Meet Vi Hart, and then watch ALL of her videos. Doodling has never been so cool and informative. There’s also a bit of commentary on mathematics education. Here is one of Vi Hart’s videos to get you hooked. You can find the rest here.

A couple years ago, I attended a meeting about a new math program that a local town was planning to implement. The switch to a new program was largely motivated by new common core standards, and the meeting was called to prep parents for the “new” mathematics that their children would encounter.

A representative from the company behind the new math curriculum ran the meeting. The representative, who I’ll call Linda, was an experienced teacher and one of the first teachers to implement this new math program.

While at the meeting, Linda gave the parents a math problem. It went something like this. A basket half full of apples weighs 20 pounds. An empty basket weighs 5 pounds. How much does a basket full of apples weigh?

After allowing the parents enough time to work out solutions in small groups, Linda asked for volunteers to share their work.

The first parent shared how she arrived at a correct answer of 35 pounds. This parent took the 20 pounds that a half-full basket weighed and subtracted the weight of the basket to learn that the apples (without the basket) weighed 15 pounds. The parent then doubled 15 to see that the full load of apples (minus the basket) weighed 30 pounds. She then added the basket’s weight of 5 pounds to arrive at 35 pounds for a full basket of apples.

Linda congratulated the parent for getting the right answer and having a correct approach to the problem.

The common core holds dear the principle that the process is as important, if not more important, than the answer. Common core standards fully acknowledge that there are multiple correct approaches to the same problem. In fact, they encourage students to explore multiple paths to each solution.

So, I was very pleased when Linda asked the audience if anyone had arrived at 35 pounds in a different way.

A parent volunteered the following approach. He doubled the weight of a half-full basket to see that a full basket plus an extra basket weighs 40 pounds. He then subtracted the weight of the extra empty basket to arrive at the correct answer of 35 pounds.

BRAVO! It was a valid approach.

Linda did not agree. She suggested that although the answer was correct, the process was flawed. Her reasoning was that there weren’t two half-full baskets in the problem, and you couldn’t just create one for the sake of generating the correct answer.

Uh-oh.

For decades, math education has misled students to believe mathematics is all about calculations and symbolic manipulations. For example, students think they learn multiplication so that they can multiply six times nine. If that was the end goal of teaching multiplication, then it might as well be taught with a trick like the following.

Even better, we could just give everyone a calculator and use valuable school time teaching other concepts.

Mathematics is not calculations and symbolic manipulations. It’s a structured approach to problem solving. This is what students should take away from any mathematics education, and students aren’t going to develop this ability if they replace true understanding with tricks and memorization.

Parents are struggling with the common core for two reasons. One, it teaches mathematics in a new way. This means new terminology and notation. It means a totally new way of thinking about the same math that these parents thought they mastered years ago.

The other reason for parents’ struggle runs deeper. Parents are having trouble understanding *why* the math has changed. They are having trouble understanding why their children can’t use tricks or just memorize the answers. For example, parents want to know why their children can’t add fractions with the “butterfly” method. The answer is because that’s not math.

Parents are struggling, and that’s an issue. What about the teachers? These teachers went through the same math curriculum as the parents. Many of our teachers also lack an understanding of the true nature of mathematics and the purpose of a math education.

The common core puts an emphasis on evaluating how students arrive at their solutions. Certain approaches are discouraged. These include tricks and memorization that undermine the students’ development of mathematical reasoning.

However, the common core is not discouraging mathematical exploration. Students are encouraged to find multiple ways to solve the same problem. The core embraces mathematical creativity.

In this new system, teachers are tasked with judging the merits of a students’ approach to each problem. The teachers must determine whether a particular approach is in line with the purpose of mathematics education. Is the approach consistent with the objective of developing the students’ ability to reason mathematically?

We’re learning that this is difficult for teachers like Linda.

The following is another example of teachers’ struggles to properly assess methodology.

Problem 1 is an attempt to see if students’ recognize that multiplication is repeated addition. The student accurately shows that 5 x 3 is 5 + 5 + 5. Why was a point deducted? My guess is that the teacher has taught the students to write 5 x 3 as 3 + 3 + 3 + 3 + 3. In other words, the teacher has asked students to always sum the second term in the product.

The teacher’s instructions are not consistent with the intention of the common core. Writing 5 x 3 = 5 + 5 + 5 is not a trick that undermines the student’s ability to understand the concepts. By not allowing this response, the teacher is stifling the students’ mathematical creativity as opposed to embracing it. The teacher is teaching mathematics as if it is mechanical or formulaic. Why? My guess is that the teacher was taught that way.

Problem 2 is a demonstration of the same issue, which shows that the teacher’s incorrect assessment of the first problem was not a fluke.

I want to be clear that these teachers and parents that are struggling with the common core are smart. The problem is not a lack of intelligence. The problem is that our educational system has been misleading.

The damage cannot be undone overnight.

The common core asks teachers to assess the validity of each student’s mathematical approach, but how can they do this if they haven’t been taught what mathematics is?

What is Mathematics? It sounds like a simple enough question. Wouldn’t you expect that a high school or college student that has been formally engaged in mathematics education for more than 10 years would have an immediate (and relatively complete) response?

They don’t.

The truth is that mathematics is difficult to define, and even worse, there are many misconceptions regarding the true nature of math.

Fordham math professor, Dr. Robert H. Lewis, did his best to address the topic in an excellent article that first appeared online in 1999 (and has been revised several times since).

Many believe that mathematics is calculations and symbolic manipulations. This is both incorrect and problematic. When a calculus student believes that the primary objective of differential calculus is to be able to take derivatives, he focuses on the “tricks” such as the chain rule, product rule and power rule. The ability to take derivatives algebraically is a good thing, but if it becomes the focus, this same student might not be able to answer, “What is a derivative?” He might also lose sight of the mathematical process that led to the creation of these differentiation “tricks” as means to be more efficient. This student might be frustrated by a lecture on limits or a proof of the chain rule.

The role of limits in calculus or the notion of a proof in mathematics are far more important for developing one’s sense of mathematics and developing one’s ability to reason mathematically. If a proper understanding of math and a capability to reason mathematically are of value, then they must be taught as well.

Dr. Lewis’s article, through analogies (or “parables”), captures the misconceptions about math and the consequences of these misconceptions.

Unfortunately, the article does not provide anything better than a definition by analogy for mathematics.

In my courses, I regularly point to examples that I believe demonstrate mathematical reasoning. However, I can’t formulate a complete definition of what this is. Like Dr. Lewis’s article, I tend to resort to analogies.

Why is it so difficult to answer, “What is mathematics?”

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