Preservice mathematics teachers are entrusted with developing their future students' interest in and ability to do mathematics effectively. Various policy documents place an importance on being able to reason about and prove mathematical claims. However, it is not enough for these preservice teachers, and their future students, to have a narrow focus on only one type of proof (demonstration proof), as opposed to other forms of proof, such as generic example proofs or pictorial proofs. This article examines the effectiveness of a course on reasoning and proving on preservice teachers' awareness of and abilities to recognize and construct generic example proofs. The findings support assertions that such a course can and does change preservice teachers' capability with generic example proofs.
Shiv Karunakaran, Ben Freeburn, Nursen Konuk, and Fran Arbaugh
Amy F. Hillen and Tad Watanabe
Conjecturing is central to the work of reasoning and proving. This task gives fourth and fifth graders a chance to make conjectures and prove (or disprove) them.
Easy-to-design puzzles that encourage mathematical reasoning and promote numerical fluency, arithmogon puzzles are simple: Add the numbers in two circles to get the number in the square. Every month, this final page of the journal highlights a quick game, puzzle, activity, or instructional strategy and suggestions for teachers of different grade bands to use the idea in the classroom.
Lisa A. Brooks and Juli K. Dixon
A second-grade teacher challenges the raise-your-hand-to-speak tradition and enables a classroom community of student-driven conversations that share both mathematical understandings and misunderstandings.
Andrew M. Tyminski
Skip counting around the room (SCATR) is a strategy that promotes numerical fluency and attention to number relationships. Variations of SCATR for students in K'grade 6 are shared.
Angeliki Kolovou, Marja van den Heuvel-Panhuizen, and Olaf Köller
This study investigated whether an intervention including an online game contributed to 236 Grade 6 students' performance in early algebra, that is, solving problems with covarying quantities. An exploratory quasi-experimental study was conducted with a pretest-posttest-control-group design. Students in the experimental group were asked to solve at home a number of problems by playing an online game. Although boys outperformed girls in early algebra performance on the pretest as well as on the posttest, boys and girls profited equally from the intervention. Implications of these results for educational practice are discussed.
Tutita M. Casa
This instructional tool helps students engage in discussions that foster student reasoning, then settle on correct mathematics.
Melanie Wenrick, Jean L. Behrend, and Laura C. Mohs
See how the NCTM Process Standards in action integrate Common Core State Standards in a second-grade classroom.
Jennifer Noll and J. Michael Shaughnessy
Sampling tasks and sampling distributions provide a fertile realm for investigating students' conceptions of variability. A project-designed teaching episode on samples and sampling distributions was team-taught in 6 research classrooms (2 middle school and 4 high school) by the investigators and regular classroom mathematics teachers. Data sources included survey data collected in 6 research classes and 4 comparison classes both before and after the teaching episode, and semistructured task-based interviews conducted with students from the research classes. Student responses and reasoning on sampling tasks led to the development of a conceptual lattice that characterizes types of student reasoning about sampling distributions. The lattice may serve as a useful conceptual tool for researchers and as a potential instructional tool for teachers of statistics. Results suggest that teachers need to focus explicitly on multiple aspects of distributions, especially variability, to enhance students' reasoning about sampling distributions.
Katie L. Anderson
Teachers share success stories and ideas that stimulate thinking about the effective use of technology in K–grade 6 classrooms. This article describes a set of lessons where sixth graders use virtual pattern blocks to develop proportional reasoning. Students' work with the virtual manipulatives reveals a variety of creative solutions and promotes active engagement. The author suggests that technology is most effective when coupled with worthwhile mathematical tasks and rich classroom discussions.