How I want to teach horizontal function transformations

Horizontal function transformations are tough to teach.  Sure, you can just tell students how they work (“everything is backwards inside the parentheses, so minus makes you go to the right”), but that just seems arbitrary and pointless to students.  Slightly better would be to let students discover by a process of induction that directionality is reversed inside the parentheses, and then get into the big discussion of Why?  My sense is that this is Dan Meyer’s preferred approach, based on his comment in this post by Kate Nowak, who is struggling with teaching this concept in the quadratic function family.  And I have nothing against Dan’s approach–letting the induction come first is fine, but it’s really the Why? that’s tough about this idea.

First of all, it’s not just the directionality that’s reversed–it’s also the order of operations.  So in sin(2x + pi), everything is shifted left by pi first, and then contracted by a factor of 1/2.  The “+ pi” is dealt with before the “2*”.  How are students supposed to discover that fact by induction, and even if they do discover it, how will that set them up to make meaning out of it?  I am not a fan of induction just for the sake of having students discover things when that discovery does not build connections or make meaning.  I mean, you could theoretically have students “discover” the rules for integer operations by letting them play around with a calculator and look for patterns, but that would still be an awfully shallow learning experience.

To return to the question of horizontal transformations, here’s an idea I’ve been cooking for a few years: teach the concept in the context of encryption.  We all know the idea of encoding by letting 1=A, 2=B, etc.  But as Julius Caesar found, it’s better to introduce a shift into your code, e.g., 1=Y, 2=Z, 3=A, 4=B, …  This shift works in the same way as a horizontal translation.  Pow!  A real-world, intuitive example (I hope) of an otherwise difficult concept.  The example even generalizes pretty easily to show why order of operations is backwards as well.  Here’s how I’d use this hook in class:

  • First, give students an encoded version of a sentence (with no shift applied), and ask them what they think it says.  I’d say something like, “Ender’s Game is Better than the Hunger Games”.  So “Ender’s” would be encoded as “5  14  4  5  18 ‘ 19”  Shouldn’t take long before someone cracks the code.
  • Then discuss why a shift would be nice–both you and the message recipient would have to agree on the shift in advance, but it would be more secure.  Maybe I’d show the video linked above about Caesar and his cipher.
  • The class would agree on a small shift, say -3.  Now, to read a message, you subtract 3 from each number, and then decode.
  • I’d ask each student to write a message and encode it so it can be decoded with a shift of -3.  They’d check their work by decoding their own message before exchanging with other people and decoding each others’.
  • Here’s the point: to encode when the shift is -3, you have to start by adding 3 to each number, since the recipient will then subtract 3.  So far, this should all be fun–little complaining about “when are we ever going to use this.”
  • Then I can introduce a graphical representation of the code.  It might look something like this:













That’s the graph of the original code.  I’d ask students for the graph of the new code, with the shift.  The shape would be shifted to the right by 3.

  • Of course, this concept would work with any original code–it wouldn’t have to be 1=A, 2=B,… to start.  It could be 1=Z, 2=Y, … or any one-to-one function to start.  This lets us talk about functions versus relations in an intuitive and meaningful way in the same lesson.
  • I developed the connection to inverse order of operations a little bit in my first attempt at this lesson last year.  Here it is on dropbox.  It was a success in the sense that students finally understood that between reversing direction and order of operations with horizontal transformations is just an application of “undoing” an expression to solve an equation.  But I thought the lesson design and delivery were a little clunky.  I plan to revise it for future use.  But if someone else likes the idea and revises it first, and then shares it with the rest of us, well, I won’t object…

That’s it!  So, what do you think?

UPDATE: the formatting seems all messed up in my dropbox .docx link, so here’s a pdf version

FURTHER UPDATE: Oops, in my original post, I had the grid showing the “graph” of the original code transposed.  That is now fixed.

From the comments:

Dan Meyer: Great to see you blogging on the regular, Kevin. We seem to be on the same page here that discovery without building connections between the discoveries and axioms is kind of useless. But using a real-world context here seems risky also. Is it possible that the cryptography connection will obscures mathematical meaning? How do you help students make the leap from Caesar Augustus to periods and amplitudes?

(My response to Dan’s question is in the comments below).


Hey education researchers! Here’s how to get teachers involved.

I have an idea for how education researchers and teachers could connect better in order to really see what principles that researchers are positing really translate robustly to the classroom. The short version is that students or researchers who want to study a learning principle could embed their experiment in one of the lessons from Dan’s summer “Makeover Mondays” series, in which Dan remakes boring problems/tasks from textbooks into more inspiring lessons.

These makeovers often raise theoretical questions for me, such as how much to ask students to discover for themselves, and when to schedule the “drill practice” (e.g., in bits and pieces during an investigation, or after it?). A good example would be the task called Shipping Routes, in which students apply/discover least common multiples to predict the first time two ferries with different round-trip times will be in phase with each other. Dan’s makeover involves sending students to this simulator to look for patterns in the scenario.

Testing learning principles in the context of these lessons seems to offer a few advantages:

  • If learning sciences researchers want to connect with the teacher community, they have to connect either before or after they do a study. Connecting afterwards means doing the study first, and then trying to get teachers interested in it. This has all sorts of pitfalls. You have to describe the debate your study informs, and people can accuse you of mischaracterizing their position or get sucked down a black whole of semantic arguments. You have to describe whether you think the results are broadly or narrowly applicable, but that part may not be heard. Etc.
  • Connecting before means getting teachers interested in the study before doing it. With this approach, you don’t have to worry as much about recreating the debate before situating your results in it. You can just describe your experiment and empirical results, and the discussion community will do that framing for you and translating from theory into practice for you. That’s what I’m suggesting here.
  • Many of the redesigned lessons often involve some kind of software interaction that could be used to log student behavior. And the lessons are commercial-quality, but you don’t have copyright issues because they’re usually licensed Creative Commons.
  • The final products of the lessons bubble up through Twitter, blogs, etc. from a community teachers. All of those teachers would be very interested in reading an article about a lesson they’ve tweeted about and given serious thought to (or plan to teach!).
  • Dan’s blog is read and commented on by opinion leaders among math teachers, such as Grant Wiggins (the co-creater of Understanding by Design) and Michael Serra (author of the Discovering Geometry textbook series that’s very popular). So it’s a robust conversation.

So what do you think, MTBoS? Is this a good idea?