Mathematical Miseducation

This article provides commentary on the traditional mathematics education paradigm, its detrimental consequences, and evidence and reasons for shifting to current reform in primary and secondary education. Battista introduces controversial criticisms to current reform movements being made in mathematics education and how the majority of American schools still follow the traditional route when teaching students math. He explains that this route leads to failure of students to retain understanding, thus wasting enormous amounts of time and money to reteach students concepts they had been taught in previous years. After reviewing a short history of the mathematics education reform movement, Battista elaborates on the constructivist view that students must personally construct mathematical ideas as they make sense of a situation.

I think like any major change in society, the reform movement in mathematics education will take time to catch on. Scientific studies on this subject have only come into play within the past quarter-century, and in the world of scholarly research, this is relatively young. The thesis of Battista’s article is not to educate readers about mathematics education reform, but to educate readers about mathematics educators who are not educating themselves about mathematics education reform. In a way, he is “preaching to the choir.” By that I mean, his audience by definition will have already agreed with his point of view. The people he is trying to persuade are not likely aware of this article. However, I think that research about research can help, through recommendation and word of mouth. Readers of this article can recommend it to their uninformed colleagues, and every bit of awareness helps.

The more teachers that take Battista’s research into account and spread the news to fellow educators, administrators, and policymakers, the more the mathematics education community can be lead in the right direction. Soon the traditional mode of teaching will no longer be mainstream, and students will be accustomed to constructing their own understanding.

References

• Battista, M. T. (1999). The mathematical miseducation of America's youth: Ignoring research and scientific study in education. Phi Delta Kappan, 80(6), 424–432.

Teachers' Support

A three-minute video of the student-teacher dialog

Jacobs and Philipp (2010) explore the possibilities of support that teachers can provide to students during problem solving. They discussed four basic types of support:

1. support that focuses on the teacher’s mathematical thinking,
2. support that focuses on the child’s mathematical thinking,
3. support that focuses on the child’s affect, and
4. support that includes general teaching moves.

When working with students on an individual basis, support that focuses on that student’s thinking is best. Teachers must adjust their instructional behavior to fit every individual student’s needs. In the example of Rex, this teacher said she would use what she learned about Rex’s thinking on the first two problems to help him solve the third, more challenging, one. Rex liked to use his fingers instead of the manipulatives that were available, so teachers should not push for use of one manipulative over another. In addition, children often think about math differently than adults do, and many can offer new ideas to the table. Provided the opportunity, teachers can learn from their students all the time.

That is not to say that the other three types are not appropriate. In a class discussion or lecture-type class, support that focuses on the teacher’s thinking is more suitable, since the teacher is providing content and modeling processes for the students. Students’ affect should always be taken into account, especially for struggling students; and general teaching moves are useful ‘go-to’ tools in everyday instruction.

This article would have major effectiveness in a traditional classroom. In this type of classroom, all students follow along with the teacher at the same pace, and individual needs are rarely attended to. Usually, problem solving is nonexistent because there is only one or two ‘right’ ways to solve a problem, and these ways are presented by the teacher. The students’ job is to remember how the teacher solved it and perform the same procedure on a similar problem in assessment. Rather, problem solving itself includes the task of figuring out on one’s own the method of solving the problem, and then carrying out that method. Once the algorithm for completing a task is known, the problem is no longer a ‘problem’ even if the answer is still unknown (Harvey, 2010b, p. 3).

Teachers could, and will, benefit from concentrating on students’ mathematical thinking when instructing their classes. By calling for more student involvement and keeping their classes stimulating, engaging, and fun, teachers can get to know their students and how they think mathematically.

References

• Jacobs, V. R., & Philipp, R. A. (2010). Supporting children's problem solving. Teaching Children Mathematics, 17(2), 98–105.

High-Level Tasks

Henningsen and Stein (1997) begin with some background information about the (relatively) new theories about thinking and doing mathematics, which depends on educators' view of the nature of mathematics. The research shows that one theory that has been increasing in popularity is one that emphasizes exploration and has a dynamic stance of education, and holds that student learning is seen as the process in which students develop their own opinions, beliefs, and perspectives of mathematics.

Activities that promote having a mathematical disposition include the following (Henningsen & Stein, 1997, p. 525). These are the processes that students should take away from their mathematical studies, not necessarily the specific content that is covered in their classes.

• Looking for and exploring patterns
• Using available resources effectively and appropriately to solve problems
• Thinking and reasoning in flexible ways
• Conjecturing, generalizing, justifying, and communicating one's mathematical ideas
• Deciding on whether mathematical results are reasonable

In order to promote these activities, teachers must situate their classroom environments such that students engage in high-level tasks. The introduction of this article includes an in-depth discussion of the importance of high-level tasks, the difficulties that may arise from implementing such tasks, and the ways of supporting such implementations.

The authors report that tasks are essential in a mathematics lesson because they give students messages about what “doing mathematics” is. So tasks teachers pose have a major influence on how students think mathematically, and thus they have an altered perception of mathematics. This brings us to the point of developing tasks with a high level of cognitive demand. But implementing these tasks in a high-level manner is not so easy. Since students are comfortable with being told what to do, sometimes teachers lower the rating of a task by making it more simple or giving students a procedure.

To avoid implementing high-level tasks in a low-level manner, Henningsen and Stein (1997) offer some factors, as outlined in Henningsen (2000, p. 245). Teachers should focus on the meaning of the content, rather than the procedures involved in calculating it. Scaffolding is another useful tool. By using students' prior knowledge and building them up to a new concept, they will more easily connect ideas together creating a richer conceptual understanding of a topic. Other strategies mentioned are modeling and self-monitoring.

The remainder of the research in Henningsen and Stein's (1997) article is shown in their empirical study, in which they demonstrate factors that support high-level thinking, reasoning, and sense-making.

References

• Henningsen, M., & Stein, M. K. (1997). Mathematical tasks and student cognition: Classroom-based factors that support and inhibit high-level mathematical thinking and reasoning. Journal for Research in Mathematical Education, 28(5), 524–549.
• Henningsen, M. A. (2000). Triumph through adversity: Supporting high-level thinking. Mathematics Teaching in the Middle School, 6(4), 244–248.
• Smith, M. S., Bill, V., & Hughes, E. K. (2008). Thinking through a lesson: Successfully implementing high-level tasks. Mathematics Teaching in the Middle School, 14(3), 132–138.