The trouble with writing about, or quoting, Liping Ma, is
that everyone interprets her words through their own frame, influenced by their
own experiences and beliefs.
“Well, yes, but isn’t that true for anyone reading
anything?” you may ask. True enough. But in Ma’s case, readers often arrive at
diametrically opposed readings. Both sides in the US Math Wars quote from her
in support of their positions.
Still, if I stopped and worried about
readers completely misreading or misinterpreting things I write, Devlin’s Angle would likely appear maybe
once or twice a year at most. So you can be sure I am about to press ahead and
refer to her recent article regardless.
My reason for doing so is that I am largely
in agreement with what I believe she is saying. Her thesis (i.e., what I
understand her thesis to be) is what lay behind the design of my MOOC and my recently released video game. (More on both later.)
Broadly speaking, I think most of the furor
about K-12 mathematics curricula that seems to bedevil every western country
except Finland is totally
misplaced. It is misplaced for the simple, radical (except in Finland) reason that curriculum doesn’t really matter. What matter are teachers. (That last sentence
is, by the way, the much sought after “Finnish secret” to good
education.) To put it simply:
BAD CURRICULUM + GOOD OR WELL-TRAINED
TEACHERS = GOOD EDUCATION
GOOD CURRICULUM + POOR OR POORLY-TRAINED
TEACHERS = POOR EDUCATION
I am very familiar
with the Finnish education system. The Stanford H-STAR institute I co-founded and direct has been collaborating with Finnish
education researchers for over a decade, we host education scholars from
Finland regularly, I travel to Finland several times a year to work with
colleagues there, I am on the Advisory
Board of CICERO
Learning, one of their leading educational research
organizations, I’ve spoken with members of the Finnish government whose focus
is education, and I’ve sat in on classes in Finnish schools. So I know from
firsthand experience in the western country that has got it right that teachers
are everything and curriculum is at most (if you let it be) a distracting
side-issue.
The only people for
whom curriculum really matters are politicians and the politically motivated
(who can make political capital out of curriculum) and publishers (who make a
lot of financial capital out of it).
But I digress:
Finland merely serves to provide an existence proof that providing good
mathematics education in a free, open, western society is possible and has
nothing to do with curriculum. Let’s get back to Liping Ma’s recent Notices article. For she provides a
recipe for how to do it right in the curriculum-obsessed, teacher-denigrating US.
Behind Ma’s
suggestion, as well as behind my MOOC and my video game (both of which I have
invested a lot of effort and resources into) is the simple (but so often
overlooked) observation that, at its heart, mathematics is not a body of facts
or procedures but a way of thinking.
Once a person has learned to think that way, it becomes possible to learn and
use pretty well any mathematics you need or want to know about, when you need
or want it.
In principle, many areas of mathematics can be used to master that way of thinking, but
some areas are better suited to the task, since their learning curve is much
more forgiving to the human brain.
For my MOOC, which
is aimed at beginning mathematics students at college or university, or high
school students about to become such, I take formalizing the use of language
and the basic rules of logical reasoning (in everyday life) as the subject matter, but the focus is as described in the last two
words of the course’s title: Introduction
to Mathematical Thinking.
Apart from the final
two weeks of the course, where we look at elementary number theory and
beginning real analysis, there is really no mathematics in my course in the
usual sense of the word. We use everyday reasoning and communication as the
vehicle to develop mathematical thinking.
[SAMPLE PROBLEM:
Take the famous (alleged) Abraham Lincoln quote, “You can fool all of the
people some of the time and some of the people all of the time, but you cannot
fool all the people all the time.” What is the simplest and clearest positive
expression you can find that states the negation of that statement? Of course,
you first have to decide what “clearest”, “simplest”, and “positive” mean.]
Ma’s focus in her
article is beginning school mathematics. She contrasts the approach used in
China until 2001 with that of the USA. The former concentrated on “school
arithmetic” whereas, since the 1960s, the US has adopted various instantiations
of a “strands” approach. (As Ma points out, since 2001, China has been moving
towards a strands approach. By my read of her words, she thinks that is not a
wise move.)
As instantiated in
the NCTM’s 2001 Standards document,
elementary school mathematics should cover ten separate strands: number and
operations, problem solving, algebra, reasoning and proof, geometry,
communication, measurement, connections, data analysis and probability, and
representation.
In principle, I find
it hard to argue against any of these—provided
they are viewed as different facets of a single whole.
The trouble is, as
soon as you provide a list, it is almost inevitable that the first system
administrator whose desk it lands on will turn it into a tick-the-boxes
spreadsheet, and in turn the textbook publishers will then produce massive (hence
expensive) textbooks with (at least) ten chapters, one for each column of the
spreadsheet. The result is the justifiably maligned “Mile wide, inch deep” US
elementary school curriculum.
It’s not that the
idea is wrong in principle. The problem lies in the implementation. It’s a long
path from a highly knowledgeable group of educators drawing up a curriculum to
what finds its way into the classroom—often to be implemented by teachers
woefully unprepared (through no fault of their own) for the task, answerable to
administrators who serve political leaders, and forced to use textbooks that
reinforce the separation into strands rather than present them as variations on
a single whole.
Ma’s suggestion is
to go back to using arithmetic as the primary focus, as was the case in Western
Europe and the United States in the years of yore and China until the turn of the
Millennium, and use that to develop all of the mathematical thinking skills the
child will require, both for later study and for life in the twenty-first
century. I think she has a point. A good point.
She is certainly not
talking about drill-based mastery of the classical Hindu-Arabic algorithms for
adding, subtracting, multiplying, and dividing, nor is she suggesting that the
goal should be for small human beings to spend hours forcing their analogically
powerful, pattern-recognizing brains to become poor imitations of a ten-dollar
calculator. What was important about arithmetic in past eras is not necessarily
relevant today. Arithmetic can be used to trade chickens or build spacecraft.
No, if you read what
she says, and you absolutely should,
she is talking about the rich, powerful structure of the two basic number
systems, the whole numbers and the rational numbers.
Will that study of
elementary arithmetic involve lots of practice for the students? Of course it
will. A child’s life is full of practice. We are adaptive creatures, not
cognitive sponges. But the goal—the motivation for and purpose of that
practice—is developing arithmetic
thinking, and moreover doing so in a manner that provides the foundation
for, and the beginning of, the more general mathematical
thinking so important in today’s world, and hence so empowering for today’s
citizens.
The whole numbers
and the rational numbers are perfectly adequate for achieving that goal. You
will find pretty well every core feature of mathematics in those two systems.
Moreover, they provide an entry point that everyone is familiar with, teacher,
administrator, and beginning elementary school student alike.
In particular, a
well trained teacher can build the necessary thinking skills and the
mathematical sophistication —and cover whatever strands are in current favor—without having to bring in any other mathematical structure.
When you adopt the
strands approach (pick your favorite flavor), it’s very easy to skip over
school arithmetic as a low-level skill set to be “covered” as quickly as
possible in order to move on to the
“real stuff” of mathematics. But Ma is absolutely right in arguing that this is
to overlook the rich potential still offered today by what are arguably (I would
so argue) the most important mathematical structures ever developed: the whole
and the rational numbers and their associated elementary arithmetics.
For what is often
not realized is that there is absolutely nothing elementary about elementary
arithmetic.
Incidentally, for my
video game, Wuzzit Trouble, I took whole number arithmetic and built a game around it. If
you play it through, finding optimal solutions to all 75 puzzles, you will find
that you have to make use of increasingly sophisticated arithmetical reasoning.
(Integer partitions, Diophantine equations, algorithmic thinking, and
optimization.)
I doubt Ma had video
game instantiations of her proposal in mind, but when I first read her article,
almost exactly when my game was released in the App Store (the Android
version came a few weeks later) that’s exactly what I
saw.
Other games my colleagues and I have designed but not yet built are based on different parts of mathematics. We started with one built around elementary arithmetic because arithmetic provides all the richness you need to develop mathematical thinking, and we wanted our first game to demonstrate the potential of game-based learning in thinking-focused mathematical education (as opposed to the more common basic-skills focus of most mathematics-educational games). In starting with an arithmetic-based game, we were (at the time unknowingly) endorsing the very point Ma was to make in her article.
