Monday, February 25, 2013

When does "by hand" graphing or processes matter?

I am finishing up teaching polynomials to my Algebra 2 classes. We are discussing the Rational Root Theorem (i.e. p's and q's) right now. When I presented the material on Friday, I went through the whole process - finding the factors of p, finding the factors of q, finding the p/q values and testing using synthetic division. As much as I like doing synthetic division, I had forgotten how frustrating this process is for students - it is tedious and they know by now that they can find the zeros by finding the x-intercepts of the graph.

So, as I reflected over the weekend and into this morning, I decided when I assess them on this learning target, I am only going to ask them to identity the factors of p and q and the p/q values. I am not going to ask them to fully find the zeros of the polynomial from that list. The more I thought about it, the more it became clearer to me that if they were going to have to find zeros from a polynomial students would have access to technology (such as a graphing calculator or Desmos). As I was working with students today and reflecting on this learning target, I kept coming back to this question: How do we determine when it is important to have students do the processes by hand versus using technology instead? Is it really important to have students find all the zeros by hand? I had to (granted, the first easily available graphing calculators came out when I was a senior in high school) - so why shouldn't my students have to? (note - I know that's not a good reason why, just throwing my thoughts out there.)

This question is not new to me. I have wrestled with it on and off throughout my teaching career. When students have graphing calculators that can do things like graph and find zeros and maxima and minima, etc., this question returns often. However, this year, I have a mixed bag as far as who has a graphing calculator and who does not. With having (4) computers in the classroom, Desmos has been a nice addition and is far less clunky in identifying intercepts and extrema. From the bits and pieces I have caught from tweets, Desmos continues to improve and everything I have seen from them shows that they are incredibly responsive to its users (and teachers!) Bob Lochel addressed the TI vs. Desmos issue in his blog post this weekend "An Open Letter to My TI Friends." I have to say that even though I have not received quite the training and benefits that Bob has from TI, I found myself really agreeing with every point that he made in his Dear TI letter. (Go read it if you haven't already - it's worth it!) But once again, not everyone has access to the technology in my classes. We are not a 1-to-1 school, I don't have a class set of iPads or tablets or even graphing calculators. My classes range from about 30% to 50% of my students having a TI graphing calculator. I have students who do have smartphones, but they are not allowed to use them in school. I'm already starting to think about next year and how I'm going to deal with the whole technology issue. Do I have my students all get TI graphing calculators, full well knowing that many of them will not use them after my class? Do I skip the graphing calculators and find a way to work with Desmos, knowing that I have 4 computers in my class and that's it? Do I try to find funding for a class set of graphing calculators? Tablets?

But I digress from my original query. I have taught for 21 years and this is the same question that I had when I first started teaching with TI graphing calculators then. At what point do you push aside the by hand processes and let the technology take over? Are you shorting students mathematical learning by doing this or is it enhancing it? How do you structure lessons so that the technology enhances the mathematics rather than glosses over it? I'm curious to see what you all think. Please share your thoughts in the comments. Thanks!


Karl Fisch said...

I struggle with this as well (although I'm only teaching Algebra 1 so it doesn't come up quite as often).

What I try to do is figure out whether the task is mathematically meaningful and whether doing it "by hand" enhances their understanding. It's been a long time since I taught synthetic division (mid 1990's), but I wonder what value that has for most students. It seems to me that whatever value there is in finding the zeros of a polynomial, it's more important to understand what they are and why that might be important than which method you used to find them.

As far as your quandary about equipment, I think that's at least partially a "schooly" problem. In other words, your viewing it from the perspective of the artificial limitations of a math class in high school. If any of your students go on to actually use this they will not have those limitations.

So it comes back to - is it mathematically meaningful to do synthetic division by hand. I'm not sure that it is, but I don't think I know enough to say that for sure.

KFouss said...


I totally agree with you. Some of the skills we learned in high school, college, or even taught are now seemingly outdated. Rational root theorem? Descartes Rule of signs? Why bother? Finding those values when we have tools at our fingertips that can do so much more seems a waste of time.

As for what to do with the calculator choice for next year, I'm in the same boat. I love Desmos for graphing, and there are a ton of free apps the kids could use on their phones/iPads/laptops at school (because my school encourages students to bring in their devices). But what happens when a student goes to take the ACT? They can't use said technology and need a TI... And need to know how to use it. I hate requiring something that I know they really don't need and costs $85+, but until the higher ups decide that the other options are viable for testing, I think we're stuck.

Sorry I can't help... Just wanted you to know that I feel your pain!


Anonymous said...

This is an issue I think every math teacher struggles with...

In this particular case, I think it may be useful to think about what the Rational Root Theorem is actually good for instead of just what it CAN be used for. Is it good for finding all the roots of a polynomial? NO. Like you said in your post, it's an INCREDIBLY tedious process, especially when you can just graph it on a graphing calculator. And then there's the fact that only a small fraction of polynomials will have ANY rational roots at all. So to me, having kids go through this process is pretty pointless.

For that reason, combined with the fact that proving the rational root theorem involved some really complex math that my kids will never see/ use, I skipped the rational root theorem entirely. I wish I wouldn't have though. Next year, I'll use it for more conceptual questions. For instance I might give them a polynomial and a list of rational roots, which they have to say are either possible or not based on the rational root theorem... Last year I did a graphing/hand hybrid approach. They had to write out the possible zeros. They then used graphing calculators to determine which of those they needed to test, so they were only testing a small subset of the possible zeros. This made it much more tedious and a bit more conceptual.

cheesemonkeysf said...


I got such a different perspective when I started taking math classes with a number of different Russian professors.Their whole perspective on graphing was completely different from anything I had experienced in my American mathematical upbringing.

First and foremost, they ALWAYS insisted on clarifying the difference between SKETCHING the graph of a function and "graphing it." In their pedagogy, sketching the graph of a function was about CONCEPTUAL UNDERSTANDING, never about procedure. They wanted us to think about sketching the graph as having a first-hand relationship with the visual representation, not merely about computation. Every class I took forced me to rethink my whole relationship to graphs, and over about two years, I realize now, I stopped caring about the calculator-generated graphs when I thought about functions and visual representations. They're now such completely separate actions for me -- sketching and "graphing" -- I have to try consciously to re-link them together in my mind.

As a learner, this gave me a really different feeling for multiple representations, so now, as a teacher, I find that I approach the teaching of graph-sketching with an enthusiasm that my students find perverse at first... but then slowly, the feel of the graph shapes enter their bodies in a different way too. And I know I have done my small part for conceptual understanding. :)

Now I find myself approaching the pursuit of graph sketches as a kind of quest for predictions about functions, shapes, and transformations.

I am a huge believer in "the life made by hand," so perhaps that informs my understanding and teaching. The abstract procedures and rules served me so poorly as a learner, but developing my own authentic relationships with the shapes and behaviors of curves really rocked my world.

- Elizabeth (aka @cheesemonkeysf)

Lsquared said...

I feel your pain. Us college teachers wonder these things too. I've had this discussion (with myself and other teachers) several times this year, and these are my (admittedly really vague) conclusions at the moment...

I don't know where to decide "this detail isn't worth the extra time" and of course you have to decide that somewhere (do you do rational roots but not Des Cartes rule of signs? Do you factor, but not do rational roots? do you give problems in already factored form to graph? etc.) I do know that in teaching first semester calculus last year (college) I had more students than ever that didn't have a strong intuition about rational functions (what's a missing point? How do you get a graph from a function without a calculator?) It's not that they _do_ a lot of problems in the class where they don't have access to a graphing calculator, but they need the mental framework of having done graphing from roots and asymptotes to understand what's going on and why things work. So, I don't know where to draw the line, but I think that spending time graphing by hand (by finding roots and asymptotes--not by plotting points) is really helpful for understanding calculus.