Errors are Opportunities

Errors are Opportunities.  This is the most important norm in that I teach my class.  I’ve been thinking about and procrastinating this post for some time.  My thoughts about the topic were tipped over the edge this week by two separate events.  The first was a quotation given in a post by Dan Meyer about feedback:

If you can both listen to children and accept their answers not as things to just be judged right or wrong but as pieces of information which may reveal what the child is thinking you will have taken a giant step towards becoming a master teacher rather than merely a disseminator of information.

JA Easley, Jr. & RE Zwoyer

The second event was the unexpected way in which my students responded to my introductory unit on Tiling Pools this week.  I asked my students to evaluate an error and they responded in a way that resulted in a major paradigm shift for me as a teacher.  Truly, my students are my best teachers.

I begin the year with a unit that centered on the classic Tiling Pools Problem . I first learned about this problem from Leslie Butler, a exceptional math teacher in Davis School District, Utah.  My approach to the problem is heavily modeled after hers and most of the credit should go to her.

I start by giving students the basic synopsis: We’re going to build a border around a 5 ft square pool using 1 ft square tiles. We define the term ‘square’ and then I ask them to use their mini-whiteboards to show me the number of tiles needed for the border.  Invariably, I see the following responses: 20 tiles, 24 tiles, 25 tiles.  I write all of these responses on the board and, one at at time, ask students to discuss with their group the thinking behind each of the responses.

Beginning with 20 tiles, students recognize that this answer is a result of thinking about perimeter of a 5 ft square as 5 x 4. Unless a student brings it up, I’m careful not to comment on whether this strategy is correct or incorrect at this point. Instead, I immediately ask students to discuss the 24 tiles and the strategy that resulted in this answer.  After a minute, I cold call on a student to describe this strategy.  Most students recognize that the 24 is the perimeter (5×4) with an additional four tiles added on to form the corners of the border.


24 tiles needed to build a border around a 5 foot square pool. (Sorry for the very rough sketches.)

I keep calling on students to explain, re-explain and summarize until all students understand this strategy: tiles equals 4 times the side length plus four [t = 4s + 4].  I don’t get into equations with them on the first day, because I don’t want the equations to narrow their thinking.

Once all students accept that 24 is the correct number of border tiles, I ask them to take a moment and discuss the 25 tile response.  I always leave this one for last because it emphasizes that errors are opportunities.  How often do we stop exploring a problem once we’ve arrived at the correct response? I do it all the time.  How often do I find myself sequencing student responses, saving the ‘best’ and ‘most correct’ response for last? More often than I’d like to admit.  Students aren’t immune to these tactics. If I really want my students to believe that “The answer is not the most important part” then I need to get out of the habit of stopping when I find the ‘right’ answer.  In this lesson, that means that I have them discuss the 25 tile answer last.

Most students are able to tell me that this answer comes from multiplying 5 by 5.  By asking them when this would be a good strategy, I can usually push them into telling me something about area and tiling the bottom of the pool.

25 tiles needed to tile the bottom of the pool. (Sorry for the rough sketch)

25 tiles needed to tile the bottom of the pool. (Sorry for the rough sketch)

Once we’ve explored all three responses, I ask students to use what they learned to tell me how many tiles I would need to build a border around a 10 foot square pool. For the 10 foot pool, most students will write 44 on their whiteboards, but a few will write 48.  Again, I wrote these two answers on the board and asked students to describe to me the thinking behind each of them. A common error in students’ thinking is that since the side length doubled, the number of border tiles should double.  I call this the trap of proportional reasoning.  Since proportional reasoning is emphasized so heavily in 7th grade, it is a trap that a large number of new 8th graders fall into. In my regular ed classes, they identified this strategy and were able to describe why it was an incorrect answer (even though the side length doubles, the number of corner tiles in the border won’t double).

Here’s where things got really interesting this year: my afternoon honors classes used a completely different strategy to describe the 48 tiles error.  They described it as, “Maybe they multiplied the whole side instead of just the 10 part.”  After some prodding, I was able to get them to explain that by ‘the whole side’ they meant that they were talking about the ‘whole side of the border’, not the whole side of the pool.

10+2 = 12 tiles along the length of the border of a 10 foot square pool.

10+2 = 12 tiles along the length of the border of a 10 foot square pool.

When asked them what the problem was with this strategy, the class explained that each corner is being counted twice.

I illustrated what they were talking about like this, and then said “I like this idea of using the whole side of the border.  Is there a way that I could tweak this strategy so that it works? I want to keep using the whole border side, but it’s giving me to many border tiles.” With some encouragement, a few students tell me that we could subtract four at the end, since the border tiles are being counted twice.

I then asked students to calculate the number of border tiles needed for a 98 ft square pool, using whichever strategy they preferred. The majority of my students used the first strategy [t=4s+ 4] to come up with the 396 border tiles, but a few of them used the new strategy [t=(s+2)*4 – 4].  I wrote and talked out the work for multiplying 98*4 and then adding four and asked students which strategy it matched.  I then wrote and talked out the work for (98+2)=100*4= 400 – 4 = 396 to demonstrate how to use the other strategy.  I then had them find the border tiles for a 23 foot pool.  Many more of them chose to use the [t=(s+2)*4 – 4] strategy this time because they could see its advantages (it is much easier to compute 25*4 than 23*4).

I took a moment to emphasize to students how important it was that we took the time to really think about that error of 48 tiles for a 10 foot pool.  Because we examined the error, we were able to develop a new strategy that is much easier for certain sizes of pools.  The error was an opportunity.

I then asked students to calculate the number of border tiles for a 99 foot pool.  They were allowed to either use a previous strategy or come up with a new one that fit the problem better.

Here is what I learned from this:  When the need arises, the strategy will present itself.

It was easy for students to create the t = (s+1)*4 strategy based on their knowledge of a previous strategy.

(99 + 1)*4=400 border tiles for a 99 foot square pool

(99 + 1)*4=400 border tiles for a 99 foot square pool

This is exactly opposite of how I have taught this lesson in previous years.  I used to wait for a student to develop a strategy, and then I would have students practice the strategy by giving them pool sizes that fit the strategy well.

What does this mean for me as a teacher?

My job is to create the need.  When the need arises, the strategy will present itself.

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