How can substantive conversation be
promoted in a traditional secondary
mathematics classroom?
Winnacunnet High School
University of New Hampshire
May 7, 2012
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I. Rationale
When I entered graduate school, I had three years of classroom teaching
experience. As I expanded my knowledge of best practices through my methods
classes and education coursework, I was eager to get back into the classroom and
put into practice all that I had learned. I came to the University of New Hampshire,
because I wanted to grow as a teacher, and I spent a lot of time reflecting on what I
would do differently as a result of all that I now knew.
When I arrived at Winnacunnet, this was at the forefront of my mind.
Problem-solving, metacognition, and classroom community had become passions of
mine, and I wanted them to be integral parts of my teaching. I was really surprised
to find that many of my students did not. The transition from educational ideals to
educational reality was a huge challenge for me. Many of my students did not want
to engage. They wanted to come to school, put in the minimal effort required, and
not think about my class again until the next day. Their attitudes and behaviors
were not bad, and I had no significant discipline problems. Most of my
studentssimply wanted to come to class, receive the content from me, and do some
minimal practice. I found it very difficult to implement my ideals with classes that
themselves desired minimum engagement.
I realized that student engagement was in fact the key to my dilemma. If I
could find a way to get my students participating more fully in class, I believed I
would have more success in encouraging problem-solving and promoting
metacognition. I decided to research ways to transform my class into a learning
community that would investigate critical questions, weigh points of view, form
positions, present and defend their work, and collaborate with peers. I was seeking
a pedagogical transformation. My research led me to communication.
The Importance of Communication in a Mathematics Classroom
The National Council of Teachers of Mathematics (2000) named
communication one of the five key process standards for mathematics, declaring it
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“an essential part of mathematics and mathematics education.” (NCTM, 2000, p. 60)
As stated in NCTM’s Principles and Standards for School Mathematics, instructional
programs should enable students to use communication in order to organize and
consolidate their thinking as well as to analyze and evaluate their own thinking and
the strategies of others. Additionally, these programs should enable students to
communicate their thinking coherently and precisely using mathematical language.
The benefits of communication in the classroom are abundant. The process
helps to build both meaning and permanence for ideas. By making ideas public, they
become tools for the learning of the community. The explanations by peers help
students to develop their own understandings. Exploring multiple perspectives or
approaches help participants to sharpen their thinking skills and discover
connections among mathematical concepts. In justifying their solutions to others,
students gain better mathematical understandingand develop a language for
expressing mathematical ideas and an appreciation for the necessity of the precision
of mathematical language. (NCTM, 2000)
Students who have opportunities, encouragement, and support for
speaking, writing, reading, and listening in mathematics classes reap
dual benefits: they communicate to learn mathematics, and they learn
to communicate mathematically. (NCTM 60)
An additional benefit of communication is that it promotes metacognition.
Reflection and communication are intertwined. As teachers pose thoughtful
questions and provoke students to reexamine their thinking, students gain
proficiency in organizing and verbalizing their thoughts. Communication for the
purpose of reflection can become a natural part of classroom discourse and
mathematical learning (NCTM, 2000).
II. Literature Review
There is an abundance of educational research available on the subject of
classroom discourse. To best inform my action research, I narrowed my search to a
more precise definition of productive and engaging classroom conversation, best
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teacher practicesfor promoting classroom discourse, criteria for analyzing the
presence of this type of conversation within the classroom learning community, and
challenges to facilitating substantive conversation.
Defining Effective Classroom Communication
Researcher Fred Newmann (1995) definessubstantive classroom conversation
as a discussion that goes beyond the facts and experiences to an interaction about
the ideas of the topic. Such substantive conversation is further characterized by
evidence of higher order thinking. For example, students compare and analyze
ideas, synthesize information, or raise questions in order to clarify or to further
their understanding. Secondly, during substantive conversation students interact
directly with each other rather than withsolely the teacher, and they give sustained
attention to a topic. Rather than correcting student responses, the teacher
encourages students to support their thinking with evidence or reasoning. Finally,
substantive conversation builds on previous comments as students work together
toward a collective understanding of the concept (Newmann F.M., Secada, W.G., &
Wehlage, G. G 1995).
Newmann and Wehlage (1992) go on to say that a high level of substantive
conversation is evidenced by three features. The first is considerable interaction
between students about the ideas of the topic which includes indicators of higher
order thinking. Another feature of substantive conversation is the unscripted
nature of the exchanges. Participants explain themselves, respond directly to
comments from previous speakers, and ask questions. Finally, the dialogue builds
coherently on the students’ ideas.
Best Teacher Practices
Research by the National Research Council (20005) suggests several shifts in
teacher practices thatpromote a classroom community of discourse. Rather than
asking questions that focus on the answers, teachers should ask questions that focus
on mathematical thinking. They should probe student thinking, model or expand on
explanations, and physically fade from the center of the classroom. Additionally,
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coaching students in their participatory roles can be helpful. For example, a teacher
can suggest that everyone have a question ready once a classmate has presented a
solution to a problem. (NRC, 2005)
Joan Moss and Ruth Beatty (2010) take this idea further by investigating how
to promote the inclusion of all students in the classroom discourse. There exists an
imbalance in students’ participation levels in most mathematic classrooms. The
contributions from the higher performing students are generally given more weight
by both the teacher and the class than those of the lower-performing students. Moss
and Beatty’s research found that an effective way to address this hierarchy of
students’ mathematical achievement and status is to apply the principle of
democratization of knowledge. This principle supports the idea that all
contributions in the class are both valued and can be improved, and it can best be
achieved when the teacher reduces his/her role in student discussions. Because the
teacher no longer provides the answer, there is an increase in student efforts to
provide explanations
Additionally, Moss and Beatty (2010) found that epistemic agency, the
responsibility that the group assumes for the ownership of ideas, was greatly
improved when the teacher was removed as the primary source of mathematical
ideas. The students take responsibility for contributing ideas. They monitor each
other’s contributions, asking for both clarification and revision. The teacher and
textbook are no longer the sole authorities of mathematical ideas. Students build
the knowledge together.
In order for such a classroom community to be successful, there must be a
balance between the process and content of the discourse. In other words, the ways
in which student ideas are elicited must be balanced with the substance of the ideas
of the discussion. One method to facilitate such balance is provided by Miriam
Sherin (2002). Her “filtering approach” begins with soliciting multiple ideas from
students to facilitate a student-centered discourse. During this time, students are
encouraged to elaborate on their thinking and to compare their ideas to those of
their classmates. The teacher then filters these ideas, bringing the students’ foci to a
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subset of the mathematical ideas that have been raised. The final step is to
encourage another student-centered discourse.
Communication can also play an integral role in the successful
implementation of other process standards. For example, as part of an exploration
of how teachers can orchestrate pedagogically sound, active problem solving in the
classroom, Jinfa Cai and Frank Lester (2010) concluded that desirable discourse in
mathematical teaching was crucial. Specifically, students should provide
explanations that are grounded in conceptions of the situation rather than strictly
procedural. Additionally, in desirable classroom discourse, teachers should
persistently probe their students’ thinking, particularly when their responses were
in the more basic terms of numbers and operations.
Criteria for Measuring the Existence of Communication
Researcher Kimberly Hufferd-Ackles (2004) provides a theoretical
framework to support teachers in their efforts to develop their classroom into a
math-talk learning community. She defines amath-talk learning communityas “a
community in which individuals assist one another’s learning of mathematics by
engaging in meaningful mathematical discourse.” (81)The components of this
framework are questioning, explaining mathematical thinking, sources of
mathematical ideas, and responsibility for learning. The presence of these
components can be assessed by an analysis of: evidence of mathematical
community, teacher actions, and student actions.
Research shows that the progression through the levels of a math-talk
community is generally concurrent for these components. The first level, level 0,
represents a traditional, teacher-directed classroom where the teacher’s questions
seek the correct answer, and the teacher is the source of ideas and affirmation. In
level 1, the teacher begins pursuing students’ mathematical thinking but is still the
lead in classroom discussions. Learning to elicit meaningful student comments
through questioning is a critical first step. In level 2, the teacher is coaching
students to take on more important roles in the learning community and is stepping
out of the central role in classroom discussions. In level 3, the teacher is facilitating
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a math-talk learning community by coaching and assisting the students as they take
on the lead roles (Hufferd-Ackles, 2004).
This trajectory can be examined by exploring the progression through each
of the components. The first component, questioning, focuses on the questioner in
classroom conversations. It is important to use questions to assess both what
students know and how they think. Once student thinking is brought out through
questioning, it is a part of the discourse space and can be assessed and built upon by
others. Another goal of questioning is a shift from teacher as exclusive questioner to
students questioning each other, as well. This questioning should not be exclusively
to find answers, but predominantly to uncover the mathematical thinking behind
these answers (Hufferd-Ackles, 2004).
The second component, mathematical thinking, revolves around explaining.
This is most profoundly influenced by the social climate that develops to support
student explainers. As students learn to express their thinking, scaffolding by
supportive classroom colleagues is important. Additionally, the teacher coaching
the student to explain his/her solution or thinking rather than elaborating or
summarizing for him/her is a critical shift. Classroom expectations should shift
from providing numerical answers to sharing mathematical ideas (Hufferd-Ackles,
2004).
The third component of the math-talk community framework issource of
mathematical ideas. A key shift for growth in this component is to elicit student
ideas as the teacher presents the content. This uncovers previous knowledge and
misconceptions and exposes the students’ developing understanding of the material.
As students try to understand the meaning of the mathematics rather than simply
imitate the teacher, their thinking becomes part of the mathematical content. They
can explore content and suggest multiple methods. These methods then form much
of the content (Hufferd-Ackles, 2004).
The final component, responsibility for learning, results from students
becoming increasingly invested in both their own and in each other’s learning. Key
indicators are students desiring to ask questions, volunteering to go to the board,
and assisting each other. Rather than students believing they must listen to and
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imitate the teacher to learn mathematics, they have the confidence and engagement
to verify the correctness of each other’s work, choose their own language to convey
their ideas to the class, and comment, clarify, and expand on other’s work and
ideas.The whole class acts as the teachers, assisting others in understanding. The
teacher supports the discussion while allowing students to take on the central role
of explaining (Hufferd-Ackles, 2004).
Challenges
Cai and Lester (2010) documented several roadblocks to orchestrating
classroom discourse. One such factor is the amount of time allocated to solving and
discussing a problem. Teachers often wait less than one second after asking a
question before intervening again. A second factor is the amount of opportunity
teachers are allowing their students to struggle with challenging problems.
Productive struggle is necessary for learning problem solving, yet many teachers in
the United States take over the thinking and reasoning by telling students how to
solve a problem.
Additionally, there are many dilemmas faced by teachers in trying to
implement a discourse community resulting from the students. For example,
students often become disengaged when faced with more challenging problems.
Often it becomes more difficult to manage the direction of a class discussion when
teachers open the classroom to student ideas. Because it is more difficult to predict
where a lesson will go, it is more difficult to prepare for their role in instruction.
(Hufferd-Ackles et al, 2004)
III. Methodology
In order to investigate how substantive conversation can be promoted in a
traditional secondary mathematics classroom, three investigations were made. The
mathematics faculty of Winnacunnet High School was surveyed to shed light on
thecurrent philosophiesand observations of secondaryteachers regarding
conversation in the classroom, three classes completed a questionnaire to ascertain
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the current beliefs and attitudes of high school students, and three lessons were
planned, executed, and analyzed to determine the presence of substantive
conversation in my classroom communities. The three resulting data sets were than
analyzed in order to identify possible hindrances, impetuses, and trends of
substantive conversation.
Teacher Survey
In order to assess the current place of substantive conversation in the
mathematics classrooms of WHS, a survey was given to the teachers that asked them
to consider the place of substantive conversation in their teaching philosophy and in
their classes.
The first part of the survey asked teachers to reflect on whether or not
substantive conversation was a part of their teaching philosophy and how the level
of the class being taught affected this. Additionally, teachers were asked to consider
how the pace of the curriculum and the academic strength of the students either
promoted or impeded academic discussion. Teachers were given a statement as
asked to agree or disagree given a likert scale.
The second part of the survey asked teachers to consider the existence of
Newmann’s (1995) specific evidences of substantive conversation in their classes.
To prevent teachers from having to make generalities about courses whose learning
communities might operate differently, a response space was provided for
accelerated/honors/advanced courses and another for preparatory/remedial
courses. Additionally, teachers were asked to consider the existence of these
evidences in both whole class and small group discussions.
Student Questionnaires
All students in three classes were asked to complete a questionnaire during
class. The Algebra 1A is composed entirely of freshmen. The two Algebra 1B
classes are mostly freshman, but also include a few sophomores and a junior. In all,
fifty-one students completed the questionnaire. Students included their names so
that correlations between the academic strengths of the student, the classroom
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community of which they were a part, the level of the math class, learning style
differences, and their responses could be investigated.
Most questions pertained to students’ comforts and experiences with
academic class discussions in general. Students were asked to reflect on how they
felt about participating in discussions and what they felt hindered or facilitate good
class discussions. Students were also asked to consider how they viewed whole
class and small group conversations specifically in mathematics classes, both past
and present.
Lesson Planning and Analysis
Three lessons were planned and analyzed using an adaption of Learning
Sciences International’s lesson plan and substantive nature analysis (2007). The
lesson plan created by Learning Sciences International is more appropriate for a
discovery- based classroom. Therefore, it was adapted to facilitate substantive
conversation within a more traditional environment. Possible student approaches,
questions to be posed, and possible student misunderstandings are added to the
traditional planning. The lessons were then analyzed, once by myself and twice by
other teachers observing the lesson.
IV. Data
Teacher Surveys
Ten of the thirteen mathematics teachers responded, with some teachers
considering each of their courses separately. The analyses of agreement were
rounded to the nearest tenth of a percent.
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WHS Teacher Survey Part I Results:
Strongly
Agree
Agree
Strongly
Disagree Disagree
20%
70%
20%
40%
30%
60%
20%
20%
10%
30%
60%
0%
20%
50%
30%
0%
1. Substantive conversation is a part of
my personal educational philosophy.
10%
0%
2. The amount of substantive
conversation desired depends on the
10%
level of the class.
3. The amount of substantive
conversation possible depends on the
0%
level of the class.
4. There is little time available in the
curriculum to allow for learning to occur
through substantive conversation
5. Planning for a lesson involving
substantive conversation requires a
different approach to lesson planning.
Summary: Most mathematics teachers at WHS value substantive conversation as
part of their educational philosophy. While the majority of teachers felt that the
amount of substantive conversation possible is dependent upon the level of class,
the group was fairly divided on whether the amount they desired depended on the
level. Additionally, most teachers felt that facilitating such conversation in class
requires a different approach to lesson planning and were fairly evenly divided on
whether our curriculum allows enough time for it.
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WHS Teacher Survey Results Part II:
Percentage of Teachers Who See these Evidences in their Classes
Small Group Preparatory
Make Distinctions
Whole Class Preparatory
Apply Ideas
Forming Generalizations
Ask Questions
Small Group Advanced
Explain Thinking
Build on Ideas
Whole Class Advanced
0
100 200 300 400 500 600
A breakdown of all six evidences can be found in the appendix.
Summary: Teachers at Winnacunnet High School believe substantive conversation
is very present in the accelerated/honors/advanced mathematics classes. Most
believe all these evidences are present in whole class discussion, but over a third of
the teachers felt that students do not explain their thinking in this large group
format. In small groups discussions, however, the teachers overwhelming believe
these students engage in substantive conversation, including explaining their
thinking.
By contrast, the teachers were fairly evenly divided on the presence of
substantive conversation in the preparatory and remedial classes. While most
teachers believe these students both ask questions to clarify or expand their
understanding and make distinctions between mathematical concepts during whole
class discussions, the presence of both these evidences is believed to decrease in
small group work. Explaining thinking to peers is again the weakest of the
evidences, in both whole class and small group discussion.
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Student Questionnaire
The first question asked students how they felt about participating in
classroom discussions. The results for the two tracks are shown in the charts below.
Algebra 1A
Algebra 1B
dislike
like
nuetral
depends
Reasons students gave for enjoying participating in whole class discussions
included:
1. Being able to get a sense for how well their peers were understanding the
material
2. Making the class more enjoyable
3. Learning about what not to do
4. Improving understanding
5. Being able to show off
Reasons students gave for not enjoying participation in whole class discussions
included:
1.
2.
3.
4.
Not being comfortable with classmates
Nervous about saying the wrong answer
Not feeling as smart as their peers
Having difficulty focusing
The second question asked students if they have any classes this academic year
that had good class discussions. About eighty percent of both tracks believed that
they did, with Foundations of Democracy and English being the most frequently
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mentioned. Also listed multiple times by students were math, Spanish, physical
science, and journalism. The follow-up question asked students to consider what
makes a good academic discussion in class. Their responses focused mostly on topic
choice and class participation. Studentsfelt that topics which were current and
relatable helped to promote good class discussions, as did those that allowed for
multiple point of view or correct answers. Good class participation, whether it be
asking questions, expressing an opinion, or disagreeing, was considered critical to
students.
Students were also asked to reflect on roadblocks and impetuses to productive,
meaningful classroom discussions. The roadblocks were, without exception, results
of the behaviors of their peers. Being a distraction, bullying, calling out answers,
and not taking the work seriously were highest on the list. The impetuses were
more general. For example, student participation and listening skills were
frequently mentioned. Additionally, a facilitating teacher and a positive, motivated
class were considered helpful in creating such discussions.
When asked in the forth and fifth questions to reflect on whole class discussions
and small group work specifically in math classes, student responses were similar,
regardless of track or learning style differences. Students were positive about small
group work, particularly if they were able to work with peers of their choosing.
Students find it helpful to work collaboratively and in safer environment in which to
raise questions and to make mistakes. Responses to whole class discussions were
much more varied. Some students find them very helpful while others find the
format overwhelming. The larger role of the teacher in whole class discussions was
considered helpful, as was the opportunity to hear other people’s thinking. The
multiple voices and possible distractions were considered detriments to student
understanding. Several students also felt there really is not very much to discuss in
mathematics, saying “They are not too interesting, because they are about numbers
and equations,” and that there was “nothing to discuss. There’s only one solid
answer to a math question.”
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Lessons
LESSON ONE
Wednesday, March 22, 2012
Algebra 1B
Lesson Three of the Functions Unit
_______________________________________________________________________
Goals:
1. Assess student understanding of domain & range
2. Solidify students understand the multiple representations of functions.
3. Connect this unit with the linear functions unit from trimester 2.
4. Have students participate in substantive conversation
________________________________________________________________________
Warm-Up: (10 minutes) Provide answers for last night’s homework and answer
questions.
Lesson:
1. Quiz on domain & range (20 min)
2. Substantive Conversation Activity: (40 minutes)
* Goal: Students will solidify their understanding of functions.
Students will understand how different representations can show the
same function.
Students will connect this unit to what they already know about linear
functions.
* Problem: Who Shares My Function? (Tufts, 2009)
* Students will form groups by finding the other representations for their
function.
* Instruction Techniques:
* Model during yesterday’s direct instruction how one function can be
represented as a table, graph, and equation.
* Allow students to work in groups. Lead a quick, all-class discussion after
students have found their correct groups sharing how students found the
other representations for their function.
* Questions:
1. What questions should we ask of each other as we try to locate the other
representations for our function? What else will help us? x- or yintercepts, increasing/decreasing, slope
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2. How will you know when you have found another representation? What
tests could you run? sketch a graph given the data table/equation
Closure:
1. Are all lines functions?
* Misunderstandings:
* Students may assume that, because a function is linear, it is their function.
In fact, all functions in this activity are linear.
* Students may look at the same representations to see if they are the same
function. This should be quickly dismissed as a possibility given the data
provided.
* Students may confuse slope with y-intercept in their equation.
* Students may use a positive rather than a negative slope & vice-versa in the
story.
* Students may not understand what the slope of their line could represent.
* Students may mix-up the dependent & independent variable.
* Possible Approaches students may take:
* Students may take their data and create a graph or an equation
and
then match this to another student’s.
* Students may hold up their information and simply wait for students to
find them.
* Students may identify a part of the function such as the y-intercept or slope
and seek a function that matches it.
*Accomodations: Pair a student who will have difficulty with another, stronger
student for the function search.
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Evidence of Substantive Conversation
Who Has my Function Activity
Observations by Rebecca Carlson, teacher
Block 1 Block 3
Making distinctions:
Applying ideas:
5, 3 students used the tables to test inputs/outputs on graphs & equations.
2, 3 students used the equation’s intercepts as a way to weed out possibilities.
3 students compared their graph’s slopes to find the correct equation.
1 student sketched a graph from her equation and sought out its replica.
Forming generalizations:
Raising Inquiries:
2, 1 students had similar graphs and asked the group to help them find the
difference.
Students explaining themselves:
3, 1 students felt they had found the right group that was already complete and had
to explain why they had the correct graph.
3 students with tables explained why they matched the points on a graph.
Students building on another’s ideas:
2 students worked this way as they tested inputs for a data table.
2 students worked together using slope then intercept to decide between graphs
and equations.
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LESSON TWO:
Thurs., March 29, 2012
Algebra 1A
Lesson 7.3Multiplication Properties of Exponents
Warm-Up: Check homework for completion while students work. (10 min)
1. Simplify:
𝑠 −3
𝑏 −2 𝑎0
𝑑6 𝑐 −3
1
2. 𝑘 ? = 𝑠3
________________________________________________________________________
Wrap-Up from yesterday: (10 min)
1. Provide students w/ the correct answers.
2. Allow remainder of 10 min for questions.
________________________________________________________________________
Lesson: Multiplication of exponents(50min)
(Mention that we returning to 7.2 Scientific Notation later when we can do
something more interesting with it.)
Goals: Students will discover the Product of Powers Property.
Students will discover the Power of a Power Property.
Students will discover the Power of a Product Property.
Students will engage in substantive conversation.
1. Products of powers with the same base.
* Instruction Techniques:
* Ask students to simplify 𝑥 5 ∗ 𝑥 2 . Allow students a couple of minutes to
work on it individually. Ask for volunteers to share their answers. Facilitate a
discussion on which answers are correct, perhaps by creating camps or a
debate.
* Possible Student Approaches:
* Students use expanded form.
* Students use substitution to test scenarios.
* Students use past knowledge that they can/cannot defend.
* Questions:
* Who has a solution they would like to share?
* What do y’all think of this thinking?
* Does anyone have anything to add to this solution?
* Does anyone have a counter-example?
* Does anyone have a different solution/approach?
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* How might 0 as an exponent or base affect this idea?
* Can we generalize this to a property?
* Guided Practice Opportunity on a w/s
2. Repeat process forthePower of a Power Property with (x4)2.
3. Repeat process for the Power of a Product Propertywith (xy)3.
________________________________________________________________________
Independent Practice: p. 464 (2-54 even)
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Evidence of Substantive Conversation
Observations by Meaghan Donohue, Spanish Intern with a math education
background
Making distinctions:
P explained the difference between evaluating & simplifying.
K added that n ≠ 0.
Applying ideas:
M applied the idea of a zero exponent to the warm-up.
Moving the negative to the other side of the fraction bar.
“When are we going to use this?”
Forming generalizations:
“Does 22 * 43 = 85 ?”
T: “x3*x3 = (x3)2” “Wait, x3 * x3…you just said was…oh, x6”
L: “Isn’t (xy)3 = x3y3 because it is just xy*xy*xy?
Raising Inquiries:
“What does that mean?”
J: “Isn’t it true that ….It can’t be a negative fraction?”
L: “When you’re simplifying and you have a fraction is that the simplest form?”
“x2 * x3 = ? Don’t you just add them?”
“What would a3 * b2 be?”
Students explaining themselves:
P: Explained how to remove the fractions from the numerator and denominator
fraction
B: explained that there are 2 halves in 1 for 2/(1/2).
S explained his reasoning using expanded notation.
Students building on another’s ideas:
T built on L’s idea for the warm-up.
J & P agreeing with each other. K disagreeing and coming around.
Comment: You did great! There was a lot of awesome conversation, and students
seemed very engaged. I like that you had students have different answers and you
put them on the board, and they defended their answers.
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LESSON 3:
Tues., April 3, 2012
Algebra 1B
Writing an Equation from the Graph of a Transformed Parent Function
Warm-Up: Have students complete the “Chapter 5 Transformations” worksheet.
Call on students to share their answers.
________________________________________________________________________
Lesson:
Goal: Students will solidify their understanding of transformations.
Students will review finding the domain for these functions.
Students will apply what they know about transformed functions to find the
equations from graphs.
Students will engage in substantive conversation.
1. Sketching the graphs
* Instruction Techniques:
* Give students about 10 minutes to sketch the graphs. Have students use
only their resources (PF sheet & green key). They should not yet work
together.
* Assign pairs of students a problem to discuss and then present to the class in
pairs.
* Possible Student Approaches:
* Students use their parent function graph & transform it.
* Students still rely on a table of values.
* Students use past problems as models.
* Students include/leave off points.
* Questions:
* How did you recognize the transformation/parent function?
* What do y’all think of this thinking?
* Does anyone have anything to add/change to this graph?
* Does anyone have a different graph to share?
* How did you know in what order to do the transformations?
2. Writing the Equation from the Graph
* Instructional Techniques
* Lead a whole class discussion on finding the equation.
* Have students put some answers on their whiteboards.
* Possible Students Approaches:
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* Some students might create the linear equations using what they know
from the last unit.
* Some students might have difficulty dealing with both transformations and
select only one.
* Students might need to reference their PF sheet to recognize how the
functions changed.
* Questions
* How did you know which PF/transformation it was?
* How did you know to go up/down vs right/left
* What do y’all think of this thinking?
* Does anyone have anything to add/change to this equation?
* Does anyone have a different equation to share?
* What did you need to discuss/come to agreement on when you first shared
your responses in pairs?
________________________________________________________________________
Independent Practice: Algebra 1B Transforming Functions
22
Evidence of Substantive Conversation
Observations by Stephanie Goupil, Math Teacher/Cooperating Teacher
Making distinctions:
Class discussed various transformations.
Answering, “How did you know it was horizontal instead of vertical?”
As M examined the equation of a transformation, he answered, “Why reflected?
Why to the right? Why up two?
.
Applying ideas:
L: Able to apply all concepts and provided all functions & transformations.
J: Could apply knowledge of transformations to graphs.
M: Could apply knowledge of transformations to graphs.
L: “It needs to go under the square root to be inside the function.”
Applied knowledge of symmetric shapes to identify missing parts of graph.
Forming generalizations:
K: “We knew it was a quadratic because of the small 2.”
Raising Inquiries:
R: “Isn’t it a fraction that squishes?”
L: “What is a stretch again?”
T: “ How do you know how to find what kind of function you have?”
Students explaining themselves:
When asked who can defend their solution, Alex stated that it goes through the yaxis.
Students building on another’s ideas:
Comment: Stephanie tracked student participation in the whole class discussions.
11/19 students volunteered answers individually. 3/19 asked questions. 8/19
students participated in the conversation after being called upon. 4 students tended
to volunteer significantly more than their peers. 1 student did not participate at all.
23
Summary of lessons:
In order to assess the level of substantive conversation the classes achieved
during these lessons, I applied the evidence gathered and my personal observations
to Hufferd-Ackles’ Levels of the Math-Talk Community: Action Trajectories for
Teacher and Student (2004). Progression through the levels from 0 to 3 is achieved
as the classroom community grows to support students acting in central roles and
shifts from a focus on answers to a focus on mathematical thinking. The two B-level
Algebra classes were found to be very similar and are therefore summarized in one
table.
Algebra 1B
Questioning
level 1
Teacher questions
begin to focus on
student thinking
and focus less on
answers. Teacher
begins to ask
follow-up
questions about
student methods
and answers.
Teacher is still the
only questioner.
As a student
answers a
question, others
wait passively or
wait for their turn.
Explaining
mathematical
thinking
level 1
Teacher probes
student thinking
somewhat. One or
two strategies may
be elicited.
Teacher may fill in
explanation
herself.
Source of
Responsibility for
mathematical ideas learning
level 2
Teacher follows up
on explanations
and builds on them
by asking students
to compare and
contrast them.
Teacher is
comfortable using
student errors as
opportunities for
learning.
level 1
Teacher begins to
set up structures to
facilitate students
listening to and
helping other
students. The
teacher alone gives
feedback.
Students give
information about
their math
thinking, usually as
it is probed by the
teacher (minimal
volunteering of
thoughts). They
provide brief
descriptions of
their thinking
Students exhibit
confidence about
their ideas and
share their own
thinking and
strategies even if
they are different
from others.
Student ideas
sometimes guide
the direction of the
math lesson.
Students become
more engaged by
repeating what
other students say
or by helping
another student at
the teacher’s
request. This
helping mostly
involves students
showing how they
solved a problem.
24
Algebra 1A
Questioning
Explaining
mathematical
thinking
level 2
level 2
Teacher continues Teacher probes
to ask probing
more deeply to
questions and also learn about
asks more open
student thinking
questions. She also and supports
facilitates student- detailed
to-student talk.
descriptions from
students. Teacher
open to and elicits
multiple strategies.
Source of
Responsibility for
mathematical ideas learning
level 2
Teacher follows up
on explanations
and builds on them
by asking students
to compare and
contrast them.
Teacher is
comfortable using
student errors as
opportunities for
learning.
level 2
Teacher
encourages
student
responsibility for
understanding the
mathematical ideas
of others. Teacher
asks other students
questions about
student work and
whether they agree
or disagree and
why.
Students ask
questions of one
another’s work on
the board, often at
the prompting of
the teacher.
Students listen to
one another so
they do not repeat
questions.
Students exhibit
confidence about
their ideas and
share their own
thinking and
strategies even if
they are different
from others.
Student ideas
sometimes guide
the direction of the
math lesson.
Students begin to
listen to
understand one
another. Weh the
teacher requests,
they explain other
students’ ideas in
their own words.
Helping involves
clarifying other
students’ ideas for
themselves and
others. Students
imitate and model
teacher’s probing
in pair work and in
whole-class
discussions.
Students usually
give information as
it is probed by the
teachers wit some
volunteering of
thoughts. They
begin to stake a
position and
articulate more
information in
response to
probes. They
explain steps in
their thinking by
providing fuller
descriptions and
begin to defend
their answers and
methods. Other
students listen
supportively.
25
V. Conclusions
By taking a very purposeful approach to facilitating substantive conversation,
I was able to improve the level of conversation in my classes. Using the adapted
lesson plan, I was able to shift my role from the sole source of knowledge and
questioning and improve student participation in these roles. Both the teachers’
and students’ responses are helpful in addressing the many challenges which still
remain.
Confidence
As predicted by the research of Hufferd-Ackles (2004), progression through
the levels of a math-talk community by my classes has been generally concurrent for
all four components. Additionally, as would have been hypothesized from both the
teacher surveys and the student questionnaires, my A level class possesses stronger
substantive conversation than both my B level classes. The majority of students in
my A level class enjoy participating in classroom conversation, while the majority of
my students in the B level classes are either ambivalent about or dislike
participating in them. Based on both the students’ class placements and my
personal observations of their work, it is fair to assume that students in the A level
class have more confidence in their mathematical knowledge than those in the B
level classes. Students who either dislike or were hesitant to participate in class
discussions often said it was because they did not feel comfortable with the topic or
material. They also said that good academic discussions in class were often the
result of informed students.It therefore follows that these perceived
strengths/weaknesses in mathematics affect students’ comfortable levels in
participating in classroom discussions.
This proves to be a major roadblock as I work to facilitate a math-talk
learning community. It is particularly in these lower level classes where I feel the
greatest need to increase student engagement. In order to scaffold students, I need
to implement strategies such as:
26
1.
Starting with a problem 95% of the classroom can answer (De Frondeville,
2009). This creates an intellectually safe classroom where almost all students
have an opportunity to participate and to build confidence.
2.
Using problems that allow multiple approaches (De Frondeville, 2009). This
allows students to fall back on past concepts and to be open-minded to
different ways of thinking.
3.
Setting conversation ground rules so that all students feel intellectually safe
and have an opportunity to participate (Learning Sciences International,
2007). This can help to address some of the roadblocks students listed in the
questionnaires, such as students saying the answers when they were not called
upon.
4.
Let students practice their explanations, for example by first discussing their
thinking in pairs (LSI, 2007). This gives all students the time to think and to
build confidence in the expression of their thinking.
Metacognition
Another interesting theme that emerged through all the methodologies was
metacognition. Of all the evidences of substantive conversation, teachers felt that
students least often explained their thinking to peers.Additionally, some students
responded that it is difficult to have discussions in math classes, because there is
nothing to discuss. Answers are simply right or wrong. Both of these data points
suggest a lack of thinking about their thinking. Furthermore, during class
discussions, I find that students often have difficulty explaining their reasoning
beyond reiterating the steps they took in solving a problem. Interestingly, learning
about other students’ thinking was one of the key benefits to classroom
conversation according to the student questionnaires. In order to increase the
substantive conversation in my classes, I need to improve my students’
metacognitive skills. Steps to achieving this include:
1. Modeling. I need to reflect aloud my own thinking when working problems
with the class.
2. Asking questions that elicit the verbalization of student thinking.
27
3. Increasing the opportunities for student reflection, for example by
journaling or think-pair-shares.
Independent Learners
As a teacher, one of my primary goals is to assist my students in becoming
independent learners. I found that promoting substantive conversation in my
classroom had the added benefit of promoting students’ academic independence.
In my new role, I was acting as a facilitator and coach, supporting my students as
they become responsible for their own learning and begin the journey of becoming
life-long learners and contributors.
The first step to this important shift was resisting their needs for my
affirmation of their work. Before students go to the board, they often ask me if their
work is correct. While working through a problem at their desk, they will often
pause and ask me if their work so far is correct. The benefits of avoiding providing
them with a response are two-fold. Students’ metacognitive skills are improved as
they learn to question whether or not their solutions are reasonable. When
students ask me if their answer is correct, I often respond, “Well, does it make
sense?” Additionally, students are finding ways to discover the correctness of their
work in other ways. They will more often checktheir own solution, for example by
performing a mathematical check or by conferencing with a classmate.
Small Group Work
While students from all tracks enjoy working in small groups, teachers feel
that the students from the higher tracts are more productive than those in the lower
tracks during small group work. I, too, found that while my B-level students really
enjoyed working in small groups, it was difficult to optimize their effectiveness.
Students overwhelmingly prefer to choose their own groups, stating that it is
uncomfortable talking with someone you do not know or like. Self-selected groups
create the obvious problem, however, of socializing rather than working. Since the
teacher cannot be with all groups at the same time, how does one keep the students
conversations academic? Additionally, students complained that sometimes a group
28
member will not participate and will then take credit for others’ work. The
problems inherent in small group work are a topic I look forward to investigating
further.
Questions for Further Exploration
1. How can teachers make small group discussion most effective?
2. How can teachers help students to see the topics discussed in mathematics
class as relatable or current?
3. What classroom norms/expectations promote a classroom community
where students feel intellectually safe?
4. How can technology be used to facilitate substantive conversation?
29
Appendix
Student Questionnaire
1. How do you feel about participating in whole class discussions?
2. Do you have any classes this year that you believe have really good
discussions?
3. What do you think makes a good academic discussion in class?
4. How would you describe small group work in a mathematics class?
5. How would you describe whole class discussions in mathematics class?
6. What do you feel are roadblocks to productive, meaningful classroom
discussions?
7. What do you feel are impetuses to productive, meaningful classroom
discussions?
30
Teacher Survey Results: Accelerated/Honors/Advanced:
Whole Class:
Students often make distinctions between
mathematical concepts during
conversations .
Students often apply mathematical ideas
as they discuss the answers to problem.
Students often discover new knowledge
by forming generalizations.
Students often ask questions to clarify or
expand their understanding.
Students often explain their thinking to
their peers.
Students often build upon each other’s
ideas.
Small Group:
Students often make distinctions between
mathematical concepts during
conversations .
Students often apply mathematical ideas
as they discuss the answers to problem.
Students often discover new knowledge
by forming generalizations.
Students often ask questions to clarify or
expand their understanding.
Students often explain their thinking to
their peers.
Students often build upon each other’s
ideas.
Strongly
Agree
Agree
Strongly
Disagree Disagree
25%
62.5%
12.5%
0%
25%
62.5%
12.5%
0%
25%
50%
25%
0%
25%
62.5%
12.5%
0%
0%
62.5%
37.5%
0%
37.5%
50%
12.5%
0%
Strongly
Agree
Agree
25%
75%
0%
0%
37.5%
62.5 %
0%
0%
37.5%
50%
12.5%
0%
37.5%
62.5%
0%
0%
37.5%
50%
12.5%
0%
25%
62.5%
12.5%
0%
Strongly
Disagree Disagree
31
Teacher Survey Results: Prep/Remedial
Whole Class:
Students often make distinctions between
mathematical concepts during
conversations .
Students often apply mathematical ideas
as they discuss the answers to problem.
Students often discover new knowledge
by forming generalizations.
Students often ask questions to clarify or
expand their understanding.
Students often explain their thinking to
their peers.
Students often build upon each other’s
ideas.
Small Group:
Students often make distinctions between
mathematical concepts during
conversations .
Students often apply mathematical ideas
as they discuss the answers to problem.
Students often discover new knowledge
by forming generalizations.
Students often ask questions to clarify or
expand their understanding.
Students often explain their thinking to
their peers.
Students often build upon each other’s
ideas.
Strongly
Agree
Agree
Strongly
Disagree Disagree
9.1%
63.6%
27.3%
0%
9.1%
45.5%
36.4%
0%
18.2%
27.3%
45.5%
9.1%
18.2%
54.5%
18.2%
9.1%
0%
45.5%
45.5%
9.1%
0%
63.6%
36.4%
0%
Strongly
Agree
Strongly
Disagree Disagree
Agree
9.1%
54.5%
27.3%
9.1%
27.3%
18.2%
54.5%
0%
18.2%
36.4%
45.5%
0%
27.3%
36.4%
36.4%
0%
9.1%
27.3%
54.5%
9.1%
9.1%
36.4%
45.5%
9.1%
32
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