What does it mean to think and behave as mathematicians? Over the past decade, pressure from the top has defined mathematical thinking narrowly, in terms of facts and procedures. For example, high-stakes assessments in states like California favored short problems; procedural thinking; and scant attention to extended problem solving, modeling, and producing or critiquing chains of reasoning, says Alan Schoenfeld of the University of California–Berkeley and the Mathematics Assessment Project.
To date, 45 states, the District of Columbia, 4 territories, and the Department of Defense schools have adopted the Common Core State Standards for Mathematics. The math standards represent a shift from the focus on skills and procedures of the past decade toward a conceptual understanding of mathematics, balancing practices with content and asking students to explain mathematical reasoning.
Structurally, the K–8 standards are organized by grade levels, while high school standards are organized by conceptual categories—number and quantity, algebra, functions, geometry, modeling, and probability and statistics—showing the body of knowledge students should learn in each category. Model pathways (in Appendix A of the math standards) describe how high school math courses might be organized around a progression of content and concepts.
The standards, which were developed in 2010, aim to bring content and practices into better alignment. They draw on major works like the National Council of Teachers of Mathematics' Curriculum and Evaluation Standards for School Mathematics (1989) and Principles and Standards (2000) and the National Research Council's Adding It Up (2001) to make their case.
For some, this realignment will require a radical shift in pedagogy.
"It's not traumatic if you've been teaching for the broad spectrum of mathematical processes," Schoenfeld says. "If you've had a skills focus imposed upon you, then it's going to require a significant amount of support to learn how to run classrooms in which kids have the opportunity to talk mathematics and engage in complex problems, instead of sets of exercises."
The eight Standards for Mathematical Practice are a pedagogical North Star of sorts:
- 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 make use of structure.
- Look for and express regularity in repeated reasoning.
"We want math students to act like mathematicians," explains Kathleen Dempsey, coauthor of McREL and ASCD's mathematics-focused Common Core Standards Quick-Start Guides. "The practices guide [students] in demonstrating their understanding."
Discourse Instead of Drills
The Standards for Mathematical Practice describe the ways that students should be able to engage in mathematics, on a regular basis, says University of California–Los Angeles professor and co-executive director of the California Mathematics Project Kyndall Brown.
These practices could create an important change for students, especially in the ways low-income students of color experience math, according to Brown. As UCLA's Institute for Democracy, Education, and Access documented in a 2007 study, low-income students are already more likely to be in overcrowded classrooms with underqualified teachers and have less access to advanced classes. If mathematical practices and content were to be integrated appropriately, it would mean more student-centered classrooms, Brown says.
"Students can't be passively taking notes or working on problems, and teachers can't just give all the information to students," he says.
Schoenfeld agrees: "You're never going to develop mathematical reasoning if all you're doing is imitating the procedure that the teacher shows you.
"You have to have the opportunity to make a hypothesis, see what works, have people question you, and refine your argument. All of that means a lot more discourse in mathematics classrooms."
Phillip Daro, director of the Strategic Education Research Partnership, which works with the San Francisco Unified School District, says that a student's ability to explain his mathematical thinking can be an accurate predictor of achievement. Those explanations must include why the student believes something is true and why it makes sense.
Brown's tentative optimism about the new standards is tied to his experience working in the Los Angeles Unified School District during the 1990s. Back then, Brown witnessed the positive effects created by the combination of the National Council of Teachers of Mathematics standards; the accompanying California framework for implementation; aligned curricula; and a significant investment from the National Science Foundation, which provided professional development for implementing the standards.
As a result, the Los Angeles Unified School District saw an increase in students in every demographic group taking and completing college prep mathematics.
"Those experiences led me to believe that if the standards are implemented with fidelity, they will provide access and equity," Brown says.
That's a big "if." Funding, teacher supports, and assessments will all have to line up with the intentions written into the standards.
The Test Is Yet to Come
Two federally funded assessment consortia are charged with developing assessments of Common Core State Standards: Smarter Balanced Assessment Consortium and the Partnership for Assessment of Readiness for College and Careers (PARCC).
Out of the gate, this is an improvement on the current patchwork of 50 different assessment sources, Schoenfeld says. Both Smarter Balanced and PARCC have released sample assessment items on their websites. One major point of difference between the consortia is that, rather than a single score, Smarter Balanced will judge students on four aspects of their work: concepts and procedures, problems solving, producing and critiquing reasoning, and mathematical modeling.
"It's a radical shift in goals," Schoenfeld says. "If Smarter Balanced continues this way, we'll have a mandate to teach the way we've wanted to, ever since the National Council of Teachers of Mathematics standards were written."
Amitra Schwols, Dempsey's coauthor on the Common Core Standards Quick-Start Guides, cautiously shares Schoenfeld's enthusiasm, but also points out an important challenge.
"We now have essay questions where one performance task is allotted two hours and requires persistence, modeling, and explaining your thinking," she says. Although that's promising, she says, at the same time, teachers may be sweating, wondering "How am I going to get my kids to this level?"
Focused and Formative
At the instructional level, Schwols and Dempsey advise teachers to focus on the standards for practice, critical areas, and connections. As teachers plan their units, they should think about how to integrate mathematical practices within each domain of the math content standards.
For high school students, teachers should use the model pathways in Appendix A of the standards to focus on the critical areas. Schwols and Dempsey recommend focusing on 4–6 critical areas per high school year.
At the high school level, teachers may now be asked to teach less familiar topics—advanced courses that were previously relegated to advanced placement or college prep courses, according to Dempsey. That could pose a challenge, but teachers need not go it alone.
"Building-level professional learning communities, especially around learning content in the critical areas, will be essential," Dempsey says.
Appendix A is also a good resource for seeing the connections across grade levels (K–8), and conceptual categories (high school). By looking at the connections among the content, students will experience concepts as extensions of previous learning and begin to see math as an integrated whole, Schwols says.
In California, Schoenfeld is leading a project with the Oakland Unified School District that focuses on formative assessment lessons as the linchpin for developing teachers' abilities to cultivate mathematical thinking, not rote performance of isolated skills and procedures. These formative assessment lessons are designed to reveal students' understanding, or lack thereof, and give teachers diagnostic tools to intervene.
Provide Support
"If you just give teachers the formative assessment lessons and a video tape or two showing how an effective teacher uses them, they're almost guaranteed to fail," Schoenfeld says. "What you really need is a strong, school-based support network."
Schoenfeld's crew works with Oakland teachers over the summer and throughout the year to practice, experience, and reflect on formative assessment lessons. Schoenfeld notes that although ground-level supports are crucial, school leaders need to understand what the Common Core math standards pedagogical shifts should look like, too.
His team brings administrators into lesson demos and gives them a guide for what classrooms using the lessons should look like.
Schoenfeld believes that classroom observation tools can be framed around five dimensions of activity that should be present in a math classroom:
- Quality of mathematics.
- Cognitive demand (pushing kids without leaving them out at sea).
- Equity (methods for all students to engage).
- Agency and authority (Do kids get to talk math and develop identities as mathematicians?).
- Assessment (in particular, formative assessment).
With this context, observations can be a productive lens for targeting ongoing professional development and supports for math teachers.
Getting students to develop a deeper understanding of how math works, over a range of problems, will be challenging. Daro poses the shift as moving teachers from teaching kids to get the answer to a problem to using a problem to teach the mathematics of a particular unit.
Making this shift will take much funding, time, and support. Brown emphasizes that it will, in turn, create opportunities for school-based collaboration and allow teachers to talk about the students they have and how to implement the standards within the context that they teach.
Also, assessments will need to match the standards, and assessment directors will have to fight the pressure to opt for cheaper, multiple-choice bubble tests, as opposed to teacher-scored essay tests.
