July 2017

  • For many teachers, summer is a time of reflection and professional learning. This month, we will be showcasing some of NCTM’s many member resources for helping teachers reflect on their practice more broadly, from planning homework policies, to strategies for setting personal goals for improving teaching, to ways to improve collaborations with colleagues.

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    Have you read Principles to Actions: Ensuring Mathematical Success for All? In this book, NCTM defines and describes “the principles and actions that are essential to strengthen mathematics learning and teaching for all students.” The principles and actions target teachers, specialists, coaches, administrators, policymakers, and parents. If you have read it, summer is a great time to think about how you can share the principles and actions with others in your community. If you have not read it, summer would be a great time to do so. The eBook is only $3.99 for members.

    Whether you are just starting to explore Principles to Actions or are ready to share it with others, a great starting point is the Principles to Actions Toolkit. As an NCTM member, you can access “The Case of Ms. Bouchard and the Hungry Caterpillar,” and see all eight Teaching Practices in action through video, transcripts, and a reflection guide.

    This summer, take some time to explore the case of Ms. Bouchard, delve into the teaching practices, and think about how you can work with colleagues next year to explore the teaching practices in your own classrooms.

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    Grades 3-5

    Did you know that NCTM prepares Research Briefs for teachers? You can use them to help make the case to an administrator or parent about the importance of particular practices and to ensure that you are implementing best practices in teaching.

    Parents, administrators, and often the media ask teachers about the importance of teaching with problem solving and how to do so successfully. NCTM’s Research Brief on the subject shows how problem-solving and skills fluency are not either-or discussions: We can use problem solving to successfully promote understanding, strategy, and fluency.

    For more information on successfully implementing rich tasks that allow students to grapple with concepts before they are instructed on how to solve them, see these two short articles based on using the Problems of the Week from NCTM's Math Forum. This information can help bring problem-solving research to the day-to-day classroom level:

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    Grades 6-8

    One of NCTM’s most popular articles, “Never Say Anything a Kid Can Say!” is definitely worth a read or a re-read this summer. Middle school teacher Steven C. Reinhart shares how he implemented subtle shifts in his practice that led to big changes in classroom culture and student learning. Reinhart recommends changing at most 10 percent of your classroom practice each year. His article is a wonderful starting point for helping choose some areas to focus on in the fall. Maybe he will advise against carrying a writing instrument or touching students’ pencils as you walk around the room or repeating students’ answers to questions. Reinhart recommends picking a few things, making them habits, and observing how those changes impact the classroom.

    While you are reflecting on classroom discourse and culture this summer, you might be interested in connecting Reinhart’s tips to the research on classroom discourse. The Research Brief “Strategies for Discussion” helps put good discussions into the context of learning for understanding.

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    Algebra 1

    For many students, homework in an algebra 1 course is a hurdle to being successful in mathematics. Since algebra 1 is the first high school style course that many students encounter, they may find that their study habits are not particularly effective. The homework assignments they are receiving also may not be structured effectively to help students become productive, independent learners. NCTM has prepared a Research Brief on effective homework strategies to accompany its Brief on what research says about homework.

    One of the most consistent findings in the Brief is that effective homework includes a mix of practicing previous content; exploring problems, to prepare for upcoming instruction; and putting into practice the day’s focus. Mathematics Teacher and Mathematics Teaching in the Middle School both contain excellent articles about homework (just search “homework” on nctm.org), including such ideas as these:

    • Varying problem types to include error analysis, sorting groups of problems rather than or before solving them, explaining procedures learned in class, and writing their own word problems to go with problem contexts (from “Making Homework Matter to Students” by Lee Walk and Marshall Lassak, MTMS May 2017)
    • • Giving students both problems and answers and requiring them to fill in the reasoning in between (from “Algebra Homework: A Sandwich!” by D. Bruce Jackson, MT March 2014)
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    Algebra II/Pre-Calculus

    One key idea that students encounter, especially beyond algebra 1, is that of algebraic proof. Polynomials have many interesting patterns and connections to number theory. Proving that the patterns students find will always continue, which is at the heart of algebraic reasoning, will help students work on their algebraic fluency.

    NCTM’s Research Brief “Algebraic Reasoning in School Algebra” presents two examples of how algebra instruction can go beyond procedures to help students develop algebraic reasoning skills. How might the Brief help you look for opportunities to add more algebraic proof to your own classroom?

    Perhaps when you are back in school this fall, you might meet with other algebra teachers in your school to explore this topic further. One of NCTM’s Professional Development Guides contains reflection questions about a Mathematics Teacher article on teaching algebraic proof based on common patterns in student thinking. The article “Learning from Student Approaches to Algebraic Proofs” would make a great discussion topic for a math department meeting or an Algebra 2 PLC meeting.

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    Have you ever used an NCTM article as part of your math department or Geometry PLC meetings? Did you know that NCTM collects articles and publishes reflection guides and tools for professional development (PD)? Over the summer, familiarize yourself with NCTM’s Professional Development Guides, and pick one to bring back to school with you. For geometry teachers, “Key Ideas and Insights in the Context of Three High School Geometry Proofs” contains a great set of proof tasks and explicit links to interactive geometry software. The combination of readings, activities, designing tools, article, and reflection guide provides everything you need for a successful PD session or set of meetings about introducing proof so that it makes sense to students.

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    Probability and Statistics

    Did you know that NCTM's research guides span policies, pedagogies, and key conceptual understandings? NCTM has produced two Research Briefs about how students learn particularly important concepts from probability and statistics.

    In “Statistics,” researchers address how students think about the concept of average. They describe younger students’ informal notions of average and how students invent concepts of “average” or “typical” when looking at messy data. The researchers show how some of these informal concepts are statistically productive and can be built on to develop powerful statistical concepts of average. The Brief points out that measures of center in statistics only make sense in the context of variability (if data did not vary, we wouldn’t need to do any work to find the typical data point).

    The second clip, “Variability,” helps to distinguish two meanings of this term: variability in a phenomenon over time and variability in measurements of a single phenomenon. The Brief explores how students’ understanding progresses for each type, providing teachers with ideas about what to look for in formative assessments and what kinds of experiences students need to make better sense of variability.

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    Be sure to also check out these additional resources and tools for your classroom.

    Math Forum - Problems of the Week

    Get your weekly dose of math problems and puzzles from the Math Forum. You will also find more math resources and tools, as well as a Math Forum: Problems of the Week Blog, furthering discussion.

    Problems of the Week Samples

    Brain Teasers

    Want quick ideas for great back-to-school icebreaker classroom activities? We've got you covered. Challenge your new students and mathematics enthusiasts alike with these staff-picked puzzles. In need of more? Browse the entire Illuminations library and discover what's in store in this amazing resource.

    Sliding Triangle

    Golden Ratio

    Heart Shaped Words

    Browse Illuminations