By Kayla Chandler, Nicholas Fortune, Jennifer N. Lovett, and Jimmy Scherrer
The new math standards ask for more than content memorization and application. They require math literacy and new assessments that measure it.
The Common Core State Standards for mathematics promote ideals about learning mathematics by providing specific standards focused on conceptual understanding and incorporating practices in which students must participate to develop conceptual understanding. Thus, how we define learning is pivotal because our current definition isn’t aligned with how we teach and assess. But how do we reconcile the two? What does this look like? And what does this imply about how we assess learning?
Consider for a moment that you are going to buy a new car. You already know you want a shiny black one with all the bells and whistles, but because your commute to and from work is quite a few miles, you want a car with decent gas mileage. So you look at a U.S. Department of Energy web site devoted to fuel economy (fueleconomy.gov). Here, you find tools that help you compare cars and calculate fuel cost, as well as articles on whether hybrids can really save you money. But, rather than take someone else’s word, you decide to consider the data for yourself. You already know a hybrid car is probably going to have better gas mileage, but you cannot really afford that. You just want a car that does not have terrible gas mileage. What exactly will you need to know? How will you determine which cars are even worth considering for purchase based on your needs?
Answering questions like those above require mathematical literacy. You need skills to help you reason through and analyze the problem at hand. In this case, you need to be able to make sense of the fuel economy data by reasoning quantitatively. You will have to consider the context of the situation and use it to continually evaluate the reasonableness of your decisions. Otherwise, analyzing the data just becomes a bunch of computations with no meaning. Unless you can make sense of the data, considering it on your own is pointless. You otherwise would have to believe someone else’s opinion of how they interpreted the data in order to make your decision, which, consequently, might not be based on a viable argument.
Therefore, the goal of K-12 education must be about more than learning specific content objectives. Education must enable students to learn the skills necessary to be productive members of society. The Common Core mathematics standards are aimed at providing students with the content and reasoning skills they need to be successful. They were developed to provide a more coherent and focused picture of what students should understand and be able to do throughout their study of mathematics. Ultimately, the objective is not only to prepare students for college but also the world around them by emphasizing mathematical understanding and application.
One way the mathematics standards’ authors envisioned this could happen was through applying the Mathematical Practices. These practices aim to ensure that students experience processes important to mathematics, such as making sense of problems and constructing viable arguments. But current assessments do not reflect these ideals. In the September 2014 issue of Kappan, James Popham argued that current assessments “are faulty indicators of how well [students] have been taught” (p. 47). We agree that current assessments are poor indicators of teaching, but when mathematical literacy includes the ability to participate in mathematical practices, current assessments are poor indicators of learning.
Traditionally, learning has been measured by focusing on basic knowledge and skills. Students are required to remember facts and apply procedures. Conversely, the mathematics standards suggest that mathematical literacy also includes conceptual understanding and participation in mathematical practices. Thus, different perspectives need to be taken when crafting measures of these other criteria.
Assessments must align with instructional goals and classroom routines (Shepard, 2000). In classrooms, this means teachers give students opportunities to engage in practices common to the field of study. For example, when studying mathematics, teachers create environments and activities that enable students to experience work that mathematicians would do. Then, as students have opportunities to participate in these environments, they begin to understand the types of practices mathematicians engage in and also the mathematics itself. This is how all educators could view learning. After all, many of us already have learned this way before.
Remember the first time you entered a Starbucks? Did you know how to order the drink you wanted? Did you know Starbucks had its own language? It was probably a little overwhelming. And, the fact that the regular customer behind you in line was tapping his foot and sighing as he checked his watch likely didn’t help! But, hopefully, you had an understanding barista and made it through your first ordering experience. But, despite the difficulty and shame you felt ordering so slowly, the latte was so tasty you decided to continue visiting Starbucks. Each time, ordering your favorite drink became a little easier until one day, you had it down to a science: “I’ll have a venti, nonfat, light whip, stirred pumpkin spice latte.” After that, it was not long before you realized you had become the impatient regular in line who was upset that the customer at the counter did not seem to know how to order a drink. And, suddenly, you laugh at yourself as you realize the irony of it all.
Learning how to order at Starbucks is an example of learning to participate in the practices of a certain community. You not only had to learn the language of the community to obtain your desired beverage, you also had to learn what was acceptable behavior for that community. By participating in the Starbucks community, you learned that being able to order efficiently was a valued practice and eventually expected that behavior of others. In our Starbucks scenario, you went from being a peripheral participant to someone who could participate fully in that community (Lave & Wenger, 1991). Perhaps you have done the same at Chipotle or as you learned to play video games. If you stop to think about the various communities you are part of — work, friends, sports, etc. — you can probably remember what it was like to be the newcomer and how you became a full participant of that group over time. Why should education be different from that?
Measuring classroom learning
What does learning this way look like in the classroom? Let’s consider our new car example. Perhaps a mathematics teacher is having students complete a project that requires constructing viable arguments and critiquing the reasoning of others. In this project, students are responsible for completing tasks such as identifying the job they expect to have after college, talking with parents and other adults about the types of monthly bills they might expect to have, buying a car, and considering the cost of living for their desired location. At each step, students must generate explanations that peers will review and critique. Thus, students are in a situation that simulates problem solving in everyday life.
Within this larger project that focuses students on attending to argumentation is the consideration of which car to buy. Students not only must consider cars of their dreams, but they also have to reason quantitatively to weigh the practicalities of how the car they choose fits their situation. Thus, students must decide, based on all of the components in their problem, which car best fits their needs. Since some students may have to live farther from work in order to afford the home they want, they could encounter a problem similar to the fuel economy issue introduced earlier. At this point, students will encounter the mathematics involved in their decision and be expected to generate an argument that considers the context of their situation. Students also will be expected to analyze the reasoning of others and determine the effectiveness of a plausible argument.
Tasks like the one described above have no right answer. So how do we know if students learned anything from engaging in them? We propose that learning could be measured from a situated point of view. Using this perspective, we can consider students’ participation in practices of value to mathematics. This means that learning occurs when students have more effective participation in these practices. And, because the content is tied to these practices, “a lack of understanding [the content] . . . prevents a student from engaging in the mathematical practices” (National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010, p. 8). This means that examining how a student participates in the Mathematical Practices can reveal what the student understands or does not understand about the mathematical content.
Implications for assessment
This change in perspective of learning has drastic implications for how we assess learning. In our previous example, we suggested that the Mathematical Practices could be the framework that guides the teachers’ insight on how to assess students’ immersion into the mathematics community when situated in the car-buying task. Overall, the Mathematical Practices can be a vehicle that will support mathematical literacy in today’s learning environments. By these standards, students should be able to:
- Make sense of problems and persevere in solving them;
- Reason abstractly and quantitatively;
- Construct viable arguments and critique the reasoning of others;
- Model with mathematics;
- Use appropriate tools strategically;
- Attend to precision;
- Look for and use structure; and
- Look for and express regularity in repeated reasoning (National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010, pp. 6-8).
In so doing, they move from peripheral participation to central participation in the community of mathematics and are also mathematically literate. Consequently, since the Mathematical Practices aim to ensure that students experience processes important to mathematics, teachers can use them as a situated assessment framework.
Mathematics is not the only field that can support and implement situated assessment frameworks. The Common Core standards for English and language arts and the Next Generation Science Standards also contain standards that suggest that learning is more than remembering and applying skills. Members from these fields should look at their standards to determine what practices are valued in that field to develop frameworks to use for situated assessment.
However, teaching will not change until we change assessments. Historically, regardless of calls for reform, we resort to teaching to the test. Thus, we need to carefully consider how current assessments should change. Common Core mathematics assessments should attend to the connection between content and the Mathematical Practices. After all, students cannot successfully participate in the Mathematical Practices without knowing the mathematical content. Only when we measure learning from a situated perspective and incorporate the Mathematical Practices into assessments can we hope to see the Common Core mathematics standards realized by developing mathematically literate students.
Lave, J. & Wenger, E. (1991). Situated learning: Legitimate peripheral participation. New York, NY: Cambridge University Press
National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. Washington, DC: Authors.
Popham, W.J. (2014, September). The right test for the wrong reason. Phi Delta Kappan, 96 (1), 46-52.
Shepard, L.A. (2000). The role of assessment in a learning culture. Educational Researcher, 29 (7), 4-14.
KAYLA CHANDLER (email@example.com) is a doctoral candidate and graduate research assistant in mathematics education in the STEM Education Department at North Carolina State University, Raleigh, N.C., where NICHOLAS FORTUNE is a graduate student and graduate research and teaching assistant and JENNIFER N. LOVETT is a doctoral candidate and a graduate research assistant. JIMMY SCHERRER was an assistant education professor at North Carolina State University and a 2011-12 PDK Emerging Leader. Dr. Scherrer passed away in August 2015.
© 2016 Phi Delta Kappa International. Phi Delta Kappan 97 (5), 60-63.