kitchen table math, the sequel: question

## Tuesday, January 15, 2008

### question

This is a copy of an email I sent out to my local math list shortly before the last algebra test:

I found a nice group of “linear function” word problems today (answers included - pdf file).

The kids in Math A are doing these problems. They aren’t classic Algebra 1 word problems, so if your child needs extra practice they’re not easy to come by.

As I understand it, algebra 1 used to be focused on setting up and solving linear equations. The classic Algebra 1 word problems were number problems, consecutive integer problems, distance problems (2 trains leave a station), coin & age problems, & mixture & work problems.

Today algebra 1 tends to be focused on the idea of linear functions. That’s why our kids did all those “function machine” problems back at the Main Street School. In a linear function problem you are given a pair of “solutions” and asked to write an equation to model the situation. My favorite linear function word problem thus far is the Celsius – Fahrenheit conversion:

The graph of an equation to convert degrees Celsius, x, to degrees Fahrenheit, y, has a y-intercept of 32°. Given that water boils at 212°F and at 100°C, write the conversion equation.

To write the equation you use the two coordinated pairs you already know:

When Celsius is 0 degrees, Fahrenheit is 32
When Celsius is 100 degrees, Fahrenheit is 212

These two points are coordinated pairs on the graph of the function:

(0,32)
(100,212)

You derive the slope of the linear equation from these two points; then you plug in one of the coordinated pairs and derive the y-intercept; et voila.

y = 9/5 x + 32

Or:

F = 9/5C + 32

I've never been able to remember how to convert from Celsius to Fahrenheit, and knowing how to use the two known conversion points to produce the formula is great.

I'm curious about people's reaction to this shift in content. One of our regulars is skeptical; he sees this as a move to turn algebra into statistics. Another sees this as another instance of pushing advanced topics lower down in the curriculum. Functions used to be introduced in calculus; now kids see "function machines" in 3rd grade. (And, of course, both motivations could be involved - both and more.)

Any thoughts? Anonymous said...

Functions are a huge, huge topic in academic math. Our 1960s algebra program that we are using covers both linear and quadratic functions. After algebra they study trig functions and in general the study of functions in academese is called "Analysis." A mathematician who specializes in functions is an Analyst (which means something completely different in the ivory tower than it does for the rest of us peons, kind of like the way the word "rigor" means something different to the mathematicians than it does to us) If you look at the titles of Pre-Calc books in the 60s and 70s they were called such things as "Introduction to Analysis." And now someone is going to have to step in and explain this to me, but it seems to me that they call it "Analysis" when it's the proofy study functions (but since 60s and early 70s math was proofy they called it Analysis) and it's called Calculus when the theorems are applied for calculational purposes to solve science and physics problems rather than studying the math involved theoretically.
The functions are all about getting the kid ready for Calculus and putting him on the engeering/physics road.

As an aside, our (us personally and our kid, not as society)emphasis on functions is from the perspective of sets and the relationship between ordered pairs, which is what motivated the teaching of all the set theory back in the 60s to begin with. Today set theory is some sort of supplemental topic that comes up in the last chapter of the book, there seems to be little or no connection made to functions and kids are trained to think of a function being defined by what the picture looks like, rather than any formal definition...not that they don't get a formal definition of the function...at some point, maybe in the 12th grade, but for the five years before their idea of a function is motivated by a picture of a line on graph...probably after that too. I probably got the formal definition of function in college, but whenever I wonder if something is a function or not I think "Can I draw a vertical line through this graph...etc, rather than seeing if it meets the test of the formal definition. In other words, relying on the pictures taught me to think fallaciously, and the TA taking a five second aside to give us a formal definition did NOT undo the years of looking at pictures. I suffered when I was forced to think about it correctly in chapter negative two of an Abstract Algebra text I picked up.

As usual, I suspect that ultimately the beef with functions in algebra isn't so much with the topic being covered at an early age but that they are skipping over all the necessary steps that make that topic understandable to the students...formal defintions will left out because they are "too difficult to understand" (read: we don't want to take all the time it really takes to develop this appropriately) Anonymous said...

Another sees this as another instance of pushing advanced topics lower down in the curriculum. The topic of functions was in algebra since the '60s. Sputnik motivated the US to start cranking out engineers and physicists as soon as possible. What may be happening is that it's taking up more space than it used to.

Functions used to be introduced in calculus; now kids see "function machines" in 3rd grade. They used to do set theory in 3rd grade as well. I've heard of fractals in the 3rd grade, the other day I ran across a lesson plan that claimed to be teaching Peano arithmetic to kindergartners! (It amounted to the kids counting out objects one by one but the teacher who wrote up the lesson plan did it with a straight face.)

I suspect someone is getting complaints that the kids "can't do functions" and so whatever it was that didn't work in the 10th grade is now being pushed to a younger age and expected to work. An analogy would be teaching the long division algorithm in the second grade because the fifth grade teachers say "the kids can't do long division" And that may be true, but it will turn out that they can't do long division because they don't know their multiplication table cold. Anonymous said...

Okay, so I'm being told that I'm still thinking about this wrong. "A function isn't the relation between two sets it is a set."

That makes my head hurt and it's much more fun to draw pretty lines on graphs.

SteveH said...

"As I understand it, algebra 1 used to be focused on setting up and solving linear equations."

"Today algebra 1 tends to be focused on the idea of linear functions."

I had functions in algebra 40 years ago, but I don't remember enough to know whether it was developed properly. I do remember wondering why they switched so cavalierly between a functional and explicit representation.

One could have

F = 9/5C + 32

or

f(C) = 9/5C + 32

Both are really the same animal, almost. A function form, however, is used for a limited set of problems. It maps one value into another - one to one. You could also have a function that maps two or more variables into one value.

However, you can't use a functional form to define a circle like this:

x^2 + y^2 - 16 = 0

For each value of x, there are two y values. This is not a one-to-one mapping. This form of an equation is called an implicit form. A functional form is called an explicit form. You can also have something called a parametric form.

You can use whichever form best suits your need. Here are the three forms for a linear equation:

Explicit or functional
y = mx + b
or
f(x) = mx + b

Implicit (a, b, & c are constants)
ax + by + c = 0

Parametric form
x(t) = x1 + (x2 - x1)t
y(t) = y1 + (y2 - y1)t

Where t varies from negative infinity to positive infinity, and the line passes through the points (x1,y1) and (x2,y2). 't' also varies from 0 to 1 between the two points.

Note (ironically?) that the parametric form uses two functions (using the variable 't') to define one line.

Each of these forms for describing a line has its uses. This is NOT something I was taught in algebra. I wish it was.

"Functions used to be introduced in calculus; now kids see 'function machines' in 3rd grade."

No, functions were (and should be) introduced in algebra.

I remember my son doing "function machines", but it had little to do with an algebraic understanding of functions. I think it had more to do with some strange attempt at discovery or guess and check. You know, those things that fuzzies use in place of learning real math.

The shift in content I see has more to do with replacing real algebraic mastery of basic techniques like expanding, factoring, and solving equations, with a more descriptive or picture-based approach. Algebra by graphing calculator and pictures, not symbols. Anonymous said...

I'd bet money the shift in content has to do with the graphic calculator, nothing else.

Catherine Johnson said...

Interesting -- thanks so much. (More comments to get pulled up front.)

Allison could be right, too, although it's definitely the case that the various K-5 curricula teach "function machines" now.

Actually, I don't know that, come to think of it. C. had SRA Math, not Trailblazers. SRA had lots of function machines.

C's face lit up when I pointed out to him that the "x / y" coordinated pair charts he was making were the exact same thing as his 4th grade function machines.

Instructivist said...

Fuzzy math is obsessed with patterns. Aren't patterns function machines?