A pair of new studies has found that the stereotype of the aloof professor—you know, the one that is accomplished in her field but I’d like to see her come teach the kids in my school—might be, surprise surprise, a little unfair. Experts, it turns out, are not necessarily such bad teachers after all.

Researchers found that the superior content knowledge of mathematics professors (8 assistant professors and 7 full professors) relative to secondary teachers was associated with a significantly greater amount of conceptual explanations, as opposed to “product oriented” (answer-getting) explanations—and these conceptual explanations resulted in the superior performance of students receiving them.

Study 1

In the first of two studies, researchers gave a group of secondary school teachers and a group of mathematics professors this problem, along with the diagram that I have recreated here, and asked participants in each group to provide a written explanation for a hypothetical student.

**Just imagine an 11th grade student shows you the following mathematical word problem:**

At a school, a school director wants to build a new archway of the form of a rectangle and an arched semi-circle (also have a look at the figure). As the costs for the material are fixed, the perimeter is set to a predetermined value. How should the vaults be designed, to get the largest area as possible?

**Please provide an explanation about the mathematical background of this word problem. Please write a coherent explanation, that the student could understand the explanation without any additional material. Please write complete sentences.**

The professors and secondary teachers were scored on content knowledge and pedagogical content knowledge, and their explanations were rated according to the proportion of “process-oriented” statements they contained (statements related to conceptual knowledge) as well as the proportion of “product-oriented” statements (statements related to rules and algorithms without referring to conceptual knowledge). Participants’ explanations were also compared with regard to word count, total number of statements, proportion of omissions of steps, definition statements, and reading level.

Results of Study 1

As expected, mathematics professors scored significantly higher with respect to content knowledge and secondary teachers scored significantly higher with respect to pedagogical content knowledge.

Yet, while both groups produced explanations with roughly similar word and statement counts, nearly identical reading levels, and similar proportions of omissions and definition statements, the professors’ (a.k.a., the experts) explanations contained more than twice the proportion of references to conceptual knowledge (29% to 12%). By contrast, more than half (52%) of secondary teachers’ explanations, on average, contained answer-getting, “product-oriented” statements, compared with just 36% for the mathematics professors.

Further analysis showed that, within groups, “only the instructor’s content knowledge predicted the level of process-orientation [conceptual orientation], whereas pedagogical content knowledge did not account for the process-orientation of their instructional explanations.”

Study 2 and Results

But were the more conceptual explanations from the professor experts more successful with students? To find out, in their second study researchers gave groups of students the problem shown above, along with one of the different explanations, in their regular mathematics classroom settings. Students were then given an application test consisting of a problem very similar to the example problem above as well as two near-transfer problems.

The different student groups were very similar in both their prior knowledge and in their ratings of the difficulty of the learning phase (reading the problem and the explanation). Yet, students given the experts’ conceptual explanations outperformed those given the secondary teachers’ “product-oriented” explanations (43% to 34%). And both of these groups roundly trounced a group given the example problem with no explanation at all (19%).

Discussion

I’ve highlighted three interesting points from the general discussion section of this paper. The first is a rather sensible conclusion, in line with the results of the experiments:

This second highlight is a fascinating note, suggesting that the greater proportion of “product-oriented” statements among secondary teachers’ explanations may be due, in part, to those teachers being influenced by what their students demand:

Lachner, A., & Nückles, M. (2015). Tell me why! Content knowledge predicts process-orientation of math researchers’ and math teachers’ explanations Instructional Science DOI: 10.1007/s11251-015-9365-6

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