1. Making Sense of Teaching:
Attending to Classroom
Interaction in Mathematics
Vilma Mesa
AMATYC Webinar
March 22, 2012
Funded in part by NSF Career DRL 0745474 to Vilma Mesa. The opinions expressed here are those of the
author and do not reflect the views of the National Science Foundation
2. Today…
Definition of instruction
Looking inside the classroom:
Who participates? How frequently?
What patterns of interaction occur?
What types of questions get asked?
For each: definition, examples, poll questions,
what the analyses give us, findings
2
3. Instruction
When I say ‘instruction,’ what comes to your
mind?
Type all the ideas using the chat…
3
4. Instruction
We attend to four elements:
• The teacher,
• The students,
• The mathematics,
• The environment where instruction happens.
4
5. Instruction is what teachers
and students do with the content
over time.
The environment shapes these
actions.
5
6. Instruction
environments
teachers
students
content
students
time
(Cohen, Ball, Raudenbush, 2003)
6
7. Inside the Classroom
I look at how the interactions happen.
By interaction we mean:
What is said and by Whom.
Three different tools of analysis:
• The classroom maps
• The patterns of interaction
• The types of questions that teachers ask
7
9. chalkboard P
TV screen
CLASS A
C answer
question
Af
Headphones
Af As Arrives @ 9:45
Leaves @ 10:18
whiteboard
Af I Af
As
L
Fell asleep @
10:18
I
SB
TV
10. whiteboard
CLASS B
answer
question
P
*2
c Af
Af
As H
L
Ar
*2
Af
SB
Na
10
11. chalkboard
CLASS C answer
question
C
Af Af Af
As
1 2 3 4 5 6 7
chalkboard
As
8 9 10 11 12 13 Whispers to
him during
quiz @ 8.45
Surveys up and
down aisle
during quiz
Ar Ar
14 15 16 17 18 19
JK
20 21 22 23 24 25 26
Arrives @ 8:40 Arrives @ 8:55
12. chalkboard
TV
CLASS D
answer
question
As
test
whiteboard
Na Af
Af Af
VM Af
TV
13. Who asks more questions, the teachers or the
students?
a. Teachers
b. Students
c. Both the same
13
24. Questions We Can Answer
• Who talks
• How is the interaction distributed in the class
• Frequency of questions and answers
• Origin of questions and answers
• It is not exact but provides a good overview of the
class interaction
24
25. Findings
• Substantially more participation in community college
than in other settings (university, four-year college)
• Teachers ask questions and give students
opportunities to answer them
• Patterns vary by type of course
• Lots of exchanges:
Teacher Student Most By Whom
Developmental 209 43 Females
Trigonometry 80 17 Males
(Mesa, 2010)
25
27. IRE/F
• Teacher initiates the interaction (I)
• Student responds (R)
• Teacher evaluates or gives feedback to the
students’ response (E/F)
Variations:
• Funneling: The teacher seeks a specific answer, so
he or she keeps asking more and more specific
questions.
• Fill-in-the-Blank: The teacher lets the students finish
the statements.
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28. Elijah
1. T: So the 8x3 – 8x3 is gone. 18x2 – 8. Several: 0
24x2…
9. T: What we wanted. 25x – (-
2. F: -6.
18x)...
3. T: -6x2.
4. F: Plus 25.
10. F: Plus 18.
5. T: Bring down the 25x. All right. So x 11. T: Exactly which turns it into…
into -6x2… 12. M: 43.
6. F: -6x. 13. T: 43?
7. T: -6x because this is -6x2 divided by 14. F: 43x.
x. That’s what that part is. So -6x
times this quantity. -6x2 – 18x. Don’t 15. T: Yes, this becomes 25x + 18x
lose your negative because now we because the double negative
get into all the negative signs. turns into an add. Bring down
(writes on board 4 seconds) -6x2 – (- the 42x into 43x. Plus 43. So
6x2)… 43x + 129.
28
29. Elijah
What type of patterns do you mostly see here?
a. IRE/F
b. Funneling
c. Fill in the Blank
d. Other
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30. Gabriel
T: Questions? F1: That’s if it’s an increase. Is it
M: I, ok, I did this percent increase. actually an increase?
I did it the way you always tell T: Tell me again, you’re comparing…
us to do it and I got the wrong
M: Well the minimum wage was $3.35
answer.
and it was $5.45.
T: How do you know the answer’s
F2: But you’re trying to compare two
wrong?
things that are in different units.
M: Well because um, I got 10% This is in 1989 units, this is in 2006
and that’s not right because I units. You can’t compare things
plugged into the equation and I that are in two different units. You
got a wrong answer. have to compare 2006 dollars to
T: Ok. Well… 2006 dollars.
M: It’s supposed to be new equals T: M, what do you think about what
(1 + r) times old. F1 and F2 said?
30
31. Gabriel
What type of patterns do you mostly see here?
a. IRE/F
b. Funneling
c. Fill in the Blank
d. Other
31
32. Elena
1. T: How many cards are an 10. T: There’s 17?
ace or a spade? 11. F: 17.
2. M: 13. 12. F: What about...
3. T: An ace... 13. T: But we already, we
4. F: 17? talked about this, you
5. T: An ace, how many can’t count the…
aces? 14. Several: Ace of spades.
6. Several: 4. 15. T: Ace of spades twice.
7. T: 4. How many spades? 16. M: 16.
8. F: 13. 17. T: So sixteen.
9. F: 17.
32
33. Elena
What type of patterns do you mostly see here?
a. IRE/F
b. Funneling
c. Fill in the Blank
d. Other
33
34. Think about your own way of interacting with
your students, say in a developmental class.
Which patterns are you more likely to use?
a. IRE/F
b. Funneling
c. Fill in the Blank
d. Other
34
Hinweis der Redaktion
We use a definition that has been useful for us to organize our inquiry. In it, instruction involves these four elements, and it is defined as, or more schematically as follows
We use a definition that has been useful for us to organize our inquiry. In it, instruction involves these four elements, and it is defined as, or more schematically as follows
So instruction is defined as the interaction between teachers students and mathematical content; this can be studied over time, and these interactions are embedded within particular environments. This model serves to see how things happen: we can look at just what the teacher does, or just at what the students do, or at how the mathematics is presented. We are interested in all the arrows, and we like seeing whether there are changes over time. The environment for us represents different types of colleges. The setting of the college, for example whether the college is small or large might imply that you interact with the students differently because your classes are either too small or too large. Or your school might not have resources like technology or might offer only on-line courses. These different conditions may influence how these interactions unfold over time. Another point that is important to highlight is that this is only a representation for analytical purposes. This model helps us see through any classroom no matter whether it is a lecture or a seminar. It is an abstraction. So how have we used this?
I am going to show you four maps of four different community college math classrooms. You will have about 15 seconds to observe and see if you can make sense of each map. You can take notes as needed. After showing you the four maps, I will ask you a few questions about what you saw and possible inferences you can make. For your convenience, as I ask the questions, I will show a slide with the four maps side by side. Ready? Are there any questions?Ok, here is the first map
Let me explain the symbols. The light triangle represents the teachers. The blue circles with arrows represent male students. The green circles with an arrow represent female students. When it is possible to identify the ethnicity of the student, this is noted. Show the African American student and an Asian student.The symbols of speech bubbles represent ANSWERS, the stars QUESTION. These are noted close to the person who formulated them. Other symbols note where the observer was (SHOW SANDY) or other elements in the room, the board, the tables, a projector, a TV, a computer. Arrows mean movement.
Let me explain the symbols. The light triangle represents the teachers. The blue circles with arrows represent male students. The green circles with an arrow represent female students. When it is possible to identify the ethnicity of the student, this is noted. Show the African American student and an Asian student.The symbols of speech bubbles represent ANSWERS, the stars QUESTION. These are noted close to the person who formulated them. Other symbols note where the observer was (SHOW SANDY) or other elements in the room, the board, the tables, a projector, a TV, a computer. Arrows mean movement.
We poll we discuss, what is the evidence that they have?
Class D, appears to be the most interactive, because most students speak up.
It looks like C, because fewer students speak and when they speak there is little that they say
Males participate more in all the classes, but notice that in three classes there are fewer women! Relative to men, though it appears that they participate less
Expected answer: D, more females, group work, get ideas from the participants too.
I am just curious, interesting would be to see why!
What other features do you see in these maps? What do they tell you about the different classes?
There are other questions we can’t answer with this analysis alone. We turn now to patterns of interaction
Questions?I’ll show you four excerpts from transcripts from classes I have observed. I want you to indicate which would be representing what type of interaction.
This is part way of a division of polynomials problem: FIND PROBLEM
Should be c, from lines 1, 7, 5, 9 the teacher asks students to fill in the expected response
A class on using percentages, a college everyday math class
None. It shows a different type of interaction, one that would be preferable.
Should be b, from lines 4 to17, specifically when the teacher completes what the student is about to say: can’t be counted twice
Poll question to examine own teaching patterns. Reasons?
Categorization is highly context dependent.
A class on using percentages, a college everyday math class
Recent calls for reform in undergraduate mathematics suggest that including students in mathematical discussions can be beneficial for their learning. Yet, how to manage this is not straightforward. In this webinar we will present preliminary findings from ongoing research on classroom interaction, attending in particular to the types of mathematical questions asked by teachers and students and the interactional moves that teachers propose. Using transcripts and other records, participants will be able to code for some of the features we have attended to and discuss the affordances and challenges of sustaining high-quality discussions of mathematical ideas.
