Reflections from Teaching: It’s More Than Just Doing
“Students learn more when their teachers know the content, and when they can anticipate student misconceptions.” -Daniel Willingham
Today in class as my student were presenting the solutions to the surface area and volume questions they had been assigned and an interesting opportunity came up. So I decided to do a little experiment.
In order to understand what I would like to share, we have to quickly refresh on some basic arithmetic when dealing with surface area:
(1) To calculate the lateral surface area of a right pyramid, one can relate this to the perimeter: L.S.A. = 1/2 x Perimeter of base shape x Slant height of pyramid. Note that this formula is equivalent to summing the various triangles that compose the lateral faces.
(2) To calculate the lateral surface area of a right prism, one can also relate this to the perimeter (in a much easier to see way): L.S.A. = Perimeter of base shape x Height. Note that this formula is equivalent to summing the various rectangles that compose the lateral faces.
Now that we have that under our belts, here is what happened. One student was calculating the volume and surface area of a right square-based pyramid (no diagram was given – but we had side length of 5ft and height of 12ft). The following was given as a solution:
L.S.A. = (4s) x h = 4 x 5ft x 12ft = 240 square ft.
After the question was complete, I asked the class to analyse the response. I mentioned that often as a teacher of mathematics it is useful to be a detective (I call this Maths CSI). Where is the error? What went wrong? How was the student thinking and what should we say to help with the misconception? After several minutes, they were unable to find the error – mind you, they were looking at four calculations instead of one, so maybe the scope was too large. In our case it turns out that the student calculated the lateral surface area of a square-based prism instead of a square-based pyramid.
To me, this brought back the discussion of factual/procedural knowledge versus pedagogical content knowledge. My teachers could very well utilize the formulas given, make substitutions and explain exactly how they arrived at their solutions. That is, their factual/procedural knowledge was quite strong. But, when I asked them to reflect on what went wrong in this particular example, this was pushing them a bit too far out of their comfort zone.
Often I hear the argument that if our future teachers only took calculus (or some other high-level mathematics course) then they would be good teachers. I find this argument rather empty – I’m not convinced that more factual/procedural knowledge necessarily leads to a more effective teacher. I am certain we have all had instructors who were quite knowledgeable – yet lacked the aspects that would make them an excellent teacher. Now, I am certainly in favour of teachers knowing the progression of mathematics – that is I would like my high school teachers to have seen some university/college level calculus and algebra courses. But do our elementary level teachers have to take calculus to be good teachers? I feel this is out of the scope of beneficial progression for them to see. Would it not be more beneficial for a teacher teaching K-4 to understand deeply the progression of mathematics to Grade 8 instead? And those teaching middle school to know deeply the progression up to precalculus?
Rather than taking calculus, would it not be more beneficial for our future teachers to take a mathematics course in pedagogical content knowledge where they study aspects such as the progression of mathematical concepts, common misconceptions students have, questions to ask to understand how students are thinking, and how to know when students are ready for deeper learning? I would argue yes. Prominent researchers such as Dan Willingham, Deborah Ball and John Hattie remind us of the important role pedagogical content knowledge of teachers plays in the success of students. This is not to say that factual/procedural content knowledge has no place – in fact, I would argue that this type of knowledge is necessary before teachers can develop worthwhile pedagogical content knowledge. However, it may also be important to ask the question “How much factual/procedural content knowledge is sufficient to help teachers acquire pedagogical content knowledge required for success in the classroom?”
For the Love of Maths 
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That’s really quite strange! Or at least unexpected. That suggests a good understanding of the difference between slant height and height, so I’d lean towards your error guesses as well.
How about a deal with a local math teacher (at the level for which the pre-service teachers are training)? The university class will mark a stack of assignments for that teacher, once every week or two, and be able to assess and discuss student understanding. The trick would be a strong relationship between university instructor and grade school math teacher. Requires trust and good communication.
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Bryan,
I find your analysis interesting. Because of the approach used, you say your student found the area of a square based prism instead of a square based pyramid. You’re totally right, but personally, I wouldn’t have gone there.
I feel like your Maths CSI analysis here is procedural in nature. Student used wrong formula -> Check which formula student used -> determine that student found lateral area of the incorrect shape, because student used the incorrect formula.
When I looked at the student work, I saw one of two mistakes:
1) Student multiplied base by slant height, but forgot the nature of triangular faces and so forgot to divide by 2.
2) Student may have used pyramid height instead of slant height. That may just be an unclear transcription of the student’s work though.
Either way, the mistakes I saw were more conceptual than procedural. I didn’t even think about which formula they chose – I just thought about the nature of their calculation.
I don’t think my analysis is any more “correct” than yours. It’s more like a blindspot of mine. Because I don’t love memorizing procedures and formulae, I don’t use them often, so I look for conceptual difficulties. On the other hand, students often memorize and use formulae, and so the error may well have been choosing the wrong formula.
Very interesting.
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This is a cool exchange. I agree that my analysis doesn’t get to the heart of the students own misconception since I didn’t follow up with any particular questions. (I usually ask one or two questions to try to zero in on the misconception – ie. is it procedural or conceptual?)
The interesting bit which I have not added here was that the next part of the question asked for the full surface area, and the student had that one bang on. Got the slant height using Pythagoras’ Theorem and everything! Totally peculiar, right?! Knowing that, I might lean towards a misunderstanding of “lateral surface area” or not fully understanding the textbook formulas (since they are in terms of perimeter, which is perhaps less common to use).
Either way we chat about it, I think it is certainly beneficial for pre-service teachers to be put into situations where they need to try to figure out what the student’s misconception is (whether it be procedural, conceptual or both in nature). The big question is how do we effectively do this?
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