Three Views of Math

On the way home a couple weeks ago, I heard a story on the radio about why we learn math that's 500 years old.  The basic premise is that there are three ways to teach math:

1. Economic:  The important thing is the calculation.  In this paradigm, the answer to 4+5 is always 9.
2. Philosophic:  The important thing is the relationship between numbers.  Here, 4+5 could be 9, or 6+3, or 3-squared, or square root of 81, or ...
3. Artisanal:  The important thing is the units/application.  What does the 4 and the 5 mean?  If it's 4 feet and 5 inches, adding them up won't make sense.

I accept that these are the historic reasons for teaching and learning math.  As civilization developed, people needed to keep accounts, measure fields, and as they had leisure time, pursue math philosophically.  

One of the difficulties with teaching math in the present time stems from the fact that we want to include all three of these paradigms.  Unfortunately, students are not always clear about what view to use and what we expect of them.  For example, many of my students have told me they like math because "there is only one right answer: it's right or wrong."  And most of their background has been about performing calculations to get the right answer.  When I ask them to think more about the numbers, and get philosophical, some get overwhelmed because 4+5 now has lots of answers, some they know, and some they have never seen.  As they try to wrap their minds around that, we throw "word problems" at them where they have to worry about units and whether or not the answer makes sense.  (Never mind that I have seen some published problems where the answer "97 watermelons" is supposed to make sense.)

All mathematics is an abstraction, which means the way we view math focuses our work on some aspects and ignores others in order to better understand/use/apply the ideas and procedures.  Different methods of abstraction focus on different aspects.  So the economic paradigm above is an abstraction of all mathematical ideas, focusing on the calculations.  Limiting our understanding of math to this paradigm is good if we want to produce accountants, but bad if we want to produce engineers.*   Similarly, limiting math to the philosophical or the artisanal paradigm also gives us an incomplete view of math; each abstraction loses details.

So I will continue to try to teach using all three paradigms, and my apologies to my students who think I'm "doing too much".  Perhaps I am, but I promise you it's because I believe in your ability to push past your own limits.

*My apologies if that sounds insulting to accountants, I don't intend that.  My father-in-law was an accountant, and brilliant at his job.  Unfortunately, I did not understand his work very well despite his trying to teach me some basic accounting ideas late one New Year's Eve.  That's a story for another time.  The point is that accountants and engineers use math in very different ways, and teaching only one way is limiting.

New School Year!

So I spent most of last week back at school, making sure that my classroom is ready for Monday and that I have interesting and engaging lessons for the first week.  I enjoyed talking to colleagues who I had not seen in a couple months and rediscovering lesson notes I made last year.  There were of course beginning of the year meetings and activities and the all-staff picture.

But it's my gradebook with the names of all my students that keeps drawing my attention, and I keep checking it to see who else might be added to my classes.  I'm excited to see that more girls than usual have signed up for Computer Science.  A student I enjoyed teaching last year is on my roster for a different class this year.  I have students from all grade levels in my classes, and I get to teach some ELL students again.

All the names in my gradebook represent so many possibilities, so much potential, so many opportunities.  I look forward to learning and growing with all my students, and I can't wait to meet them on Monday.

Geeking out a little

I came across this TED talk from Adam Spencer, and I had to post it.  It tickles my math brain, my computer science brain, and my humanity brain, and it made me smile.

Teaching Bravery

This summer, I have been participating in some workshops about computer science, in preparation for teaching a new class in the fall.  In one session, we were referred to the TED Talk video from Reshma Saujani, the founder of "Girls Who Code".  In the talk, she referred to work by Carol Dweck, and what she said about teaching girls to be brave, not perfect, resonated with me and what I have seen in math and computer science classes.

At one point in the talk, Saujani talked about boys' responses to problems: "There's something wrong with my program" versus girls' responses to problems: "There's something wrong with me."  This may be over-generalized, but it highlights how girls, and, I think, many underrepresented groups in STEM fields, respond to difficulties.  As a white male, I automatically belong to the "STEM Club", so any difficulties I experience are not part of who I am; the difficulties are part of my process.  Unfortunately, those belonging to other groups not part of the "STEM Club" may start to believe something is inherently missing in their make-up or that STEM fields are not for them.  Just as unfortunately, those of us in the club can also start to believe this.  And it's nonsense!

Students tell me all the time that they're not good at math, and it's just not true!  Just because you have to work at something or just because you don't understand an idea quickly does not mean you are not good at it, or that you shouldn't try it!  Much of math, computer science, life even! is figuring out the next step based on limited information.  And even when you figure out your next step, you have to realize that it might not be correct and you have to do it again, and THAT'S OKAY!  Perfection is not expected nor encouraged.  Growth and improvement and moving forward are.

I teach because I want to help students embrace the struggle, because in the end it's not the math (or English or History, or ...) concepts that are the most important.  (Yes, I know, that's what's usually being graded, which makes it important, and that's my current struggle.)  Some of what I hope my students take from my classes is a willingness to work hard in the face of challenge and to wrestle with difficulty cheerfully.  With these, math (and life!) are infinitely more enjoyable.

"Success is staggering from failure to failure with no loss of enthusiasm."

The Personality Myth

I heard a piece on WBEZ's Invisibilia program, The Personality Myth, and the story resonated with me because its point was that personality is mutable.  The cells in our bodies are constantly being replaced, our brains are always being rewired, and our memories of events, even big important events, change over time.  What makes us who we are is not a fixed, unchangeable entity, but a mutable and constantly growing set of attributes that we actually have a lot of control over.

In the Invisibilia piece, a woman working with prisoners finds that her experiences, along with a conscious decision on her part, changed how she thinks about "good people" and "bad people": while there are amazingly good actions that people take as well as horrifyingly evil actions, people themselves, because of their mutability, are not so easily categorized.

All this reminds me of my reading and experience around growth mindset, and the saying "Success is never final and failure is never fatal; it's courage that counts."  What has already happened, whether good or bad, can be learned from and, as amazing, changeable people, we can choose the next steps on our path.

The implication for me and for my students is that regardless of past experiences with math, we have the ability to learn new strategies and techniques, and develop deeper understandings.  Right now, for example, I am in the middle of a two-week workshop getting ready to teach the Computer Science Principles class.  While I have some experience with CS, I am being asked to think in some new ways, and I am really enjoying this experience.  I'm not completely comfortable with the material yet, and the experience of learning new things has me both tired (brain work takes a lot of energy) and exhilarated (each new idea sparks lots of other ideas and questions for me).

At a deeper level, the Invisibilia piece reminds me that I have to be careful about categorizing my students.  Regardless of their past experiences and views about math, regardless of their apparent energy level for the topic, their gender, sexual orientation/identity, race, culture, year in school, or the thousand other attributes that make them who they are at this moment, my job is to recognize their humanness, honor what makes them unique, and help them determine the person they will become.

New Problems to Think About ...

Last night, I attended the January meeting of the Metropolitan Mathematics Club of Chicago (MMC), at which Steve Viktora talked about word problems through the ages,  He showed examples of word problems across cultures and times (going back 6000 years!), and highlighted some common themes he found.  Here's an example of a type of problem Steve called a "hydraulic problem"; these types of problems deal with some sort of public works, like building dikes, storing grain, or marking out land, and they appear in documents from Mesopotamia, Egypt, and China going back thousands of years.  This one is from the 4th millennium BCE from the Sumerian city of Shuruppog, and is the oldest known word problem:

A granary of barley.  One man received 7 sila [of grain].  What are its men?  [i.e. How many men can be given a ration?]  

Of course, to solve this, one needs to know that capacity of a granary was 2400 gur, and one gur was equal to 480 sila.  (Any Sumerian bureaucrat could tell you that.)

Steve presented other hydraulic problems.  This one got everyone talking, and there were a number of different, interesting, and elegant ways that people solved it.  There were also some good questions about what happens if the base is not a leg of the triangle.  I don't remember the  time or place for this problem:

A triangular piece of land [in the form of a right triangle] is divided among six brothers by equidistant lines constructed perpendicular to the base of the triangle.  The length of the base is 390 units and the area of the triangle is 40950 square units.  What is the difference in area between adjacent plots of land?

There were abstract problems from Islamic cultures, temple problems from Japan, puzzles from Medieval Europe, and many others.  They were fun to see, and we got to work on a few, which is always a good time.  (Yes, I like discussing and solving problems.  I am a Math Geek, but you already knew that.)

After Steve's presentation, a few of us were talking, and Carol said she was teaching a Geometry class, and asked the students to come up with four integers that could represent the sides of a rectangular solid and the length of one of its interior diagonals.  Nice problem, and if you are studying the Pythagorean Theorem and Triples, you can figure out that if the dimensions are 3, 4, and 12, the diagonal is 13.  One of her students said, "Sure, and if you pick and two consecutive numbers for the base, and multiply them to get the height, then the diagonal will be one more."  Carol thought that was an interesting hypothesis, and matched the expected solutions, but cautioned that it might not really be a general solution.  Then she did the algebra ...  So what do we have?  A way to find "Pythagorean Quadruples"?  I love math!  Paul added that when he was teaching the triples, he listed several out for his students:  3,4,5;  5,12,13;  7,24,25;  9,40,41;  11,60,61 ... and hoped they would see some patterns that could help generate more triples.  (There are all sorts of patterns here.)  Paul said one of his students noticed this one, which I had not seen before:  4 = (1/2)(3+5); 12 = (2/3)(5+13); 24 = (3/4)(7+25);  40 = (4/5)(9+41);  60 = (5/6)(11+61).  Okay, mind blown.  Did I mention that I love math?  MMC Dinners are always worth the price of admission.