Comparing Methods

[mmm4444bot]

Mark

I know that your way is the traditional way to teach algebra: try to keep the number of variables to one as long as possible. I disagree that that is the best way to teach it. (What do I know: I never taught anyone's kids except mine, and then only until my son's math studies started to go beyond mine.) The traditional way essentially requires kids to do substitutions in their heads as they formulate the problem. I'd rather teach kids the mechanics of systems of linear equations right up front and then teach them to do word problems as a three-step process of (1) identifying relevant variables or unknowns and symbolizing them with letters, (2) translating the conditions of the problem from words into equations using those symbols, and (3) solving the resulting problem in pure math.

If I have identified the bees travelling north as n and the total number of bees as b, finding the equation that n = (1/5)b is very easy. If I am trying to start by formulating a single equation in b, I have to keep a lot of things in my head at once.

There is nothing wrong with presenting rigor, Jeff! I respect your post, in that thread.

If a student were to benefit from any particular approach, then my hope for them is that they're exposed to it (and put effort into it).

It could very well be that this original poster needs a bridge, or sequence of bridges, to arrive at a point in their thinking where they've chewed the givens into some new form that makes the solution strategy obvious to them. Step-by-step substitutions may very well build those bridges, for them.

In virtual tutoring-environment, we lose interpersonal communication and all of the non-verbal clues that go with it. Sometimes, people trying to get help appear as little more than a blank page, to me. When people volunteer nothing, maybe a generally-good (initial) response is for various contributors to toss their ideas at the blank wall and let the original poster decide how to clean up and proceed. Hopefully, some of what they read clicks.

I have no teaching experience in mathematics. I have no formal training in the science of education, either. I rely on life experience and my gut, when considering replies. (As of late, I should also rely on putting some pencil to paper, too!)

I'm glad that your post above bumped that thread. I noticed and fixed a typo in my equality. :cool:

 

[Bob Brown MSEE]

Blank pages are like blank stares

In virtual tutoring-environment, we lose interpersonal communication and all of the non-verbal clues that go with it. Sometimes, people trying to get help appear as little more than a blank page, to me. When people volunteer nothing, maybe a generally-good (initial) response is for various contributors to toss their ideas at the blank wall and let the original poster decide how to clean up and proceed. Hopefully, some of what they read clicks.

This is a very though problem for a Help Forum and first time OP? (what the heck does OP stand for?). Most first-time students resort to on-line help, in a crunch. Self imposed, but none-the-less a crunch. They want us to do ALL the work, the better of them will try to learn from the example if we do. When we do not, we are likely to lose them -- for ever. We could do one of the following to mitigate this ...
1) Just work the problem for first timers (unfair to others and most of them)
2) Be extra friendly, help them believe (my approach -- not sure it helps)
3) Make rules clear and accessible (this site is good here)

I'm not sure any of these are effective for first timers. At the point where trying to conform is harder than getting help elsewhere, they leave.

 

[Bob Brown MSEE]

My kid is also a blank page -- to write on

I never taught anyone's kids except mine,

My son's technical/maths elementary education was home-schooled.
Altogether different environment than this Message Board experience.

My son was accepted as a full-time student at CSU at the age of 12.
First Semester: Tested out of 4qtrs of freshman physics, tested out 2Qtrs of Circuit analysis.
First Year: placed 30th in Putnam. Won CSU Most Valued Math student
He now heads internal tools development at Google at GooglePlex.

I'm not sure there is a best way for math training. I do know the following are powerful.
1) LOVE the child.
2) Strong concept of where the child seems to be going.
3) Be very careful about what's next.
4) Full Credit -- child's rapid discovery is natural, we kill it.
5) Very young: Attention span is the key issue.
6) Older: Always leave them wanting more.

Sorry for the authoritative tone. But as Monk says, "I could be wrong..., but ..., of course I'm not."

 

[JeffM]

There is nothing wrong with presenting rigor, Jeff! I respect your post, in that thread.

If a student were to benefit from any particular approach, then my hope for them is that they're exposed to it (and put effort into it).

It could very well be that this original poster needs a bridge, or sequence of bridges, to arrive at a point in their thinking where they've chewed the givens into some new form that makes the solution strategy obvious to them. Step-by-step substitutions may very well build those bridges, for them.

In virtual tutoring-environment, we lose interpersonal communication and all of the non-verbal clues that go with it. Sometimes, people trying to get help appear as little more than a blank page, to me. When people volunteer nothing, maybe a generally-good (initial) response is for various contributors to toss their ideas at the blank wall and let the original poster decide how to clean up and proceed. Hopefully, some of what they read clicks.

I have no teaching experience in mathematics. I have no formal training in the science of education, either. I rely on life experience and my gut, when considering replies. (As of late, I should also rely on putting some pencil to paper, too!)

I'm glad that your post above bumped that thread. I noticed and fixed a typo in my equality. :cool:

Mark

My post sounded both more antagonistic and more defensive than I intended. Let me try again.

We require kids to study math for the excellent reason that it is useful for many practical purposes in science, technology, and business. In those practical realms, the problem is seldom presented to us in mathematical format. Therefore, it is absolutely necessary to present word problems to kids in order for the effort of teaching algebra to have a useful result. (I sometimes comment at another blog that is advocating eliminating algebra as a required course because so few kids actually learn it!!!!!)

But kids seem to find word problems much harder than the mathematical manipulations required to solve pure math problems. That is, they can solve equations much more easily than they can set them up. That suggests to me that it would be beneficial to teach algebra with an eye to making the formation of relevant equations as intuitive as possible. It further suggests to me that a much earlier introduction of solving simultaneous linear equations by the method of substitution might result in better results. Of course, we have to deal with kids who are being taught by various methods; we must deal with the world as it is, not as it might be.

With all that said, I agree with you wholeheartedly that many of the posters are likely to benefit from seeing multiple approaches to the same problem. One approach may click.

 

[JeffM]

Bob

My kids no longer have to worry about my deficiencies as a teacher unless they elect to inflict my instruction on their kids.

PS OP means original poster.

 

[Deleted member 4993]

1) LOVE the child.
2) Strong concept of where the child seems to be going.
3) Be very careful about what's next.
4) Full Credit -- child's rapid discovery is natural, we kill it.
5) Very young: Attention span is the key issue.
6) Older: Always leave them wanting more.

Three more things (and I know Denis-the-hockey-puck will concur):

7) Practice

8) Practice

9) Practice

My grandfather (who taught me math) used to say (in Bengali) "if you don't get it by doing 10 problems - do 100 problems....and 500 more..."

But then we did not have TV, I-Pod, nothing. Very deprived childhood - only books, balls and sticks, poetry and theater ... and such.....

 

[JeffM]

Hmmmm, and Hmmmmm, and Hmmmm again

3a + 5 = a + 13
Subtract 5 from each side:
3a + 5 - 5 = a + 13 - 5
3a = a + 8
Subtract a from each side:
3a - a = a - a + 8
2a = 8
Divide each side by 2:
2a / 2 = 8 / 2
a = 4
YES; agree that above is "nice" when student is "introduced" to solving.

BUT no student in his/her right mind will keep doing above once
they "understand" what's going on:
3a + 5 = a + 13
3a - a = 13 - 5
2a = 8
a = 4 ; over and out!

Same thing with "number of variables"; why use 2 or 3 of them
if 1 will do....once the student understands, of course.

If a student "sees" how to set up an equation in one variable, then of course they will and should do so. But the situation that I am discussing is when they do not "see" how to do a word problem. It is then that a systematic method is more than "nice."