Developing Mathematical Reasoning Through Collaborative Discovery
Dr. Wendy Hageman Smith, Asst. Professor of Mathematics Education
Longwood University
Abstract: Carefully designed activities enhance classroom discourse as a
means of collaborative discovery. I provide two field-tested examples
developed under a grant funded by the Department of Education, showing
how such activities implement the NCTM Standards in the application of a
constructivist theory of teaching, and explaining how middle school teachers
can design such activities to develop mathematical reasoning in their
students.
Mathematical reasoning is not developed with the application of a single activity. Rather, it is an ongoing
process that develops over time, throughout students’ experiences in the classroom and while working on a
variety of assignments (homework, projects, activities). The NCTM Standards (2000) states that reasoning is
part of doing mathematics and involves examining mathematical patterns to see if any generalizations can be
made based on an emerging pattern. Based in their own explorations students need to make conjectures about
any generalizations they might see and test those conjectures. Meanwhile, students should discuss their
reasoning with others regularly and explain their rationale for making those assertions. This is an integral
part of learning mathematics with understanding (Fonkert, K., 2010).
Because of the longitudinal character of the development of these skills, it is important to create a
classroom environment that is conducive to these kinds of discussions as a regular part of the classroom. In
the application of any mathematics learning activity it is important for teachers to develop their ideas of what
it means to teach mathematics in such a way that they can support the process standards within their
classrooms every day. The importance of a classroom environment to stimulate and develop productive
classroom discourse cannot be stressed enough; it is not enough to give the students activities and let them
“loose on the activity.” Picollo‘s et. al. (2008) research has highlighted that teacher-guided dialogue is an
essential component of the classroom environment.
The first part of this paper contains a reading I developed for teachers. It is a guide in the form of an
operative list for developing the type of classroom Piccollo et.al. describe in their research. The reading is
based in my own research and synthesizes that research for the teachers (Hageman Smith, 2003).
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This introduces the teacher to the habits of mind about teaching mathematics that can help them develop a
classroom environment that promotes the NCTM (and Virginia SOL) process standards:

Problem Solving

Reasoning and proof

Communication

Connections

Representation
Then I will share experiences in an attempt to demonstrate how some every-day classroom activities can be
turned into a process-rich environment for students to learn mathematics.
A Constructivist Theory of Teaching Mathematics
This theory of teaching is based on constructivism, which is a philosophical theory about how it is we come
to know things (epistemology). Constructivism has been used as a framework to form cognitive theory, also
called constructivism (Steffe, L, vonGlasersfeld, E., 1995), that attempts to explain the specific mechanisms
by which we acquire knowledge. The basic tenet of constructivist theories is that all knowledge is
constructed by the individual, that is, it must be ‘formed’ in the mind and cannot simply be ‘acquired’ by
direct transmission. This is of course directly relevant to our profession as educators, for to teach well we
must be able to understand how students learn, so that we can develop and employ methods that work.
The NCTM Professional Standards for Teaching Mathematics (1991) adumbrates a framework for a
theory of teaching when it presents six elements of teaching that need to be addressed by teachers and
administrators. The standards identify broad areas of consideration for teaching; this may be understood as a
framework for teachers and administrators. The main body of the Standards then supplies the particulars
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within the following six framework elements for teaching mathematics, i.e., six key areas in which teachers
take responsibility (Standards, p. 19):
Presenting worthwhile mathematical tasks,
The teacher’s role in discourse,
The student’s role in discourse,
Using tools for enhancing discourse,
The learning/classroom environment, and
The analysis of teaching and learning.
Framed within these six key areas, the Standards offer much of what a theory of teaching should include, but
they do not themselves constitute a theory of teaching because they omit the basic tenets upon which such
standards should be based. Expanding from the six areas included in the Professional Standards, it may be
surmised that a theory of teaching must have an epistemological foundation (the nature and origins of
knowledge), in addition to addressing the respective roles of teacher and student in the learning process
(within the relevant institutional and social context), providing examples or models of appropriate
methodologies, and characterizing appropriate assessment. Hence, a constructivist theory of teaching should
begin with basic constructivist tenets (the epistemological foundation). The following list of those tenets is
based, in part, upon the differing interpretations of constructivism that can be considered common concepts
of inquiry among the different constructivist paradigms. Expanded from a list by Noddings (1990), it may be
considered an appropriate starting point to develop a constructivist theory of teaching:

All knowledge is constructed and is

Constructed from pre-existing knowledge structures and is recursive and dynamic.

Knowledge construction is inherently social in nature.
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
Dialogue plays an important part in knowledge construction.

Knowledge construction is contextually situated.
This expanded list is meant to sidestep certain pitfalls; the dead-end of defining a theory of
constructivism based solely on the tenet "all knowledge is constructed" that arise from the adoption of
‘weak’ constructivism (Ernest, P. 1991), or the idea that the teacher should just "give them the materials and
they will learn," an approach that does not account for the role of guidance by the teacher through interaction
and dialogue. What does this mean for the teacher and his/her classroom? The following paragraphs are
meant to guide, not dictate.
The first tenets: All knowledge is constructed, is constructed from pre-existing knowledge structures
and is recursive and dynamic. This suggests that classroom activities (discussions, worksheets, projects, etc.)
provide the student opportunities to use, revise, and build upon previous knowledge. This suggests a very
active type of learning that is also interactive.
The second tenet: Knowledge construction is inherently social in nature. This tenet suggests an
atmosphere that is socially interactive, socially or contextually situated, and influenced by content area
norms; when doing mathematics the students learn to emulate inquiry and communicate findings in the
manner practiced by mathematicians.
The third tenet: Dialogue plays an important part in knowledge construction. What this tenet defines
is a social learning atmosphere that is dialogue based. Students and teacher(s) converse with each other about
the mathematics concept, problem, or experiment being examined currently in the classroom. Dialogue may
be characterized by the extent to which classroom discourse resembles conversation. Conversation is judged
by the extent to which students as well as the teacher contribute to the discussion, by the seriousness with
which the teacher and other students treat others’ ideas, and how the teacher and students build on
demonstrated knowledge or revise knowledge structures during the conversations. With conversation the
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teacher is not the sole source for knowledge; each individual is provided the opportunity to communicate
ideas.
These first tenets suggest an atmosphere within the classroom. One might refer to it as playing. It
becomes important for the teacher and the students to approach learning in the same manner they approach
any active discovery: by playing. This playing can include playing with objects (as we have all done in any
playground), or playing with the ideas presented in class, the content and the application of knowledge.
The last tenet: Knowledge construction is contextually situated. What this suggests for the classroom
is that mathematical tasks, projects, and assignments required from students should have different formats
and contexts. This can take the form of worksheets, manipulative exercises, discussions, experiments,
computer exercises, papers, or any other context that is suitable for the particular mathematics concept under
consideration. "Situated" includes the environment where the activity is taking place, the project itself, and
how the students work on the problem; a student working alone, working one-on-one with the teacher (or
mentor) or another student, or a group of students.
Just as one-size-does-NOT-fit-all, neither does one type of mathematics exploration work for
building substantive knowledge for all students.
Any good mathematics exploration includes aspects from all the tenets.
The tenets imply the following operative questions for the teacher’s analysis of his/her classroom:
 Is there opportunity in the classroom to form knowledge recursively? Is a student provided the
opportunity to actively apply, question, and revise her/his ideas in the classroom?
 What environmental structures exist in the classroom to make learning opportunities social?
 Is there dialogue in the classroom? Between/among whom is the dialogue? At what level is the
dialogue?
 Are there learning opportunities in different contexts? What are the contexts? Am I exploiting as
many different contexts and situations as possible?
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Application
Recently I taught a “developing number sense” professional development class for teachers in
through a grant created to train them to develop new classroom ideas to help their students pass the state’s
standards of learning. The teachers were introduced to the teaching theory above and we did all activities in
the grant class according to the theory in order to model how activities might appear in a classroom. They
analyzed each activity according to the operative questions to check for the activity’s adherence to the
teaching theory.
When we started activities on adding and subtracting integers we played the following game using
two-color counters:
Digging Holes and Filling Them In
In this game let’s represent digging a hole with a white marker and filling it in with a black marker.
We know that if we dig a hole and then fill it in the ground will look like we never dug the hole. We call
digging the hole and then filling it in as zero, since the hole no longer exists and the ground appears the same.
This means that a white marker and a black marker cancel each other out.
This game has only three rules:
1. Addition means combine the objects.
2. Subtraction means remove the objects.
3. We can always add zero and not change the number.
First let’s dig five holes. We represent this with 5 white markers. Now let’s fill in three of the holes. We can
say that 5 holes plus -3 holes gives us how many holes? 5 + -3 = ? See figure 1.
Now, let’s dig five holes and take away three already filled-in holes from that five holes. Obviously we cannot
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remove something we don’t have; we only have five holes, not already filled in holes. To do this trick, we can
add zero filled-in holes. We can represent this by adding three white and three black to represent this; see
figure 2. Notice the 3 white and three black markers cancel each other, so we have not really added anything.
We now have the three black markers we can remove from the new group. This shows us that 5 - -3 = 8. See
figure 2.
Now let’s dig three holes and then remove five of them, in other words, fill in five of them. We can’t do that
so we have to dig two more holes. Since that is cheating, we have to add two holes and two filled-in holes
(adding zero). Now we can fill in five holes. See figure 3.
Obviously, we are modeling addition and subtraction of integers. The activity allows students to play a game,
which has three simple rules, and record results. We found by testing that a teacher can give students any
number of combinations of integers to add and subtract (until they get the pattern successfully) but should
introduce worksheets that contain groupings of addition and subtraction separately.
Groupings:
Addition:
Subtraction:
Worksheet of two integers where one is negative and one positive ex:
Worksheet of two integers where both are negative ex :
Worksheet of two positive integers ex:
Worksheet of one positive and one negative integer ex:
with
Worksheet of both negative ex:
There are countless workbooks available to use for the exercises to go along with the game. However,
playing the game is not the real investigation we are after. What the students have to do is play the game,
with several examples and record their results. The idea is to have them come up with their own rules for
adding and subtracting integers. They can do this by comparing their results to others in the class and
conjecture what will be the result. For instance, when we subtract a negative number from a positive number,
we get a larger positive number; or that when we add a negative number to a larger positive number, the
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result will be the difference of their magnitudes. We, as teachers, know these facts and can tell our students
that is what happens or we can allow them to discover it for themselves.
The activity observes the tenets of the constructivist teaching theory. There is nothing that stipulates
that you cannot play the first several rounds of the game to get your students going in the right direction.
Students will learn when lectured, but the neural paths will be formed when they play the game, record the
results, discuss the problems with others, perhaps revise their thinking, search for patterns, formulate their
conjectures, discuss their thinking, and formalize their results via a conjecture that works for everyone for
every occurrence.
I like to start with this example because it was not obvious to the teachers in the class that something
like adding and subtracting integers was a concept that readily leant itself to students conjecturing about
results. I think that the teachers are not alone.
This next example is within a content area that we would expect to see reasoning and conjecturing as
an integral part of doing the mathematics; geometry. I believe, however, that it is not obvious that in order to
write and understand geometric proofs, conjecturing is necessary – students need to see how to connect the
necessary steps to building a proof and where they come from (rather than just memorizing how to prove a
theorem). The reasoning and conjecturing does not require a proof, but it is a first step and can be done in the
middle school, long before writing a proof becomes part of the curriculum for them.
The exercise is given as an in-class investigation after students have learned to use the computer
program Geometer’s Sketchpad. They work the exercise in groups and provide their results to the rest of the
class. We then create a conjecture together after discussing what the measurements lead us to believe.
In class exercise: Vertical angles are those angles that are opposite each other when two lines meet. See
illustration below:
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Angle 1 and angle 3 are vertical angles. Angle 2 and angle 4 are
vertical angles. So, there is two pair of vertical angles for each pair of intersecting lines. For this
exercise:
a) Draw four different pairs (label the line pairs l and m, l’ and m’, etc.) of intersecting lines. Label each pair
of intersecting lines (as shown above) and the angles for each pair of intersecting lines. Hand in your GSP
work with this exercise. In the following table, record the data:
Lines:
ex:
l, m
Lines:
Angle __1___
Measure:
Angle __2___
Measure:
Angle __3___
Measure:
Angle __4___
Measure:
Angle __5___
Measure:
Angle ___6__
Measure:
Angle ___etc…__
Measure:
Angle ___ etc…__
Measure:
Lines:
Angle __9___
Measure:
Angle __ etc…__
Measure:
Angle ___etc …__
Measure:
Angle ___ etc…__
Measure:
Lines:
Angle __13___
Measure:
Angle __ etc…__
Measure:
Angle ___etc …__
Measure:
Angle __ etc…__
Measure:
b) After everyone in class writes their results, we will do this portion of the exercise as a whole class
discussion. Make a conjecture (and record it here), based on your data, about the vertical angles of two
intersecting lines (word the conjecture appropriately):
Conclusion
Although a student’s understanding may sometimes appear sudden, we know that in fact a student’s
mathematical reasoning builds over time, contextually, collaboratively, constructed on a complex of
experiences and insights. The activities presented here were used by teachers in their middle school
classrooms, and they reported that the students enjoyed the activities and that they were successful in
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building students’ reasoning about the relevant concepts. I encourage teachers to develop activities like
these, carefully implementing each aspect of the tenets of the teaching theory outlined above, for every
learning objective they strive to meet in their own classrooms.
References
Ernst, P. (1996). Varieties of constructivism: A framework for comparison. In Steffee, L., Nesher, P., Cobb,
P., Goldin, G., and Greer, B. (Eds.) Theories of Mathematical Learning. Hillsdale, New Jersey:
Lawrence Erlbaum Associates.
Fonkert, K. (2010) Student Interactions in Technology-rich Classrooms. Mathematics Teacher, v104 n4 pp.
302-307.
Hageman Smith, W. (2003). A Constructivist Theory of Teaching Mathematics: application in a postsecondary mathematics classroom. (copywrited PhD dissertation) University of Colorado, Boulder.
National Council of Teachers of Mathematics (2000) Principles and Standards for School Mathematics,
NCTM , Reston, VA.
Noddings, Nel. (1990). Constructivism in Mathematics Education. In Journal for Research in Mathematics
Education, Monograph No. 4.
Piccolo, Diana L.; Harbaugh, Adam P.; Carter, Tamara A.; Capraro, Mary Margaret; Capraro, Robert M.
(2008). Quality of Instruction: Examining Discourse in Middle School Mathematics. Journal of
Advanced Academics, v19 n3 pp.376-410.
Steffe, L.P., Nesher, P., Cobb, P., Goldin, G.A., & Greer, B. (1996). Theories of Mathematical Learning.
Hillsdale, New Jersey: Lawerence Elrbaum Associates.
Von Glasersfeld, E. (1991). Introduction. In E. von Glasersfeld (Ed.), Radical constructivism in mathematics
education (xiii-xx). Dordrecht, The Netherlands: Kluwer.
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Figure 1
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Figure 2
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Figure 3