Communicating Mathematically: Preparing literate mathematics teachers

1
Communicating Mathematically Preparing
literatemathematics teachers

• Michaele Chappell
• Mary Enderson
• Jason Johnson
• Jacob Klerlein
• Middle Tennessee State University
• Department of Mathematical Sciences
• Murfreesboro, TN

2
Communication

• Is an integral component of teaching and
  learning.
• PSSM (NCTM 2000) recommends instructional
  programs promote the organization of student
  thoughts and processes and the ability to
  communicate this information coherently. It also
  promotes using the language of mathematics.

3
The teachers role

• This process doesnt just occur
• Teachers must engage in learning experiences
  where communication is used as a tool to enhance
  their own growth of knowledge.
• Time must be given to develop this communication
  process.

4
Session goals

• Address how communication
• is used as a major theme in methods and content
  courses for prospective teachers elementary,
  middle, and secondary
• fosters communities of learners

5
Presentation Framework

• Mathematics Teaching Today (NCTM 2007) provides 4
  points on promoting discourse
• Provoking students reasoning about mathematics,
• Encouraging the use of various representations,
• Monitoring and organizing students
  participation,
• Encouraging student dialogue (pp. 46-48).

6
Provoking Students Mathematical Reasoning

• Pose Problems
• In particular, numeration in Xmania adapting D.
  Schifters summer mathematics work with teachers
  at Mt. Holyoke

7
Monitoring and Organizing
8
Monitoring and Organizing

• Physical arrangement
• Note which groups are on particular trajectories
• Present in an order so as to not give away any
  thinking

9
Encouraging Student Dialogue

• Students must share thinking in small groups
• Small groups present ideas to whole class
• Presenting groups pose questions and then
  circulate to support others

10
Encouraging Student Dialogue

• Dialogue is at least a 2-way path
• One must speak and others MUST listen
• Listening is explicit part of the course both in
  spoken word and syllabus
• Dispositions of Prospective Elementary School
  Teachers
• Supportive Nurturing more often than not
• Sharing critical thinking is less common than
  offering praise

11
MATH 1410/1420Mathematics for Elementary Teachers

• During non-routine problem solving, preservice
  teachers use varied forms of communication.
• However, talking about their strategies is
  difficult for those who are not accustomed to
  sharing their thinking aloud.
• Communication becomes full-circle as teachers
  reason through the mathematics and dialogue about
  their strategies.

12
Example Pizza Cuts Problem

• Ask Enrolling Questions
• Present the Scenario (i.e., the Problem)
• Allow Think Time
• Observe Strategies
• Address Clarifying Questions
• Reflect Extend the Problem

13
ExamplePizza Cuts Problem

• Round Square
• Oblong

14
Example Pizza Cuts Problem

• Many Strategies come into play
• - Drawing a Diagram
• - Using Trial-n-Error
• - Creating a Table of Values
• - Looking for Pattern(s)
• - Working Backwards.
• Different Strategies, Reflections, and Extensions
  create more mathematical ruckus about the
  problem, evoking more reasoning and dialogue.

15
MATH 3320Middle School Methods

• As preservice teachers work (individually or in
  groups) and discuss tasks, varied forms of
  communication are used.
• Two Course Activities
• Roundtable Problem-Solving Sessions
• Field-based Interviews

16
Example Field-based Interviews

• Methods students work and discuss selected tasks
  prior to interviewing one-three middle school
  student(s). The Cocoa Problem (Billings, 2002)
  elicits much (controversial) dialogue.

17
ExampleField-based Interviews

• 1. Thermos A contains cocoa with a stronger
  chocolate taste. If one scoop of cocoa mix is
  added to Thermos A and one cup of hot water is
  added to Thermos B, which thermos contains the
  cocoa with the stronger chocolate taste? Explain
  your answer.
• Thermos A Thermos B
• 2. Thermos A and Thermos B contain cocoa that
  tastes the same. If one scoop of cocoa mix is
  added to both Thermos A and Thermos B, which
  thermos contains the cocoa with the stronger
  chocolate taste? Explain your answer.
• Thermos A Thermos B

18
ExampleField-based Interviews

• 3. Thermos A contains cocoa with a weaker
  chocolate taste. If one scoop of cocoa mix is
  added to both Thermos A and Thermos B, which
  thermos contains the cocoa with the stronger
  chocolate taste? Explain your answer.
• Thermos A Thermos B
• 4. Thermos B contains cocoa with a stronger
  chocolate taste. If one scoop of cocoa is added
  to Thermos A and one cup of water is added to
  Thermos B, which thermos contains the cocoa with
  the stronger chocolate taste? Explain your
  answer.
• Thermos A Thermos B
• (Billings, 2002, pp. 39-40)

19
ExampleField-based Interviews

• Benefits of the Dialogue
• Clarifies method students reasoning
• Forces students to listen to each other
• Forces students to think more flexibly
• Provides basis for beginning the interviews
• Provides knowledge of what to expect during
  interviews.

20
Secondary methods

• Use of Case studies
• Expose students to real classroom scenarios
• Engage students in the process of analysis with
  respect to school mathematics and the teaching of
  it.
• Establish the professional literature (to
  students)

21

• Course ALGEBRA
• Topic INVERSE FUNCTIONS
•  
• Read the following REAL classroom episode taken
  from a High School Algebra class
•  
• Tina (teacher) Now, today I want to talk to you
  about inverse functions (erasing the board as she
  talks). Who knows what we mean by inverse
  function?
• Sue Is it like one over a function?
• Tina (laughs) I knew you were going to make that
  mistake a lot of people think that when you say
  inverse function it is like fractions and when we
  found the reciprocals of themNO, this is a
  little different (Pause)I am going to give you
  a definition and I want everyone to write this in
  his or her notebooks(She turns around and starts
  writing on the board) If you get a little
  confused by the definition, dont worry, Ill
  explain it to you after I am finished writing.
  (She writes the definition directly from the text
  and reads it loudly as she is writing on the
  board).
• Sam I dont get it!
• Tina I told you it might be hard for you to
  understand, let me finish it then Ill explain
  (she finishes writing). Okay, now its like
  this Say I give you a function like and then
  I ask you to find the inverse of this function.
  The first thing you need to do is write it as ,
  then like we did for linear equations solve it
  for x, in this case, we first subtract the 1,
  then we divide by 5, and what do we get? , this
  is the inverse of our function To see if we are
  right, all we have to do now is compose this
  function on the original one If we get x, then
  it means that we are right and that this one is
  an inverse function. So lets do it.
• Peter I dont get it why do we have to get x?
• Tina It is ok. I will review what I did in a
  minute and then do a couple more examples.
•  
• What is your reaction to this classroom episode?

22

• TASK Now focus on the following points to guide
  your analysis of the case on inverse functions.
• React to the episode from the point of view of
  the 6 principles for school mathematics (equity,
  curriculum, teaching, learning, assessment, and
  technology).
• Analyze the episode from the point of view of the
  learners. What are the dilemmas? How are their
  needs accommodated or denied?
• Analyze the episode from the point of view of the
  teacher. Why is she doing what she is doing?
  What is guiding her instruction?
• Analyze the episode from the point of view of
  PSSM (NCTM) and (particularly for Algebra).
  Which of the standards are met? Which are not
  getting enough attention? What about the process
  standards?
• If you were the teacher, what would you have done
  differently? Why? Be specific!
• Find literature that supports what Tina did and
  why. Find literature that supports what you
  intend to do you must go beyond PSSM.

23
Provoking students reasoning

• Read thru the classroom scenario
• What do you know about the classroom situation?
• What do you know about the content area?
• Analyze case with respect to the 6 principles for
  school mathematics
• Analyze case from point of view of learner
  teacher PSSM
• Literature professional reading

24
Encouraging use of various representations

• What did the teacher do for the concept?
• Why do you think she did what she did?
• What other ideas do you have on approaching the
  teaching of xxxx?
• BREAK OUT sessions (small communities)
  brainstorm other methods of presentation COME
  BACK together to share what was organized.
• Record the information generated and discuss
  pros and cons of each

25
Monitoring and organizing students participation

• Monitor individual responses, small group
  responses, and entire class (community)
• A list of items for students to focus on is
  provided by instructor. This helps organize the
  conversation as it relates to the case study.
  These items include analysis from the point of
  view of the learner, the teacher, PSSM.
• Each student is accountable for his/her responses
  but works in a small community.

26
Encouraging student dialogue

• Throughout the entire analysis of the case, there
  is student dialogue. Some of it is
  opinion-based, textbook-based, PSSM-based, and
  literature-based.
• By reviewing literature (some self-selected,
  others assigned), students are forced into
  continued dialogue related to the high school
  content and presentation of it.
• Oftentimes, the problem is finding a good
  stopping point for the dialogue!

27
Provoking students reasoning about mathematics

• Students are asked to read the article Skemp, R.
  R. (1978), Relational understanding and
  instrumental understanding. Arithmetic Teacher. 9
  - 15.
• Students are then asked to answer questions based
  on the article.
• A dialogue is then sparked to have students
  analyze their mathematical understanding based on
  Skemps idea about relational and instrumental
  understanding.

28
Provoking students reasoning about mathematics

• Students are also given the following definition
• The derivative of a function f is the
  function f ' whose
• value at x is given by f ' (x)
  ,
• provided the limit exists.
• The writing assignment Students are then asked
  to provide a relational explanation for the
  derivative.

29
Encouraging the use of various representations

• The next class meeting, the students are put into
  groups of two where they are expected to
  communicate their relational explanation for the
  derivative.
• In the same class meeting, each group is
  responsible for presenting one explanation.

30
Encouraging the use of various representations

• Typically students present an algebraic
  representation.
• rate of change - velocity, steepness of graphs
  and, increasing or decreasing temperatures.
• Hiebert and Carpenter (1992) illustrate
  understanding in terms of how information is
  represented and structured.
• Hiebert, J., Carpenter, T.P. (1992).
  Learning and teaching with understanding. In D.A.
  Grouws (Ed.), Handbook of Research on Mathematics
  Teaching and Learning (pp. 85-97). New York MPC.

31
Monitoring and organizing students participation

• As a course requirement, students are asked to
  assist in the Mathematics Calculus Tutoring Lab
  working with Calculus I.
• Students are asked to keep a journal of their
  experience. Then, submit a paper analyzing their
  work.

32
Monitoring and organizing students participation

• Before the paper is due, students are given time,
  in class, to reflect on their experience.
• By doing so, this creates an atmosphere for
  students to ask questions about what they know
  and do not know.

33
Encouraging student dialogue

• This for me is an on-going process throughout the
  course.
• When students present their relational
  interpretation for the derivative
• Open forum to discuss the progress - Mathematics
  Tutoring
• To end, students are asked to present their
  Mathematics Tutoring Lab paper.

34
Discussion/Questions

• What are others doing to promote communication at
  their institutions?
• What research questions need to be studied more
  carefully as they relate to preparing literate
  mathematics teachers?
• What have we learned about building communities
  of learners focused on communication?