Published on *Psychology Today* (http://www.psychologytoday.com)

By *Nate Kornell, Ph.D.*

Created *Nov 27 2012 - 5:58am*

Children from the United States do not score well on international math tests. The gap grows as students get older, suggesting that our educational system is the problem. But how little math do US students know? A fascinating recent article (Richland, Stigler, & Holyoak, 2012) paints a sobering picture. It may also suggest an avenue for improvement.

These studies examined community college students who, like the majority of community college students, placed into a remedial math class. This is an interesting group because they mostly graduated from high school, they passed their required math classes, and they decided to go to college. It is also a large group; there were 13 million community college students (as of 2009).

The most interesting thing about the new research is that students were *not tested using traditional measures of mathematical ability*. You can do well on many traditional tests if you have memorized enough mathematical procedures (e.g., how to compute the third angle of a triangle if you know the first two). But this test was different. The students were tested on very basic conceptual knowledge, like what addition really is. For example, a student could know the procedure for multiplying fractions without having a conceptual understanding what a fraction is—or even what multiplication is. These new studies checked to see whether students understood things like the meaning of fractions and multiplication.

The Richland et al. (2012) article reports a string of shocking findings gleaned from two other recent articles (Givvin et al., 2011; Stigler et al., 2010). Two of the questions assessed whether or not students understand what a fraction is.

- Students were shown a number line from -2 to 2 and asked to draw a line marking the approximate location of two numbers: -0.7 and 13/8.
**Percentage who answered correctly: 21%.**

- Students were asked "If a is a positive whole number, which is greater: a/5 or a/8?" Fifty percent would answer correctly if they just guessed.
**Percentage who answered correctly: 53%.**

If you’ve been assuming high school graduates fully understand how fractions work, these results say otherwise. Some fell back on procedural knowledge, probably because that’s the only knowledge they had about fractions. For example, seeing two fractions near each other apparently triggered an urge in some students to use the cross-multiplication procedure they had memorized.

Another question looked at whether the students understood addition.

- In an interview one student was asked if he could think of a way to check whether 462+253 = 715. He smartly subtracted 253 from 715 and came out with 462. So far so good. But when he was asked whether he could have subtracted 462 from 715 instead, he said he did not think so. He had been told in school to subtract the second number from the bigger number, not the first. It appears he was just following a memorized script.

Another set of questions checked to see whether students would take advantage of relationships between problems to find easy solutions. For example, the students were asked to solve the following set of problems:

- 10 × 3 =
- 10 × 13 =
- 20 × 13 =
- 30 × 13 =
- 31 × 13 =
- 29 × 13 =
- 22 × 13 =

Once you solve the problem two, problem three becomes easier; just multiply the answer by 2. Similarly, the answer to problem five is the answer to problem four plus 13. Yet 77% of the students never took advantage of any of these relationships and simply did the multiplication for each problem. They made errors in the processes that could have been caught with a cursory inspection and some very basic conceptual knowledge. Here are one student’s answers (Givven et al., 2011):

- 10 × 3 = 30
- 10 × 13 = 130
- 20 × 13 = 86
- 30 × 13 = 120
- 31 × 13 = 123
- 29 × 13 = 116
- 22 × 13 = 92

**Math as rules versus math as a system**

Many of these students do not fully understand how a fraction works, how addition works, or the fact that different problems can relate to each other. So the question becomes why? And what can be done?

One of the most enlightening questions the students were asked was not a math problem. They were asked (Givven et al., 2011) what it means to be good at mathematics. Here are some of their responses (quoting Richland et al., 2012):

- “Math is just all these steps.”
- “In math, sometimes you have to just accept that that’s the way it is and there’s no reason behind it.”
- “I don’t think [being good at math] has anything to do with reasoning. It’s all memorization.”

In all, 77% of the students seemed to believe that math was not something that could be figured out, or that made sense. It was just a set of procedures and rules to be memorized. This is, of course, exactly the opposite of true.

Where did they get this belief? Their teachers did not believe it—in fact, the faculty members who taught these students were shocked and distressed to learn about this result.

Still, the students probably develop the belief that math is all memorization in their classrooms. The second part of Richland et al.’s (2012) article deals with improving mathematical learning. It suggests that students do not see mathematics as a logical system because their teachers do not present it that way. In particular, if the teachers did a better job of pointing out connections between mathematical concepts, the students would develop a deeper understanding of what they are learning.

**International differences in instruction**

The TIMSS video studies were a landmark in educational research. These studies involved videotaping and coding teacher and student behaviors in hundreds of mathematics classrooms around the world. These studies were used to identify variables that predicted success. Many of the obvious variables were not predictive, including teaching primarily by lecturing versus group work, or using abstract versus real-world problems.

When the researchers broke problem-solving activities down into procedural activities and conceptual activities, they expected to find that the higher performing countries engaged in more conceptual problem solving. They found no such difference. But then they took a second step. They coded the data based on whether the teachers made the conceptual problems easier by converting them, for the students, in to procedural problems.

Looked at this way, it became clear that the US was an outlier (as was Australia, the only other low-performing nation in the study). Teachers in the US almost always converted challenging conceptual problems into procedural problems. In doing so, they did exactly the wrong thing. According to a seminal study by Hiebert and Grouws (2007) the two features of instruction that predict good math outcomes are

- Being explicit about the conceptual structure, and interconnectedness, of mathematics
- Allowing students to struggle to understand mathematical concepts.

By converting conceptual struggle into procedural learning, US math teaches were unintentionally depriving their students of two crucial elements of effective learning.

This finding helps make sense of the community college students’ lack of conceptual understanding. They have been taught in a way that deprives them of the chance to work through the concepts they are being taught. No wonder they see math as an exercise in memorization. All too often, that is how it is presented. And given the way we test students, memorizing mathematical procedures will get you a decent grade, and for most students that’s the bottom-line.

These results are promising one way: they suggest that US students might be able to do a lot better in math if they can develop a basic conceptual understanding. It’s not like they need to be taught general relativity. The concepts are learnable. On the other hand, teaching mathematical concept is deceptively difficult. Teachers need high quality training, and more research (and funding) is needed to make that happen.

**References**

Hiebert, J. C., & Grouws, D. A. (2007). The effects of classroom mathematics teaching on students’ learning. In F. K. Lester, Jr. (Ed.), Second handbook of research on mathematics teaching and learning (Vol. 1, pp. 371–404). New York, NY: Information Age.

Givvin, K. B., Stigler, J. W., & Thompson, B. J. (2011). What community college developmental mathematics students understand about mathematics, Part II: The interviews. The MathAMATYC Educator, 2(3), 4–18.

Richland, L. E., Stigler, J. W., & Holyoak, K. J. (2012). Teaching the conceptual structure of mathematics. Educational Psychologist, 47(3), 189–203. doi:10.1080/00461520.2012.667065

Stigler, J. W., Givvin, K. B., & Thompson, B. (2010). What community college developmental mathematics students understand about mathematics. The MathAMATYC Educator, 10, 4–16.

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