My fellow teachers of College Algebra. I want to talk to you today about domain and range. For now, let’s leave aside all the analogies, vending machines, notation and ants crawling on graphs. Let’s get to the heart of the matter with an assessment/instructional task.
This is in the spirit of Eric Mazur, but it is low-technology. It will take 10 minutes. Then you’ll need to report your results back to me. We can talk about what those results mean.
Here is what you do.
Get yourself a College Algebra class that has studying (or even one that is still studying) domain and range.
Then get yourself some index cards in two colors. We used yellow and pink. You may use whatever you have on hand. Make clear that yellow means yes, and that pink means no.
Also make clear that they will raise their cards in unison and on the count of 3. This is to prevent conforming to the majority and will result in more honest representations of your students’ understanding.
Practice this routine on some non-mathematical questions. Some that will be universal nos, some universal yeses, some that are mixed.
Now have your students consider the function , where x is interpreted as the input, and y as the output in the usual way. You are going to ask a series of questions about the range of this function.
Ask, Is 4 in the range of this function?
You should get near universal yellow. Ask someone to state the case and make a note of their argument on the board.
Ask, Is –2 in the range of this function?
My bet is that you’ll have a lot of pinks, but several yellows. Those yellows probably need clarification that we are asking about range, not domain. But don’t assume it. Ask the pinks to state their case. Ask the yellows to refute or question. Never say anything a [student] can say.
Likely, the yellows will argue that there is no number that, when squared, gives -2. They mean real number; I see no need to make a big deal out of this. But make a note of the argument. Perhaps writing something like this on the board:
You will probably need to revisit this later on.
Ask, Is in the range of this function?
This is probably all yellow and mostly unproblematic.
Now the fun begins.
Ask, Is π in the range of this function?
If you don’t get a good mix of yellow and pink here, I will eat my hat. And those pink people? They are going to tell you that there is nothing that—when squared—gives π.
Have them talk it out in pairs or threes. Then have them show cards again. And then have the pinks state their case. Nine times out of ten, it’s going to be that there is no number that can be squared to get π.
My fellow College Algebra teachers, I am not interested in your theoretical arguments about what a fabulous job you/your textbook/your online homework platform are doing at teaching domain and range. If you wish to claim that your students will not show pink for π here, the burden of proof on you is high.
Notice with your students the very important difference between:
nothing I can think of
Someone will point out that is a number, and that when you square it you get π. Highlight that contribution and estimate the value of this number.
Ask, Is 0 in the range of this function?
Probably mostly yellow, but worth asking to make sure.
Ask, Is infinity in the range of this function?
Seriously. If you don’t get, like, 80% yellow here then I do not understand your school’s placement system
Reinforce that infinity is not a number. Connect it to the notation and wrap up with one more.
Ask, Is 12 in the range of this function?
You should get nearly all yellow here. Get back to your regularly scheduled classroom activities.
Then report and discuss your results with colleagues, on Twitter, on your blog and here.
There is lots of unexpected pushback in the comments on the value of teaching domain, range and functions in a College Algebra course. I had previously thought these to be de rigueur topics in such a course. I suspect sampling bias here.
Gregory Taylor mostly reproduced my results in his own course and had some lovely mathematical conversations along the way. You should go read his account.
Wish I had a class studying domain and range. I will have to settle for vicarious enjoyment.
I will be studying domain and range later this trimester. perhaps in a month. Is that too late to participate in this?
It is never too late, mathybeagle. Never too late. I look forward to hearing what you learn.
Is your point that students don’t really understand irrational numbers? Is your comprehension of my school’s placement system something I need to
No, xiousgeonz, my point is not that that students do not understand irrational numbers, although this is probably also true. My point is that we probably fool ourselves that teaching domain and range is about getting the notation right. Students (my students) can be trained to write 0≤y<infinity, as the range of a quadratic, but they do not get what this means. That seems like a failure on our part.
But the good news is that we can design activities—uncomplicated ones—that put the important ideas out in the public realm for all of us to examine. Telling and quizzing does not tend to get those ideas out for examination, and I suspect we—as a group—do too much telling-and-quizzing and not enough talking.
My point is that the activity I describe here is fundamentally different from what occurs for many college algebra students, and that this activity reveals hidden misunderstandings.
And, no, my comprehension of your placement system is not something you need to concern yourself with. It was meant to suggest the strength of my conviction about this particular characteristic of College Algebra students (as a group)—namely that they do not understand much about infinity and are likely to think of it as the biggest number there is.
I skip most domain and range stuff. Can you help me understand why I should give it time?
I assume you don’t skip functions. Domain and range basically are a way of explaining where functions function. I think you need to know where a function is a valid thing in order to understand how to use it.
Our textbook has a whole chapter on functions that I skip. I might mention domain and range in passing, when it comes up.
For example, when explaining inverse sine, I showed how the input to sine is angles between 0 and 90 degrees (when we use the right triangle definition) and the output is a ratio between 0 and 1. I mentioned, I think, that those are called domain and range. There was the big idea of inverse function that I wanted to help students understand, so even though I was mentioning the domain and range of sine, I’m not sure I ever used those words.
We will definitely discuss the domain and range when we work on exponential functions. However, I never test on that vocabulary, nor on those concepts.
Yeah, I’m not clear that the vocabulary words (domain, range) are all that useful unless students want to be able to communicate the ideas with future math collaborators, but the concepts are important.
Wow. Interesting, Sue. How, then, do you talk about inverse with students if function, domain and range are not part of the vocabulary? I suppose your answer could be that you don’t really linger on inverse either (but maybe your answer is different from that). If inverse gets skipped, I would want to argue for its importance as one of the foundational ideas of College Algebra. Function, too. But inverse is a unifying big idea across many mathematical domains (heh).
As I think you may have implied (here: “what a fabulous job you/your textbook/your online homework platform are doing at teaching domain and range”), the textbooks don’t do a good job with this. I skip the sections that focus on functions, because I feel like there’s no ‘meat’ there. I mention how trig functions are a very new sort of function for them. I use the term function often. I don’t use the terms domain and range often. I agree that inverse is an important concept, and I doubt that I do it justice. But the book’s treatment feels close to useless to me.
Christopher, if you’d like to chat about how I could strengthen my approach to this course, I’d love to have that discussion.
Sue writes, But the book’s treatment feels close to useless to me.
I am not going to argue here. I don’t think textbooks are written in a way that builds on students’ experience with these ideas; whether those ideas come from outside the classroom (such as vending machines and speedometers), or inside the classroom (such as likely partially correct residues of previous instruction). Instead, they tend to trade in the fully processed abstractions of previous mathematicians. These abstraction are beautiful but I don’t think they are a good place to start.
Sadly, most of what I know about strengthening an approach to college algebra is here on this blog. I suppose the storyline of the semester is not written up anywhere, and I ought to do that at some point. But I really don’t have any evidence that it’s a stronger course. It is a more principled course than the one I taught a number of years ago that adhered more closely to a standard textbook treatment. I think it is more coherent and focused. But I do not have evidence that it is stronger in the sense of meeting my goals of creating lasting change in students’ thinking and mathematical habits of mind.
In that, I suppose I have quite a bit in common with my colleagues at the Common Core State Standards Initiative.
Christopher, I don’t need no stinking evidence. I trust that your years of experience show in what you do. I think a skype conversation, or google hangout if others are interested, would be a great way for both of us to strengthen our courses.
I agree with you: My course is “a more principled course than the one I taught a number of years ago that adhered more closely to a standard textbook treatment.” I’m guessing you’ve done more creative modifications than I have. But maybe not. Maybe we’ve seen different things we knew needed changing.
Working with Algebra1 students, I try to simplify the language as much as I can. The phrases I’ve found most effective are actually questions. To identify whether or not a value is in a function’s domain, I ask, “Can it go in?” To test whether or not a value is in the range, I ask, “Will it come out?” So, for y = x^2, we determine that everything goes in, but only non- negatives come out. No set notation or interval notation – I think that can wait for Algebra 2.
I also use function language and notation starting very early in the course, and the following simple technique seems to help quite a bit.
Rewrite f(x) = 8 + 5x as f( ) = 8 + 5( ) – that’s it!
We also look at the meaning of f(2) = 18 on a graph and in a table, and create a verbal model that would fit (or v.v.)
With those things in place for linear functions in Algebra1, students are better prepared when new functions are introduced.
Christopher, this reminds me VERY MUCH of an instructional format I learned from Helen Doerr at Syracuse University — it’s a combination of the Cornell GoodQuestions project and clickers (your index cards are just a low-tech version of clickers). I wrote about it briefly here.
I wish that Helen had a bigger web presence and had written up this technique, because it’s so effective at any level. (Unfortunately all I can do is beg people to go see her at NCTM conferences. She’s really a joy to watch teach.) I’ve watched her use it with a freshman calculus class, and I’ve used it in high school classes. It’s all about cleverly asking a multiple choice question that will elicit different answers from a class, where students will have reasons for those answers, and then be prompted to defend their answers.
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I don’t think I’m a college algebra teacher, but I tried the activity anyway. Results can be found here:
Like Sue, I pretty much entirely skip the “domain” and “range” vocabulary in College Algebra. I usually also skip anything along the lines of “does this graph represent a function?” or “does this equation define a function?” In my opinion (even as someone who later ends up teaching calculus to some of these students), those issues are empty, purposeless formalism. But I do teach function inverses thoroughly.
I tried this activity with my high school honors precalculus class. All 27 answered every one correctly except split decision on range value of pi. Loved the activity. I will use it again. Thank you!
Very interesting, Tonye about π. Do you think this is about the I can’t think of anything that, when squared, is π idea?
It is also interesting to me to note the difference between College Algebra students (as a group) and honors Precalc students (as a group).
I mentioned this post on my blog this week. Thanks http://ontariomath.blogspot.ca/2013/10/math-links-for-week-ending-oct-4-2013.html
I did this the other day with my college Algebra students. They totally nailed the pi question. I was proud of them. To be fair, this is a college algebra course taught as a “college in the schools” program through Southwest State. These students have the option of taking this course because they are in the top third (juniors) or top half (seniors) of their current HS class. These are smart kids.
I had a few want to make an argument about infinity being in the range. They said that infinity is “in” the range, but if the question said “included in” the range, they would have answered “no.’ Splitting hairs, I suppose.