EXEMPLARY PRACTICE

What Does It Look Like?

While I was helping my daughter Lisa with her math homework, I asked her to explain why she chose the operation she used to solve a problem. Not only did she not know, but also she did not care. She was more interested in getting the right answer by plugging in the proper formula. This is how she was taught math, and she doesn’t seem to want to change the way she learned it. I can only generalize that this is how many students are responding to attempts by teachers to create conceptual understanding. This age group is where so many students lose interest in math—just when they should be finding the beauty of it. Perhaps I shouldn’t worry too much; Lisa’s passion lies in social studies and literature. She is not a “math-brained" child, I guess. Are these children born, and not made that way?

Eugenie, pre-service teacher

We can surmise that Lisa has procedural fluency but, because she lacks conceptual understanding, she is not able to apply the strategic and adaptive competencies necessary to solve real-life problems and that this contributes to her unproductive disposition toward mathematics. Many students perceive mathematics to be a bunch of numbers that go into formulas to solve problems. More often than not, the problems they are asked to solve are not t heir problems, nor do the problems come close to anything they are interested in pursuing. Lisa’s experiences with mathematics are similar to those that I had as a mathematics student: The mathematics I learned focused on finding the teacher’s or the book’s answer to a problem. But when I studied mathematics methods at Brooklyn College, my classmates and I explored a different kind of teaching and learning. Rather than lectures about what we needed to know, Professor Dorothy Geddes invited us to experience mathematics as a dynamic discipline that sometimes required tools such as toothpicks, geoboards, or mirrors to resolve thought-provoking problems. Dr. Geddes’ definition of mathematical competence clearly went beyond numbers and computations; she included the ability to test a hypothesis, find patterns, and communicate understanding—all of which are recommended by NCTM/CCSS as essential elements for both teaching and learning mathematics today.

In her reflection, pre-service teacher Eugenie writes about her concern for getting Lisa to understand and appreciate mathematics. Unfortunately, Eugenie’s acceptance of Lisa’s dislike for mathematics as a natural outcome also contributes to the problem. Another facet of the problem is that of teachers bypassing conceptual understanding and substituting a fast-forward to rules. In her article on teaching to the CCSS, Crowley (2013) writes about what she learned from mathematics educator Ann Shannon. Shannon would describe this fast-forwarding process as teachers’ tendency to “GPS” students by giving them step-by-step directions for solving problems followed by worksheets to practice the steps. If those steps bypass concept understanding, then students likely will not be able to apply the concept to real-world situations or have the concept serve as a simpler problem whose process helps to solve more difficult problems.

What Ann Shannon would say is that in this particular situation, the students have been “GPS-ed” from problem, to solution. Just as when I drive in a new city using my global positioning system, I can follow the directions and get to where I need to go. But I can’t replicate the journey on my own. I don’t have a real understanding of the layout of the city If a road were blocked because of a parade, for example, I would be in trouble because I have no real understanding of the city’s geography

(1)

We need to shift away from GPS-ing students, which is the traditional way for teaching mathematics. We also need to shift away from believing that only “math-brained” students should be expected to understand math. The importance of this statement is reflected in the research of Carol Dweck (2015) and colleagues that show such beliefs impact student achievement. She writes:

More precisely, students who believed their intelligence could be developed (a growth mindset) outperformed those who believed their intelligence was fixed (a fixed mindset). And when students learned through a structured program that they could “grow their brains” and increase their intellectual abilities, they did better. Finally, we found that having children focus on the process that leads to learning (like hard work or trying new strategies) could foster a growth mindset and its benefits.

(1)

What happens to Lisa’s mathematics learning when her mom or her teacher believe that she does not have a math brain (i.e., a fixed mindset)? The answer depends on their response. If they think that it is acceptable for Lisa not to succeed in mathematics because she’s smart in other areas—just not in mathematics—and that there is no reason to work to enhance her mathematical understanding, then Lisa may never change her own attitude about mathematics and may not be motivated to improve. On the other hand, if they believe that Lisa can succeed with appropriate help (i.e., Lisa has a growth mindset), then her mother and teacher could work toward connecting mathematics to her strong areas of interest by thinking about how math is integrated in literature, art, science, the movies, music, politics, sports, puzzles, or other interests of Lisa’s. The teacher could also use varying teaching strategies that have appeal to a broad number of students, including Lisa.

Boaler (2015), in her book on mathematical mindsets, dispels the notion of a math brain: “Although I am not saying that everyone is born with the same brain, I am saying that there is no such thing as a ‘math brain’ or ‘math gift,’ as many believe. No one is born knowing math, and no one is born lacking the ability to learn math” (5). Helping Lisa accept the fact that everyone has a growth mindset waiting to be tapped may lead her to believe that she can solve more challenging problems if she persists. Such persistence is what the literature refers to as grit or productive struggle.

From her National Public Radio interview with Smith (2014), described in Chapter 1, Duckworth agrees that the responsibility for helping kids develop a positive disposition toward mathematics is the responsibility of schools, teachers, and parents. Duckworth adds, “I don’t think people can become truly gritty and great at things they don’t love . . . so when we try to develop grit in kids, we also need to find and help them cultivate their passions. That’s as much a part of the equation here as the hard work and the persistence.”

The PISA 2012 Results in Focus newsletter (OECD, March 2014/03) summarizes this discussion:

The bottom line: practice and hard work go a long way towards developing each student’s potential; but students can only achieve at the highest levels when they believe that they are in control of their success and that they are capable of achieving at high levels. The fact that large proportions of students in most countries consistently believe that student achievement is mainly a product of hard work, rather than inherited intelligence, suggests that education and its social context can make a difference in instilling values that foster success in education.

(4)

The reality is that every student has a unique and complex brain. Our classrooms are composed of many Lisas, with varying interests and aptitudes, but they can all learn to do and to appreciate mathematics.

 
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