Think like a mathematician
Teach students math procedures if you want them to solve equations. Instill a deep conceptual understanding of mathematics and get them thinking like mathematicians, and you’ve prepared your students to meet the challenges of the 21st century.
That’s a message presenters will send at the National Council of Teachers of Mathematics conference, which will be held in April in Boston. The Common Core State Standards and their focus on critical thinking and conceptual understanding are among the top issues that will be addressed this year, says NCTM President Diane Briars.
Most districts in Common Core states have spent the last few years aligning curriculum to make sure they were teaching the correct mathematics for each grade level. Now the focus is shifting to the Common Core’s Standards for Mathematical Practice, which describe not what should be taught, but how students should be engaging with mathematics, says NCTM presenter and math consultant Grace Kelemanik.
Students should be debating with one another and engaged in hands-on math learning, says Mark Goldstein, an NCTM presenter and vice president of teaching and curriculum at the Center for Mathematics and Teaching, a California nonprofit working on middle school Common Core curriculum.
Changing classrooms from rows of students memorizing facts to actively engaged learners in coming years will be buoyed by a slew of other advancements in mathematics instruction. They include improvements in classroom technology, the growth of 1-to-1 programs in schools, flipped classrooms, and efforts to better integrate formative assessments into daily instruction.
Indeed, from changes in K12 standards to new recommendations on what students should be studying in the first two years of college, there is a remarkable coherence of vision right now about how to transform math education, Briars says.
Diving into Common Core math
The eight Standards for Mathematical Practice in the Common Core describe math processes and ways of thinking that students should master. Many teachers have an understanding of just a few of the standards, and might be overlooking the most important ones because they are the most complex, Kelemanik says.
The three practices that could have the biggest impact are:
MP2: “Reason abstractly and quantitatively.”
MP7: “Look for and make use of structure.”
MP8: “Look for and express regularity in repeated reasoning.”
“Those three math practices actually define three different ways of thinking mathematically, or three different entry points into a mathematical problem,” Kelemanik says. “If you can develop those three different lines of reasoning, then you can develop grit. You can persevere in your problem solving.”
Students become comfortable with these habits of thinking when they are integrated into daily classroom routines Ñjust like the procedures teachers have for handing in homework or lining up for lunch.
MP7 is really asking students to understand how numbers and shapes get put together and taken apart, Kelemanik says. In a classroom routine that Kelemanik and her colleagues developed called “contemplate then calculate,” a teacher flashes a problem, covers it up after a few seconds and thenÑinstead of asking for the answer right awayÑhas students break down the problem with the person sitting next to them.
For a second- or third-grade subtraction problem such as 103-97, students might notice that 103 and 97 are both very close to 100. Instead of stacking the numbers and subtracting, the teacher asks students to find the answer with the fewest number of steps using a shortcutÑlike visualizing both the numbers on a number line.
The goal of the strategy is to get students to start thinking mathematically. It can be applied across grade levelsÑfrom simple subtraction problems to complex algebraic equations, Kelemanik says. The key is for teachers to use the process to teach students the properties that allow them to use the shortcut.
Helping English learners
The Common Core and other college and career readiness standardsÑsuch as teaching math using real-world problems and encouraging student discussionsÑalso could be a big help to English language learners, says Briars, the NCTM president.
Research shows that increased discourse in class is one of the best ways to build English language skills, says Susie Hakansson, an NCTM presenter and president of TODOS: Mathematics for ALL, an education advocacy organization.
That also means calling on English learners to answer questions in class, something that teachers might not do in an effort not to embarrass these students, Hakansson says. Finally, it’s important to maintain the rigor of the mathematics. Just because students don’t understand the language doesn’t mean they don’t know or can’t learn the content, Hakansson says.
Providing visuals that help students understand the language and developing lessons that are multi-modal are other ways to help English learners, Kelemanik says. English learners also should be allowed to communicate their thinking through manipulatives, such as cubes and tiles, and by writing, drawing and annotating texts in different colors, she says.
Developing formative assessments
Students studying to become teachers are often taught to focus only on summative testing, overlooking the importance of integrating formative assessments into daily instruction, says Skip Fennell, an NCTM presenter and former president of the organization.
Changing this mindset has never been more important, as Common Core’s emphasis on conceptual understanding means teachers need to gain insight into students’ thought processesÑnot just their ability to solve an equation, says Matt McLeod, NCTM presenter and research scientist at the Education Development Center.
The best formative assessments are developed during the lesson planning process, says Fennell, part of a group of NCTM presenters who have been working for the last two years on developing a toolbox of five formative assessment techniques for teachers to use. The five techniques are:
“Show me,” in which students demonstrate how they solved a problem
Interviewing students one-on-on or in groups about the lesson
“Exit tasks,” which should be a problem or meaty task
“Hinge questions” that relate to the key goal of the lesson.
Better preparing students for college
Two new reports analyzing changes to college math instruction also could have big implications in K12, says Briars, who will be presenting key takeaways from the reports at the April conference.
The first report is from Transforming Post Secondary Education in Mathematics, a project involving professors from top universities across the country and sponsored by the Carnegie Corporation of New York and the Alfred P. Sloan Foundation. It was released after the organization’s first annual meeting in June. The other report, from the Mathematical Association of America, will be released this year.
Universities are working on transforming the first two years of collegiate mathematics to improve students’ in-depth conceptual understanding of mathematics, Briers says.
And such transformation will likely mean an increased emphasis on statistics and modeling in high school mathematics classes. District leaders also may have to better prepare students for calculus, while also reducing the number of students who end up pushing themselves to take the advanced subject in high school because they haven’t yet mastered the underlying concepts.
“The big message we are hearing from [universities] is, âit’s not a race,’ ” Briars says. “Make sure you are having students develop deep conceptual understanding before they are moving on to take more advanced mathematics.”
Jessica Terrell is a freelance writer based in California.