TEACHING AND LEARNING MATHEMATICS IN THE SECONDARY SCHOOL

Daniel BochicchioNeag School of Education

Teaching (Instructing) requires

Knowing and understanding mathematics, students as learners, and pedagogical strategies.

A challenging and supportive classroom learning environment.

Continually seeking improvement

Guiding Principles

Provide multiple entry points into every lesson.

Employ a range of instructional modes. Use a variety of group formats and

configurations. Describe and demonstrate each

technique Describe students’ background

knowledge using different techniques.

Entry Points

Analogies: Make comparisons Deriving the Pythagorean Theorem in Ancient

Greece was like… Stories: Tell a story about a certain

experience, focusing on the choices you had to make along the way. I was offered a job and had to choose between

getting paid an hourly rate or a set salary plus commission.

Numerical Representations: Graphing data Let’s look at the levels of ozone in out atmosphere

since 2000.

Entry Points

Dramatic Interpretations: Have students role-play Act out the solving a problem involving finding

the height of an object. Essential Questions: Ask the big

questions. When do we need to measure area and

perimeter? Hands-on Manipulation: Design and Build

Using popcorn, what is the relationship between the volume of a cylinder and a cone?

Instructional Approaches

Demonstrate: Show students what successful performance looks like.

Read to Think: Read excerpts or short texts aloud as a means of introducing a topic.

Write to Learn: Have students write formally or informally to discover what they know or synthesize their learning.

Investigation: Design an inquiry that asks them to find and make sense of information.

Simulation: Provide a range of roles students can play.

Construct: Provide materials and ask students to design and create an original project.

Reciprocal Teaching: Ask students to teach what they have learned to others in a group or the class as a whole.

Problem-Solving: Place students in the middle of a problem they must solve using their understanding of the material.

Generate: Have students come up with their own questions and problems, answers and solutions.

Reflect: Ask students to reflect on their learning process to increase their understanding of what they learned as well as how they learned it.

Groups

Pairs (2) If you want quick, informal conversations

where students can compare solutions. Small (3-4)

If you want deeper discussion from several perspectives.

Large (5-7) If you want chaos.

Full Class If you want to participate in a whole class

discussion.

Before, During, & After

Before using strategies, students need to know what an effective performance looks like.

During, you need to clarify and refine students’ application of the technique.

After they finish, it is often beneficial to have students reflect on their use of the technique or strategy to help them improve their application next time.

Sentence Starters My purpose is… My top priority is… I already know that… This reminds me of… In a nutshell, this says that… I can relate to this problem because… To understand better, I need to know

more about… I like/don’t like ______, because… A conclusion I’m drawing is… At first I thought ______, but know I…

Socrates Says… What’s next? How did you get that? Can you clarify that for me? Where do we go from here? Why did you do that? What allows you to do this? When can you do that? What’s your reasoning behind that? What’s the next logical step? Who else agrees with this? Where have we seen this before?

A conceptual model of the teacher’s work to support whole-class discussions

Mathy Moves

Assessing

Broad-Based Participation

Publicizing

Valuing

Mathy Moves

Assessing

Broad-Based Participation

Publicizing

Valuing

Valuing Valuing lies at the core of the model because the other

components depend on students feeling that their contributions are significant and their participation matters. For example, Publicizing moves will not be effective unless students feel that they and their ideas are valued. The teacher must find ways to value each student and establish a climate of mutual respect. This climate must be established from day one.

Publicizing This component pertains to teacher moves that make student

ideas public. The teacher must support students in sharing their ideas so that other students have the opportunity to hear and comprehend those ideas and subsequently think about and respond to them. Without this step, there can be no discussion. The teacher’s actions can include asking students to go to the board to share their work or thinking and having a student’s idea restated (by the student, other students or the teacher).

Broad-Based Participation The teacher must actively pursue participation and

involvement from all students and support them in generating their own ideas. This component includes strategies that make openings for a range of students to offer input and that provide the time and space students need to develop their own thoughts.

Assessing Assessing moves serve two main purposes:

1) to determine where the students are in their understanding and

2) to determine whether the discussion is headed in a productive direction. Although some moves focus almost exclusively on assessing, others combine assessing with other purposes. For example, asking how two representations are related (mathy move - building meaning across representations), provides important assessment information and promotes students’ mathematical thinking.

Mathy Moves For whole-class discussions to be productive,

the teacher must create and capitalize on opportunities to develop conceptual understanding and engage students in mathematical practices. Mathy moves are used in response to the class’s ongoing dialogue and thus rely on the work done in relation to the inner circles.

Building meaning across representationsMoves that aim to develop connections between representations (but not strictly compare, contrast)

Exploring incorrect solutionsTeacher asks for explanations of the (known) incorrect solution and for students to articulate why the solution (and/or thinking) was incorrect

HighlightingHighlighting a valued practice or idea

Moving the math vertically/sequentiallyTeacher poses a question to move the discussion forward onto the next idea

Promoting justificationThe teacher asks for specific ways for approaching seeing and/or solving the particular problem

Promoting other’s perspectivesRequesting students to use another student’s method /take another perspective

Recapping/ConsolidatingTying all the pieces together so big ideas/key points are identified (where we’ve been, where we are, where we are going). Teacher summarizes students’ ideas, highlights different features of the different mathematical concepts (or has students do this)

Requesting alternative methods approaches The teacher asks if someone has another way of approaching, seeing and/or solving a problem

Requesting clarificationTeacher asks follow up questions after idea is initially stated to make sure idea is clear to teacher and/or class

Requesting compare/contrastTeacher asks student to compare two different methods, representations, ideas, approaches, etc.

Seeking new representations (for clarity)Teacher asks students (either the initial speaker or classmates) to re-represent the idea

Refocusing The teacher offers a comment, question, or remark that creates continuity in the conversation, but shifts its focus, or at least the students’ attention back to the intended learning goal.