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 Student Thinking Matters Most  

Submitted by Aimee L. Evans

Arch Ford Education Service Cooperative

And ADE CCSSM PD Team 

Teaching mathematics requires finding out how students think and what they know about a particular kind of problem before helping them develop and refine their thinking. Sometimes this requires us as teachers to put on the brakes, fight the temptation to dive into demonstrating something when we aren’t sure what strategies and ideas students possess and would use to approach it already.

A well-known article from the December 2000 issue of Teaching Children Mathematics, “Fractions: What happens between Kindergarten and the Army”, raised the specter of what happens when we impose strategies onto students without understanding how they think and connecting strategies to that thinking. “We realized that children enter kindergarten with a good deal of mathematical understanding. We feared that they emerged in sixth grade having been more handicapped than helped by the instruction they traditionally receive.” Upon posing a mixed-number addition problem to students in grades 1-5, the authors of the article found that the percent of students able to solve the problem successfully did increase across the grades. Two-thirds of fifth graders solved the problem successfully using personal strategies. However, of the twenty-two percent of the fifth graders that attempted to use “the traditional method” the teachers anticipated, none of them got a correct solution. Are students putting away sense-making in order to use traditional methods? Can these methods be taught by making a better connection to students’ intuition?

This is not a concern reserved solely for student understanding of fractions. Consider kindergartners and first graders who can use reasoning and strategies to find out what 17 – 9 is. They take away; they count up; they take away 7 to get 10 and then 2 more to get 8. But give this simple problem to second graders after they have been taught to subtract by borrowing, and how many of them will “borrow” by marking out the 1, marking out the 7 and writing 17, and never even recognize that they still have 17 – 9, the same problem they had originally? How many attempt to borrow on every subtraction problem they encounter whether they need to or not? How many borrow for what should be simple “count up” problems such as 92 – 89? Why are students blindly following a procedure rather than thinking about the problem to be solved?

Are the algorithms for subtracting whole numbers and for adding or subtracting fractions really so complex that it is their very nature that confuses these students? Or is it that the methods they are taught to solve these problems are in no way connected to their existing thinking?

David Sousa, in his book How the Brain Learns Mathematics, tells us

Preschool children use their innate but limited notions of numerosity to develop intuitive counting strategies that will help them understand and measure larger quantities. But they never get to continue following this intuitive process. When these children enter the primary grades, they encounter a sudden shift from their intuitive understanding of numerical quantities and counting strategies to the rote learning of arithmetic. …. Many children persevere with this major upheaval in their mental arithmetic and language systems despite the difficulties. Unfortunately, most children also lose their intuition about arithmetic in the process (my emphasis). (p. 41)

The thesis of Cognitively Guided Instruction, a research-based approach to using student thinking as a cornerstone to teaching number concepts and operations, reflects Sousa’s statement. The thesis is that “Children enter school with a great deal of informal or intuitive knowledge of mathematics that can serve as the basis for developing understanding of the mathematics of the primary school.”

Each of us can cite examples of the informal or intuitive knowledge to which the above statement refers. I offer a personal anecdote regarding a first grader’s ability to solve problems without instruction on the particular kind of problem. Upon hearing that she could attend camp at Sea World at ten years of age, my niece, who had just completed Kindergarten and always announced her age as six-and-a-half, was very quiet for a couple of minutes. Then she said, “That’s just in three-and-a-half years.”

As a primary grades teacher, you want students to maintain and build upon their mathematical intuition and be successful. No one teaches with the goal of handicapping his or her students’ thinking. So what is the missing link?

In their book Extending Children’s Mathematics - Fractions and Decimals: Innovations in Cognitively Guided Instruction, Susan B. Empson and Linda Levi offer the following advice regarding fraction instruction, but it applies to the teaching of almost any mathematics concept.

The basic teaching practices that support children to draw on what they understand to make sense of new content include:

·Posing problems to children without first presenting a strategy for solving the problems

·Choosing problems that allow children to craft a solution on their own

·Facilitating group discussions of children’s strategies.

Because a variety of strategies is necessary for rich mathematical discussions, teachers need to establish and reinforce routines to help students realize they are expected to solve problems in their own ways rather than by applying teacher demonstrated strategies. (p. 10)

To some, this appears to be a sort of discovery approach where teachers never teach anything; they just let students figure everything out. And yet, it is so much more thoughtful than that.

Those who train for scripted curricula and direct instruction approaches say that the teacher must model exactly what she wants students to do before asking them to do it, and that she should not expect to get from students anything she has not taught them to do. In an attempt to persuade me, someone recently told me that Robert Marzano had concluded that direct instruction was the superior teaching approach and referred me to an article in the September 2011 edition of Educational Leadership entitled “The Perils and Promises of Discovery Learning.” I read the article and concluded that the person directing me to read it had read only the first two paragraphs and stopped. Marzano deems what he calls “unassisted discovery learning” to be decidedly inferior to direct instruction, but goes on to conclude that “enhanced discovery learning” is superior to other forms of instruction. What are the distinguishing differences? In an enhanced discovery learning situation, students have obtained necessary prior knowledge, teachers scaffold experiences to provide assistance, and they use student thinking and discussion to accomplish the learning goal. In other words, teachers attend to student background and student thinking in how they guide the “discovery.” The process is very deliberate and purposeful.

All of these references coalesce into a coherent approach to teaching. We are not being asked to let students figure out everything for themselves. Marzano concludes that students need relevant prior knowledge. This tenet is acknowledged by Empson and Levi when, for example, they tell us that experience with multiplication and division problems are an important stepping stone to having students engaging in equal sharing fraction problems. Teachers following their advice pay careful attention to the strategies and thinking of their students in order to select the approaches that should be shared and discussed as a class, which conventions of vocabulary and notation to teach, and what problem to pose next. Thus they are very much guiding the discovery.

So-called direct instruction has its uses. Students will need to see bar notation for writing a fraction and be taught how to use it, just as they need to see modeled the writing of two-digit arithmetic problems vertically. They will not invent that the name of the number below the bar is called the denominator or that a plus sign is used to represent addition. However, by beginning with children’s intuition and strategies, such conventions can be introduced when they comport with student thinking rather than as a substitution for it. The same is true of vocabulary terms and of conventional phrasing in mathematical language.

Once conventional terms are introduced to students to help them explain their thinking, their true understanding of the terms develops through experience and discussion. Consider a simple term such as “triangle.” What seems a simple definition can be challenging when young children encounter triangles that are oriented in unusual ways or when wavy or curved lines are included. Is it still at triangle? Students refine and deepen their understanding when confronted with these figures and asked to reconcile their definition of triangle. It is easy for a teacher to assume children understand the term triangle when they can say it is a shape with three sides or circle the triangle when faced with a triangle and two quadrilaterals as choices. The mathematics that young children need to learn does not seem complex to adults who mastered it decades ago, but taking time to “pick at” their understanding can reveal hidden beliefs that need to be brought out or dispelled through thoughtful discussion and specially designed experiences.

As you work with young children, whether on number and operations, geometry and measurement, or early understanding of fractions, be conscious of the lens through which you are looking. Is that lens focused and tinted by knowledge of your students’ current understanding and strategies? How much effect does the way your students approach a problem today influence what you will discuss, what you will pose, and what you will ask tomorrow? Help those around you keep perspective on those pacing guides, which, in the words of Captain Barbossa from Pirates of the Caribbean, should be thought of as “more what you'd call ‘guidelines’ than actual rules.” Student thinking matters most. §