There are some rather murky philosophical questions around the concept of emergence. Emergent phenomena seem to be, in some way, “more than” the sum of their parts. I think perhaps a better way to approach the concept is by pointing out that emergent phenomena are those that are more easily understood from a holistic point of view than a reductionist one. The most straightforward example I know of is the concept of temperature. Temperature reduces to the average kinetic energy of all the tiny atoms and molecules in a physical system. But we never actually use the concept of temperature in this way. Instead, we measure temperature directly via its effects on volume and pressure, and we think of it as a property of the system as a whole.

Now consider the following Fibonacci identity:

This pattern emerges from simple combinations of Fibonacci numbers. The underlying properties of the Fibonacci numbers that lead to this pattern have nothing to do with squares (see the video I did with Gavin Ford on this identity). However, the pattern is there and it can be grasped at this higher level. It can be treated as an abstract tool without referring back to the simple combinations that constitute a rigorous proof of the identity.

Perhaps much of mathematics is the same. Rules for various derivatives can be proved by examining the underlying details of slopes of tangent lines and limits. But the rules themselves can be understood and applied without any mention of limits. We could say that the derivative rules are emergent from the interaction of more fundamental concepts in calculus.

Is this kind of mathematical emergence the same as emergence in complex systems?

Many things in this world are a matter of perspective. Often, those things can stimulate conflict and polarization. But constants are the same, no matter who we are or how we look at the world.

There are a slew of scientific constants. The ideal gas constant (R), the gravitational constant (G), and of course the speed of light (c), to name a few. And while these constants are the same for everyone, they sometimes appear different because empirical units like length and time are measured differently in different places. We could not broadcast 3.0×10^8m/s into outer space and hope that it could be understood by an alien being on a distant planet.

But we could broadcast τ (tau), the ratio of a circle’s circumference to it’s radius.  No matter where you are in the universe or what kind of measurement system you’re using, you will come up with 6.283185… for τ.

Okay, okay, there is some translation necessary. If you’re an alien you’re probably not going to have ten fingers, so you probably are using binary (τ = 110.01001…) or base twelve (τ = 6.339416…) or something (and yes, I did geek out and calculate those by hand). Anyway, my point is that τ is the same ratio however you look at it. There are some truths we all have in common no matter how big our differences seem sometimes. Happy tau day everyone!

(If you still think that π is the proper circle constant, check out tauday.com or Vi Hart’s lovely video on the topic)

I have a new student, let’s call him Sam. He is in first grade, he loves snakes and stories, but according to him, he doesn’t like numbers or challenges. He likes patterns a little bit. Sam doesn’t like it when his mom makes him do subtraction problems assigned by his teacher.

On our first lesson we read the first chapter of the book The Number Devil. It tells a story about a boy in whose dreams the number devil gives him a magical tour of number theory. We paused in our reading often for him to pull out his calculator and follow along with what the boy in the story was doing on his calculator. This frequently turned into Sam playfully experimenting with various operations on the calculator and discovering new patterns. In the beginning of our second lesson, my he asked eagerly, “But do you remember the Number Devil??” as if worried that I might forget about reading the second chapter.

Long story short, Sam clearly loves math. He is delighted by patterns, by making discoveries about numbers, and by big ideas like infinity and dividing by zero. Yet, a bit of first-grade level arithmetic in school had him convinced of the opposite. And that’s only in first grade. What would we expect to happen by middle school? By high school?

It is common knowledge that most people emerge from public school with a distaste for math. But what is less commonly recognized is that most first graders start out with the capacity to love math. All we have to do is allow them to play with it and encourage them to be imaginative about it. In fact, I think all we really have to do is refrain from forcing them to do repetitive and tedious arithmetic!

Last week I was honored to do a guest blog post for the krazydad.com puzzle site! Thank you, Jim, for the invitation.

I wanted to cross-post it here, so here’s the first bit, and a link if you’d like to read more:

In the midst of a global crisis, I’ve been noticing some silver linings and surprising upsides.  This is not to downplay the seriousness of it all or dismiss any of the real loss and suffering in our world, but I feel it is important to take hold of the good things that can be found amidst all the uncertainty and struggle. 

Being stuck at home can be seen as a chance to spend more time with family.  Being unable to do some kinds of work can free up time for projects that otherwise get neglected.  And a break from math class can be an opportunity to play with math together.

I know, I know. The words “play” and “math” in the same sentence?  Hear me out.  In my world of teaching math to homeschoolers I find that the more playful I can make math, the better.  So today I want to share with you a couple ways I’ve been playing with math with my students this week. (read more)

This summer I started four new habits. I started them a week apart because I was feeling too impatient to spread them out more. I knew it would be better to start them at least a month apart but I couldn’t bear to wait on any of them. Like Alice, I give myself very good advice.

First there was the habit of working on my side-hustle. This is not actually habit material. The tasks are too diverse. There are potential habits within the project that I could establish – posting on social media daily, for example – but “working on my side hustle every day” is too broad. Instead, what I need is weekly planning time where I plan out how to include tasks from my side-hustle into my week.

Habit number two was writing daily. I happen to be practicing that habit right now. This habit is starting to be more regular but has not gotten to the point of a daily habit.

Reading interesting non-fiction daily, however, was the easiest of the bunch to get started. This was something I was just waiting for an excuse to get used to. Deciding I was going to make a habit of it felt like getting permission from myself to spend time on it. It feels great that reading is now so easy and so much more common in my daily life.

Lastly, I wanted to get into the habit of cleaning a little extra daily, beyond what I was already consistently cleaning. I’ve realized now only part of this goal is really suited for a habit. Extra cleaning involves what I think of as “conquering new areas” of the apartment and “maintaining the conquered areas”. The maintaining part is much more suited to being habitual – there’s not a lot of planning or decision making involved and most of the component micro tasks are the same from day to day. The conquering new areas, as well as somewhat infrequent tasks like cleaning the bathroom or vacuuming, are more suited to being planned tasks. Due to my attempt to lump all the “extra cleaning” into a single habit, I haven’t managed to establish this regular cleaning habit yet – but I did learn from my first attempt!

So, my summer self successfully gave my present self two habit gifts: reading nonfiction and writing. The two that weren’t successful both taught me why they weren’t ideal habits as I originally conceived of them. Learning, learning!

I’ve been building a lot of new habits this summer. Writing, working on the Pixidoku Kickstarter, reading non-fiction, and house-cleaning. But I also started getting in the habit of not playing addictive video games before noon. I’ve noticed that my morning hours are my most motivated, productive and creative hours (and I believe this is backed up by the neuroscience) and so I wanted to reserve these hours for tasks that require high cognitive function. It’s been very successful and it’s given me a new model for how to set myself up for success in forming new habits.

Often, an obstacle to forming a new habit is that there is an existing habit that gets in the way. Before I start building my actual new habit I can first get out of the old habit. I think doing a different enjoyable but non-addictive activity every day during that time slot would be a good start. One candidate for that type of task is a task I’m already habituated to. For me, I’ve been practicing Norwegian on most days for the last few years so it is easy and fun for me at this point. If I want to form a new habit at a certain time I can start doing my Norwegian at that time first. This will prepare me to transition to the new habit.

How do you form new habits? What are the biggest challenges and strategies you’ve tried for overcoming them?

I met someone recently who was diagnosed with autism late in life.  He said that in a way, he found the diagnosis relieving because he was starting to wonder “Is something terribly wrong with me?”.  But, he says that the diagnosis hasn’t helped him figure out how to fix his very real problems.

He forms friendships but he is awkward meeting new people and in social situations.  He has trouble picking up on social norms and standards of behavior.  He struggles with the transition from friendship to more intimate connections such as romances and sexual relationships.  As a result he has been alone and celibate for 15 years.

He has access to a psychiatrist, but cannot afford the copays for a therapist who could help him learn the social skills he needs.

This struck me as an unacceptable lack of resources.  And it got me thinking.  The mental health sector seems to be taking on a lot.  I’m wondering if some of the education that goes on between therapist and client might be made more accessible.  Some of the things that my therapist teaches me could be taught in a class, and by someone with less specialized training than a psychologist.  I’m imagining low cost public classes for adults on relationships, communication, mindfulness techniques etc.  Wouldn’t that be less expensive than learning one-on-one from a trained therapist?

And wouldn’t the same be true for someone diagnosed late with autism seeking to learn how to improve his social skills?  Are there programs like what I’m thinking of?

I know there is some of this sort of thing out there.  My co-counseling organization fits into this category I think.  There are organizations that teach authentic relating techniques, groups that practice deep sharing and empathetic listening, and classes on non-violent communication.  I think we need more of this.  Being a human is complicated.  There is a lot of knowledge out there from science and from age-old cultural wisdom.  Maybe we can do a much better job at spreading that knowledge to both children and adults.

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.

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.

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?