be based in science?

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Bubbling excitement and eagerness rise in my chest.  I lean forward in my seat as if I could hear the speakers on stage better that way.  The audience fluctuates between quiet attentiveness and laughter, appreciating the conversation between the two men sitting in black arm chairs, one a cosmologist the other a neuroscientist and philosopher.

A few months earlier I had bought my tickets to the event in Portland, a live audience episode of the Waking Up podcast hosted by Sam Harris.  I had bought the tickets without knowing who the guest was going to be and I was thrilled to discover that Sean Carroll would be sharing the stage, a scientist and blogger I had admired for even longer than I had been familiar with Harris’ work.

I’m particularly intrigued by one question that the two thinkers disagree on.  Can ethics be based on empirical science alone?

Some assumptions are needed to get the enterprise of science off the ground.  We need to assume that our senses convey some information about reality, and that there is some amount of predictability to the fabric of our universe that will allow us to formulate theories that accurately predict phenomena.  I guess we don’t have to assume the latter to make the attempt, but unless it’s true there doesn’t seem to be much point.  Perhaps we are assuming other things about logic and foundational mathematics.  As we go along we sometimes discard assumptions that don’t prove useful in constructing the predictive theories we’re after.

The question is, do we need to assume anything further to get ethics off the ground?  Science can answer questions about how reality is structured and what exists.  Can answers to questions of how we should act follow from that knowledge of reality?  Or do we need ethical axioms in addition to our scientific axioms?

Sam Harris thinks we don’t need any further assumptions.  He points out that if you study conscious creatures deeply enough you will have a complete understanding of which behaviors are best for those conscious creatures.  Goodness is something you understand by looking inside brains, and brains are part of the natural world.

Sean Carroll argues that although a complete science of conscious systems may lead to an understanding of which outcomes will result from which behaviors, but that science will not be able to conclude that any of those outcomes are better than any other.  We may be able to determine scientifically that a certain action will lead to great suffering, but it is a step further to then claim that the suffering would be a bad outcome.

I have many thoughts on this which I hope to explore in future posts.  Delightfully, I have wavered back and forth between agreeing with Sam and agreeing with Sean.  What do you think?  I created a discussion on Kialo and I would love for you to join me!

 

 

Base 10 Toothpicks

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Practicing our number system with a tangible model is an invaluable way for learners to internalize base 10.  I remember how valuable base 10 blocks were for me in second and third grade.  A few years ago someone pointed out that toothpicks work great as a base ten manipulative and I’ve been using them regularly in my math tutoring.  Searching around just now to see if anyone else had blogged about this idea, I found this great blog post about why pop blocks that can be broken apart work better than base 10 blocks as a manipulative for learners who are still mastering base 10.

Here’s how you can do it with toothpicks.

Base Ten Toothpicks Supplies

 

Get about 500 toothpicks for each student who will be using them at one time.  Get some of the tiny hair rubber bands and some larger rubber bands. 

 

 

Ask learners to help you construct the base 10 toothpick set by making some bundles of 10.  Even if you already have bundles made by previous students, it’s great to start learners off with these by letting them construct at least some bundles of their own.

When hundreds are needed, they can be created by bundling 10 of the 10-bundles together with a bigger rubber band.

As with the pop blocks, a great thing about these is that they can be physically taken apart to aid in modeling subtraction and division, or to easily create numbers close to 10 or close to 100.

Plus, they fit in my cute manipulatives bag!             

Melting point musings

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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?

Playing with error

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This week I learned something new from one of my learners.  This happens quite a bit and I always love it.  He had watched a video about a method of estimating square roots.  Here’s how it goes.

Of course we start by finding the integer square root of the last perfect square below our number.  For 45 our result will be 6 plus a fraction, for 124 it will be 11 plus a fraction.  Now we just need to find a fraction to add on to our integer that closely approximates the square root.  For the numerator we use the difference between our number and that nearest perfect square below our number (45 – 36 = 9 or 124 – 121 = 3).  For the denominator we double the integer square root we arrived at in the first step.  So for 45 our final approximation is 6 9/12 and for 124 we get 11 3/22.

This brought up some intriguing questions.  Why does this method work?  In particular, what’s up with that doubling of the integer to arrive at the denominator of our added fraction?  I should have let my learner lead the way on this question, but I let my curiosity get the better of me this time.  We worked on the problem together but I had some previous experience to draw on from having proved to myself why the pencil and paper algorithm for taking square roots works.  I let that background knowledge I had lead us through it and we pretty quickly arrived at a satisfying explanation.  Can you create your own explanation?

Next we wondered: How does the error vary as you increase your starting number?  We hypothesized that the error would be smallest for numbers just slightly above a perfect square and largest for numbers just slightly below a perfect square (because those are farther away from the next perfect square below them.  We also guessed that the accuracy would get worse as our starting number got bigger.

To test this second hypothesis his first impulse was to try it out with a number near the largest square number.  Having satisfied himself that there is no largest perfect square, he settled for numbers near 10,000 (100 squared).  This produced a surprising result!  Rather than finding a significantly inaccurate estimate for 9800 (one less than 99^2) the algorithm produced an estimate that was very close to exact.  So now, we still expected numbers close to the next perfect square to fare worse in the algorithm, but that as the numbers went higher the algorithm would produce better and better accuracy.

Later in the week at home he produced this graphic to show how the error changes as the input number increases.

So cool!  Great work on his part.  Maybe next we can explain to ourselves exactly why the error falls off with larger inputs!

Puzzles in Math Class

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My main goal when I teach math is to help learners develop the ability to independently solve problems they have not encountered before.  I feel that this is a sort of master-skill in math learner because if you’re a strong problem-solver you can more easily make sense of new mathematical concepts, which in turn makes it easier for you to apply those concepts to more challenging problems and so on.

As I practiced coaching learners on the skills that make them better problem-solvers I began to favor tasks that isolated this problem-solving element.  The natural progression of this led me to logic puzzles.

I love using puzzles in a math-education context and I think it should happen more often.  When learners work on puzzles I can positively reinforce mental and emotional skills like persistence, willingness to test solutions and creative thinking to generate new approaches.  With puzzles learners get to experience cycles of struggle followed by very gratifying success.  Once learners love puzzles and are confident with them, particularly with figuring out strategies for new types of puzzles they haven’t seen before, it’s natural to frame math problems as a different kind of puzzle.

Tau

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On 6/28 this year I introduced quite a few people to the circle constant tau, as I usually do.  Surfing around on the web there were quite a few articles explaining why tau is better than pi, and there’s always the Tau Manifesto.  But none of them cut to the chase in quite the way I wanted.  There is one reason that stands out to me as by far the most important reason to start using tau as a circle constant, and that is that it makes basic trigonometry way more intuitive to learn.  Here’s why.

If you think back to advanced algebra in high school you might remember having to master radian measures of angles.  Instead of 30 degrees, you had to use pi/6.  Instead of 90 degrees, pi/2.  Your teacher may have given you a list of common angles to memorize in pi radians so that you could convert these angles easily from one to the other.  Very likely, you found this confusing and frustrating.  As you finally mastered it, you may have been able to visualize these fractions of pi as fractions of a half a circle.  Stop and ponder that for a moment… fractions of a half of a circle?

Radians measure angles by tracing out a fraction of the circumference on the unit circle.  3.14159… radians gets you half way around the unit circle.  6.2831… radians on the other hand, gets you all the way around the circle!  This is tau, equal to 2 times pi.  If you use tau as your circle constant then angles measured in radians suddenly become easy.  Now instead of thinking of fractions of a half of a circle, you just have to think of fractions of a circle!

I have a fond memory of one of the first times I introduced tau to a student I was tutoring.  We met in the evening and she told me that she had come away from her math class that day feeling confused.  They were beginning to study radians.  I asked her if she wanted to know a better way of writing these radian angles even if it was going to be different than the way her teacher was writing it.  She agreed so I showed her tau radians and how they related to pi radians.  After a very short explanation she pointedly picked up her pencil and proceeded to cross out her entire two pages of notes from class that day, in order to re-do them using tau.  Ah, the delights of elegant and intuitive notation!

Learning Arc

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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?

The Way We Talk

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When two people disagree about something, part of the disagreement is due to the two misunderstanding one another, and part is due to actual difference of opinion.  I have a hypothesis that most of the time the misunderstanding piece is far bigger than it seems.  I feel that if two people can manage to avoid talking past each other, they will be most of the way to agreement or mutual understanding.

This is why definitions are so key in fields like philosophy.  It is essential to create a shared vocabulary in order to successfully communicate about complex ideas.

So, I try to make clarification my top priority when I find myself disagreeing with someone.  What do they really mean?  What do I really mean?  Which words are doing a lot of work, and do we really have a shared definition of them?

There’s another bonus to this strategy also.  By taking the time to clarify someone’s position I’m demonstrating to them that I actually care what they think.  That will help develop the rapport and goodwill that is also important to successful communication.

I think this kind of practice is especially important in this era of Trump where people are feeling so much distance from others who don’t share their views and perspective. As Prashant Kakad wisely declared from the stage at WDS, “We need to talk to each other.”  Moreover, we need to do so in a way that will allow us to understand one another better.

Don’t try to remember, try to figure it out

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Often I see students I’m working with struggling to recall something, and drawing a blank.  Usually they remember that they had seen an example of the process that would help them with their current math problem and they’re trying to remember exactly what it was, and how it starts.  When I see this, the advice I give is, “Don’t try to remember.  Try to figure it out in a way that makes sense to you.”

I think this has two advantages.  For one, I feel like I am more likely to remember something if I am actively playing around with related information, rather than trying to recall by brute force.  Often if I start doing something, anything, on a problem, it will jog my memory and I will suddenly remember what I had forgotten.

Also, of course, I feel like the process of trying to figure out a problem anew – especially one that is somewhat familiar – deepens the understanding and helps build a quicker recall the next time a similar problem is encountered.

I mentioned this to a friend of mine, and she said she refers back to this mantra even when trying to remember simple things like where she put something or the name of something.  She runs through related information, often talking through it aloud, and that process helps her find the right mental pathways that lead her to recall the information she was seeking.

I’d be interested in hearing everyone’s thoughts about memory and recall.  What techniques work for you?  Do you think your own memory has been a strength throughout your life or has it been something you’ve struggled with it?  What helps you commit mathematical knowledge to memory?  Is it different for you than learning other topics or types of information?