Math wars: is it "drill and kill" or "discovery learning"?

Today, Barry Garelick sent me an article of his which explores the topic of MATH WARS.

You may have heard of them. Math wars refer to the debate or controversy about how math should be taught in school. There have been two main camps: those who emphasize discovery learning and calculators, de-emphasizing algorithms and memorization - and those who claim kids need to learn and memorize basic facts and algorithms such as long division.

Mathematicians have been fiercely opposing the math education reforms that are based on the discovery learning postulate.

In my opinion, I think kids surely need to memorize their basic facts, and learn the basic algorithms. Learning by discovering is a fine teaching method, but it shouldn't replace teacher telling or explaining how things work.

I guess what I feel is that learning by discovery (or explorations) can be great but you need to use it right and not let it 'rule'. It is an instructional method or a tool - not a goal in itself.

Anyway, if you are lacking in this topic, Barry's article summarizes nicely how math wars and fuzzy math began, what has been going on, and how the wars are now (unfortunately) turning into politics instead of being purely an academic debate.

Comments

Anonymous said…
If anyone plans to advance in math beyond Algebra, they are going to need to have mental access to certain formulas and procedures, or they will be incapable of "discovering" unique or difficult relationships. I agree that a balance between working with calculators and memorization is valuable. The reality though, at least in math related jobs, is that we will have a generation of mathematicians who spend 3 hours looking up the equations or processes before they can begin to "solve" anything in the field.
Anonymous said…
I do not feel memorising is too important to mathematics.

In the end, you will end up memorising, however, and if you want to get things done you will have to.
Anonymous said…
Disclosure: I am a graduate student in mathematics. I'm in a masters program at a no-name university.

What is clear among the math professors that I interact with is that younger kids cannot think about problem solving or solve even simple problems without the use of a calculator.

Ask them what is 2/3 of 10 and they are reaching for their calculator. Insist that they don't use the calculator and they look confused. They aren't able to solve the problem without technology because this notion of "exploration with technology" is, in reality, dependance on technology. The methods are supposed to teach a student to estimate the answer at a little more than 6 or almost 7, On the other hand, the exact answer of 20/3 or 6+2/3 is trivial to obtain by traditional methods, but these students can do neither. They punch it into their calcuator and blindly write down 6.66666666666667 without regard to what they've just written.

You can claim all day long that it doesn't matter because you always have the calculator, however, it is that fundamental ability to reason things through in your head that allows you to succeed in technical fields. If the calculator doesn't give these kids an answer they don't know what to do.

Ok, not everyone needs to be in a technical field, and moreover, the students that have prepared themselves properly have the ability so maybe the point is moot.

But, the fact remains, replacing memorization with "discovery" because of a belief that it helps kids become better problem solvers doesn't follow when those that have had such training aren't able to solve problems.

There is a common misbelief that you can just look things up if you don't know them. While on one level, that's true, if you don't have enough of an idea of where to look you may not even know what to look for. I witness this regularly. Students claim they can just look stuff up but because their skills are so limited they aren't able to translate what they're looking for into any kind of querry. Instead of asking questions about the mathematics, they ask for hints on getting started, or the "next hint". What happened to discovering the process of discovery? These students are far less prepared to think for themselves than those who've been forced to develop their own algorithms for negotiating a problem space. One learns this skill by being stumped and having to use one's mind to find the path to the solution. But without even the most basic of skills this becomes a random search problem in a huge solution space.

Math students often complain about "tricks" that are used to solve a problem. There is a joke in math that says, once you've seen a trick three times, it's a method. We need to practice methods in order to develop enough tools to allow us to combine them in a sophisticated way to more difficult problems. If the first tool we learn is the calculator, we become dependant on it.

I interact with schoolteachers in training every day. They are among the worst students in the mathematics department. We dread taking certain classes because they are the teacher magnets. You are certain that the class will be brought down to ridiculuously low levels and that you will hear constant whining all semester.

Now, while I stated that my university is a no-name school, it is renowned for its "teacher program" and people are actively being recruited from it. Thus, these are not the "bad teachers", rather they are what the state considers to be the best among primary/secondary schoolteachers.

My point is that while the discovery idea might actually be valid it is not interpreted well by most school teachers. They are the college students who whine the most about taking anything difficult, who constantly complain that they shouldn't need to learn abstract algebra because they're never going to teach it, etc, etc. These teachers, who, in general, lack mathematical sophistication cannot
apply a sophisticated mathematical learning idea because they are incapable of discovery themselves. What is supposed to be an elegant road to discovery, ala polya, becomes, instead, a modern version of drill where instead of drilling to remember the facts, students drill on remembering the keystrokes to obtain the facts.

Discovery is important. But it must be balanced by memorization of the basic facts. When implementing any system in our public schools, unfortunately, we must either so thorougly develop the method that anyone could implement it, or, we have to significantly improve the quality of our teachers. It is my belief that without one or the other, sophisticated methods will not succeed.

We should not forget the lessons of new math (old new math, not new new math).
Anonymous said…
Finally a math forum that isn't controlled by the powers that be! There is still a remnant of us that derive the Quadratic Formula, teach classic Euclidean Geometry where one must prove a proposition before using it, and have genuine respect for the exactness and logical precision of the discipline that they teach. What is passed off as "discovery learning" involves very little true mathematical discovery on the part of the student. For example, the bestselling Alg. I book in the country gives the Quadratic Formula as a result in Chapter 9 and then teaches factoring in Chapter 10. When taught classicly the student learns to solve equations by factoring and then completing the square. When the instructor steps back and lets the student complete the square on a general quadratic a sense of discovery and ownership results. Math educators have been using the guided discovery approach since the days of Pythagoras and Archimedes. The vacuous "constuctivist approach" does not allow for genuine mathematical discovery;the students play with blocks and are spoon-fed anything of mathematical substance. If the educrats have their way we will all be flushed out of public education and anyone who dares to offer resistance to the ruling politburo will be set aside for re-education. Have a good day.

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