Context: In the second grade classroom in which I completed my student teaching, I taught a weekly Problem Solving lesson in which students applied specific problem-solving strategies to a mathematical problem. I taught a four-week series of lessons on the strategy of “making a list” to solve a problem that required students to add more than two numbers to construct a solution. Artifacts I include in this entry are two examples of student work illustrating how students visualized the problem and applied the list-making strategy, and observation notes from my cooperating teacher on the lesson.

I understand that many students need explicit, direct teaching of specific problem-solving strategies in order to learn and develop mathematical understanding, and I create multiple successive opportunities for students to develop a targeted problem-solving skill. I support these students throughout the process of learning by referring students to a problem-solving poster that describes the steps, explicit modeling and demonstration of how to follow the steps to solve a problem, and repeated practice and review to ensure student understanding.

The oral and written directions I communicate to students in this lesson are clear, complete and organized, and I assess student understanding of directions by asking students to read the problem with me, participate in the demonstration part of the problem-solving lesson, and repeat the directions before working independently on the problem-solving task. I use positive verbal reinforcement as a communication strategy for managing appropriate behavior while I am teaching, and I continuously praise students who meet expectations for behavior.

Reflection: A tension exists in mathematics education between teaching students procedures for computation and solving problems and teaching students underlying mathematical concepts that explain why specific procedures and strategies work. Although some educational researchers argue that children must develop their own strategies and approaches to solving problems, the NCTM Standards state, “As with any other component of the mathematical tool kit, strategies must receive instructional attention if students are expected to learn them.” One of six major instructional principles in making effective adaptations in the classroom for students with learning disabilities is teaching students conspicuous strategies (Kame’enui et al., 2002). I believe that teachers can combine and implement a student-centered approach with an explicit, teacher-directed approach to teaching mathematics in the classroom.

Rather than teach to a prescribed curriculum or to a set of problems in a mathematics textbook, I am interested in further developing a problem-based approach to teaching mathematics to elementary students. Putting students in touch with real problems and creating authentic learning experiences is a way to help students develop conceptual and procedural understanding in mathematics.

I envision this happening in the following way: I pose a problem to students, observe them thinking and wrestling with the concepts in the problem, ask questions to understand their thinking about the problem and strategies for solving it, and use that information to guide future teaching. For example, if a student lacks a prerequisite understanding of place value, I will know to teach concepts of place value before posing additional problems that require knowledge of place value. If a student lacks skills to approach a word problem in a systematic way, I can develop a conspicuous problem-solving strategy to teach students and gradually fade away instructional scaffolds designed to facilitate independent use of the targeted strategy.

In a series of four problem solving lessons, I taught students the conspicuous strategy of “making a list” for solving problems such as the following:

Problem Solving Student Work 1

As I carefully observed students working with the problem and questioned individuals to learn more about how they were thinking about the problem, I noticed how some students naturally integrate what they already know with what they don’t know. Prior to this lesson, students had multiple opportunities drawing a picture to illustrate and solve a problem. Although I had demonstrated how to make a list to solve with the days of the week at the top of each list and a checkmark for the number of books under each list, I noticed one student drawing the books under each list rather than simply making one checkmark for each book. On Tuesday, since there were six books read in all, these books were drawn as being smaller. On Friday, since there were only two books read in all, one book was large, and the other was very short. This student’s need to visualize what was happening in the problem gave me valuable insight into his thinking about problem-solving.

Yet the problem-solving curriculum prescribed for me ended with students learning how to “make a list.” The targeted objective was for students to make a list to solve a problem involving three or more addends, and students certainly did meet these objectives. However, I would have liked to approach the curriculum from a problem-based perspective and differentiate the actual problems posed to different students to provide different levels of challenge. A problem-based approach to teaching mathematics would allow me to provide a richer curriculum for students, not one based on isolated concepts and skills, but based on integrating multiple strategies and concepts in authentic ways.

Maggie Lampert, in Teaching Problems and the Problems of Teaching, articulates her philosophy of problem-based mathematics teaching in the following way: “I teach by engaging my students with the big ideas of the discipline as they work on problems and discuss the reasonableness of their strategies and solutions. Using problems, I teach them that they all can learn and that they can do it in school.” (Lampert, 2001) My dilemma in teaching mathematics from a problem-based approach is: In a diverse and inclusive classroom, should I focus on teaching students the strategies they will need to solve the problems or focus on the actual problems themselves to help students develop their own mathematical understanding and problem-solving strategies?

In a classroom with students who have special needs, it is important to make the mathematics curriculum accessible in order to promote equity in education. Therefore, explicit teaching of concepts and strategies is a necessary instructional strategy to enable students to learn. However, it is possible to differentiate mathematics instruction by posing different problems with different levels of challenge and scaffolding to different groups of children. An approach to teaching mathematics by differentiating the curriculum can effectively meet the needs of students who conceptualize mathematics in different ways, as shown by the two examples of student work included here. Additionally, I think it is helpful to teach mathematics with short-term as well as long-term goals in mind: Although students may be in the early stages of developing mathematical understanding and have acquired a few specific strategies for approaching and solving problems, with highly effective teaching, they will learn and internalize many different strategies that they will be able to apply to future problems that they encounter, in the area of mathematics as well as in other academic disciplines.

Problem Solving Student Work 2

Problem Solving Observation Notes

Kame’enui, Edward J., et al. (2002). Effective teaching strategies that accommodate diverse learners. Upper Saddle River, NJ: Merrill.

Lampert, Magdalene. (2001). Teaching problems and the problems of teaching. New Haven: Yale University Press.

Principles and standards for school mathematics. (2000). Reston, Va.: National Council of Teachers of Mathematics.