25 December 2012

New Semester Resolution #1: Innovate Like Its 1992 (or, Do Your Students Still Make Graphs By Hand?)

  1. Raise your hand if your students love making graphs by hand.
  2. Keep your hand up if all of your students are proficient at producing graphs of equations (especially anything past linear functions)
Still have your hand up? Okay... Maybe this post isn't for you. Just kidding - universal appeal here. Several years ago I picked up the 1992 NCTM Yearbook at a massive used book sale held yearly in St. Louis. Every archived article in this yearbook was concentrated on using the power of calculator technology to enhance students' understanding of numbers, operations, and functions, and moving instruction beyond simply replicating graphs. Yearly we have conversations in my Algebra 2 PLC over how we're going to get our kids graphing quadratics and describing transformations of the accompanying graphs and equations.

As you may have also experienced, students often get caught up in making the graphs and sometimes never even make it to the transformations (which puts a wrench in the cogs of your lesson). Do your state standards say anything about students constructing graphs by hand from tables? Do the CCSS imply any graphing by hand? (In fact, Mathematical Practice Standard #5, Use appropriate tools strategically specifically expects that students will use technology for their graphing.)

Graphing by hand - right up there with top hats, saloons, and spitoons.
Being able to sketch a graph after using a graphing calculator or spreadsheet program is different work than what my students usually do with functions. Using technology to grab a graph and then use it in modeling doesn't eliminate necessity of my students' knowledge of the graphing, and it probably demands more.

When my students graph by hand, I usually...
  • Pick "easy" equations that have as many integers as possible in the features
  • Use "small" numbers
  • Don't require students to manipulate the scales on their axes.
  • May or may not know who actually has an understanding of "rate of change" on that function they just graphed beyond the fact that they used "rise-over-run"

When we use technology to generate graphs, students must...
  • Trace graphs to find features (intercepts, zeros, maxima, minima)
  • Have an awareness of the domain and range the want for their function ("I didn't get anything on my graph when a put in the equation")
  • Know the scale of their axes in order to judge reasonableness of their graph to the situation they are attempting to model (A strangely high/low y-intercept of a linear function over time when students input "1999" instead of years after *)
So why do we still so often require that students are able to construct graphs by hand? Is it a filter for straining the "good" students from the "poor"? Is it "good to know?" I don't know that many people could legitimately give me a case for the necessity of a student being able to make a graph by hand without any assistance from a calculator, software, or web-based utility, like the Desmos calculator (of which I recently blogged about if you didn't catch it.).

More than TWENTY YEARS later... we're still having this discussion. In fact, one of my favorite Google+/Twitter follows, +David Wees also blogged about it earlier this year and made some valid points in favor and opposition. Is one barrier to education reform/relevance that we're holding onto too many they-should-know-this-because-we-did skills in our curricula? Will this be the year you finally give in to technology's relentless march? :)

It will take some editing of my assessments, but I am going to allow (and encourage) my students to use technology to graph EVERY instance it arises this coming semester.

Are you ready for 1992? I encourage you to join me in this challenge!


  1. The only justifiable reason I can think of to have students graph by hand is that it helps them develop a spatial awareness of the position of the points on the graph, and the relationship between the function instructions and the function graph. That being said, one could easily use technology to develop this relationship. Ie, you could build the table of values for a function (by iterating through a few x values and finding the y values) and then plot these points to see that they do indeed lie on the graph of the function.

    I'm with you. I can otherwise find no sufficient reason to teach graphing "by hand." The important part for me is that the students get practice making decisions about how to construct their graph, whether this happens on paper and pencil or in graphing software, I can't see that it matters (except that in the software they are likely to run into their mistakes fairly quickly).

    1. I don't remember how I learned graphing in Alg1, but I had a graphing calc all through Alg2 and up, and all I remember of that is how useful it was going between my calc to the paper, and I didn't have the "Where's my graph?" window problem. I think many teachers see it as by hand OR with technology; I think the two go hand in hand and students need to be able to translate between the two.

      Like you say, practice making those decisions is most important, and if you neglect one for the other, you're missing out on different strengths of each method.

  2. I totally agree with this challenge and I think we do use calculators and computers fairly well with the students to help them make connections and deal with real world messy numbers. We are also liking fathom to do some modeling with real world data sets- I am going to pass this along to the math folks (and science)types to make sure tho - (I am the instructional technologist so i don't teach math but work with teachers on tech integration)
    I do have a question for you tho- because so many of our students come back from very good schools and say they are not allowed a graphing calculator, much less a computer, in their math classes (typically Calc 1 but also college algebra)- I typically go with that the graphing calculator helps build deeper, conceptual understanding and they will have an easier time of it because they prepared with it- Our own STL university doesn't allow calcs and computers in math classes or in the Olin Business School -
    Definitely we need to use the technology - and we need to prepare them -and I think we do both - just curious your thoughts-

    1. Glad to hear that you like Fathom, I've thought about playing with it several times, but I know no other stats teachers that have used it, so I don't have a clear vision for using it over Excel (or Google spreadsheet)

      You make a great point about university math, and I agree with what you said about conceptual understanding. Because I'd done so much work with graphs on a calculator by the end of high school, I was able to THINK about them more clearly because I'd seen so many. The repetitions kids are able to get analyzing and working with functions when they use technology build the understanding. Once they know why, the DOK level 1 and 2s of reinforcing manual graphing skills would be simple to add on.

      I'd used my graphing calculator hand in hand with my paper work through most of high school, but I actually scored best on the ACT exam WITHOUT my graphing calculator. I had a clear understanding of the graphs, and was able to work faster that day when my graphing calc quit on me.

      Just like most technology, if we're completely dependent on the calculator to do work, we're lost with out it. If we use it to make our own thinking more efficient, its a tool we can make without when its gone.


Thanks for sharing!