So we approach problems and ways to solve them, not algebra and then problems using it. It seems like a subtle change but the emphasis is incredibly different. In MLCS, algebra is a means to an end, not the end itself. We value algebra tremendously. But there is an overemphasis on this wonderful subject at the developmental level. I've often thought calling this area "developmental math" is a misnomer. It should be called developmental algebra, because that's what we spend 80% of the time working on. The remaining 20% is spent on numbers, geometry, and applications.

In reality, students will face problems with numbers in their lives and future classes, not necessarily algebra problems. Knowing when and how to use algebra matters more if we really do believe this subject is important. But that's not the traditional focus. It's just on the mechanics, as though linear equations and polynomials are falling from the sky waiting to be solved and factored.

Another reality is that in real life, the numbers aren't nice. Polynomials exist but they often don't factor. And graphing by

*m*and

*b*is a lovely idea but real data doesn't make it nice to do. And it's often possible to come up with an equation that is not easily solved without technology or numeric methods. We do students a disservice by shielding them from these truths. And in teaching them how to deal with them, the problems are far more interesting and students are more engaged.

Our approach is to always begin concrete and with numbers. Bring in the algebra when it will make the problem easier in terms of solving and/or organizing the details. Students will gravitate toward it often if they are not forced to use it because it is so powerful and often helpful. And sometimes algebra completely obscures the issue at hand and is overkill. Again, knowing when to use it is just as important as being able to use it.

This can be a difficult shift at first, especially if an instructor has taught traditional algebra for many years. Nearly every book has the same ordering of topics and the same emphases. So it's unusual at first to see commutative property after integer operations have been addressed. But students can still be successful with integer operations because they've been using the commutative property since first grade as a natural behavior for numbers and certain operations. It's not the name that matters; it's the understanding of the concept so that it can be used. Intuitively, students can solve many problems without the formality we often put on them.

Take Pythagorean theorem for example, a topic I taught this week. We came at the theorem and its uses from every angle. Using it requires algebra but really an understanding of numbers, something we've been working on for weeks, will suffice when solving the equations involved. When we get to equations like this, this is the approach we used since we haven't gotten to formal equation solving:

15

^{2}+ leg

^{2}= 22

^{2}

^{}Simplify the exponents.

^{}225 + leg

^{2}= 484

We need to find what adds with 225 to achieve 484. That can be found by subtracting 225 from 484. This approach uses the concept of addition and subtraction being inverse operations. The result is 259.

leg

^{2}= 259

The length of the leg will be a positive number and it should square to be 259. That is the definition of the principal square root of 259. We write that next and then use the calculator to get a decimal approximation.

leg = √259 ≈ 16.1

It was also interesting when using the calculator for this problem. Students used the square button and then the square root button. I asked them if they notice anything about the placement of those functions on the calculator we typically use, the TI-30. They are paired on one button with one function being the direct operation and one requiring the 2nd function to get to it. I told students this is not a coincidence and asked if they could figure out why. A student's eyes lit up and said, "they're inverses of each other!" Exactly! That's a valuable concept to know.

Can students solve problems like this without formal equation solving techniques? Absolutely! And it makes sense because we are working at the numeric level. When we get to the formality of algebraic manipulation, it is always easier because a strong foundation in numbers has been established.

Some students remember equation solving from a previous course and prefer to use those steps. We encourage them daily in this course to use methods that make sense. Every concept that is taught uses multiple approaches to understand it, be it visual, numeric, algebraic, or verbal. After we learn those methods, we ask students to reflect on the method they prefer going forward. Learning how they learn provides a student tremendous insight on their understanding and is very helpful throughout college.

There are many people who feel this way about algebra, that our focus could use some updating. But often they are the ones using the mathematics instead of teaching it. I think it's worth listening to the people who use math for a living. Of course not every concept has to be immediately useful to be worth learning. But with the current traditional approach, there is too little that is useful and too many topics developed in a way that is unrealistic in terms of the real world uses. For more on this idea, please watch this short video by Richard Feynman, Nobel prize winning physicist.

Spot on! But the 'formalization' cannot be overlooked. There eventually needs to be a precise development of the tools being used, which is something that often gets overlook in this sort of 'discovery learning' approach.

ReplyDeleteFor example, sqrt(x) and x^2 aren't technically inverses. They only are if we limit the domain of x^2 to all x>0. While most people don't need to know this, it is a very important concept that--if not understood-- will lead to misunderstanding later in more advanced mathematical topics.

...Before you know it, Mathematics will be set back a couple hundred years. (I jest, of course--but still) The problem is mathematics is very multifaceted. There are the pure mathematicians doing their thing, and there are those who apply it doing there's. There are those who need to trudge through it to prove themselves worthy of a degree and then never use it; and, there are those who never need it. Yet the approach to math education is very straight and narrow--partly because when we are teaching the foundational concepts in secondary schools we do not yet know where everyone is going to end up (nor do we want to limit them by sending them off in one direction or another)

Algebra is already a 'gatekeeper' (those who pass it move on and get BS degrees, those who don't find a BA degree (again I jest--sort of)). Why not diversify how we teach it to fit the goals of those learning it--the mathematicians, the scientist, the doctors, the "burger-flippers"?

A: Because then you'll quickly find yourself in a class system and we'll be right back in dark ages of feudalism.