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News from Mathnasium of Cambridge

Is Mathematical Understanding Really Necessary?

Jan 23, 2018

One of the lifelong benefits of mathematics education is the ability to solve problems. As adults we are confronted with myriad problems daily and our capacity to look at, analyze, decompose or break the problem down and come up with solutions derives directly from our understanding of math and our ability to think mathematically. “Most people think math is computation at the elementary level – drilling them in the skills,” said Jeanie Behrend, an education professor focused on math education at California State University, Fresno. “Math is really about application and problem solving.”

Note that I said our understanding of math. That is one of the differences between the Mathnasium Method and nearly all other math educational methods and curriculum - understanding and mathematical thinking or number sense. Our goal as a math learning center is for our students to achieve understanding, and ideally mastery, of the concepts they study with us. Mastery is built upon three pillars: conceptual understanding, factual knowledge and procedural skill.

Students definitely gain factual knowledge and procedural skill when they come into our center. However, all of our instruction is directed beyond facts and procedures toward conceptual understanding; the lightbulb moment and the child’s ability to think mathematically.  We do this through Socratic questioning, gently guiding them toward their own discovery of the answer, and by constantly having the child explain what they are doing and why.  As I learned recently at the Mathnasium Convention in Atlanta, “if you can’t say it, you don’t know it.”

Many people even educators do not understand conceptual understanding themselves. Far too many think that if students know the definitions and the rules or algorithms then they possess conceptual understanding. “It’s easy (for educators) to focus on memorization of facts and memorization of procedures without really identifying the important mathematics” behind them and teaching those concepts to students.

Memorization has never been the best way to learn math, but it was often enough to meet many of the old standards. And knowledge of procedures is no guarantee of conceptual understanding.

Doing math is an operation. It's about applying mathematical procedures, or a sequence of steps and rules, such as addition, subtraction, multiplication, division, estimation, and measurement to solve an algorithmic or story problem correctly and successfully.  It's all about the reproducing and applying facts and rules to achieve or attain that correct answer because, in the end, that's all that matters - getting the correct answer!  This “mindless mimicry mathematics,” as the National Research Council calls it, has come to be accepted as the norm in our schools. This still-dominant Old School model begins with the assumption that kids primarily need to learn “math facts”:  the ability to say “42” as soon as they hear the stimulus “6 x 7,” and a familiarity with step-by-step procedures for all kinds of problems — carrying numbers while subtracting, subtracting while dividing, reducing fractions to the lowest common denominator, and so forth.

More than 70 years ago, a math educator named William Brownell observed that “intelligence plays no part” in this style of teaching. Even now, most students are still being taught math as a routine skill. They do not develop higher order capacities for organizing and interpreting information. Thus, students may memorize the fact that 0.4 = 4/10, or successfully follow a recipe to solve for x, but the traditional approach leaves them clueless about the significance of what they’re doing. Without any feel for the bigger picture, they tend to plug in numbers mechanically as they follow the technique they’ve learned.  Drill does not develop meanings.  Repetition does not lead to understanding.

As a result of the standard approach to math instruction, students often can’t take the methods they’ve been taught and transfer them to problems even slightly different from those they’re used to seeing.  For example, a seven-year-old may be a whiz at adding numbers when they’re stacked or arranged vertically on the page, but then throw up her hands when the same problem is written horizontally.  She may possess a rich informal knowledge base derived from working with quantities in everyday situations that allows her to figure out how many cookies she would have if she started out with 16 and then received nine more – but regard that understanding as completely separate from the way you’re supposed to add in school (where she may well get the wrong answer).

Math educators are constantly finding examples of how kids can do calculations without really knowing what they’re doing.  Children, given the problem   (274+274+274) ÷ 3  set about laboriously adding and then dividing, missing the fact that they needn’t have bothered — a fact that would be clear if they really understood what multiplication and division are all about.

Another example frequently observed is a question like this  “A school bus holds 36 children.  If 1,128 children are being bused to their school each day, how many buses are needed?”  If you divide the first number into the second, you get 31 with a remainder of 12, meaning that 32 buses would be required to transport all the children.  Most students do the division correctly, but fewer than one out of four got the question right.  The most common answer is “31 remainder 12”

This sort of robotic calculation doesn’t reflect a mental defect in the students but the triumph of the back-to-basics, drill-and-skill model of teaching math.  And that’s not just one person’s opinion.  Analysts of National Assessment of Educational Progress data for the Educational Testing Service observed that students can “recite rules” but often don’t “have any idea whether their answers are reasonable.”

For example, many children learn a routine of “borrow and regroup” for multi-digit subtraction problems, however, they do not understand why that procedure works or how the “rule” came about. That would require mathematical understanding of what we call the Law of Sameness as well as Place Value!

Another common conceptual problem is understanding that an equal sign ( = ) refers to equality— that is, mathematical equivalence or balance between the two sides of the equation. By some estimates, as few as 25 percent of Canadian sixth-graders have a deep understanding of this concept. Students often think the equal sign signifies “put the answer here.”

The idea of reading for understanding is clear enough (few adults, after all, spend their time underlining topic sentences or circling vowels), but how many of us have had any experience with math instruction that emphasizes understanding?  We think of math as a subject where you churn out answers that are either right or wrong, and we may fear that anything other than the conventional drill ‘n skill methods will leave our kids unable to produce the correct answers when it comes time for them to take a standardized test.  Indeed, it asks a lot for people to support, or even permit, a move from something they know to something quite unfamiliar.

If you want to grasp the poverty of your own education in math, I offer you the following challenge: explain long division. Explain it to a child, to an adult, to yourself–but really explain it. Use words to describe not the process, but the reason for the process: why each number goes where it does; why you subtract, or divide, or bring down; why the process works. It won’t be easy. I maintain that if you had been educated properly in math, it would be.

That is what mathematical understanding looks like. The ability to justify, in a way appropriate to the student’s mathematical maturity, why a particular mathematical statement is true or where a mathematical rule comes from.

At Mathnasium of Cambridge, our primary concern is not to confuse doing math with thinking mathematically. With the opportunity to use computers and calculators always at our fingertips, it is thinking mathematically that will give our children lifelong skills that will support them in every endeavor they attempt.