So here are the things that are important to me in my classroom and on my blog (in no particular order):
- Seeking to understand,
- Basing our arguments and claims on evidence,
- Asking honest questions,
- Looking for meaning, and
- Using only names and labels that make useful distinctions.
Despite my intentions, I do not honor each of these on a 24/7 basis.
example 1
See, for example, my claim that “Everyone knows that there are 2 Pop Tarts in a pack.” gasstationwithoutpumps called me on it. I had no evidence for that claim (principle 2 above), and making that claim seemed to suggest that I wasn’t seeking to understand the perspectives of readers (principle 1). I took my well-deserved lumps.
example 2
I recently found myself having to backpedal in College Algebra when a student was explaining her thinking. I basically cut her off and finished what I thought was her thought. It was not. I was not really seeking to understand her in that interaction, nor was I asking her honest questions. I realized these things, apologized and told her that I was going to stop guessing what she was thinking and listen instead.
Why this matters
We teachers tend not to think about how much power we really have to set the tone in our classrooms. I was reminded of this when a College Algebra student recently asked plaintively, “But what does it mean?” And when one student asked another in another course, “Wait…did you get your question answered?”
And, by contrast, when two students in the next class (not mine) were watching me erase the previous class’s whiteboard, on which was written “1/16+1/16+1/32=5/32”. The first student couldn’t fathom how the right-hand side of the equation resulted from the left. The second student explained, and told the first, “You have to kind of dumb yourself down to see it.”
Those first two examples are the result of my hard work setting the tone in my classroom. The last examples are reminders that this tone doesn’t just happen on its own.
And I am sure the same is true of our virtual forums as well.
Getting down to business
I knew I’d get a few readers riled up about cross-multiplying this weekend.
If an algorithm can be political, this is the one.
Cross-multiplication is to algorithms what Herman Cain, Michael Moore and Ron Paul would be to politics if they were all one political entity. Its alleged misdeeds are numerous. Its defenders are passionate. It never, ever retires and goes away.
gasstation wrote to defend the algorithm:
I think that the cross-multiplying approach becomes more useful when there are variables in the denominators
I wanted to encourage principle 2, Basing our arguments and claims on evidence, and did so with a light-hearted jab.
I’m gonna go ahead and forward this to the OMT Dep’t of Specificity. They’ll work it over to decrease the vagueness of the claim. Maybe even give us an example to chew on.
Michael Paul Goldenburg provided an example:
Say we’ve got 3/x = 1/6.
Now we have something to work with. But I’m not buying that this is the sort of thing gasstation meant.
It can’t be, can it? There was nothing about the algorithm I proposed that stipulates we need to compare denominators. Nope, my algorithm is couched in making comparisons among numbers with meaning. That proportion above? It represents two triangles, one with a short side of length 3 units and an unknown side length for the long side; the other with short side of length 1 unit and long side of length 6 units.
Scale factor from small triangle to large is 3.
If what we’re building is an algorithm based on meaning (principle 4 above), this hasn’t convinced me to change my blueprint. I need better evidence. Someone’s got something chambered, I’m sure. Let’s see it.
postscript
In the spirit of principle 5 above, I hereby and respectfully express my request to refrain from the following practices:
- Taking or implying sides in the Math Wars of yore,
- Lumping together groups of teachers in order to make claims about origins of problems in the teaching and learning of mathematics, and
- Using the term constant of proportionality as though it were something different from slope.
Overthinking my teaching 
I won’t presume to put words in gasstation’s mouth, but a couple of proportions that just don’t scream out for the scale factor method to me would be:
x/(x+2) = 3/4
(x+1)/(x+2) = (x+3)/(x+5)
Of course, some people seem to define proportion as an equation between two ratios, where three out of the four quantities are given. Scale factor would always make sense with that restriction.
And I have to respectfully disagree with your PS3. It’s not always x and y that have the have the proportional relationship. Just last week, I was teaching a lesson with differential equations, with set ups like “the rate of growth is proportional to the the population and difference between the population limit and the population” yielding dy/dt = k y (L – y). So talking about slope in this context begs the question “slope of what?”
Kate:
Right. Calculus. Lovely.
You could have busted me with the simpler “rate of growth is proportional to the population,” right?
So do you think this is the origin of constant of proportionality being in the seventh grade Common Core? And if so, is it justifiable there? “They’ll need it for calculus” is a pretty lame reason to put something in the seventh grade curriculum, since a pretty small percent of students ever really will need it for calculus.
In other words, Kate argues convincingly that constant of proportionality is different from slope in Calculus. Is there a principled difference in seventh grade? Or might it be better to save the generalization of slope to constant of proportionality for a time when we need the distinction?
Hey, it was just fresh in my mind. Also the reason I spouted off the logistic growth model, instead of exponential. 😉
And, no, I don’t think that a topic that doesn’t come up until calculus should determine seventh-grade language, but are you sure that y = k x is the only proportional relationship that they will encounter for awhile? Not y = k x^2, or y = k/x? I’m not involved with middle school math at all, but it seems like back in the day I was introduced to direct and inverse proportionality at around the same time. Does that not happen anymore? Because the constant of proportionality is only the slope with direct proportions.
Actually, I thought that Michael Paul Goldenburg did a pretty good job of clarifying what I meant. I’ve been vaguely troubled by this post for most of this week, but unable to put my finger on precisely what was bothering me.
I think it was this: “That proportion above? It represents two triangles, one with a short side of length 3 units and an unknown side length for the long side; the other with short side of length 1 unit and long side of length 6 units.”
You’ve assigned a geometric meaning to an algebraic formula that makes your approach look reasonable. Fair enough—if the algebraic formula came from that geometric problem. If the algebraic formula came from some other problem, then forcing that geometric interpretation onto it is harmful, rather than helpful, as it requires multiple changes of abstraction: from the original problem to the algebraic formula to the geometric interpretation back to the algebra, then back to the original problem.
I rarely have my algebra problems come from geometry (and I’m a lot better at algebra than I am at geometry), so forcing a geometric interpretation in order to get a scale factor seems awkward to me. Talking about scale factors in geometry makes a lot of sense, but seems like an awkward way to simplify algebraic equations in other contexts.
gasstationwithoutpumps:
I couldn’t agree more. But it’s not just geometry that gets us the scale factor. I chose that since it was the context of my proportion in the original post (and I chose geometry there because I was struggling to work with the given numbers in any other way).
The only challenge Michael Paul Goldenburg‘s proportion seemed to present to the original post was putting x in the denominator: 3/x = 1/6. Let’s tweak it so that we don’t have a ‘1’ anywhere. Let’s make it 3/x = 2/6.
In a purchasing context, two dollars get me six oranges; how many oranges can I get for three dollars? Scale factor still applies, although not in a strictly geometric sense. I’m scaling my money up by a factor of 1 and 1/2, so I need to scale the oranges too; I get 9 oranges for three dollars.
The same critique applies to cross-multiplying (only even more so):
If you cross multiply, you get 18=2x. What’s the meaning of the 18? Taking a units-based approach, it’s 18 orange-dollars.
Of course that’s not right. It’s not 6 oranges that I’m multiplying by 3 dollars. It’s a scalar of 6 that I’m multiplying by 3 dollars. The scalar of 6 was chosen conveniently so that my denominators would be the same. That 18=2x means “The price of 18 oranges is the same as the price of 2x oranges (and we’re ignoring the price, which would be the denominator, and which incidentally is 6x).”
Alternatively, we could think about it the other way. Maybe we’re making common numerators, and 18=2x means “The number of oranges we get for 18 dollars is the same as the number of oranges we get for 2x dollars (and we’re ignoring what this number of oranges is, which would be the numerator, and which incidentally is 6x).”
In any case, I am more convinced by Kate MacInnis‘s proportions, including: (x+1)/(x+2) = (x+3)/(x+5). The scale factor reasoning here doesn’t resolve the fractions. I’ll end up with (x+1)=(x+3)*((x+2)/(x+5)). In this case, I still don’t want to cross-multiply with students, though. I’d much rather get common denominators, then notice that equality implies the numerators are equal.
Like “FOIL” I contend that cross-multiplication is a meaning-free mnemonic device for which we have better instructional alternatives.
I agree with you on FOIL. I never learned that mnemonic, and when it was explained to me it seemed more difficult than the distributive law and much less useful.
It seems to me that the fundamental idea in handling equations with fractions is multiplying both sides of the equation by the same thing. Common denominators, which are essential for addition, are likely to confuse students when multiplication is involved. Cross multiplication is a shortcut that may be dangerous to teach to kids who don’t understand the fundamentals, but is certainly easier to manipulate mentally than fractional scale factors.
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