kitchen table math, the sequel: getting ahead of ourselves

Monday, April 18, 2011

getting ahead of ourselves

from purplemath:
  • Given that f(x) = 3 x 2 + 2x, find [f(x + h) – f(x)] / h.
This is actually something you will see again in calculus. I guess they're trying to "prep" you for upcoming courses when they give you exercises like this, but it's not like anybody remembers these by the time they get to calculus, so it's really a lot of work for no real purpose. However, this type of problem is quite popular, so you should expect to need to know how to do it.
I'm a big fan of purplemath.


Allison said...

--but it's not like anybody remembers these by the time they get to calculus,

No, people DO remember. They may not remember they remember, but their brain is quite comfortable thinking about this if it's had time to do so.

-- so it's really a lot of work for no real purpose.

This flies in the face of what we know about learning. Learning comes from getting enough practice with stuff like this so that when you see it presented towards a new goal, your brain can connect synapses together. That's why we can make things make sense.

Catherine Johnson said...

Makes sense --- but do these function problems 'crowd out' more standard algebra 2 problems students need to become proficient in?

It's not enough to work with function notation and do simpler functions?

(I'm asking - I don't know.)

Allison said...

I don't know who is teaching what. Certainly I wouldn't teach this example of function notation as a starting point, nor would I teach it in lieu of teaching simpler ones. But what we teach is so weak, so artificially depressed. We should be teaching the key elements to mastery so that undertaking this problem is a workable amount of hardness, not in vain.

Independent George said...

The link is broken, so I don't know the context of where this appeared, but this is actually really important, and really, really easy to screw up if it's not taught incrementally and carefully. Again - it depends on where & where this is being taught, and how it's followed up, but getting functions right is useful not just in calculus, but computer programming.

Catherine Johnson said...

link should be working now - sorry!

Catherine Johnson said...

I'm wondering about a possible cart-before-the-horse phenomenon...where much more difficult material is 'taught' but not really taught, while the material you should be learning is given short shrift

(Obviously I can't make a judgment, but the phenomenon is real -- we've experienced it ourselves.)

Allison said...

Barry certainly talks about that phenomenon in his essays. It's a feature, not a bug, according to ed school. It's part of his "let's move on" piece, iirc--we won't teach the underlying skills, we'll just kinda introduce a just-in-time attitude to solving math.

But it's difficult to tell how things get implemented without seeing them implemented. It's a good idea to motivate work with functions; it's good to explain what you're doing and where you're going so a student can learn why they are being asked to do the simple stuff. Then, with those whys and motives, a good teacher is going to teach skills in a step by step fashion to show that This Hard Problem isn't so hard at all, and this hard problem has value, which is why we need to get to where we can solve it.

Can many teachers do that? Do they see all of those pieces? Do they see if their students are connecting the dots? Do they even know why they should teach something, let alone how to explain why to a student at the student's level?

I looked at the purplemath site and understand less of the context than before. There seems to be a teacher talking, and then suddenly the teacher isn't the teacher but says someone else wants you to solve these problems.

Worse, their notation and explanation is awful. "You can evaluate functions at variables or expressions, rather than merely at numbers" is so muddled that it's best to throw out the explanation.

f(x) and x are not defined. Domain and range aren't defined. Is x a parameter or a value? If you handwavily act like it doesn't matter, you'll understand precious little math as you go forward. What is the domain for x? What is h? Is it a parameter or a value? etc.

Independent George, you're right about the importance of understanding functions and evaluation of functions, and it's not limited to computer science. Understanding operators acting on vectors can't happen without understanding functions, so basically every engineering discipline rests on this as well as all of physics and math.

Anonymous said...

I think that the point is that even if you aren't studying the details of this sort of problem now, you should be able to do this calculation now. If you can't do the calculation, this is a red flag that you need more work with functions before you are ready to move to calculus. This is a very nice "formative assessment" problem: Have you learned the skills that you will need in the future?