## Problem-Based Learning

Today I was a fourth grade teacher at Robeson Elementary. I was subbing for a teacher who has used me several times in the past, and also has written one of my professional letters of reference. I was curious to see how her class would be this year, because last year’s class was awful, and I don’t say that lightly. She herself would often comment on how they were her worst class ever, and most days in her room were simply a struggle to prevent arson, defenestration, and worse.

Fortunately, this year’s class is totally different. While there were a couple of boys who were quite a challenge, I found most of the class to be quite delightful, which meant that I could really teach, rather than simply referee fights and hope that nobody would draw blood during the day. (Incidentally, two students today needed band-aids for different bumps they got, but neither was because somebody caused them injury.) The main focus of my teaching today was math, which is one of the few subjects I have found I love teaching, despite the fact that, throughout my own public education, I detested math. Oddly so, I imagine, since I was actually quite good at math–despite what my wife will say. (In her defense, I am terrible at doing mental math. In my defense, I am terrible at doing mental math, but I don’t think anybody needs to be particularly adept at mentally adding up long lists of numbers!)

The math lesson today focused on understanding the relationship between base-1o place values. One may initially think that this is an easy concept to teach. It is easy to tell students what the place values are. It is much more difficult to get them to understand why they are that way. A current trend in education, and particularly in mathematics, is problem-based learning. The quick summary of this pedagogical style is that it is better for students to discover concepts on their own, rather than be told them. While I believe this is true in many fields, I do not believe it is true in mathematics. A recent discussion with a friend of mine who is earning a PhD in applied mathematics and secondary education actually affirms this notion. It turns out that research has indicated that problem-based learning in math is not more effective that lecture-based learning and, in fact, it may be less effective. Which means that what we are doing may not be for the best.

Luckily for me and my students, I do not believe in problem-based only learning. I believe in briefly presenting an idea, then allowing students to further develop their understanding through problem-based learning. Sometimes this is done by giving them relevant problems and seeing if they can correctly solve them. And sometimes it is done by presenting a scenario and asking the students to explain what is wrong. I used the latter method this morning, and it worked quite well. This is a (not very good) illustration of what I presented to the students:

By way of explanation, the symbols are representations of base-10 blocks. From left to right, we have the large cube (that was 1 whole unit), the flat (1/10th), the long (1/100th), and the cube (1/1000th). For those who are not aware, base-10 blocks don’t have any block larger than the large cube. So the question posed was two parts: one, what’s wrong? and two, how do we fix this? The students quickly identified that there cannot be two digits in one place value column. Their solutions for how to fix it were quite creative. Eventually, they decided that they needed to come up with a symbol to represent a long bar consisting of ten large cubes. They decided to do this by taking the large cube symbol and using a long rectangle instead of a square. It would still have the inverted L-shape in the top right corner to indicate the large cube concept.

This was a perfect example of merging lecture-based instruction (I explained that they were working with base-10 place values, and led a brief discussion about what the place values are), and problem-based learning, in which the students “discovered” a way of representing the concept of the ten (a numeric value that is all sorts of messed up when trying to understand base-systems and numerals and zero). So, all in all, my day as fourth grade teacher went quite well!

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