Education 413 - Fall 2012

Instructor: Dr. Yvonne Lai education413the logogram for 'at'umichelectronic mail punctuationedu
Co-instructors:Annick Rougee
Rachel Snider


Featured records of practice

To illustrate the ideas treated in the course, we will use a few records of practice as points of reference.

Display/hide more information and links to these records.

TIMSS Video Source: Featured records:
  • AU2 Congruence - An eighth grade mathematical lesson on congruent triangles. The lesson is 47 minutes and was taped in a girls' school. The class has 26 students.
Inside Mathematics Source: Records:
Source: Records:
kate_nowak_logo Source:
  • Function of Time is a blog on teaching by Kate Nowak, who has been a high school teacher for 8 years in Syracuse, NY.
  • Completing the Square - A post about an Algebra 2 lesson on Completing the Square. The lesson features a designed sequence of examples from which the procedure of completing the square is developed.
math teacher mambo Source: Records:
  • Cell Phone task - a post about an Algebra 1 lesson on linear equations and inequalities.

Course Organization

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Setting mathematical work Explaining mathematical concepts, procedures, and solutions
Eliciting thinking, assessing understanding, and handling error Establishing norms and routines for classroom discourse for mathematical work

Setting mathematical work

Setting mathematical work refers to the work you to do set yourself up to teach a mathematical task, such as doing the task to familiarize yourself with its mathematical demands, how you plan to help students get to work on the task, and how you will wrap things up.

In this course, we will rehearse these three components of setting mathematical work:

(Setting up the task)
What you say and do before the students begin work on the task or one of the problems in the task.
(Launching discussion of the task)
What you say and do to transition between different parts of the task, or between different modes such as individual work, group work, small group discussion, or large group discussion. Launches happen after the students have begun the work and before conclusions. Sometimes there may not be a launch, and sometimes there may be multiple launches.
(Concluding work on the task)
What you say and do to end the task. They might highlight the mathematical point of the task, summarize notation, observe progress toward long-term goals, or use the work just done to look forward to future work. Teaching a task should almost always include a conclusion. There may be multiple parts to the conclusion of a task with multiple problems.


For more information about the records of practice, see the section above.
[beginning time - end time]
Notes on example
  1. AU2 Congruence (10 Q)
  2. AU2 Congruence (Worksheet)
  3. AU2 Congruence (Textbook)
  4. Quad Fns: Intro (Part A)
  5. Quad Fns: Prob 1 (Part A)
  6. Quad Fns: Prob 2 (Part A)

In (1) there's a fuzzy line between setup and work. You can either consider the oral reading of the questions to be part of the setup for the students' individual written work, or to be part of the work as students have to process what's just been said. The important thing here isn't splitting hairs on what exactly is setup and what is work, but rather to think through the implications of the decision to deliver the questions aloud (instead of, for example, on a handout or on an overhead).

Clip (2) illustrates a setup for a complex task. Note how the teacher uses this setup to preview what will come up in the task, who should work together, expectations for revising work, what materials the students have in front of them and how to use them.

Note how the teacher uses a part of the task as the setup as well as how she uses the blackboard in clip (3).

In (4)-(6), note how the teacher's setup includes observing how various students begin their work, as well as expectations for notebook organization. Note also how the teacher motivates the tasks for the students.

  1. AU2 Congruence (10 Q)
  2. AU2 Congruence (Worksheet)
    (part 1 of discussion)
  3. AU2 Congruence (Worksheet)
    (part 2 of discussion)
  4. Quad Fns: Intro (Part B)
    [00:00:00 - 00:00:25] (whole task)
  5. Quad Fns: Intro (Part B)
    [00:03:07 - 00:03:10] (y-intercept)
  6. Quad Fns: Prob 1 (Part A)
  7. Quad Fns: Prob 2 (Part A)

For each of these clips, watch for what mathematical query the teacher uses to open discussion. (For example, in (4), the teacher asks a student to "share how they began the problem".) How do these mathematical queries help the class get to the mathematical point of the task?

  1. AU2 Congruence (10 Q)
  2. AU2 Congruence (Worksheet)
    (part 1 of discussion)
  3. AU2 Congruence (Worksheet)
    (part 2 of discussion)
  4. AU2 Congruence (Worksheet)
    [00:37:09-00:37:48] (whole task)
  5. AU2 Congruence (Textbook)
  6. Quad Fns: Intro (Part B)
  7. Quad Fns: Intro (Part B)
  8. Quad Fns: Intro (Part B)
    (whole task)
  9. Quad Fns: Prob 1 (Part D)
    (conclude Katie, Stefon, Miguel, then whole task)

In these conclusions, note what kind of conclusion is given (does it summarize key mathematical points, look ahead to other work, observe progress on long-term goals, compare and contrast situations, ... ?).

If the conclusion discusses mathematical content, make sure you can identify (a) what is being explained and then (b) observe the quality of the explanation. How global or local are the statements (that is, if you listen to the words, do they apply only to the task at hand, or could they apply to more general forms of the kind of problems represented by the task?) Are there multiple representations used and compared explicitly? Does the conclusion develop a generalization from examples done as part of the task? What mathematical terminology is used?

The organization of these ideas is based on work of the University of Michigan School of Education Math Methods Planning Group. The comments on quality of explanation are based on the Mathematical Quality of Instruction instrument.

Features of good explanation

There are many features of good explanation; besides correctness, completeness, organization, and coherence, we will rehearse for the qualities of:

(Connection to meaning)
Explanations address "why" rather than just "what" or "how" steps were carried out. An explanation of a computation not fitting this feature might only explain what numbers were used and how they were added and multiplied together, rather than the meaning or motivation behind any substitutions or manipulations.
Developing generalizations
(Developing generalizations from examples)
Develop generalizations from examples or experiences rather than stating the generalization before examples. An explanation not fitting this feature might begin with the general statement before giving any examples, or may only give the general statement without motivating it afterwards or beforehand with any examples.
(Richness of use of representations)
Features sustained and careful links between different representations or ideas, and be explicit rather than tacit about the linking. An explanation not fitting this feature might feature only one representation, or if it features more than one representation, it features only cursory links between the representations and the idea.
Global explanations
(Use of global explanations rather than only local explanations)
Global explanations are explanations that could apply to a class of problems as opposed to just one. For example, a global explanation for solving an equation such as 5x=20 might describe using "dividing by the constant the x is multiplied by", as opposed to saying "dividing by 5". Of course there are places when a local explanation is appropriate; however, global explanations tend to be more useful when concluding work on a problem, as they are a way of capturing the point of the problem.
(Analyzing conditions and conclusions, including conditions of use)
Discusses the conditions that a problem implies for a solution to it, or the conditions for an if-then statement to hold, or conditions for where it may make less or more sense to apply a particular procedure. Explains how the conclusions coming from mathematical reasoning on a task relate back to the conditions of the problem given or the if-then statement used or the procedure used.
Aligning claims
(Aligning students with claims and giving them the responsibility for defending it.)
If part of the explanation comes from students, asks students to state what it is that they are reasoning about, and positions them to defend their statements.
Mathematical language
(Purposeful use of mathematical language)
Uses relevant mathematical terminology whenever the opportunity arises; models good usage of mathematical language.

For an explanation to be mathematically rich, it does not necessarily have to satisfy all the above features. However, because these are features that take practicing to carry out well, and, when appropriate, can greatly enhance the quality of mathematical communication, we will work on adhering to all these features as much as possible.

Preparing for good explanation of something (e.g., concept, procedure, solution) involves:

Identifying the core principles of the something, and how to connect your explanation to those core principles
Having a good set of examples to illustrate the something
Connecting students' experiences or prior knowledge to the something. The experiences might come from a task that you design for the lesson where you introduce the something.
Familiarizing yourself with the various representations of the something that may arise in textbooks, in student work, or other sources available to your students
Familiarizing yourself with common errors that arise when encountering the something
The first four features for quality of explanation are taken from the Mathematical Quality of Instruction (MQI) instrument. Here are an overview of the instrument (pdf), more information about its background, and a version of the rubric for the instrument. The characterization of "aligning claims" is based Stein (2001)'s article "Putting Umph into Classroom Discussion".

Eliciting thinking, assessing understanding, and handling error

We focus on rehearsing the following methods for eliciting, assessing, and handling.

Eliciting thinking


Assessing understanding


Handling error


Establishing norms and routines for classroom discourse for mathematical work

Routines and norms we will rehearse establishing are