Proof! It is the heart of mathematics as individuals explore, make conjectures, and try to convince themselves and others about the truth or falsity of their conjecture. In fact, proving is one of the main aspects of mathematical behavior and “most clearly distinguishes mathematical behavior from scientific behavior in other disciplines” (Dreyfus et al. 1990, 126). By its nature, proof should promote understanding and thus should be an important part of the curriculum (Hanna 1995). Yet students and teachers often find the study of proof difficult, and a debate within mathematics education is currently underway about the extent to which formal proof should play a role in geometry, the content domain in which reasoning is typically studied at an intensive level (Battista and Clements 1995).
Denisse R. Thompson
Suzanne Davis and Denisse R. Thompson
In 1994, the National Council of Teachers of Mathematics issued a position statement titled “Algebra for Everyone … More Than a Change in Enrollment Patterns” (NCTM 1994). Success in addressing this call will involve changes on many fronts. Although curriculum changes will be an essential ingredient, teachers at all levels will need to assume responsibility for fostering algebraic thinking. This article describes the efforts in one large county in the Southeast to provide professional development and encourage dialogue among elementary, middle, and high school teachers about issues related to algebraic thinking. This dialogue is essential in helping teachers appreciate their role in the development of algebra as a K–12 concept. By constantly weaving algebraic ideas into the curriculum at all levels, the district's teachers hope to give students the necessary background to pursue successfully the formal study of algebra.
Gwendolyn Johnson and Denisse R. Thompson
According to Principles and Standards for School Mathematics, two standards that should be addressed in all grades are (1) Algebra and (2) Reasoning and Proof (NCTM 2000). Both involve generalization and are, therefore, naturally related (Lannin 2003). There is often much discussion about the need to engage students with tasks related to algebra; however, it is unclear if as much attention is given to engaging students with reasoning and proof, although this topic is also fundamental to mathematics. As Hanna (2000) noted, “Students cannot be said to have learned mathematics, or even about mathematics, unless they have learned what a proof is” (p. 24).
Rheta N. Rubenstein and Denisse R. Thompson
A tool used in reading theory is adapted to help mathematics teachers ask good questions that help students interpret displays of information.
Denisse R. Thompson and Gladis Kersaint
University teacher educators typically wear many hats. Their many roles may include providing professional development (i.e., workshops) for practicing teachers on various topics (e.g., use of technology, teaching English Language Learners); teaching mathematics or methods courses to teachers or teacher candidates enrolled in undergraduate or graduate teacher education programs; and working with various groups or entities related to policy issues in mathematics education (e.g., teacher licensure, high-stakes assessment). Although mathematics teacher educators contribute to the field in various ways, a perception exists that university faculty are in an “ivory tower,” having few or ancient connections to schools. In some cases, their credibility may be questioned because of the time that has elapsed since they were in a classroom full time.
Sarah K. Bleiler and Denisse R. Thompson
Measuring student understanding of math concepts in this manner offers insight into the robustness of their knowledge, particularly of the Common Core State Standards for Mathematics.
Daniel B. Hirschhorn and Denisse R. Thompson
If one topic is likely to be stressed by algebra and geometry teachers, it is reasoning. In algebra classes, students are constantly being asked to show their work and justify their simplifications, often without formal connection to proof concepts or the proof process. In geometry classes, students are expected to learn how to write simple proofs. However, evidence shows that students are not learning these reasoning skills. In the 1985–86 National Assessment of Educational Progress, Silver and Carpenter (1989, 18) found that “many eleventhgrade students are confused about the fundamental distinctions among mathematical demonstrations, assumptions, and proofs.” Most students thought a theorem was a demonstration or an assumption. Senk (1985) found that only about 30 percent of students mastered proof wTiting in geometry, despite being enrolled in a year-long course emphasizing proof. Thompson (1992) found that roughly 60 percent of precalculus students were successful at trigonometric-identity proofs, more than 30 percent could complete number-theory proofs dealing with divisibility, and less than 20 percent could handle indirect arguments or proof by mathematical induction.
Using Rubrics in High School Mathematics Courses
Denisse R. Thompson and Sharon L. Senk
Recommendations in the Curriculum and Evaluation Standards for School Mathematics (NCTM 1989) and in the Assessment Standards for School Mathematics (NCTM 1995) encourage teachers to incorporate into their curriculum and assessment practices more tasks that require students to construct their own responses, as opposed to primarily using tasks for which a response is provided, such as true-orfalse or multiple-choice tasks. Constructed responses enable students to demonstrate their depth of understanding of mathematics and give teachers greater insight into their students' knowledge of concepts. But when students are required to write about mathematics or explain their solution strategies, teachers want to know how to score such responses. Teachers have therefore become more interested in issues related to rubrics. A rubric is a set of guidelines for evaluating students' responses to one or more tasks. A general rubric is a broad outline that indicates vatious levels of performance and the factors that teachers should consider when specifying performance levels; a task-specific rubric interprets the general rubric for a specific task and specifies the particular mathematical aspects of the task that determine each level of performance (NCTM 1995; California Mathematics Council 1993).
Denisse R. Thompson and Rheta N. Rubenstein
Do your students speak mathematics, or do they think that the mathematics classroom is another country where they must use a foreign language? Are they sometimes confused or overwhelmed by new vocabulary? Do they misuse words, forget key terms, or ignore important distinctions between words? Do they ask, “Where did anyone ever get a strange word like asymptote?” or “I forget, is twelve a factor or a multiple of twenty-four?”
Rheta N. Rubenstein and Denisse R. Thompson
Attempts to sensitize teachers to challenges students have with mathematical symbols, and suggests instructional strategies that can reduce student difficulties.