Monday, November 4, 2013

The Educational Power of Elementary Arithmetic

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.

That happened with the book that brought her to most people’s attention, Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States, first published in 1999. And I fear the same will occur with her recent article "A Critique of the Structure of U.S. Elementary School Mathematics," published in the November issue of the American Mathematical Society Notices.

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:



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 classroomoften 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 goalthe motivation for and purpose of that practiceis 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 favorwithout 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.


Rebecca Hanson said...

Dear Keith,

I really must take issue with this comment:


It pays no respect whatsoever to the horrors of what is now going on in England. Key politicians have developed a primary mathematics curriculum based on the naïve idea that the more abstract content you give children and the younger you give it to them the better they will be at maths. Instead of paying any attention to those with relevant experience and their own expert advisers they have chosen to completely ignore such people and instead to viciously attack them in the press as being 'enemies of progress'.

The primary curriculum which we are to teach from January is not remotely age-appropriate and is not like any other national curriculum. It is statutory and teachers are required by law to teach it by year group. This is overseen by an exceptionally brutal regulator.

To simply say that if you're a good teacher it won't bother you is very out of touch. There's a full report here:

BTW my classes had all those structures LiPing Ma found in China (like 3 conceptual methods of solving division of mixed numbers rather than procedural methods). More here if you're interested:


Keith Devlin said...

Rebecca, Thanks for writing. I am vaguely aware of what is happening in the UK from reports in The Guardian, but surely, unless the UK has imposed a very totalitarian regime of local control, good teachers will prevail. No?

Which is not to say I don't understand your frustration and anger at having a national education policy controlled by someone totally ignorant of education.

Cathy Kessel said...

You write: "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.”

One way of reducing the variability of interpretations might be to compare definitions for certain words. In the case of this post, I think your meaning of “curriculum” might refer more to selection of topics or subject (e.g., arithmetic) and not include much emphasis on how that content is construed or organized (e.g., construing arithmetic as a collection of algorithms without rationales). That meaning of “curriculum” seems plausible for the US, given traditions of local control and standards (and accountability) that focus on performance. It seems less accurate as a description of its meaning in countries where curriculum objectives place more emphasis on sequencing and topics taught, and policy-makers give more information about intended meanings of topics via textbooks and teachers’ guides, and where teachers are afforded opportunities to share meanings via practices like lesson study and demonstration lessons.

Keith Devlin said...

Cathy, Indeed my primary focus is the US educational system.