cheesemonkey wonders

cheesemonkey wonders

Sunday, June 26, 2011

WARNING: This post contains math education heresy

I am so tired of math teachers and teacher educators telling me memorization doesn't work that I am willing to take a reckless step into the fray.

My purpose in this post to demystify this dangerous misunderstanding and to say that memorization of basic facts not only can work but indeed does work and works well -- as long as several essential conditions are met.

First, a definition. When I talk about memorization, I am NOT suggesting that one can achieve mastery by rote memorization of the proof of the Fundamental Theorem of Galois theory.

Rather, I am referring to an active process of integrating certain basic, rudimentary facts into one's mind and body, both backwards and forwards. By this, I mean that, given a vocabulary word, one can produce the definition, and similarly, given the definition, one can produce the vocabulary word. Or given a basic multiplication fact, one can produce the product and similarly, given a composite number, one can produce its basic factors.

 I am talking about filling in basic math facts. Expanding critical vocabulary. And solidifying basic mathematical skills that are often missing in the high school math student. 

Facility with this kind of memorization is the sine qua non to serious foreign language study as an adult, since it is simply not practical to put yourself into the way of enough quality adult conversations to absorb all the vocabulary one will need to read, write, think, and speak another language with the basic fluency that is required if you are going to be dropped into another culture. But it IS possible to approximate that vocabulary and bring its user closer and closer to a level of acceptable fluency.

This is how I both learned and taught Italian and Latin, as well as a number of other languages as a Comparative Literature scholar. It is also how the Peace Corps trains its Corps members in preparation for their overseas, language-intensive assignments. 

I was schooled in the Rassias Method, a highly dramatic, intensive, and effective technique of drilling students in the language classroom to approximate and accelerate the contexts of listening and speaking another language. It does so through very strategic, high-energy, rapid-fire, and theatrical drilling and practice techniques.

Approximating contexts is important because, as Skemp puts it, purely instrumental learning without any relational context is just pointless. But I believe many of my colleagues and math education instructors have misunderstood this critical distinction. The way I read it, Skemp is not suggesting that there is NO room or role for instrumental learning. He is asserting that instrumental learning is insufficient without relational learning as well.

This intersection between instrumental and relational learning is where the Rassias Method really shines. One thing I used in my math classes this past year was the Rassias strategy of "flooding" students (my term, not Rassias') with productive opportunities in order to burn those facts and skills into their minds and bodies. But even more important than the drill itself is the process of breaking down student inhibition in the classroom.

This strategy is key.

Discouragement is always accompanied by inhibition. And the only way I've ever found to break down inhibition in this regard -- my own as well as that of my students --  is to insist on lots and LOTS of participation and practice -- with no chance of opting out.

The Rassias Method taught me to use a technique that blends rapid-fire drill with micro-contexts and an often humorous dramatic flair to create a heightened emotional charge in the classroom in which anyone could be called on at any moment to produce anything that is being asked for. It encourages learners to engage, to enjoy, and to stop worrying about producing the right answer because it creates dozens and dozens of chances to produce the right answer. It accomplishes this goal by flooding learners with basic language demands, all the while heightening drama, motivation, and interest in success while simultaneously lowering the stakes of failure. 

To put it another way, trying becomes more important than succeeding -- because eventual success is assumed.

Here is an example of how I used this in teaching my Italian language classes at Stanford.

One of the biggest hurdles in learning Italian is mastering its complicated matrix of prepositional contractions. Wikipedia has a reasonable summary of this matrix here:

In Italian, a number of key basic prepositions are ALWAYS merged with the direct article preceding the noun that is the object of that preposition. So for example, to say that something is "on the table," you need to merge the preposition for "on" (in this case, "su") with the direct article "la" ("the") that precedes the noun "tavola" ("table"): in other words you need to say that something is "sulla tavola" ("on the table') instead of "su la tavola."

Practically speaking, this roughly six-by-eight matrix of prepositions and direct articles needs to be absolutely second nature for a speaker who wishes to be able to produce and recognize the right prepositional contraction for the job.

Basically, the prepositional contractions are the times tables / multiplication facts of the Italian language.

To get students using these, one Rassias technique I used involved a little plastic elephant, whom I named Signor Elefante, which I held in different positions with respect to a festive-looking cardboard box and drilled my students, asking, "Dov'è Signor Elefante?" ("Where is Signor Elefante?"). Or as we say in edu-speak, I used situational motivation (for a good discussion of situational motivation, see Wilhelm and Smith, "What Teachers Need to Know About Motivation," Voices from the Middle, Vol. 13, No. 4, May 2006).

Sometimes Signor Elefante was "nella scatola" ("in the box"), sometimes he was "sulla scatola" ("on the box"), sometimes he was "vicino alla scatola" ("near the box") or "lontano dalla scatola" ("far away from the box"). Sometimes he was "sotto la scatola" ("under the box") or "alla porta" ("at the door"). Occasionally he was "sulla lavagna" ("on the chalkboard"). He got himself into some pretty wacky prepositionally contracted situations. But after a lot of practice and inhibition-destruction -- as well as their own practice at home with flash cards -- locating Signor Elefante in time and space became more and more natural for my students. They got themselves over this major linguistic hurdle and developed their own relationship with the prepositional contractions.

They blended instrumental learning techniques with relational learning to generate understanding and fluency that was more than the sum of its parts.

In my math classes last year, I found that many of my lowest-achieving students responded well to this kind of approach to remediation. Even my higher-achieving students responded well to this approach. In fact, the most important the thing I discovered this past year is that many students have no idea how to practice basic techniques... but they get excited by the results they can achieve once someone shows them how memorize and assimilate bits of information like this.

Now since in spite of my best efforts I am bound to be misunderstood and misquoted, I'll restate this as plainly as possible: I'm not talking about using memorization with higher-level thinking and problem-solving. I am, however, talking about using these techniques as a strategic intervention to help students remediate and give them the tools and techniques they will need to fill in the gaps and potholes that riddle their elementary mathematical preparation. 

Confidence with the basics is a necessary condition to cultivating curiosity and persistence about mathematics. I speak from my own experience as well as that of my students.  Mastery of basic tools and techniques, combined with a lowering of inhibitions, is a foundation upon which confidence and curiosity can grow. And that can be the basis for a turnaround to success in high school mathematics -- regardless of where students are starting from.

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