Algebra in 2nd Grade?

Back in February, a teacher in Montana made EdWeek headlines because she was teaching algebra to second graders and had been doing so for five years. Why all the oohs and aahs?

Gregorio C. Sablan, CNMI Congressional Delegate, got it right. “Pre-algebra” starts early, or should.

At Broadwater Elementary School in Helena, Montana, algebra starts in second grade, where students learn the basics behind mathematical theory and reasoning to prepare them for high-level math classes in middle and high school.

Elementary math is supposed to prepare students for high-level math classes is middle and high school. Students should not need a dedicated pre-algebra class. When I was a kid, pre-algebra did not exist. Now it is part of every school's math course line-up. The author of a pre-algebra text wants students to build math reasoning skills.

But that doesn't always happen. Many teachers treat pre-algebra as a last chance for students to get those elementary math procedures down pat. Problem is, a student can be A+ in procedures and still not get algebra. In fact, students who are competent with procedure often believe they are good at math. It's not their fault. Our education system has been telling them for years that grades equal understanding. So if they get a good grade in math, naturally they conclude they are good at math.

Math has been misnamed. What passes for math in schools is often non-math. “Carry the one” is not a mathematical explanation. Students get good grades in non-math believing it's math. No wonder algebra is such a shock. Math reasoning skills actually matter in algebra.

Still a student with a good memory can get by, at least until they meet a new math monster, calculus. However, since middle and high school math also fail to teach math reasoning, students get to take pre-calculus, another relatively recent addition to course offerings. Without a major change of emphasis, pre-calculus prepares students no better for calculus than pre-algebra prepared them for algebra.

By now pre-calculus students have so internalized non-math that they complain to the instructor, “Just tell us how to get the answer. We don't want to know why.”

The Place of Place Value

Perhaps one of the most important foundational concepts in mathematics is place value. As the Massachusetts Department of Education rightly observes, “The subtly powerful invention known as place value enables all (my emphasis) of modern mathematics, science, and engineering. A thorough understanding removes the mystery from computational algorithms, decimals, estimation, scientific notation, and—later—polynomials” (Massachusetts Department of Education (2007). In fact, it is when students first meet polynomials in algebra, that the lack of a proper grounding in place value becomes painfully apparent. Most likely a significant number of the difficulties that students experience with math may be traced to place value.

I reviewed the state standards of various states with regard to place value. I looked for an explicit reference to “regrouping,” the current term for what we used to call “borrowing” and “carrying.” Personally, I prefer to call it “filling the cup” and “dumping the cup.” In the same vein, I like to call the “ones” place the “loose ones.” My survey of state standards resulted in a mixed bag. Some states require students to do little more than name the place value of a particular digit. Other states expect students to use various means to model place value. Alaska asks students to not only perform the operations of addition and subtraction, but to explain those operations.

State standards have their utility, but apparently whatever the specific state standard, students are able to follow the regrouping recipe without having any real understanding of why the recipe works. In fact, adults of all ages add and subtract by mindlessly following the recipe. Most adults, and of course, all children could do with a solid grounding in place value.

I have a number of activities I use to make place value explicit. Tomorrow I will tell you about an activity I like to call “The Chocolate Factory.”

Are You Good at Non-Math?

One of the most persistent issues in math education has been the reliance on non-mathematical explanations of mathematical principles. For example, we tell students that when multiplying positive and negative numbers “two negatives make a positive.” Such an explanation clarifies nothing about how the numbers behave or why an ostensibly English grammar rule should apply to math.

What is worse, we tell students who successfully master such non-math explanations that they understand math, or that they are good at math, when really what they are good at is non-math. Young children have no way to distinguish non-math from math. They believe, because we have told them, that they are learning math, when in fact they are learning non-math. If it does not catch up to them earlier, it often catches up to them in algebra class where historically “A” students may find themselves inexplicably failing to understand the subject material.

Children rely on adult teachers to initiate them into the joys and delights of math, but often teachers make math a difficult subject, usually because they themselves understand non-math rather than math. After all, if numbers are running around, it must be math, right? Even sadder are the number of elementary teachers who lack an interest in acquiring what math education researcher Liping Ma called “the profound understand of fundamental mathematics” even while believing that they “know” math.

Many colleges of education and community colleges have sought to address the serious weaknesses in the mathematical understanding of elementary teachers by either requiring, or at least offering, coursework in mathematics for elementary teachers. I am quite sure a survey of professors teaching such required courses would report remarkable levels of student resentment at being forced to take a class in something they think they already know, to “jump hoops” as they say . Some of these students may wake up and get motivated to learn the math concepts. Some seethe inwardly as they pass the class. However, most students will pass the class and eventually be certified to teach regardless of their attitude toward or understanding of the vital core subject of mathematics.

Only later, once they are in the classroom, will they be likely to regret the squandered opportunity to finally get math. They may grow to appreciate the professor who tried to give them the gift of mathematical understanding, a gift they resisted at the time.