Many math instructors teach math concepts the same way they were taught – by using gimmicks to memorize rules and procedures, applying rules and formulas without understanding “why” they work, and not really caring why as long as the result was a correct answer. These methods may work well when simple computation is the goal; however, when students are asked to apply their isolated procedural understanding to solve real-world situational problems, they are often frustrated.
“Teaching for conceptual understanding” is all about the “why” along with the “how.” The rigor of the College and Career Readiness Standards for Mathematics (CCRSM) requires that conceptual understanding be balanced with procedural skill and real-world application. But what does “teaching for conceptual understanding” actually look like?
The conclusions of a study published in the Harvard Educational Review identifies four observed hallmarks of teaching for conceptual understanding. The article is posted on the Teaching Now blog from Education Week Teacher.
“The researchers identified four hallmarks of what it means to teach mathematics conceptually:
- Using mathematical language. Teachers should use academic language purposefully. That practice helps students develop a technical understanding of the meaning of the words. A past study found that children who were exposed to explicit number names in pre-kindergarten showed a consistently stronger performance in math than their peers for up to six years later.
- Using visual representations. These representations help students make sense of mathematical concepts and understand mathematical structures and the relationships between quantities. Ideally, researchers say students should be able to flexibly use different representations for the same mathematical concepts.
Pressing students for explanations. Doing so allows students to further develop their understanding by working through obstacles and contradictions and reaching for connections across strategies.
- Teachers should establish classroom norms, researchers say, where a good explanation is a mathematical argument and not simply a description of the procedures, and errors are further opportunities to learn.
- Using story problems. Problems with illustrative contexts connected to the real world can help develop students’ understanding. When done in a way that challenges students’ thinking, the story problems give students a familiar metaphor, they are interesting, and they enhance transfer of learning.”