While half of this activity tends to digest itself easily, the interesting debates take place when students need to compare graphs which contain skew or unusual peaks. In particular, the activity reveals many misconceptions surrounding boxplots – short, dense tails and longer, less-dense tails are often switched.

The beauty of programs like Fathom, and the TI Nspire also does a nice job, is that we can more from one representation to the next quickly. In the end, I often wonder why dotplots, which would seem to most easy to digest for younger students, are often marginalized for the sake of funky boxplots.

It shows the limitations of both graphs and tables/numbers.

I also appreciated Jodi’s note that “model selection” is a distinct skill from “model implementation.” Computers are awesome at one and rather more limited at the other. Added to the post.

Bridget, thanks for linking your post. It’s so helpful to see that one task filtered through all three representations. Their upsides and downsides shine right through.

Also: no idea how I hadn’t subscribed to your blog before now. Got that fixed.

David Srebnick:

I think that when using multiple representations, the first uses should include a discussion of the advantages and disadvantages of each representation.

Beyond that discussion, a drum I’ve been beating throughout this “headache” series is that students should experience those advantages and disadvantages. Viz: it’s better to experience the power of a product firsthand than hear about that power from a salesperson.

I do feel that is our job to set up situations where students can have discussions on what’s best and why. The trick here is finding the right problems/scenarios that lend to these sorts of discussions.

I was able to collect a wide variety of representations that I wrote about here: https://elsdunbar.wordpress.com/2015/07/17/intenttalk-book-study-leads-to-questions-about-effective-vs-efficient/

I think the task I used yielded the kind of work that I can use with future students in thinking about “best” representation.

First I create a few data sets of test grades from fictitious teachers. All have about the same average, but the distribution of grades is different.

I ask students: Given these averages (90, 89, 87), which class did best on the test? When I press students, they usually start asking questions like, “but maybe 3 students in the 90% class failed, and no one failed in the 87% class.

I repeat the process with a couple more of the following: median, mode, box & whisker, and possibly some other visual representation.

I think that when using multiple representations, the first uses should include a discussion of the advantages and disadvantages of each representation. Then, later on, things like “write an equation for this graph” can be framed in terms of ‘make a new representation.” Also, given a word problem it might be good to ask “which representation is the best one for this problem?”

Now what is the value at x = 1.37?

Now they see that the equation is quicker and more accurate than the graph — even when inside the graphed region.

Or draw two lines that do not meet at integer values. Where do they meet, exactly? Hence that simultaneous equations are better in some situations than graphs.

But again, we can draw y = log x crossing y = x^2 quicker on our graphics calculator than we can solve it.

(Of course y = x^2 doesn’t cross y = log x, but they only know that if they graph it!)

Tables are great when you need to generate data from a scenario – you have a situation that has been given to you and you need a place to start. Creating a table for some initial data helps you see the patterns in whats happening and helps you make littler predictions about where the data is going. If I want students to appreciate tables, I give them a visual pattern or a scenario problem with a starting condition and a rate of change, then ask them some questions about what will happen.

Graphs are great when you’re given several random data points that, even when arranged as a table, don’t indicate a clear pattern. Sometimes plotting these visually helps you predict what other points could be missing, or what other points exist as the pattern continues. This is especially true for situations whose solutions depend on two variables, such as only having 30 dollars to spend on item A that costs 2 dollars and item B that costs 1 dollar. When I want students to appreciate graphs, I give them one of these situations (which usually lends itself to standard form of an equation, but they don’t know that) or I give them several data points and ask them what’s missing in the pattern. This is easier to see when organized visually and you you a particular shape to your points rather than a random collection.

Equations are the most efficient way to make predictions about patterns – if you’re given an equation, there’s no reason to have any other representation. Equations are useful for predicting far into the future for your data – maybe you can figure out the first few terms of your pattern, but trying to generate the 100th term is inefficient. Using an equation is like being omnipotent with a set of data. When I want students to appreciate equations, I give them a scenario but ask for a data point in the absurd future where the table or graph necessary to find the point would be too large and unwieldy to use.

The order I’ve presented these in this comment is also my typical order for presenting these representations to students: tables are useful at the beginning to generate data; graphs are useful once you have lots of it that may or may not be organized and may be missing some points, and equations are good for predicting the future.

A curious consequence might be: it’s not particular situations that necessitate one representation versus the other; rather, its what data you choose to give them at the beginning and what you ask them to do with it that makes one representation more valuable than another.