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August 2011, Volume 105, Issue 1

 FEATURES Cornered by the Real World: A Defense of MathematicsSamuel OttenOur answers to students’ questions about the relevance of what we teach might paint mathematics into an undesirable corner. Student-Generated Questions in Mathematics TeachingColin FosterExploring even something as simple as a straight-line graph leads to various mathematical possibilities that students can uncover through their own questions. Errors in Mathematics Aren’t Always BadSheldon P. GordonWe tell students that mathematical errors should be avoided, but understanding errors is an important tool in developing numerical methods. Second Look:Learning from Student Errors Delving into Limits of SequencesBeth Cory and Ken W. SmithThrough these calculus activities, students reach an understanding of the formal limit concept in a way that enables them to construct the formal symbolic definition on their own.

 Departments Reader ReflectionsReader Reflections - August 2011 Sound Off!Sound Off!: The High School Mathematics Curriculum—What Can We Learn from History? Media ClipsMedia Clips: Facebook Tops 500 Million Users//Life by the Numbers? Mathematical LensMathematical Lens: A Mathematical Pyramid Scheme Calendar ProblemsCalendar and Solutions - September 2011 Activities (for students)Activities for Students: The Hands Project Problem SolversProblem Solvers: The 5-Horse Race Technology/Technology TipsTechnology Tips: Constructing the Square Root of a Complex Number Delving DeeperDelving Deeper: Beyond 1 + 2 = 3 ... For Your Information/Products/PublicationsFor Your Information - August 2011 The Backpage: My Favorite LessonThe Back Page: My Favorite Lesson: What’s the Difference?

 Second Look - Learning from Student Errors Sharing Teaching Ideas: Why Is the square root of 25 Not ±5?The value/s of the square root of a positive integer. How students may incorrectly acquire the belief that the square root of 25 is plus or minus 5 and describes teaching methods to be used to convince students their belief is incorrect. Includes graphs to show how the misconceptions occur. A Function or Not a Function? That Is the QuestionAnalyzing and discussing examples and nonexamples of functions can help address students’ misconceptions about this important concept. Connecting Research to Teaching: Quadratic Functons: Students Graphic and Analytic RepresentationsAnalyzing one student's work on four translation tasks. Through separate interviews and attempts to determine a possible reason for the nature of the student's understanding of the vertex and the coefficients b and c of a quadratic function in standard form, the authors call attention to cognitive obstacles and how to address these challenges. Connecting Research to Teaching: Habits in the ClassroomCommon practices that have the unexpected consequence of narrowing the student's view of the concept of function. Illuminations: There Has to Be a System for This Sweet ProblemWe are confronted with problems on a regular basis. Some of these are easy to solve, while others leave us puzzled. In this lesson, students use problem-solving skills to find the solution to a multi-variable problem that is solved by manipulating linear equations. The problem has one solution, but there are multiple variations in how to reach that solution. The Teaching PrincipleEffective mathematics teaching requires understanding what students know and need to learn and then challenging and supporting them to learn it well. The Learning PrincipleStudents must learn mathematics with understanding, actively building new knowledge from experience and prior knowledge.