Why Can’t Students Use Prior Knowledge – Redux

March 31, 2011

OK, I get it – here is a snapshot of what I saw along with my comments on the work:

Student work on a system of equations

Solve by graphing -

Upon closer inspection, I saw that the only real error in the student’s work was a sign error in solving the second equation for y. She wanted to have the two equations in the familiar form y=mx+b, and then find three points for each line. That was her method – and if she hadn’t made that one error, her solution would have been completely correct. I was thinking “Gee, if she were lazy like me, she would not have first manipulated the given equations into slope/intercept form, maybe not made that mistake, and then would have had the correct graph and solution.”  But for what she did, her solution was actually correct.

So, now as I think about it, I see that this student is actually a great lazy mathematician – she has a method that works for her – she can’t use extra energy to apply different methods for each situation – she can find a solution quite nicely, thank you,  just doing it her way. And I should not have a problem with that – she found some points that were on each line – plotted those points – drew the lines – and gave her answer.

Then I noticed that she used the same x values for each line – WOW! What a great economy of time that works out to be – she doesn’t need to try and think of x values that give nice y values – she is happy to just plug away and plot points. Nice.

Why Can’t Students Use Prior Knowledge in Novel Situations?

March 21, 2011

In my intermediate algebra class, we spent almost 2 weeks studying lines, slopes, and intercepts. Students were instructed on several techniques for graphing lines: plotting random points, using the x- and y-intercepts, and finally using the y-intercept and two other points. So, students came away thinking that these are three different methods of graphing a line when actually they are the same thing – using points to draw lines. And they end up doing a lot of calculations that only encourage arithmetic errors of many types.

They were then asked to solve a system of equations graphically.  One student very painstakingly expressed each of the given equations in the slope/intercept form and THEN substituted values for x to calculate y. She didn’t get two things, at least, that I see.

First, expressing each equation in slope-intercept form makes it unnecessary to then plot points and get the graph. Once you know the slope and the y-interept it is easy to draw the line.

Second, if your plan is to choose a value for x and then calculate y, there is no reason to use the slope-intercept form to begin with. Just plug and chug in the given equation.

So, what could I have done differently to make sure that my students made that connection? Because I’m pretty lazy and I’m always looking for ways to make problems easier – I assume my students are the same way. But the fact is they have a lot of energy but expend it most inefficiently in their problem-solving.

Teaching Slope as Rise/Run

March 17, 2011

Our lesson today in the basic concepts of algebra course had to do with the slope concept. We had already spent some time on linear functions and investigated the notion that the rate of change of a linear function is constant – and that is what the slope of the line is. Side note: a few students were confused about the difference between a constant function and a constant rate of change – that can be a topic for future discussion here – I can’t afford to digress because as a lazy mathematician, I may never get back to the current topic – slope of a line.

So, we were going about it quite happily and figuring out our rises and our runs – focusing for today only on lines with positive slope. All of our rises were positive, an our runs were also positive. We worked with pictures like this one:

Line showing slope as rise/run

Positive slope -positive rise - positive run

Then our author did something that threw at least one student for a loop. He put this picture up as an illustration of the rise/run process for figuring out slope:
Positive slope - negative rise - negative run

Positive slope - negative rise - negative run

My very astute and inquisitive student wondered out loud: “Shouldn’t the rise and the run both be negative?”

Well, the lazy mathematician would have to agree – the rise and the run as pictured should indeed both be negative. To have our picture match what we lazy mathematicians do, we need to have both the rise and the run as negative numbers – else we get confused.

Why the lazy mathematician?

September 26, 2009

I often tell my students that mathematicians are motivated to do the things they do out of laziness.  To avoid writing long and complicated expressions, we invent all sorts of notational devices as a sort of shorthand – a shorthand that must then be taught to others who may want to learn a little about the mathematics we develop. It is a type of code or language and if others want to communicate with us as mathematicians, I explain, then they must first learn and be comfortable with our symbols.

But mathematics encompasses more than just manipulation of symbols (apologies to all you abstract algebraists and number theorists out there – I have utmost respect for your symbolic machinations!). To me it is about establishing a world view that is concise and precise – and so the laziness comes from not so much a lack of energy or a desire to avoid hard work as from a perspective that says “Let’s try to get this done with as little effort as possible.”

So, a goal in this blog is to show you a little about how we do that – the secrets and apparent tricks we employ to make our work easier. Unlike the magician who wants to keep his tricks to himself, I want to show you the tricks and help you to avoid the traps. I will be sharing the things I find that give my students trouble with the hope that you can learn how to avoid those troubles – minimizing the effort you must expend by showing you how to avoid those blind alleys and false directions you may have traveled down before. But always remember, an occasional trip down the wrong path often leads to a surprising discovery – we’ll take that trip together and it won’t be so scary!