## Multiple Representations v. Best Representation

This is from a worksheet I assigned during my last year in the classroom:

There are lots of good reasons to ask students for multiple representations of relationships. But I worry that a consistent regiment of turning tables into equations into graphs and back and forth can conceal the fact that each one of these representations were invented for a purpose. Graphs serve a purpose that tables do not. And the equation serves a purpose that stymies the graph.

By asking for all three representations time after time, my students may have gained a certain conceptual fluency promised us by researchers like Brenner et al. But I’m not sure that knowledge was ever effectively conditionalized. I’m not sure those students knew when they could pick up one of those representations and leave the others on the table, except when the problem told them.

Otherwise, it’s possible they thought each problem required each of them.

The same goes for representations of one-dimensional data. We can take the same set of numbers and represent its mean, median, minimum, maximum, deviation, bar graph, column graph, histogram, pie chart, etc.

So here is the exercise. Take one representation. Now take another. Why did we invent that other representation? Now how do you put your students in a place to experience the limitations of the first representation such that the other one seems necessary, like aspirin to a headache?

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Ok. First is bar chart, second is box plot.

All situations in statistics require some data, and the best data is that which students compile themselves. For this comparison a single set of data is best presented as a bar chart, but compare the data from five or more distinct groups of subjects, same measure, and the multiple strip bar chat is a bloody mess. Five box plots above the same numberline, and so much more is revealed, at a small cost of loss of detail.

I used to think that box plots were a waste of time until I saw the above usage.

The same is true of physical representations. I am thinking of many algebra growth problems that involve squares and growing patterns. It is valuable to ask students to go through the actions of adding squares to watch a pattern grow through the addition of tiles. This action can help them have the physical experience of a rate of change. But this representation also has its drawbacks. It is clearly cumbersome and not efficient.

I think of them all as connected to making predictions about data – certain representations lend themselves to different ways in which data is presented, and certain representations help make predictions about that data.

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.

Jodi:

So very true. This skill seems to be neglected in our classrooms. Computers can take one representation and switch to others over and over again, much faster than humans can. If switching back and forth is your only skill, I can easily replace you with a \$100 calculator from Target. And the calculator will be faster and more accurate.

But if I’m training students to be problem solvers who are smarter than computers, the “which representation is needed here” is a much more important question. I’m not aware of a computer that can answer that question.

Draw a simple line on a graph.

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 = x2 quicker on our graphics calculator than we can solve it.

(Of course y = x2 doesn’t cross y = log x, but they only know that if they graph it!)

BTW: Essential reading from Bridget Dunbar also: Effective v.Efficient.

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. More here.

1. #### Howard Phillips

August 27, 2015 - 5:24 pm -

Ok. First is bar chart, second is box plot.
All situations in statistics require some data, and the best data is that which students compile themselves. For this comparison a single set of data is best presented as a bar chart, but compare the data from five or more distinct groups of subjects, same measure, and the multiple strip bar chat is a bloody mess. Five box plots above the same numberline, and so much more is revealed, at a small cost of loss of detail.
I used to think that box plots were a waste of time until I saw the above usage.

2. #### Shaun

August 27, 2015 - 5:46 pm -

Great point. This made me think of the progress made as we developed the number line, coordinate plane and the complex plane.

3. #### Clara

August 28, 2015 - 1:43 am -

I pose a Problem of the Week through our math classes at my school. I think I will use your question – as is, with teachers; slightly modified to students, asking them to provide the “limitations” of each representation. Hmmmm.

4. #### Sarah Lanahan

August 28, 2015 - 5:40 am -

Ooo, I love this! I’m an IC at a high school and I’m literally in the middle of working on creating PD for my Algebra I team on multiple representations of linear equations and slope. I’m so pulling this into that PD. Thanks so much!

5. #### Kim Morrow-Leong

August 28, 2015 - 6:03 am -

All representations offer advantages and constraints. In that list I include mathematical symbols. They offer precision, but they take away the context of a problem. On the other hand, a graph can offer trends but it may sacrifice precision.
The same is true of physical representations. I am thinking of many algebra growth problems that involve squares and growing patterns. It is valuable to ask students to go through the actions of adding squares to watch a pattern grow through the addition of tiles. This action can help them have the physical experience of a rate of change. But this representation also has its drawbacks. It is clearly cumbersome and not efficient.
But each of these examples is referring to the process of learning. I think you’re talking about the process of communicating results of a mathematical investigation or exploration. The question is, how is that different? Is one description all about learning the mathematics and the other about communicating to others who need to know the results of your mathematics? At this point I think about the exercise of writing. Clearly we write to communicate, but isn’t there value in practicing writing techniques in order to assure that you can do them first (like ending a comment with a question in order to spur thinking)?

6. #### Kyle Pearce

August 28, 2015 - 9:40 am -

You make a great point here. I have been teaching a lot of grade 9 math courses recently and one of the Big Ideas is four representations of a linear relationship. We do it so much throughout the year that students have no issues with it. The problem is, students often forget (or never know) why we were doing it in the first place. Recently, I have been allowing students to use any representation they want to solve problems, but I often find myself going back to the old “now, show me the 3 other representations…” but for no good reason.

Thanks for challenging me to take this one on. I’m excited to tackle it over the coming months.

7. #### Jodi

August 28, 2015 - 12:45 pm -

So very true. This skill seems to be neglected in our classrooms. Computers can take one representation and switch to others over and over again, much faster than humans can. If switching back and forth is your only skill, I can easily replace you with a \$100 calculator from Target. And the calculator will be faster and more accurate.

But if I’m training students to be problem solvers who are smarter than computers, the “which representation is needed here” is a much more important question. I’m not aware of a computer that can answer that question.

Seems like we should present students with various situations which would require one of the three representations, and let them argue over which one works best. Marketing, political data bending, and infomercials are the three categories I can think of off the top of my head.

“You’ve just published a huge education bill and you don’t want people to look too closely at how you will be evaluating educators because you know it will cause dissent. Should you explain your process using a giant table, a graph, or a complex equation?”

“You’re marketing a new leaner hamburger through TV and internet adds. To advertise your caloric content versus your competitors, should you include the table, graph, or equation?” (Or perhaps a fourth option: the verbal explanation)

8. #### Daniel Schneider

August 28, 2015 - 5:37 pm -

I think of them all as connected to making predictions about data – certain representations lend themselves to different ways in which data is presented, and certain representations help make predictions about that data.

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.

9. #### Chester Draws

August 28, 2015 - 8:53 pm -

Draw a simple line on a graph.

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!)

10. #### David Srebnick

August 30, 2015 - 4:58 am -

I do something similar with my eighth grade class.

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?”

11. #### Bridget Dunbar

August 30, 2015 - 5:17 am -

I’ve spent a lot of time thinking about what you’re talking about here over the past couple of years. There has always been a push to show each of the representations–thinking that students will independently make the leap to making decisions about which is best for a given situation. But they don’t.

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

12. #### Dan Meyer

August 31, 2015 - 8:14 am -

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.

13. #### Bob Lochel

September 1, 2015 - 12:15 pm -

I’d like to share an activity I use within the 1st week of AP Stats to help students understand the different persepctives boxplots, dotplots and histograms reveal. Download the graphs here is you want to play along: https://www.dropbox.com/s/bkgzxm2hy6njrst/Chapter%201%20-%20Comparing%20Graphs.doc?dl=0
Students are given 12 graphs, and are asked to use appropriate terminology for center, shape and spread to descibe each.
Next comes the big reveal….while there are 12 graphs, they come from only 4 distinct data sets. For each data set, one of each type of graph was produced, and the challenge is to locate the correct trios.

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.

14. #### Eric Newman

September 20, 2015 - 6:40 am -

This reminds me of The Vicki Mendoza Diagonal lesson, Dan. I’m gonna gave the kids graph the crazy. Then graph the hot… and sit back and watch the looks on their face as this reveals nothing spectacular. They usually see the need for some kind of way to display both data sets on the same chart pretty quick! One of my favorite lessons of the year!