mathematical knowledge and teaching practice relationships betwe

Mathematical Knowledge and Teacher Practice 1

RUNNING HEAD: MATHEMATICAL KNOWLEDGE AND TEACHING PRACTICE

Relationships between Mathematical Knowledge for Teaching and Teaching Practice: The Case

of Proof

Michael D. Steele and Kimberly Cervello Rogers

Michigan State University

To appear in Journal of Mathematics Teacher Education

DRAFT: Do not cite or reproduce without permission from the authors.


Abstract

Teachers of mathematics orchestrate opportunities for interactions between learners and subject

matter. Therefore, mathematics teachers need rich, multidimensional content knowledge for

teaching mathematics, which incorporates knowledge of the subject matter, students, and

teaching. Studying this mathematical knowledge for teaching (MKT) necessitates more than a

unidirectional assessment. In this study, the mathematical knowledge for teaching reasoning and

proving of two secondary mathematics teachers was investigated through classroom observations

and clinical assessments. Results indicate that using MKT as a frame for examining classroom

practice, in addition to assessing the MKT a teacher possesses in a clinical setting, provides an

in-depth and innovative method for investigating MKT. The comparison of the two cases also

identifies student positioning as a key mediating factor between MKT and opportunities to learn.

Implications for using MKT as a lens for examining practice in teacher education are discussed.


It is widely acknowledged that the work of mathematics teaching draws upon a deep and

broad knowledge base, including knowledge of mathematics, pedagogy, and student learning

(Fennema & Franke, 1992; National Board for Professional Teaching Standards, 1997; National

Council of Teachers of Mathematics [NCTM], 1991; Shulman, 1986). Several studies suggest

that the nature, depth, and organization of teacher knowledge influences teachers’ presentation of

ideas, flexibility in responding to students' questions, and capacity for helping students connect

mathematical ideas (e.g., Ball, 1988; Stein, Baxter, & Leinhardt, 1990). Mathematics teachers

are a special class of users of mathematics; the knowledge they need to teach mathematics goes

beyond what is needed by other well-educated adults, including mathematicians (Ball, Bass, &

Hill, 2004; Ball, Lubienski, & Mewborn, 2001; Ball, Thames, & Phelps, 2008). As described by

Hill, Rowan, and Ball (2005):

Mathematical knowledge for teaching [MKT] goes beyond that captured in measures of

mathematics courses taken or basic mathematical skills. For example, teachers of

mathematics not only need to calculate correctly but also need to know how to use

pictures or diagrams to represent mathematics concepts and procedures to students,

provide students with explanations for common rules and mathematical procedures, and

analyze students’ solutions and explanations. (p. 372)

Further, Hill and colleagues link higher levels of MKT to greater student learning gains,

underscoring the importance of teachers’ MKT and its impact on what students learn in the

classroom (Hill et al., 2005).


MKT can be investigated from a variety of perspectives. One approach involves creating

conceptual frameworks for MKT by mathematical topic, drawing on research on the teaching

and learning of that topic (Stylianides & Ball, 2004). Assessments, interviews, and interventions

can then be designed to measure teachers’ topic-specific MKT. Results can be related to previous

studies of student and teacher knowledge; however, the question of the impact and relevance of

teachers’ MKT on classroom practice remains open. A second approach investigates classroom

practice directly in order to describe teachers’ MKT: “Practiced-based research provides the

impetus for rethinking what we recommend for teachers to know… or what aspects of

teachers’… knowledge we consider important to examine” (Stylianides & Ball, 2004, p 36). In

this approach, practice is the starting point, with aspects of MKT visible in-situ. The practice-

based nature of the second approach addresses the question of utility of findings; however, the

results of such investigations are less likely to inform the literature base related to teacher

knowledge of specific mathematics content.

Most previous studies investigating MKT have chosen one path or the other. This

investigation blended both perspectives to examine teachers’ MKT related to proof, a content

area that has historically posed challenges both for students and teachers, yet is fundamental to

mathematics. Using a framework grounded in research on teaching and learning of proof, MKT

was assessed through written and interview instruments. In addition, teachers were observed

teaching a lesson related to proof, which was then analyzed using the framework. The goal of

this study was to examine relationships between MKT and practice using clinical instruments

and classroom observations.

Investigating Mathematical Knowledge for Teaching Proof


Recently, there has been a push in the mathematics education community to make proof1

more central to school mathematics (NCTM, 2009), because of its importance to both the

discipline of mathematics and learning mathematics with understanding. The increased emphasis

is reflected in research studies and curriculum frameworks calling for proof to pervade students’

work in K-16 mathematics (e.g., NCTM, 2000, 2009; Yackel & Hanna, 2003). Despite

widespread agreement that proof should be a central feature of all students’ mathematical

experiences, a growing body of research shows that K-16 students and their teachers have

difficulties with proof (e.g., Healy & Hoyles, 2000; Knuth, Chopping, Slaughter, & Sutherland.

2002; Stylianides, Stylianides, & Philippou, 2002, 2004, 2005). Teachers often favor empirical

arguments over deductive proofs, finding them more convincing or easier to follow, and are

more likely to choose an empirical argument for classroom use (Knuth, 2002a; Steele, 2006;

Stylianides & Stylianides, 2009). Teachers are more likely to identify arguments with familiar

forms (e.g., two-column) as proof even if they do not follow the reasoning (Knuth, 2002a). They

also consider proof a specific topic of study reserved for high-achieving students rather than a

practice fundamental to all students’ learning of mathematics (Knuth, 2002b; Steele, 2006).

These conceptions of proof are at odds with both the role of proof in mathematics as a way of

thinking and reasoning, and calls to position reasoning and proof as “a natural, ongoing part of

classroom discussions, no matter what topic is being studied” (NCTM, 2000, p. 342).

The results of studies on students’ and teachers’ knowledge of proof were used to create a

framework for teachers’ MKT related to proof (MKT-P), drawn from the work of Knuth (2002a,

2002b) and Steele (2006) and shown in Table 1. We do not claim that this framework represents

the complete body of mathematical knowledge for teaching proof; rather, we suggest that it

1 Our conception of proof in this article encompassed the acts of mathematical reasoning leading to informal and formal proof arguments, consistent with Stylianides & Stylianides’ (2006) conception of “reasoning-and-proving.”


provides a fruitful starting point grounded in previous research for investigating teachers’

mathematical knowledge2.

[[INSERT TABLE 1 ABOUT HERE]]

Four main components of teacher knowledge related to proof are represented in the

MKT-P framework. The first component is the ability to define proof. While there is not a

canonical agreed-upon definition for proof even amongst mathematicians, recent research has

centered on four key characteristics: a proof is a mathematical argument that is general for a

class of mathematical ideas, and establishes the truth of a mathematical statement based on

mathematical facts that are accepted or have been previously proven (Knuth, 2002a; Steele,

2006; Stylianides, 2007). Second, teachers need to be able to identify whether a mathematical

argument is or is not a proof, which includes having a set of criteria to identify proofs,

comparing one proof or proof-like argument to another, and recognizing proof across a variety of

representational forms based on mathematical criteria as compared to surface-level features.

Research identifies four mathematical criteria that can be used to link the definitional features of

proof to the features of a specific mathematical argument: level of mathematical detail, the

mathematical method used to generate the proof, the extent to which the proof explains why the

mathematical statement is true, and the extent to which the proof is general as compared to a set

of empirical examples (Knuth, 2002a; Steele, 2006). Level of detail refers to the extent to which

the explanation provides fine-grained steps or explanations of how particular mathematical

properties or ideas are applied in the course of proving. For example, in evaluating a

mathematical argument, a teacher might consider whether showing discrete steps such as the use

2 In particular, the framework addresses what Ball, Thames, & Phelps (2008) refer to as common and specialized content knowledge. Issues more commonly identified with pedagogical content knowledge, such as common student misconceptions, were beyond the scope of this general framework, as specific student thinking and misconceptions are highly dependent on, and interactive with, the mathematical content represented in the proof.


of commutative property in reorganizing an equation are necessary or not for the argument to be

considered a proof in a particular context. Valid method refers both generally to the mathematical

method used (e.g., direct proof, proof by contradiction) and specifically to the validity of the

facts, principles, and properties used in the proof. The explanatory power of the proof is a

measure of the extent to which the proof as a whole explains to a reader why the statement being

proven is indeed true or not true (c.f. Hanna, 1991). The generality of the proof represents how

well the argument moves beyond specific empirical examples, either to a completely general

case or a generic example (Balacheff, 1988).

In addition to being able to define and identify proofs, teachers need to be able to

construct mathematical proofs that are mathematically correct, that provide an appropriate level

of descriptive detail for steps taken, that make use of mathematical representations, and that are

sufficiently general. The final element of the MKT-P framework is more specialized to the work

of teaching, as compared to the previous three. For teachers to position proof as more than just a

specific type of exercise, teachers need to understand the role of proof – both in the mathematical

domain and in the K-12 classroom (Hanna, 1995; Knuth 2002a, 2002b; Steele 2006). Five roles

have been identified in previous research and are described in Table 2.


Mathematical Knowledge for Teaching in Action

As teachers enact lessons on proof, different aspects of the MKT-P framework are

operationalized in the classroom. Different classroom actors (students, teacher, and others

outside of the classroom) take on a variety of roles with respect to proof, ranging from observing

another actor to the creation and validation of a proof. Student’s roles are particularly important,

as they influence their goals and what they do or do not learn through their engagement (Nasir,


2002). Students who are conditioned to particular classroom roles sometimes act in response to

the conditioning, such as using a more complex procedure with later problems due to a

perception that they are harder, rather than relying on their mathematical knowledge (Boaler,

1999). The role of “outside others” in the classroom can be highly influential, and includes talk

about other similar doers of mathematics such as other students, mathematical authorities such as

mathematicians, and even the voice of the mathematics textbook (Herbel-Eisenmann & Wagner,

2007). As the leader of the classroom community, the teacher influences the roles of all actors

and the location of the mathematical authority. In examining how aspects of the MKT-P

framework are operationalized, it is important to consider the roles in which classroom actors are

positioned with respect to proof.

Considering the role of the teacher with respect to proof, Stylianides (2007) urges

teachers to actively engage in and provide opportunities for students to work with mathematical

arguments leading to proof. He argues that students need models of and opportunities to work

through arguments and proofs while participating in a learning community that distinguishes

between empirical arguments and mathematical proofs. Strong mathematical knowledge for

teaching proof is crucial for teachers to be able to structure such opportunities for their students

(Stylianides, 2007). However, there has been little research focused on examining how this

knowledge is operationalized in the classroom. Therefore, this study investigated the following

three research questions:

1. What aspects of a teacher’s mathematical knowledge for teaching proof are evident in

clinical assessment settings?

2. What aspects of a teacher’s mathematical knowledge for teaching proof are evident in

classroom settings?


3. How do the aspects of a teacher’s mathematical knowledge for teaching proof in each

setting (clinical and classroom) relate to one another?

Method

The results described in this article are part of a larger study investigating 25 teachers’

knowledge of proof, five of whom were followed into their classrooms. In this phase, high

school teachers were recruited to teach a lesson of their choosing related to proof and to engage

in a series of clinical assessments of their mathematical knowledge for teaching proof. Consistent

with a naturalistic approach to studying practice, teachers were asked to identify a lesson related

to proof that they would normally teach rather than inserting a special lesson. The study was

designed to capture a set of qualitative case studies – snapshots of teachers’ mathematical

knowledge for teaching proof both in the clinical and classroom settings. For this report, we

selected two contrasting cases of a novice and expert teacher in different settings, but with

relatively strong content knowledge and similar goals– to introduce their students to proof-

writing for the first time.

Participants

Steve3, a secondary mathematics teacher at the end of the student teaching phase of his

teacher preparation program, taught heterogeneously grouped geometry classes in a small rural

district. Lucy, a certified secondary mathematics teacher in her ninth teaching year taught in a

mid-sized suburban district and was observed teaching an average ability-track high school

Algebra I class. Steve and Lucy were selected as the focus cases for this report because of the

contrasts in experience and district, and because their observed lessons represented early

experiences with proof for their classes.

Data Collection 3 All names are pseudonyms.


Each teacher’s lesson was videotaped; a short interview preceded and followed each

lesson in which teachers about their goals for the lesson and their reflections on it respectively.

After the observation, teachers completed a written assessment and semi-structured interview

assessing aspects of the MKT-P framework. The written assessment drew on items from

previous research (Harel & Sowder, 1998; Steele, 2006; Winicki-Landman, 1998) and evaluated

teachers’ abilities to create proofs by completing two proofs justifying common geometric

formulas (area of a parallelogram and area of a triangle), and their understanding of key

mathematical ideas related to proof. The teachers were given unlimited time to produce the two

proofs without preparation or access to outside resources. These particular proofs were selected

for several reasons: the geometric principles used were common enough so as to be accessible to

the average teacher; there are several possible approaches to making the mathematical argument;

and both proofs invite the use of multiple representations. The interview asked teachers to define

proof, describe its role in mathematics and in the K-12 classroom, and classify a set of eight

explanations as proofs or non-proofs and their criteria for those decisions. The eight explanations

were drawn from previous research on proof and varied with respect to the form and substance

(proof/non-proof argument) of the argument. The instruments, along with results from the larger

study, can be found in Steele (2006)4.

Data Analysis.

All audio and video artifacts were transcribed. Interviews and written assessments were

used to answer the first research question. The follow-up interviews were coded with respect to

the MKT-P framework. Teachers’ proofs were coded to assess mathematical accuracy,

generality, and explanatory power of the proof. Teachers’ work and talk in identifying proofs and

4 All teachers in the project taught at least one section of geometry. To guard against effects from content knowledge unrelated to proof, a geometry context was selected for assessment tasks. Because the aspects of proof described in the MKT-P framework are content-agnostic, these items were appropriate to measure MKT related to proof.


non-proofs was assessed for mathematical correctness and coded with respect to the features of

the proofs to which teachers attended in making their choices. For each teacher, lines of

transcript coded as evidence of each aspect of the MKT-P framework were aggregated to portray

the degree to which each teacher attended to different aspects of the framework. This method

allowed us to assess the relative extent to which teachers talked about different aspects of the

framework within each teacher’s data set.

Transcripts of videotaped lessons were used to answer the second research question.

Transcripts were parsed into intervals using a simplified version of thematic analysis (Lemke,

1983). A first pass of each transcript marked intervals when the task being worked on by the

class changed; for example, when the class moved from naming steps to solve an equation to

considering the Pythagorean Theorem. A second pass added interval boundaries each time a

different aspect of proof became the topic of discussion, as defined by the MKT-P framework

(see Figure 1 for an example)5. Each interval was then labeled with the talk format during that

interval (e.g., lecture, small-group work, whole-class discussion), the primary actors (who did the

majority of the work/talk), and the aspect(s) of the MKT-P framework being discussed.

[[INSERT FIGURE 1 ABOUT HERE]]

Consistent with a situative perspective on learning, we recognized that the roles taken by

students, teachers, and others in the lesson episodes might impact how each teacher’s

mathematical knowledge for teaching influenced students’ opportunities to learn. As such, we

described the roles taken by students, teacher, and outside others (e.g., mathematical authorities,

textbook) for each interval. Descriptions of the roles were developed using a constant-

comparative method (Glaser, 1965). One-third of the data was coded by both authors to establish

5 In many intervals, more than one aspect of proof was discussed simultaneously. In these cases, intervals were coded with multiple aspects of the MKT-P framework, and a new interval was started when a new aspect was added or an aspect dropped from the discourse.


reliability of interval parsing and MKT-P coding. Inter-rater reliability of at least 85% was

achieved, with all disagreements resolved through discussion.

To answer the third research question regarding the relationships between MKT-P on

assessments and in the taught lessons, two case studies were constructed. The aspects of MKT-P

evident in the clinical assessments and in the lesson are described, followed by an analysis of the

alignment between the aspects visible in each setting.

Results

We present the results in the form of case studies describing each teacher’s observed

lesson, written and interview assessments, and how each setting makes visible aspects of the

teacher’s mathematical knowledge for teaching proof. In each case study, we begin by describing

the aspects of the MKT-P framework salient in the clinical context, continue by examining the

classroom context, and conclude with a cross-context analysis.

The Case of Steve

Steve was a student teacher in the last weeks of a 10-week student teaching practicum

during his last semester in a teacher preparation program at a small public liberal arts college.

Steve’s rural district had recently embarked upon a mathematics reform project, which included

a new text, heterogeneously grouped classes, and 75-minute block scheduling. Steve’s mentor

was a 30-year veteran teacher with a reputation for strong student achievement. Steve and his

mentor routinely provided students with rich problems with a variety of representations, making

use of a text that supported student-centered learning.

Steve’s MKT-P in the assessments. Table 3 summarizes the aspects of MKT related to

proof that were most salient in Steve’s interview. With regard to the definition of proof, Steve

stressed the importance of a proof as a mathematical argument, accentuating the logical, step-by-


step nature of proofs. Steve explained, “proofs serve the purpose of training the brain to think in

a sequential order, going from one step to the next step.” When asked to classify arguments as

proofs and non-proofs, Steve identified seven of eight explanations appropriately. In

distinguishing between proofs and non-proofs, Steve focused most heavily on whether the proof

used a valid method; he also attended to specific concrete features of the proof. Steve’s focus on

the method echoes his emphasis in defining proof on importance of the step-by-step nature.


Steve’s work on the two proof-writing tasks was mathematically rigorous. Although

Steve made some initial assumptions that limited the generality of his argument, the arguments

contained multiple representations and a logical flow with detailed explanations. Steve was able

to provide coherent mathematical arguments with justifications supporting his statements.

When talking about the role of proof in mathematics, Steve spoke sparingly about a wide

variety of roles. He discussed proof as verifying the truth of a known statement, explaining why a

mathematical statement is true, communicating mathematical knowledge, and creating new

mathematical knowledge. In considering the role of proof in the mathematics classroom, Steve

focused on proof as a tool explaining why a statement is true and systematizing the domain.

Steve’s attention to systematizing the domain resonated with his focus on the step-by-step nature

of proof. As he articulated, “the good thing about proofs is the way they teach you to think. It

teaches [students] to go from one step to the next step to the next step and that’s important,

having that line of thought.” This role was reiterated in his written responses when asked to

describe the reasoning and proving experiences middle school students need:


Middle school should focus on developing the ’step-by-step‘ mindset needed for proofs.

They should work with problems that make them show each step of their solving method

or have them follow multiple directions in a problem to arrive at a solution.

Steve’s MKT-P in the classroom. Steve’s observed lesson was not his students’ first

exposure to a proof, but it was the first time that they engaged in proof-writing as compared to

reading through a proof on their own. Steve chose to go about this by spending the class period

co-constructing a proof found in full in the text. The lesson’s main activity was completing a

proof of the Law of Cosines, a property students were familiar with and had been using to solve

problems. The class time devoted to this proof was divided into 19 intervals, shown in Figure 2.


After some initial administrative details (Interval 1), the students were instructed to write down a

set of statements for a two-column proof of the Law of Cosines, provided in their textbook. The

task for the class was to fill in the reasons related to the statements, emphasizing an aspect of

proof that was pervasive throughout the lesson: proof as a mathematical argument. Steve

explained that although the students had already been using the Law of Cosines and the

Pythagorean Theorem, they would now use the Pythagorean Theorem and other mathematics

facts to provide justifications for “where the Law of Cosines comes from,” focusing students on

a conception of proof as verifying the truth of known ideas based on mathematical facts.

During intervals 4 through 17, the proof of the Law of Cosines was completed. Steve

initiated the analysis of each step of the proof, and students were encouraged to generate

justifications for each step. The aspects of proof salient in these intervals include proof as based

on mathematical facts and the sequential structure of the proof as a mathematical argument.

Between steps, Steve provided commentary regarding the proving process and mathematics in


general (intervals 5, 7, 9, 12, 14, and 17). Steve provided evaluative, non-mathematical

comments on the difficulty of the task and that proofs are something that students might struggle

to create on their own. These intervals were categorized as other, as there was no explicit role or

aspect of the definition of proof that was being discussed at that time.6

The final two intervals (18 and 19) provided closure. With the proof completed, Steve

reiterated the fact that this proof could help the students explain why the Law of Cosines was true

(interval 18). Then, a student questioned the structure of this proof – why the proof did not state

“a given.” Steve focused on the role of proof as a mathematical argument consisting of a series

of steps, which usually included stating the given information, and he recommended that the

students continue using that particular structure for their proofs.

Steve, his students, and outside others all took a variety of roles in the course of the

lesson. The textbook provided the structure for the proof, and Steve personified it and frequently

positioned it the source of mathematical authority. Students were initially observers of the proof,

but as the lesson progressed, they took on roles as creators of the reasons for each step. A

summary of the roles taken by the actors in Steve’s lesson can be found in Table 4.


Steve’s discussion of the proof’s first step exemplifies the roles of the teacher, student, and

textbook:

This first step, let’s take a look at it. It says c squared equals the quantity a minus x

squared plus h squared. Alright? And where’d they get that from? Here’s my c, here’s a

6 While these comments did relate to the proof and the comment at hand, they seemed intended to motivate students by suggesting that they were doing something challenging. Such comments lead us to consider the ways in which a teacher might situate proof as a part of the broader mathematical activity of the classroom to students – an interesting idea that is beyond the scope of this report.


minus x, here’s h. What do you think we could write for a reason for that? What are they

doing, basically?

The textbook (the they) provided the statements for the proof and served a source of

mathematical authority in this case. The students observed the statement provided by the text,

and Steve asked them to provide justifications, positioning them alternatively as observers and

creators. Steve moved between the roles of communicator, observer, validator, explainer, and

mathematical authority.

Typically, once the students generated ideas for justifying a step, the teacher would

assume a role as the validator: for example, Steve would make statements such as, “What is this

guy? [referring to a step in the proof] I heard it. The Pythagorean Theorem right? ... So here’s my

reason, Pythagorean Theorem.” Although Steve had previously asked the class for a reason “we

[emphasis added] could write for that” step in the argument, once the Pythagorean Theorem had

be offered as a suggestion, Steve then determined and wrote what he called “my [emphasis

added] reason.” This pattern continued throughout the lesson, with students providing

justifications and Steve validating them. The notable exceptions occurred during Steve’s

commentaries on the proof, when Steve stepped out of his role as validator and observed the

mathematics that had been done with his students, noted in part by his use of we in the following

excerpt from interval 12:

Steve: Look at these reasons we have so far. Pythagorean Theorem, substitution,

definition of cosine, multiplication. Is this that tough?

Students: No.

Steve: No. This is a pretty easy proof. It’s kind of crazy how we get something so

complicated from something so easy, isn’t it?


Alignment across settings. Proof defined as a mathematical argument was prominent in

Steve’s written assessments and was a salient aspect of the observed lesson. Proof defined as

being based on mathematical facts was not as evident during Steve’s interview, but was also a

salient aspect of proof throughout the lesson. Notably, during the post-lesson interview, Steve

explained that he was pleased with how the lesson went because he thought that the students

“really got… the notion of where [the Law of Cosines is] coming from and how you can derive it

from even just the Pythagorean Theorem.” Steve’s comments indicate that he intended the lesson

to help students verify the truth of the known mathematical statement. Interestingly, the role of

proof as verifying truth surfaced once toward the beginning and again toward the end of the

lesson, when Steve inquired if after completing the proof the students could “kinda see where the

Law of Cosines comes from.” Although this role of proof was not emphasized throughout the

lesson, Steve saw it as the intended goal.

The observation allowed significant insights into Steve’s concept of proof as compared to

simply the written assessment. Observing Steve’s practice added dimension and nuance to the

aspects of proof that he identified in the clinical setting. For example, the idea of proof as being

based on mathematical facts was a significant feature of the lesson but not a prominent topic in

the interviews. In addition, the observation showed a number of aspects of proof that Steve

valued in the written assessment but did not capitalize on or make visible in this particular

lesson. Two areas stand out – the role of proof as systematizing the domain, and the variety of

methods and representations that proof can take. Steve valued the notion of systematizing the

domain in his written interview – describing the ways in which axioms, definitions and theorems

can be built to prove new mathematical ideas, which then can in turn be used to further the

knowledge base. Steve’s proof of the Law of Cosines contained many of these definitions and


theorems with which students were familiar; however, the fact that students had already been

treating the Law of Cosines as established truth may have limited the work that this proof could

do with respect to understanding how axioms and properties build new theorems. With respect to

representations and methods for proof, Steve’s written responses contained rich mathematical

arguments and indicated that students should use multiple methods in solving problems. During

the lesson, a canonical and pre-determined sequence for proving the Law of Cosines was

presented; neither alternative explanations nor multiple representations were used in this lesson

around the two-column proof. The fact that Steve moved on with new instruction following this

lesson suggests that in this case, establishing multiple methods for proving the Law of Cosines

was not a priority.

The Case of Lucy

Lucy was a nine-year veteran teacher in a mid-sized suburban high school where she had

taught for a number of years. Lucy was a leader in mathematics at her school, teaching a wide

variety of courses from algebra to precalculus, and was pursuing a Masters degree at a local

university. Lucy’s school used a traditional course sequence and textbook with ability-group

tracking; class periods were 90-minute blocks. She was committed to improving her instructional

practice, and in particular investigating ways to support all of her students in high-level thinking

and reasoning. The lesson observed was near the end of the year in an average-ability Algebra I

course. Similar to Steve’s lesson, Lucy intended this lesson to be students’ first formal work in

constructing a mathematical argument that would constitute a proof. Lucy conceptualized the

lesson as a way to connect students’ informal experiences with proof to the more rigorous

experiences they would encounter in Geometry next year.


Lucy’s MKT-P in the assessments. Lucy’s written work on the proof assessments and her

interview captured a wide range of the aspects of proof in the MKT-P framework, as shown in

Table 5.


In describing what high school students should know and do related to proof, Lucy emphasized

that they should “experience many forms of proof… they may not master it, but exposure to

high-level thinking is good.” This both resonates with and differs from the findings of Knuth

(2002b), in which teachers saw proof as high-level thinking but did not see it as appropriate for

all students. Lucy’s definition of proof focused on proof as being based on mathematical facts

and general. Lucy’s work in identifying proofs and non-proofs focused heavily on one particular

feature: whether or not the proof used a valid method. She correctly classified six explanations,

classifying a transformation-based proof as a non-proof, and being undecided about a

mathematical argument which rested heavily on the tenet that the shortest distance between two

points is a straight line. Her issue with this argument again centered on the idea of whether the

method was valid: “I didn’t know if this was a formal proof, I guess [I would want] more

indication that they were drawing on prior knowledge of distances, or actually pinpoint a

theorem.” While Lucy did focus at times on concrete features of the explanations and how

familiar the explanations were to her personally, she focused much less on these surface-level

features as compared to Steve.

In discussing the role of proof in mathematics, Lucy spoke nearly equally about all the

roles but did not mention systematizing the domain. When asked about proof’s role in the

classroom, Lucy focused on two roles in particular: proof as explaining why and as

communicating mathematics knowledge. Lucy’s discussion of these two roles suggested that she


valued proof as a means for students to discuss mathematics and develop shared understandings

of mathematical ideas.

The two proofs that Lucy produced were both in the two-column format, using symbolic

statements and short reasons referring to formal names of mathematical properties. Both proofs

had minor flaws preventing them from being completely general (e.g., starting the proof of the

area of a triangle by bifurcating a rectangle rather than a parallelogram). The proofs used highly

technical language, and would not hold a high explanatory power for a naïve audience. However,

the statements and reasons were typical of a rigorous two-column proof, with one step flowing

from the next in a logical manner. Taken together, these responses show Lucy attended to proof

as both a way of making an argument about the truth of a mathematical statement and as a tool

for facilitating understanding and participating in conversations in the mathematical community.

Lucy’s MKT-P in the lesson. Lucy’s lesson was designed to be an introduction to proof

for her Algebra class. Lucy saw this lesson as establishing proof as a type of mathematical

argument, which would in turn support their work in Geometry during the next academic year.

The 90-minute period was spent on individual, small-group, and whole-class discussions related

to proof, parsed into 43 intervals; the aspects of proof evident in each interval are shown in

Figure 3.


Lucy’s class began with a warm-up activity that provided students with two solved linear

equations with the steps out of order (Intervals 1-18). The task was to reorder the steps and give

reasons for them. The focus was on forming a mathematical argument based on mathematical

facts; a shift occurred, however, beginning in Interval 10 when Lucy brought the class together

to review the steps. In Intervals 10-18, Lucy flagged many important ideas related to proof: that


proof is a means of communicating mathematics knowledge and explaining your thinking to

others. Students expressed some surprise about these aspects of proof, as evidenced by this

exchange from interval 17:

Student: This is proof?

Lucy: In essence, yeah. It’s an introductory to. . . how we would start proof writing.

Lucy used the notion of proof as a way of thinking and communicating as a transition to

considering three examples that highlighted different aspects of proof (Intervals 20-24). During

this transition, students questioned why they needed to show statements and reasons; Lucy noted

that the statements and reasons allowed them to communicate with other mathematical thinkers

beyond their classroom using a standard language (Interval 22). The first task focused on using

the associative property to determine the truth of a general statement, [b + (–c)] + c = b

(Intervals 25-29). In this segment, students were asked to provide steps and reasons using a list

of axioms and properties found in their text, again highlighting proof as a mathematical

argument that is based on mathematical facts. At the end (Interval 29), Lucy recapped the work

by returning to the role proof plays in communicating mathematics knowledge and explaining

why.

Lucy’s second task presented students with six Pythagorean triples and asked if these six

examples proved the Pythagorean Theorem (Intervals 30-33). This task was intended to highlight

the idea of proof as general as compared to a set of empirical examples. The third task had

students working in groups to solve a one-variable algebraic equation and provide reasons

(Intervals 34-40). The equation was written such that multiple solution paths were possible, and

Lucy emphasized to students that they may have used different orders and steps, but as long as

they explained their thinking and justified it using mathematical facts, they were fine (Interval


37). She ended the lesson by talking about why they did this work, and previewing the different

forms7 of proofs that they will encounter (Interval 40) while emphasizing that all these

representations explain why a mathematical idea is true. Her final statements in Interval 42

position proof as a way of thinking mathematically: “This was a different concept for you; this is

not just plugging in numbers and solving, it’s plugging in numbers and solving then supporting

why you did it. And it’s [an] introduction to reasoning.”

In Lucy’s class, students were positioned in a variety of roles over the class period – as

creators, explainers, authorities, observers, and validators. Lucy engaged in several of these roles

as well, often positioning herself along with the students as mathematical authorities as shown in

Table 6.


The role of ‘others’ in Lucy’s class is notable: others were only visible in seven of the 43

intervals, taking on roles as mathematical authorities. This usually took the form of consulting a

list of properties in the textbook to label the justification for steps in the proof.

Alignment across settings. In the assessments of Lucy’s mathematical knowledge for

teaching proof, she showed an understanding of and facility with proof as traditionally

represented in high school mathematics classrooms – chains of symbolic statements matched to

reasons described by formal names of axioms, properties, and theorems. Her ability to construct

such a proof (despite minor flaws in generality) without access to a text or other reference shows

great fluency with this form. In her interview, Lucy showed that her knowledge of proof

extended beyond a specific form – she noted the importance of proof as based on mathematical

facts and its generality in describing what a proof is, but also emphasized the roles of proof as an

7 Lucy’s description of forms of proof included both physical forms (paragraph proof, visual proof) and methods for proving (proof by counterexample, induction).


explanatory tool and as a means to communicate knowledge. Lucy’s lesson further illuminated

dimensions of her MKT represented in the clinical interview, affording her opportunities to link

several aspects of proof together through her activity selection and enactment with students.

Lucy’s lesson, in which the goal was to engage students in their first explicit encounters

with proof, could have easily focused on a narrow slice of proof, such as proof as a mathematical

argument and as based on mathematical facts. In fact, Figure 3 shows that these roles were her

starting point. She made frequent and explicit connections, however, to other roles of proof

mentioned in her interview: explain why and communicate mathematical knowledge. When

identifying proofs and non-proofs in the interview, Lucy focused largely on the nature of the

method used. Her classroom instruction showed this focus as well – discussing with students

how the steps fit together and the importance of justifying responses based on what they already

know. What is notable about Lucy’s treatment of method is the way in which it was integrated

with her positioning of students. Lucy frequently assumed the role of mathematical authority

together with her students, asking them to do more than just provide the reasons for a set of

steps, but instead to create the steps themselves and provide reasons in ways that made sense to

them. She blended the formal language of proof with the informal language of her students, at

one point discussing the commutative property in a way that was more familiar to the class and

empowered them as having the understandings necessary to create proof:

Donna’s infamous property’s on here, the one she likes to say “the backwards property” –

the commutative, right? Four plus three IS three plus four? She likes to say, “Oh, it’s

written backwards.” You guys, I know you remember that… it’s commutative, and it’s on

there. So these are all things you’ve seen.

Discussion


In this exploratory study, two teachers’ mathematical knowledge for teaching proof was

analyzed in clinical and classroom contexts, with attention to similarities and differences in the

ways in which their MKT was evident in each context. These two particular teachers form an

interesting comparative set, as both were teaching lessons designed to serve as an introduction to

proof-writing. The importance of MKT in facilitating student learning is well established in the

field. By carefully examining MKT in both a clinical and applied context, we sought to

investigate the mechanisms by which teacher knowledge plays out in an enacted lesson. In this

section, we look back on our analyses of Steve and Lucy and discuss the affordances of

examining MKT in multiple settings, how roles taken by classroom actors might interact with

MKT, and implications for teacher education.

Mathematical Knowledge for Teaching: Comparing Assessments and Practice

In examining the cases of Steve and Lucy, aspects of their MKT for proof that were

evident on the assessment played out in different ways in the classroom. Steve’s MKT-P on the

assessments showed a strong grasp of the definitional aspects of proof, with less attention to the

role of proof both in mathematics and in the classroom. This is not surprising, given that Steve

was completing his bachelor’s degree and student teaching at the time of the study. Steve’s

lesson reflected his knowledge of the defining characteristics of proof, showing a strong focus on

proof as a mathematical argument and as based on an established set of mathematical facts.

Steve’s lesson only touched briefly on the role of proof he discussed the most in the interview –

proof as a tool for explaining why – yet he indicated in his post-lesson interview that he had felt

that this was his area of focus. The patterns of positioning in Steve’s class may suggest a reason

for this discrepancy: Steve positioned his students as creators, but only in the sense that they

provided the reasons behind a predetermined set of steps. Given that the steps came from the


textbook (positioned as the source of mathematical authority), Steve’s ability to enact the role of

proof as explaining why a statement is true was compromised; the steps that are presented from

the textbook do the bulk of the intellectual work of explaining instead. Despite the limited

evidence for this role of proof in the lesson, Steve’s lesson and assessments showed a great deal

of consistency in the ways in which they made visible the definitional aspects of proof.

The more experienced teacher, Lucy, enacted a lesson in which the connections between

her MKT-P across the clinical and classroom contexts were stronger and more nuanced. Lucy’s

assessments showed strength in creating and identifying proofs, and in the defining

characteristics of a proof. Like Steve, she saw proof’s role in the classroom as helping students

explain why a mathematical idea is true. In addition, she saw proof as a tool for communicating

mathematical knowledge. The aspects of proof salient in Lucy’s lesson were much like Steve’s

lesson at the start, with a focus on the structural aspects of proof as a mathematical argument that

was based on known facts. The lesson diverges from Steve’s in that the work on the particular

tasks transitions into a recurring discussion of proof’s generality (Intervals 25-32), and the roles

of proof as an explanatory tool (Intervals 11-18, 30-43) and as communicating mathematical

knowledge (Intervals 10-30). Lucy was able to integrate a wide variety of conceptions of proof’s

structure and role into her lesson. Moreover, her positioning of students as observers, creators,

and explainers gave her students first-hand experiences with these aspects of proof.

Researchers who have investigated MKT have identified clinical assessments and

classroom observations as contexts for investigating teachers’ mathematical knowledge. These

case studies underscore the importance of looking across both contexts to illuminate the ways in

which teachers draw on their mathematical knowledge to make the set of instructional decisions

that result in a particular lesson and ultimately, student learning. Both teachers had particular


ideas that were core to their conceptions of proof and the work that students could and should do

related to proof. Both Steve and Lucy demonstrated relatively robust knowledge of proof and

fluency with a variety of aspects and roles of proof, yet enacted lessons that represented that

knowledge in very different ways. Their selections of classroom tasks provided opportunities for

particular aspects of this knowledge of proof to be made available to students. The means by

which each teacher enacted the task further impacted the aspects of proof made available for

students to learn, including the ways in which students were positioned with respect to the task.

These differences in positioning (Tables 4 and 6) illustrate the powerful ways in which

positioning can mediate teachers’ MKT in providing students with opportunities to learn for

students. The choices that Steve and Lucy made with respect to how students were positioned

provided different opportunities for students to engage with proof. For example, Lucy’s opening

task, in which students ordered steps solving an equation, could have been enacted as an

individual rote warm-up. Instead, Lucy encouraged students to defend their responses, placed a

focus on the reasons behind each step, and subsequently raised issues related to proof’s

explanatory and communicatory power for students that may not have been initially visible

simply from the selection of the task. Knowing that the proof’s roles as communicating new

mathematics and explaining why were key aspects of Lucy’s MKT-P provides insight into the

pedagogical decisions that she made with the task. Positioning students as mathematical

authorities has the potential to open up spaces for teachers to mobilize aspects of their MKT that

may not be available in lecture settings where the teacher functions as the authority, as well as

opening up spaces for students to develop more robust conceptualizations of what it means to

know and do mathematics.


It should be noted that while this study began to explore some of the complex

relationships between mathematical knowledge for teaching in clinical and practical settings,

there are a number of limitations on the generalizability of the findings. First, the small sample

size (both of teachers and lessons) provide us with only a snapshot of each teacher’s practice

with respect to proof. A study of a larger set of lessons related to proof would provide a more

robust portrait of the ways in which a teacher plans for and enacts a range of content related to

proof over time. In addition, the stated goals from each teacher for their lesson were relatively

thin. Teacher goals are often tacit and not represented in great depth in lesson plans or short

interviews (Leinhardt, 1989). In this study, we elected not to press teachers to describe their

goals with respect to the MKT-P framework so as to best capture their intentions without leading

them towards specific aspects of proof that they may not have considered. Future work might

benefit from a more nuanced consideration of teachers’ goals related to proof during planning,

and structured reflection on those goals using lesson artifacts.

Despite the limitations, these case analyses allow us to examine the ways in which

teacher knowledge of a specific content thread impacts and connects a various research-related

constructs that have evolved separately – studies on teacher knowledge of proof (e.g., Knuth,

2002a, b), studies of task selection and implementation (e.g., Stein & Lane, 1996), and studies of

authority and positioning (e.g., Wilson & Lloyd, 2000). Analyses such as these cases will allow

the field use existing research frameworks related to classroom practice in a way that seeks to

unpack the complex relationship between MKT and teaching practice. We conclude by

discussing ways in which this type of analysis might be useful in teacher education.

Mathematical Knowledge for Teaching as a Frame for Analyzing Practice


Despite differences in experience, prior knowledge, and school settings, there were a

number of similarities between Steve and Lucy. Both performed well on the clinical assessments,

and the MKT-P analysis of Steve’s lesson has a pattern that is very similar to the opening of

Lucy’s lesson (see Figures 2 and 3). This raised for us a question: given that Steve is a student

teacher and Lucy is an experienced teacher, what might facilitate Steve’s growth into a teacher

more like Lucy over time? Given the strong results in the clinical setting, one can rule out the

notion of teaching Steve more mathematics. His fluency with proof, its definition, and its roles

suggests that Steve’s content knowledge is sound.

One possible experience in which teachers might engage would be to articulate the

important facets of a complex mathematical idea such as proof, and then analyze their practice

with respect to that frame. By exploring the mathematical idea first, teachers would have the

opportunity to consider deeply what the important aspects of the mathematics might be that they

would want students to learn – in essence, constructing an ad hoc framework for the

mathematical knowledge for teaching the topic. Using this framework, teachers could be given

opportunities to analyze classroom practice, noting the places in the lesson where they see these

critical aspects coming out. This sort of analysis would allow teachers to see patterns across their

lesson in how a mathematical idea is developed, and to think about how to develop other aspects

of that mathematical idea in future lessons. It could also support teachers in selecting classroom

tasks that highlight particular aspects of proof. For example, Steve states that one of his goals

was to verify the truth of the Law of Cosines; however, his task choice constrained the

opportunities students had to consider this role of proof. Affording teachers opportunities to use

the MKT-P framework as a tool for task selection prior to a taught lesson might bring these

mismatches into relief.


At the preservice level, one of the challenges for prospective teachers is to examine the

complex relationships between content and practice. By using a content-oriented frame to

analyze teaching practice, prospective teachers would have opportunities to make links between

deep mathematical knowledge for teaching and pedagogical practice. This type of activity

resonates with the notion of a content-focused methods course (cf. Markokvits & Smith, 2008;

Steele, 2008; Steele, Smith, & Hillen, 2011) that seeks to develop mathematical knowledge for

teaching through carefully-structured investigations into a slice of mathematics and the teaching

of related mathematical ideas.

The close examination of the mathematical ideas first could also serve to contextualize

issues of pedagogy and positioning. Considering the mathematical opportunities to learn in a

lesson might naturally lead to questions about how one might expand those opportunities. As

seen in these two cases, some aspects of MKT are made more available to student when they are

positioned as mathematical authorities. By tracing the opportunities to learn mathematics first

and comparing them to a broader MKT framework, the issue of positioning has the potential to

come into stark relief as a mediating factor between teacher knowledge, teacher practice, and

student learning. This comes into focus when considering the seeming contradiction between the

prominent role of communicating mathematical knowledge in Lucy’s interviews and class

analysis and the fact that students were not positioned as communicators between one another

(see Table 6). Rather than focusing on communication between students in the class, Lucy

positioned the collective work of the class a means to communicate with other doers of

mathematics. In setting up the discussion of first task, Lucy notes that, “We are all math students

at Douglas High School and we solved this problem a particular way... students at other high

schools [might] call the rules and steps differently, but they are the same definitions [and


properties].” This stance has the effect of identifying the work of the class as collective, with

Lucy even including herself in that collective through the use of “we” as compared to “you,” in

dialogue. In this way, the focus was on making an argument that could be communicated to

others rather than making the argument and communicating about it in students’ first attempt at

proof-writing. Lucy came back to this role in her closing remarks on the activity:

You can see… in those two paragraphs why we have a need for some language that is

universal. Ok? Meaning that, what I say at Douglas High School is what they are saying

in Fulton Square High School, is what they are saying at Iona High School, is what they

are saying in… [students shout names of other local districts]

Lucy’s work of introducing proof in this way fulfills a key condition of an instructional

dialogue – establishing and problematizing the object of inquiry; in this case, proof (Leinhardt,

2001; Leinhardt & Steele, 2005). It is clear to Lucy’s students that a primary reason why we

might want to engage in the work of proof is to communicate with other learners. This allows a

dimension of Lucy’s knowledge of proof – communicating mathematical knowledge – to be

salient in the class and connected to the work at hand of creating a proof. Steve begins the work

of setting up the proof, and only after several minutes of work on the proof tries to establish a

frame: “…we’re going to start with the Pythagorean Theorem and we’re gonna kinda justify

where the Law of Cosines comes from.” Steve’s framing identifies the goal of the activity –

prove the Law of Cosines given the Pythagorean Theorem as a starting point – but does not

problematize why one might engage in the work of proof more generally. One wonders if this

narrow framing may explain why the focus of Steve’s lesson stays largely within the mechanics

of proof – proof as a mathematical argument based on known facts – rather than operationalizing

his knowledge of the roles of proof in mathematics. Explicit attention to the idea of establishing


and problematizing the object of inquiry in teacher preparation may support teachers in using a

wider frame, in turn providing opportunities to explore a wider range of aspects of a

mathematical idea like proof.

Mathematics education research has sought to understand the knowledge that teachers

need to teach mathematics in ways that support meaningful student learning. Researchers have

made great strides in developing, validating, and analyzing measures of teacher knowledge that

go beyond content or general pedagogical practice and investigate the knowledge base unique to

the work of mathematics teaching. Their work has raised a critical next question for the field –

how does that mathematical knowledge shape the work that teachers do in enacting lessons with

their students? Using a subject-specific framework for mathematical knowledge for teaching and

investigating teachers’ knowledge in clinical and classroom settings can give researchers and

teachers a window into the ways in which teacher knowledge influences the work that they do

with students and ultimately students’ opportunities to learn mathematics.


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Table 1. Mathematical Knowledge for Teaching Proof (MKT-P) Framework

Component of proof knowledge Criteria for evaluating component

Defining proof • Proof as a mathematical argument

• Proof as based on mathematical facts

• Proof as valid for a general class

• Proof as establishing truth

Identify proofs and non-proofs

• Distinguishing proofs from non-proofs

• Link features of an argument to aspects of the

definition of proof

• Recognize proof across representations

Create mathematical proofs Be able to write proofs in a variety of forms

Understanding the roles of proof in

mathematics

• Proof as verifying truth of a known fact

• Proof as explaining why

• Proof as communicating mathematical ideas

• Proof as creating new mathematical

knowledge

• Proof as systematizing the domain


Table 2. Five roles of proof in mathematics

Verifying truth Checking or confirming the truth of a known idea

Explaining why

Unpacking the thinking and reasoning behind why a

statement is true – includes providing reasons to support a

conjecture (Hanna, 1995)

Communicate mathematical

knowledge

Helping others understand a mathematical idea;

disseminating knowledge to other doers of mathematics

Creating new mathematical

knowledge

Developing new mathematical ideas, confirming

conjectures, building mathematical ideas

Systematizing the domain Imposing a logical structure on the mathematical domain;

organizing and cataloging results with respect to axioms

and other prior knowledge (Knuth, 2002a)


Table 3. Aspects of the MKT-P salient in Steve’s assessments.

Defining Proof Identifying Proofs and

Non-Proofs

Role of Proof in

Mathematics

Role of Proof in K-12

Mathematical argument

(24)

Establish truth (19)

Valid method (21)

Concrete features (19)

Explain why (5)

Verify truth (3)

Communicate knowledge

(3)

Create new mathematics

(3)

Explain why (11)

Systematize the domain

(8)

Numbers in parentheses represent coded lines of interview transcript


Table 4. Actors and their roles in Steve’s class (depicted by number of intervals)

Students Teacher Outside Others

Creator 7 1 0

Validator 1 5 0

Mathematical Authority 0 9 12

Observer 10 3 0

Explainer 0 1 0

Communicator 0 2 0


Table 5. Aspects of the MKT-P salient in Lucy’s assessments.

Defining Proof Identifying Proofs and

Non-Proofs

Role of Proof in

Mathematics

Role of Proof in K-12

Based on mathematical

facts (18)

General (14)

Valid method (47)

Concrete features (15)

Familiarity with Proof (11)

Generality (11)

Explain why (13)

Create new mathematics

(10)

Verify truth (10)


(9)

Explain why (27)


(16)

Numbers in parentheses represent coded lines of interview transcript


Table 6. Actors and their positioning in Lucy’s class.

Students Teacher Outside Others

Creator 19 9 0

Validator 1 12 0

Mathematical Authority 10 28 7

Observer 7 0 0

Explainer 9 2 0

Communicator 0 0 0


On the board: Mr. M Alright, does everybody kinda got a start on this? Ok. Up here, is [points

to board] just some Joe Schmo triangle up here. Alright? Triangle ABC, we’ll call it. And what happened is we drew an altitude in it. [points to altitude segment] And what it did was split the triangle into two right triangles. Ok, we’ve got triangle AB- we’ll call this [marks point] point X. I don’t know if it has a name, does it? [looks at student textbook]

St What? Mr. M The middle point. No. I’m gonna call it X. This point right here is X.

[labels point] Ok. So we’ve got triangle ABX and we’ve got triangle AXC [gestures at each triangle] - it split it up into two. And what we’re gonna do with this proof, is we’re going to start with the Pythagorean Theorem, and we’re gonna kinda justify where the Law of Cosines comes from. Alright? So I know when I write the Law of Cosines up there, everybody freaks out, because it’s this giant thing. Was it as hard to remember as you thought it was?

Sts No. Mr. M No, Ok? And you remember how we saw that it kinda looked a little bit

like the Pythagorean Theorem? That’s because it comes from the Pythagorean Theorem, so we’re gonna use that today to MAKE the Law of Cosines. We’re gonna make it, and hopefully when we’re done, you’ll kinda see where it came from, and if you had a hard time remembering it before, maybe it’ll be a little easier this time. Ok? So like the rest of our proofs, I’ve got the left hand side up here. That kinda guides us along and helps us see the step-to-step.

Interval A: Role of proof: Verifying truth

Mr. M This first step, let’s take a look at it. It says c squared equals the quantity a minus x squared plus h squared. Alright? And where they get that from [points to AB] here’s my c, [points to BX] here’s a minus x, [points to AX] here’s h. What do you think we could write for a reason for that? What are they doing, basically? Does it look familiar? What if I called a minus x something else? Like, b? What if it was c squared equals b squared plus h squared? Would that help?

St Yeah Mr. M What is this guy? St Substitution [whispered] Mr. M I heard it. The Pythagorean Theorem, right? Isn’t this just a different way

[points to statement 1] of writing the Pythagorean Theorem? We’ve got our hypotenuse [points to AB], c squared, equals one of our legs [points to BX], a minus x squared, plus [points to AX] h squared? Isn’t that the Pythagorean Theorem?

St Yup Mr. M Yeah. So here’s my reason [writes Pythagorean Thm in right column],

Pythagorean Theorem. Am I in over anybody’s head yet? Anybody havin’ any problems? Ok. Pretty easy so far, right?

Interval B: Definition of proof: Based on known mathematical facts


Figure 1. An example of interval parsing.


Figure 2. Aspects of the MKT-P Framework in Steve’s class by Interval.


Figure 3. Aspects of the MKT-P Framework in Lucy’s class by Interval.

mathematical knowledge and teaching practice relationships betwe

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