As students’ mathematical thinking develops, and they encounter more advanced mathematical topics, they are often expected to “behave like mathematicians” and engage in a number of mathematical practices, ranging from modeling and conjecturing to justifying and generalizing. These mathematical practices are distinct from specific content students might learn because they are characteristics of broader behavior, rather than mastery of a single concept or idea. However, these practices represent indispensable components of what it takes to become a successful mathematician.
We think that addressing how students develop the ability to engage in mathematical practices receives far less attention than how students develop content knowledge. In several places in the mathematics education literature, there is a distinction that suggests two different aspects of mathematical knowledge. On the one hand, such knowledge involves knowing mathematical content (such as interpreting the result of dividing a by b), and on the other hand, it entails knowledge of broader mathematical practices (such as generalization or problem solving across domains). The distinction between these two types of mathematical knowledge is reflected in the foundations of the Common Core State Standards for Mathematics (CCSSM), which distinguish between content and practice standards in order to explain how mathematical practices span specific content goals (National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010).
Here, the standards for mathematical content (SMCs) are characterized as “a balanced combination of procedure and understanding” (CCSSM, 2010) about certain content, such as the rational numbers, systems of equations, or geometric theorems. On the other hand, the CCSSM highlights eight broader standards for mathematical practice (SMPs), which “describe ways in which developing student practitioners of the discipline of mathematics increasingly ought to engage with the subject matter as they grow in mathematical maturity and expertise throughout the elementary, middle and high school years” (CCSSM, 2010). These practices include activities like Make sense of problems and persevere in solving them, Model with mathematics, and Look for and make use of structure.
We recently had a conversation with pre-service and in-service teachers in a professional development seminar focused on aligning teaching practices with the CCSSM. One teacher asked, “Am I supposed to teach students how to generalize or model, or am I supposed to teach them about exponential functions, or both? Do I approach each type of standard in the same way? When can I say I have taught them mathematics?” These questions highlight the issues the teachers identified surrounding the nature of mathematical knowledge, as well as their struggles with interpreting and coordinating two kinds of standards in their classroom.
Even more, this tension between teaching content and practices extends beyond the K-12 Common Core. In community colleges and universities (and, indeed, in our own teaching as university mathematics education professors), mathematics instructors must attend to incorporating mathematical practice in their teaching, and it is not very clear how to do so effectively.
In our initial exploration of these issues, we talked with seven university-level mathematicians about a) their interpretation of the practices of modeling, problem solving and justification, b) their own experiences with developing these practices for themselves, and c) how they incorporated these practices into their instruction. Here we share excerpts from interviews with two mathematicians, briefly discussing two preliminary themes.
First, the mathematicians we interviewed suggested that mathematical practices are important for students’ mathematical development and can, in theory, be taught and learned. For example, when asked about the teaching of practices (as opposed to specific content), Mathematician 1 said, “So bearing in mind that I don’t know how to teach any of this [the practices], in some ways being able to do these sorts of things is way more important than knowing how to find the intercept of a graph… you can probably get by in life without knowing how to multiply two numbers; you can go get a calculator. But if that stops you from learning problem solving, that’s a huge problem.”
Mathematician 2 also suggested that, in general, the practices are particularly important for students to learn but remain closely linked to content, saying “I think what you really want are the practice [standards], and the way you typically try to do that is by teaching content, but something in the way you teach the content hopefully will teach the standards for practice. So I feel like the goal is the standards for practice, but I have a hard time imagining having anyone doing that in any meaningful way without doing it through some content.”
These comments suggest that the mathematicians value the mathematical practices and feel they are important for students to learn. However, while they view the practices as important, the mathematicians also noted that the practices also pose unique challenges for how they might be developed and measured.
We asked the mathematicians to reflect on the distinction between SMCs and SMPs. In response, Mathematician 1 noted, “Well, it’s a heck of a lot easier to assess content, and these standards are designed with assessment in mind if I’m not mistaken.” We also had the following exchange with Mathematician 2, in which he highlights the inherent difficulty in assessing mathematical practices.
M2: I don’t mind that division in the standards. I think it’s probably trickier to measure the standards for practice, but I don’t mind putting them out there. Just because they’re hard to measure, to me doesn’t mean that it’s not important to include them as a stated goal.
Int: What makes them harder to measure?
M2: Well I just think it’s a little easier to ask a question and determine from a student’s response whether or not they know a certain piece of content. Where, depending on the situation, it might be easier or more difficult to draw out of them a certain ability they have to approach a problem. Like I can ask a student about whether or not they know the Pythagorean Theorem, and I think it’s not going to be a very long, difficult discussion for me to figure out if they’ve heard of it before and if they can apply it in a problem. Where if I give them a problem about the Pythagorean Theorem and I’m trying to look for how they problem solve or how they model things, that feels a little more context dependent…there might be issues with when that student displays that sort of behavior. They might be very good at that, but not in a geometric context or something. I wouldn’t necessarily say they lacked that skill just if they didn’t bring it to bear on that problem, so I just feel it’s a little harder to say conclusively.
To summarize, these mathematicians suggested that teaching and learning of mathematical practices are very important, but they are difficult to develop and measure. Teachers at all levels are thinking harder about how to teach and assess their students’ learning of mathematical practice. At the K-12 level this is seen explicitly in the inclusion of standards of mathematical practice in the CCSSM, but the same issues pertain to postsecondary classrooms as well. We encourage readers to think about how to incorporate both content and practices as they teach mathematics, perhaps explicitly having conversations with colleagues and with students about this important distinction.
In the next post on this blog, Ben Braun will discuss specific ideas for assessment that can help with some of these issues.
National Governors Association Center for Best Practices, Council of Chief State School
Officers. (2010). Common Core State Mathematics Standards. National Governors Association Center for Best Practices, Council of Chief State School Officers, Washington D.C.