*By Yvonne Lai, University of Nebraska – Lincoln and Heather Howell, Educational Testing Service*

Consider how you would respond to two different versions of a question. In the first, you are asked to solve a high school mathematics problem. In the second, some high school students’ solutions to that problem are shown to you. You are asked to assume the role of the students’ teacher and to evaluate the mathematical validity of the students’ different approaches. What knowledge, if any, do you need in the second situation that you don’t need in the first situation?

Some would argue that the second situation is just about knowing math. If you, yourself, can solve the high school mathematics problem correctly, and you are very capable in high school mathematics, then this should be enough to evaluate a high school students’ solution. Yet others might say that this question is about teaching. If you can’t interpret students’ work, you can’t judge it accurately. Still others might say that this question targets something in between straight math and teaching. We would say that this scenario assesses a blend of all of these things that previous scholars have named *mathematical knowledge for teaching *(MKT). We ask the reader to join us in considering, as some have argued, why MKT is a form of applied mathematics – and why mathematicians have a stake in thinking about MKT in this way.

To begin, consider this example:

This question illustrates a coordination of mathematics and teaching. The intended answer is that only Matt’s method is valid. Jing found the slope between two points and assumed the slopes between all points were the same, without verifying that the data were actually linear. Matt’s method, without producing the function, verifies that all the given points fall on the line. The key idea to note is that the assumption of linearity is not warranted by the information given in the problem. Jing makes this assumption, and Matt does not. Solving the question requires knowing that the linearity assumption is not warranted. Solving the question also requires making sense of what Jing’s and Matt’s written work might mean. The linearity assumption jumps out to an experienced teacher as the important point that is likely to trip students up.

The question asks the respondent to think simultaneously about mathematics and teaching in a way that taking conventional mathematics coursework may not prepare a prospective teacher to do.

In a recent validity study (funded by the NSF __[1]__) designed to support a measure of teachers’ MKT, we asked secondary mathematics teachers to respond to this question. We saw variations of all of the above perspectives in their responses. Some focused on the mathematics, others on digging into what the two students might have been thinking, and many on something in between. Notably, respondents who did not attend to the linearity assumption were likely to favor Jing’s response, often describing it as more “complete” or “sophisticated”. In a number of cases, respondents stated that Matt needed to check all three points because linearity was not given, but also indicated they would have given at least partial, if not full, credit to Jing’s answer. After all, Jing derived the equation, whereas Matt only verified it. Yet, from the perspective of strict mathematical correctness, these judgments seemingly hold the two students to different standards. Imagine these respondents bringing this understanding to real students in a real classroom. Students might deduce from this that Matt’s approach is mathematically wrong. They might generalize that only solutions that involve algebraic manipulation of formulas are valid. They might pick up the implicit message that apparent sophistication is more important than whether a method effectively and efficiently addresses a given problem. They might be genuinely confused by the apparently different standards of mathematical validity. And finally, they might take from this that linearity can be assumed when convenient, or simply not have an opportunity to learn that such assumptions need to be established and/or verified as a general practice. In other words, these responses indicated something important: the knowledge these teachers did or did not demonstrate was likely to *matter* in their teaching, for both mathematical and pedagogical reasons.

Scholars have acknowledged the importance of MKT at the elementary level, even mapping out programs of study designed to provide elementary teachers with an opportunity to acquire this knowledge. It has been less clear whether secondary teachers-in-training need to study anything beyond the content of conventional mathematics and pedagogy courses. The study from which the opening question was taken demonstrated that that there *is* MKT at the secondary level that (1) can be measured and that (2) differs from what teachers are likely to learn in conventional mathematics courses. And we suspect that secondary mathematics teachers-in-training have few opportunities to learn this MKT as part of their teacher preparation programs.

While it seems clear that MKT learning opportunities belong in teacher preparation programs, it is less clear whether they belong in mathematics courses, pedagogy courses, or elsewhere. One could argue, as some have done, that MKT is applied mathematics; applied in the context of teaching. We ask the reader to join us in considering why one might think of MKT as a form of applied math in this way and present three arguments for why mathematicians teaching mathematics to teachers-in-training might have a stake in thinking about it this way.

- MKT is, at least in part,
*math*. Although a teaching perspective makes the linearity assumption in the opening example jump out to an experienced teacher, attending to such assumptions is also*mathematically*crucial. Often, mathematical ideas are pedagogically key because they are mathematically key. Because students are still learning how to engage in mathematical thinking, teachers need to do more than just think mathematically themselves; they also need to know how to call attention to what they are thinking and why. This involves understanding the mathematics in more than tacit ways. It is not enough for a teacher to avoid the error of assuming linearity; a teacher needs to know to recognize the linearity assumption as important, pedagogically. But both kinds of understanding are grounded in the mathematical idea that linearity cannot be assumed. - As with other forms of applied math, learning components in isolation may be less useful than learning them in context. Finding solutions to differential equations doesn’t get you far as a mathematical biologist if you haven’t had a lot of practice using them to approximate natural phenomena. In the validity study, we saw striking cases in which respondents with strong mathematics backgrounds and who demonstrated strong, specific, but decontextualized mathematical knowledge in other parts of the interview did not apply that knowledge appropriately to contextualized problems. In other words, we should not assume that studying mathematics in traditional courses necessarily or automatically equips teachers to call on and apply that knowledge appropriately in their work.
- We all have a long term stake in improving teachers’ MKT. Teachers were once students and learned mathematics in school. Their students are future teachers. The authors of
*Mathematical Education of Teachers*2 urge us to remember Felix Klein’s “double discontinuity,” that a teacher-in-training doesn’t see how university and high-school mathematics connect, and upon entering the classroom, doesn’t see what they learned in university has to do with teaching. As part of the second discontinuity, prospective teachers who are able to use quite sophisticated reasoning or sound mathematical habits of mind, such as checking assumptions in advanced math classes, might be capable of applying those skills to high school mathematics that they will be teaching, but simply have never had the occasion to do so.

What might we do to create learning opportunities for preservice teachers to learn MKT as applied math? We recently convened a group of mathematicians and mathematics educators to take on this question. Participants considered questions like the opening one as examples of the MKT we would ideally expect teachers-in-training to have opportunities to learn in teacher preparation. Workshop materials included a set of 55 such questions. (We encourage you to ask us more about these questions! They were designed with the intention of giving more examples of what MKT is like, especially at the secondary level. Copies of these questions can be obtained by contacting Heather).

When we began this workshop, we had hopes that these questions could serve as potential in-class or homework questions in both mathematics and pedagogy courses. The workshop provided some encouragement that our hopes could be realized, and that as a bonus, the questions could seed productive conversations among colleagues. For instance, our participants suggested that questions could help in identifying mathematical ideas that might be missed by both mathematics and pedagogy courses because they do not fall squarely into either. They suggested using the opening example as a context to discuss what constitutes mathematical validity. Our participants further suggested that the questions could be concrete examples for conversations about the tension between the notion that in mathematics, one wants to “never say something you have to take back,” yet in teaching, one should not “say something a student can’t take in;” and that questions can be mini-cases that could develop into extended cases of teaching and mathematics. Some participants are making plans to use these questions in their programs.

Based on the results of our validity study, and the promising suggestions arising from the workshop, we are now turning our attention to supporting the use of these questions to supplement coursework, and to forming professional communities focused on learning and teaching MKT through the use of questions such as the opening example. We invite your thoughts and comments about your experiences teaching MKT as a form of applied mathematics, especially at the secondary level, and would welcome requests to share materials.

[1] This material is based upon work supported by the National Science Foundation under Grant #DGE-1445630/1445551 Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

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