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How tough is it for a math teacher to know if his or her students understand basic math?

Tougher than you think, according Marilyn Burns, a veteran math educator and founder of Math Solutions, a unit of Scholastic Inc.

She found that an alarming number of middle-school students couldn’t subtract 998 from 1000 in their heads. Instead, they cross out zeroes and subtract digit by digit without really understanding or thinking through what they’re doing. Burns says teachers too often assume that students who compute properly understand what they are doing, but often these students are unable to apply simple math in the real world.

“Right answers can mask confusion and wrong answers can hide understanding,” says Burns.

Burns’ latest math solution? Teachers in grades 4 through 8 should spend oodles of time interviewing students individually to understand each student’s reasoning and catalogue the answers in a new online assessment tool, the Math Reasoning Inventory.

The good news is that the site is free (funded by the Bill & Melinda Gates Foundation, which is also among the funders of The Hechinger Report) and that anyone, even an overly involved helicopter parent, could do it with his or her own children. But Math Solutions is marketing it to teachers.

Burns has come up with a series of clever questions—some to be done in a child’s head, others with pencil and paper—that quickly show whether a child understands basic subtraction, division, fractions and decimals. There’s never one right way to solve a problem. Burns offers several reasonable paths for how a child could have come up with a right answer. What you don’t want is a blind following of formulas or an inability to articulate how a student figured out that 7/12 is greater than 2/5. No one should be wasting time converting the denominators to 60!

Here’s an example:

What is 99 + 17?

The most important part isn’t the correct answer (which is 116). Rather, it’s when the teacher asks a student, “How did you figure out the answer?” and really listens to the answer.

In this problem, a student might offer one of these approaches:

• Counted on by 1s
• Added by using a standard algorithm (here, Burns means stacking the numbers on top of each other and beginning with the ones column, adding 9 plus 7, writing 6, carrying the one, then adding the tens column, 9 plus 1 plus the carried one, writing 11 next to the 6)
• Added 90 + 10, 9 + 7, and then 100 + 16
• Added 99 + 10 and then 109 + 7
• Added 100 + 17 and then subtracted 1
• Changed the problem to 100 + 16
• Gave another reasonable explanation
• Guessed, did not explain, or gave a faulty explanation

All but the final scenario and first two approaches show that the student understands addition.

I asked Burns why she couldn’t come up with a truly online assessment tool, where students could work by themselves with a computer and then a computer would spit out an assessment that says whether or not the students get math. She said she’s never found a way other than “human interaction” to prod reasoning from a student.

I wonder, though, whether some young, gifted mathematicians are so fast that they don’t know, or can’t articulate, how they solve problems. The student might bomb Burns’ assessment but get every answer right—and maybe he or she would even end up at MIT one day.

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