Creative Education
Vol.06 No.10(2015), Article ID:57292,33 pages
10.4236/ce.2015.610101

Mathematical Content Understanding for Teaching: A Study of Undergraduate STEM Majors

Xiaoxia A. Newton1, Rebecca C. Poon2

1Graduate School of Education, University of California, Berkeley, CA, USA

2Cal Teach Program, University of California, Berkeley, CA, USA

Email: xnewton@berkeley.edu

Copyright © 2015 by authors and Scientific Research Publishing Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY).

http://creativecommons.org/licenses/by/4.0/

Received 18 February 2015; accepted 16 June 2015; published 19 June 2015

ABSTRACT

This paper investigates the nature of mathematical understanding that is needed to teach three foundational early algebra topics. These three topics include dividing fractions, linear equation in two variables and its graph, and quadratic function and its graph. Data from a sample of undergraduate STEM majors in a major research university affirm the importance of developing what Shulman (1999) calls “far more effective mathematics courses in U.S. undergraduate program” in order to equip future mathematics teachers with profound mathematical content understanding for teaching fundamental mathematics (Ma, 1999) .

Keywords:

Mathematical Understanding, Mathematics Teacher Education, Pre-Service Content Training

1. Introduction

This paper investigates the nature of mathematical understanding that is needed to teach three foundational early algebra topics. Subject matter knowledge plays a central role in teaching (Buchmann, 1984). Teachers’ understanding of mathematics is central to their capacity to carry out instructional activities such as using instructional materials wisely, assessing students’ progress, and making sound judgments about presentation, emphasis, and sequencing (Ball, Hill, & Bass, 2005) .

Despite the fact that subject matter knowledge is central to teaching, content knowledge rarely figures prominently in research or teacher preparation programs (Ball, 1990; Harel, 1993) . Shulman (1986) regarded content as “the missing paradigm” in research on teaching. To address this “missing paradigm” on teaching, Shulman proposed several types of knowledge teachers need in order to teach. One critical type of knowledge is “pedagogical content knowledge” (PCK), which he defined as “the ways of representing and formulating the subject that make it comprehensible to others” (p. 9). Since then, educational scholars have attempted to measure PCK and link it to student learning (e.g., Ball, Hill, & Bass, 2005; Baumert et al., 2010 ).

Scholars have conceptualized PCK in different ways. For instance, Ball and her colleagues further elaborated Shulman’s PCK as consisting of different dimensions such as knowledge of content and students, knowledge of content and teaching, and knowledge of content and curriculum (Ball, Hoover, & Phelps, 2008). Ma (1999) calls teachers’ subject matter knowledge “profound understanding of school mathematics”. Schoenfeld and Kilpatrick (2008) regard this kind of content understanding as “knowing school mathematics in depth and breadth”.

These scholars’ work has crystallized the concept of PCK. However, there are several directions in which further research is needed. First, it is important to delineate the depth and breadth of the relevant knowledge for key topics in the curriculum (Schoenfeld & Kilpatrick, 2008) . Second, most work on mathematics teachers’ subject knowledge has focused on elementary to middle grades (e.g., Ball, Hill, & Bass, 2005; Ma, 1999 ), but not much work has focused on foundational algebra topics. Finally, Ball and her colleagues (2008) stressed that further work was needed on specialized content knowledge “in order to understand the most important dimensions of teachers’ professional knowledge” (p. 405).

Our work builds on these scholars’ research, but extends the existing research base in two important ways. First, it makes a deliberate effort to focus on three critical and foundational mathematics topics for learning algebra. Secondly, it intends to make explicit the key mathematical ideas underlying each topic that are the foundation for what Shulman (1986) calls “the content of the lessons taught”. The three topics we focus on are dividing fractions, linear equations in two variables and their graphs, and quadratic functions and their graphs. These topics have been chosen for several reasons. First, they are foundational topics in early algebra; yet they are challenging for teachers to teach well and for students to grasp. Second, there exist substantial misunderstandings (or incoherent portraits) of these topics in many textbooks. Finally, these topics occupy a significant part of the curriculum from grades 5 through algebra 1.

This paper is structured as follows. We begin with a review of relevant research literature concerning teachers’ mathematical knowledge for teaching. Building on prior research, we outline and elaborate a framework for examining the nature of mathematical understanding of the three topics that are the focus of this study. We then describe various aspects of the inquiry methods we have used to investigate a sample of undergraduate STEM majors’ mathematical understanding based on our framework. Following this, we present our findings on the characteristics of these STEM majors’ mathematical understanding of the three topics. Finally, we discuss the implications of our findings for pre- and in-service mathematics teachers’ content training.

2. Review of Literature

In his 1985 presidential address at the annual meeting of the American Educational Research Association, Shulman (1986) described content as “the missing paradigm” in research on teaching and argued that critical questions had not been asked. These questions include: Where do teachers’ explanations come from? How do teachers decide what to teach, how to represent it, how to question students about it, and how to handle students’ misunderstanding?

To address this “missing paradigm” on teaching, Shulman proposed several types of knowledge teachers need in order to teach. One critical type of knowledge is “pedagogical content knowledge” (PCK), which he defined as “the ways of representing and formulating the subject that make it comprehensible to others” (p. 9). Since then, educational scholars have attempted to elaborate what PCK may entail and link it to student learning (e.g., Ball, 1990; Ball, Hill, & Bass, 2005; Ball, Hoover, & Phelps, 2008; Baumert et al., 2010; Schoenfeld & Kilpatrick, 2008 ).

One theoretical framework of proficiency in teaching mathematics came from Schoenfeld and Kilpatrick (2008) . Their framework consists of several dimensions, the first dimension being “knowing school mathematic in depth and breadth” (p. 2). Schoenfeld and Kilpatrick (2008) argue that proficient teachers’ knowledge of school mathematics is both broad and deep. The breadth focuses on teachers’ ability to have multiple ways to conceptualize the content, represent the content in various ways, understand key mathematical ideas, and make connections among mathematical topics. The depth, on the other hand, refers to teachers’ understanding of how the mathematical ideas grow conceptually from one grade to another. With knowledge that is both broad and deep, teachers will be able to prioritize and organize content focusing on big mathematical ideas and to respond flexibly to students’ questions (Schoenfeld & Kilpatrick, 2008) .

The characteristics of content understanding outlined in Schoenfeld and Kilpatrick’s framework are similar to the ideas rooted in a series work by Deborah Ball and her colleagues (Ball, 1990; Ball, Hill, & Bass, 2005; Ball, Hoover, & Phelps, 2008) . Ball and her colleagues call the kind of content understanding described by Schoenfeld and Kilpatrick, “mathematical content knowledge for teaching” (Ball, Hill, & Bass, 2005; Ball, Hoover, & Phelps, 2008) . In her earlier work, Ball (1990) proposed four dimensions of subject matter knowledge for teaching that mathematics teachers need to have. These dimensions include: (1) possessing correct knowledge of concepts and procedures; (2) understanding the underlying principles and meanings; (3) knowing the connections among mathematical ideas; and (4) understanding the nature of mathematical knowledge and mathematics as a field (e.g., being able to determine what counts as an “answer” in mathematics? What establishes the validity of an answer?, etc.).

In the work that followed, Ball and her colleagues (Ball, Hill, & Bass, 2005) defined “mathematical content knowledge for teaching” as being composed of two key elements: “common” knowledge of mathematics that any well-educated adult should have and mathematical knowledge that is “specialized” to the work of teaching and that only teachers need know” (p. 22). From their descriptions, it seems that “common” mathematical knowledge is equivalent to being able to do mathematics, whereas “specialized” mathematical knowledge consists of being able to do mathematics and to know why (i.e., the reasoning that teachers need to know in order to teach students).

The notion that there is content knowledge unique to teaching is further expanded in their most recent work. Ball and her colleagues (Ball, Thames, & Phelps, 2008) proposed a sub-domain of “pure” content knowledge unique to the work of teaching, which they call specialized content knowledge (italics emphasized by the authors). The distinction between specialized content knowledge and other knowledge of mathematics, according to the authors, is that the former is needed by teachers for specific tasks of teaching (e.g., responding to students’ why questions) and is not intertwined with knowledge of pedagogy, students, curriculum, or other non-content domains. While appealing, Ball and her colleagues point out that this specialized content knowledge needs further work. This sentiment is echoed by Schoenfeld and Kilpatrick (2008) who suggest further work be done to delineate the depth and breadth of the relevant knowledge for key topics in the curriculum. Our work is an attempt to describe what the specialized content knowledge may entail for three key early algebra topics.

The premise of our work is that mathematics taught to students at the K-12 level differs significantly from that taught in university mathematics courses. This stance has been emphasized by research mathematicians who have worked with the mathematics education community (Wu, 2011a) and educators alike (e.g., Ball, 1990 ). It supports the argument that the mathematical development of topics to K-12 students must be sensitive to their knowledge base and mathematical sophistication (Wu, 2011a) . These ideas put forward by research mathematicians echo, to some extent, those proposed by educational scholars (e.g., Ball, 1990; Ball et al., 2008; Ma, 1999; Schoenfeld & Kilpatrick, 2008; Shulman, 1986 ) in terms of the specialized content knowledge for teaching K-12 students. In addition, the argument has some empirical support. As Ball and her colleagues found (Ball et al., 2005) that students of third grade teachers who did well on their measure of mathematical content knowledge for teaching performed better than those with teachers who did not do so well on their measure.

In contrast to Ball and her colleagues who attempted to measure teachers’ content knowledge for teaching, most scholars typically measure the adequacy of secondary mathematics teachers’ content knowledge by the amount of college mathematics coursework they have completed as undergraduates (i.e., college major in mathematics signals extensive mathematics coursework). The implicit assumption is that typical college-level mathematics courses teach future teachers what they need to know in order to teach at the 6-12 grade level. As pointed out earlier, this assumption has been challenged by teacher educators (e.g., Ball, 1990; Shulman, 1999 ) and mathematicians (e.g., Wu, 2011a ).

Empirical studies on the relationship between teachers’ college mathematics coursework and their students’ mathematical performance have produced mixed results. Some studies have found that teachers with degrees in mathematics have a significant positive effect on high school students’ mathematics test scores (Goldhaber & Brewer, 1997, 2000; Rowan, Chiang, & Miller, 1997) . Other studies have shown that teachers with extensive college-level mathematics coursework do not necessarily know how to explain fundamental K-12 concepts such as the reasoning behind the invert-and-multiply rule for division of fractions (Ball, 1990; Borko et al., 1992). These findings lend further support to the argument that the content of college mathematics course has little to do with the content taught at K-12 level.

The paradox begs the question: What is the nature of mathematical content understanding that is needed in order to teach middle school and high school mathematics? What does this mathematical content understanding look like? This paper addresses these questions through an empirical study of a sample of undergraduate STEM majors at one of the research universities in the west coast of the United States. Specifically, we ground our work in the three mathematical topics identified previously.

What might this specialized content knowledge look like for the mathematical topics we focus on in this study? We elaborate this in the next section, but first we outline in Table 1 the characteristics that exemplify this specialized content knowledge from mathematical perspectives.

The three characteristics are derived from the five “fundamental principles of mathematics” proposed by research mathematicians (e.g., Wu, 2010a ) and the key ideas proposed by researchers whose work focuses on mathematics education (e.g., Schoenfeld & Kilpatrick, 2008 ). These characteristics of content understanding are consistent with and reflect the mathematics education community’s call for a profound understanding of school mathematics for teaching (e.g., Ball, 1990; Ma, 1999; Schoenfeld & Kilpatrick, 2008 ). One point we want to emphasize is that we describe some of the relevant knowledge, acknowledging that there are various ways to conceptualize the content, and more than one way to approach the teaching of it ( Cochran-Smith & Lytle, 1999 ).

3. Content Understanding for Teaching: Dividing Fractions, Linear Equations, and Quadratic Functions

In this section, we elaborate what exemplary content understandinh explicit content training of mathematics topics that they are expected to teach at the K-12 level. Otherwise, STEM majors will resort to the way they were taught as K-12 students when they become teachers one day. For example, the UC Berkeley math department is one of the few that offer math content courses specifically focusing on grades 6-12 content for math majors who are interested in pursuing teaching as a career. We need policies that promote college math departments’ involvement in the training of future math teachers.

In addition, our findings have implications for using teachers’ college math coursework as a proxy measure of math teachers’ content knowledge as many empirical studies have done. As pointed out previously, empirical studies on the relationship between teachers’ college math coursework and their students’ mathematical performance have yielded mixed results. One possible explanation might be that having advanced mathematical knowledge at college level does not necessarily equate having deep understanding of K-12 content, which is necessary in order to translate this deep understanding into effective classroom practices in terms of engaging K- 12 students around substantive mathematics. Therefore, instead of using proxy measures such as college math coursework, directly measuring math teachers’ understanding of K-12 content they teach may help to produce consistent results on the relationship between teacher mathematical knowledge and students’ achievement.

Finally, our study findings have implications for the professional development of in-service teachers. Since most teachers did not have the opportunities to learn the content knowledge they need to teach from their college math courses, they typically resort to the way they were taught as K-12 students (Adams & Krockover, 1997; Lortie, 1975) . To improve the quality of teachers’ content understanding, we need in-service professional development activities that focus explicitly on the content knowledge they are teaching.

By outlining and investigating the nature of mathematical knowledge future teachers need in order to teach K- 12 students, we hope information generated from the study is relevant for a variety of stakeholder groups, including policy makers who are concerned with the quality of students’ mathematical learning, K-16 educators and administrators who are concerned with students’ under-preparation in quantitative skills for college and work, scholars whose research focuses on issues related to K-12 mathematics education and the STEM labor market, curricular developers who are interested in designing intervention materials for struggling mathematics students, and the general public who is interested in mathematics educational issues.

Acknowledgements

We would like to thank professors Hung-Hsi Wu (Department of Mathematics) and Alan Schoenfeld (Graduate School of Education), both at University of California, Berkeley, and professor Jeff Shih (Department of Teaching and Learning) at University of Nevada, Las Vegas for their comments on earlier drafts of the paper. In addition, we would like to thank anonymous reviewers for their constructive feedback. Last but not least, we would also like to thank the Cal Teach faculty and the Berkeley Science and Math Initiative research group for their support of this study. Opinions reflect those of the authors.

References

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Appendix A Linear Equations in Two Variables and Their Graphs

What is the assumption or key mathematical concept underlying proportionality mathematics problems? Proportionality occupies a significant role in algebra because it is used to teach students how to solve word problems dealing with speed, rate, and so on. One important amiss, however, lies in the fact that students are never taught about the key mathematical idea underlying such problems, i.e., making explicit the assumption we make in order to solve proportionality mathematics problems. We used scenario questions 2.1 and 2.2 to probe participants’ awareness of this issue. Since question 2.2 asks respondents to come up with a story, an exemplary response is one that is explicit about the underlying assumption needed to solve the problem. The discussion here focuses on question 2.1. Below is an exemplary response to this question:

The people who wrote this problem made an assumption. Can you see what it was? They assumed that Abe runs at a constant speed. Does that help you solve the problem?

“Einstein” might only need this much information. But he might need more, in which case the teacher might continue as follows:

So what does running at a constant speed mean? First, I would ask students how we calculate the average speed for Abe, which is:

Average speed ran during any time interval from t1 to t2 =

To say that Abe is running at a constant speed, by definition, means that the average speed for any time interval is a constant (or fixed) number, which we’ll call “v”.

Note the respondent immediately focuses on the key idea (i.e., the key to solving this problem is to assume that Abe runs at a constant speed). Then the respondent continues the line of reasoning by asking for the meaning of “running at a constant speed”. To do this, the respondent invokes a related idea, i.e., average speed, because this concept is the basis for defining constant speed (i.e., “To say that Abe is running at a constant speed, by definition, means that the average speed for any time interval is a constant (or fixed) number”). Here the respondent is very deliberate about using the definition as a basis for helping students “see” what reasoning makes it possible to set up proportionality problems, as shown below:

“Let D1 be the distance traveled from time = 0 to t1 and D2 be the distance traveled from time = 0 to t2. Then:

By the definition of division, this is equivalent to

Let t represent time (t > 0) and D represent the distance traveled from time = 0 to t. Then we can rewrite the above equation as:

Furthermore, if we consider the time interval 0 to any time t, we get:

This is a linear equation in two variables, t and D, without the constant term. It’s important to see that the key idea used to arrive at this equation is the definition of constant speed.

Building on the definition of constant speed, the respondent uses definition again (i.e., definition of division) to re-write the equation into a form that helps them to connect the definition of constant speed and linear equations in two variables without the constant term. This is the basis for the so-called proportionality reasoning that is used to solve a wide variety of problems involving speed, rate, etc.

Finally, the respondent is keenly aware of how what they are learning at the 8th grade might be connected with the mathematical concept they will learn later, i.e., function.

[Note: Later when students are introduced to the linear function, I’d also help them make the connection that the underlying mathematical concept for problems of constant speed or rate is a linear function in one variable without the constant term.]

Overall, the respondent is very purposeful in his or her reasoning and shows deliberate effort at using definition as a basis for logical reasoning and making explicit the interconnectedness of mathematical ideas at different levels of complexity.

Why can the slope of a line be calculated using any two distinct points on the line? Slope features prominently when teaching linear equations in the K-12 curriculum. Much of the emphasis in the textbook focuses more on the mechanic calculation than on the conceptual understanding of the concept of slope or the connection between linear equation and its graph. We used scenario question 2.3 to investigate respondents’ understanding of the concept of slope and the connection between linear equation and its graph.

An exemplary response to this scenario question begins with the definition of the slope of a line:

The key mathematical idea underlying this question is that the slope of a line can be calculated using any two points on the line (i.e., independence of any two distinct points on the line). So how can we help students learn this key idea? Before I use P1, P2, P3, P4 as shown in the picture, I would first review with students how the slope of a line is defined: given a line and assuming it slants upward (as the picture shows), let’s take a point P on the line, go 1 unit horizontally to point R, then go upward (or vertically) and let the vertical line from R intersect the given line at point Q. Then the definition of slope is the length of segment QR (i.e., |QR|).

Here the respondent is laying a foundation for what comes next by precisely defining the slope of a line and showing this on the graph. Note how the respondent expands the definition and stretches students’ thinking by posing the next question:

But how are we certain that this vertical length |QR| is the same for any point P we choose on the line? In other words: if we take another point P' on the line, go 1 unit horizontally to point R’ and then go upward to intersect the line at point Q', how do we know that)?

To answer this question, students need to invoke their knowledge of similar triangle. This is an important step towards defining the slope precisely and completely, as the respondent points out:

I would expect the following explanation from students: (corresponding angles on parallel lines) and, so by the angle-angle-side criterion, and, thus,. Therefore, the slope is independent of the point P and it makes sense to talk about the slope of the line.

With the definition complete, the respondent adds complexity by posing the following question: “Can we find another, more flexible way of finding the slope of a line, without having to measure 1 unit horizontally from a point on the line and then the vertical distance up?”This step builds on the previous step of defining the slope of the line but uses similar ideas (i.e., similar triangle), as shown below:

To answer this question, let’s do the following: let P, Q, R be as before (i.e., P is any point on the line used to define the slope of the line) and now suppose we take any other point on the line, call it S. From S, draw a vertical line and let it meet the horizontal line PR at point T.

So now look at the two triangles, ∆PQR and ∆PST. What can we say about them? Hopefully students would recognize that they are similar triangles; if not, I’d tell them but ask them to prove (explain) why the triangles are similar (by the angle-angle criterion: right angles formed by perpendicular lines and corresponding angles on parallel lines).

After establishing the fact that, I would then ask: what can we say about the relationship between the sides of the triangles? One of the things I would expect students to mention would be:

Then I would guide them to manipulate the above equation into the following:

At this point, I would ask students what they observe. Hopefully they would recognize that, since |PR| = 1, the left side of the equation is equal to line segment |QR|, which is the slope of the line. In other words:

Of course, the respondent is very purposeful about why they are doing this exercise:

From this exercise, I would hope students reached the following conclusions:

1. The slope of the line can be calculated using points P (the point we used to define the slope) and S (any other point on the line).

2. We can calculate the slope of a line by dividing the length of the vertical line segment by the length of the horizontal line segment of.

Because we had shown earlier that the point P used to define the slope is arbitrary (i.e., can be any point on the line) and we had defined S to be another arbitrary point on the line, then the conclusions above can be generalized into the following:

1. The slope of the line can be calculated using any two distinct points, P and S, on the line.

2. We can calculate the slope of a line by dividing the length of the vertical line segment by the length of the horizontal line segment of.

This purposefulness brings mathematical closure to students and we see how the respondent is very deliberate in scaffolding key ideas throughout the process. Having shown the underlying key ideas, the respondent then goes back to the original question (i.e., using P1, P2, P3, and P4) and has students work out the proof on their own:

To reinforce these main ideas, I would have students work in groups or pairs to prove (using similar triangle properties) that the slope of the line calculated by P1, P2 (in the original graph above) is the same as the slope calculated by P3, P4. Once they finish working in groups, I’d have a whole-class discussion and ask students to show how they did the proof. Below is an example of what I’d expect:

Draw in the horizontal and vertical lines through points P1, P2, P3, P4 and let them intersect at points Q and R as shown below:

We claim that the two triangles formed, DP1P2Q and DP3P4R, are similar. The reason is: |ÐP1QP2| = |ÐP3RP4| because both equal 90˚ and |ÐP1P2Q| = |ÐP3P4R| because they are corresponding angles on parallel lines. Then, by the angle-angle criterion, DP1P2Q ~ DP3P4R. By the key triangle similarity theorem, we can then

say, and by multiplying both sides of the equation by and, we get.

That means the slope calculated by P1, P2 is the same as the slope calculated by P3,P4. Therefore, the slope can be calculated by any two distinct points on the line.

Looking at this exemplary response overall, we see that the respondent is mindful of the purpose of each activity, focuses on the key ideas and scaffold these key ideas in a coherent way, starting with the definition, using it as a basis for subsequent logical reasoning, and leading students from simple ideas to more complex ones, from specific examples to general cases.

What is the connection among different forms of linear equations? A routine exercise students in early algebra classes do is to memorize different forms of linear equations (e.g., standard form, point-slope form, two-point form, intercept-slope form, etc.). This rote learning deprives students the opportunity to understand the connections among different forms of linear equations. For instance, students trained in rote learning are not given the opportunities to ponder the following questions: why can a linear equation be written in different forms? What information is given in each form of the same linear equation? What is the connection among different forms? Can one go from one form to another? Knowing the answers to these questions will deepen students’ understanding of linear equations. We used scenario question 2.4 to examine study participants’ understanding of these questions.

The exemplary response to the scenario question used to examine study participants’ understanding of these questions is purposeful and focuses on the key mathematical ideas, as shown below:

The key idea is that students should not need to memorize different forms of linear equations (so that they mechanically match a particular given piece of information with a particular form). The key is to understand the connection between a linear equation in two variables and its graph, so that students can use any given information to figure out the eqation of a line; or vice versa, given an equation, the students should know what its graph looks like.

To help students understand these key ideas, the respondent again resorts to using definition as a basis for logical reasoning:

The following ideas are central to understanding the connection between a linear equation in two variables and its graph. So before addressing the question raised in the scenario, I’d want students to know the following facts:

- Definition of the graph of a linear equation: collection of all ordered pairs (x0, y0) that satisfy the equation (where a ¹ 0 or b ¹ 0).

- The graph of a linear equation is a straight line; every straight line is the graph of some linear equation. [This is something that can be proved; though I might not do the proof with students right away, I’d at least point this out so that students know it can be proved.]

- The slope of a line can be calculated using any two distinct points on the line.

Having laid this foundation, the respondent begins a carefully orchestrated process of leading students to discover the key mathematical ideas:

With these basic understandings about a linear equation in two variables and its graph, I would first ask students to graph, by hand, the line. Then I’d ask them to work in groups or pairs to:

- Calculate the slope and the y-intercept (i.e., when x = 0)

- Show on the graph where/what is: slope and y-intercept

The purpose for this exercise is to get students to feel comfortable connecting algebraic calculations and graphical representations, especially as they relate to the slope and y-intercept.

Note the respondent begins with a concrete example and is very deliberate about the purpose behind this exercise: preparing students for what comes next:

Then I’d give students a line (say it’s the graph of, just to make it easier), and ask them to figure out what the equation for the line is, if:

1) We know one point and the slope

2) We know two points

3) We know the y-intercept and slope

The specific steps may go something like this:

(1) Given slope and a point:

Teacher prompt:

Suppose the slope of a line is 3 and a point on the line is. What's the equation of the line?

Expected student response:

We’ve learned that the slope of a line can be calculated by using any two distinct points on the line. We know one point is given:. Let be any other point on the line. Then the slope is. But we’re also given that the slope is 3, so. By the definition of division, we get .

[The interim step, , will be used later when I show them the generic point-slope form and have students compare this step to the generic slope-intercept form.]

(2) Given two points:

Teacher prompt:

Suppose a line contains points and. What’s the equation of the line?

Expected student response:

Let be any other point on the line besides or. Using the same reason as before (i.e., any two distinct points give the slope of the line), we get:

(3) Given slope and y-intercept:

Teacher prompt:

Suppose the slope of a line is 3 and the y-intercept of the line is −2. What’s the equation of the line?

[Recall the earlier task where students used the graph to connect the y-intercept to the y-coordinate of the point where the line intersects the y-axis. The point of intersection is where is the y-intercept of the line.]

Expected student response:

We’ve learned that, because −2 is the y?intercept, then is a point on the line. Let be any other point on the line besides. Using the same reason as before (i.e., any two distinct points give the slope of the line), we get:

[The alternative interim step, , will later be used to compare to the generic slope-intercept form,].

The purpose of this exercise is to get students used to figuring out the equation when different pieces of information are given. I would forbid them to use the forms as given in the textbook; instead, I would ask students to figure out the equations based on their understandings of the definition of the graph of linear equation and the connection between linear equations and their graphs (listed at the start of this response).

Throughout these exercises, the respondent connects ideas that students have learned in the previous steps with what they are doing now. In addition, the respondent is purposeful about emphasizing key mathematical ideas and using definition as a basis for logical reasoning. The process of discovery is coherent, going from simple to complex and from specific to general:

Once students figured out the linear equations based on different pieces of given information, I’d ask them to arrange their equations in the forms as shown in the scenario questions (now I’d show them the forms which are popular in textbooks), compare their equations with the general cases (i.e., the symbols) and see what they think. For example, to get into the form, students would add y and subtract 2 from both sides of the equation.

I would then talk about the significance of the constants in the forms and. Students should easily recognize the first form from case (1) above: given a line with slope m and point and letting be any other point on the line, we get:. So, that’s why this form is called the point-slope form.

When we consider the form, students should easily see that the b is the y-intercept because y = b when x = 0. However, what is the constant m? (Note: we don’t know that the m in this form represents the same number m in the point-slope form above) To answer this question, we do the following:

Say the equation of a line is: where m and b are constants. Let and be two distinct points on the line. Because the two points are on the line, by definition, and. Then what is the slope of the line?

Note that the final step is possible because. Therefore, the m is the slope of the line, and that’s why is called the slope-intercept form.

Other things I’d look for in their understanding at this point:

- The standard form is the most general form. It includes cases when b=0 (i.e., vertical line), which cannot be represented in the other two forms. However, this form doesn’t immediately show the slope.

- The slope-intercept form is a special case of the standard form where b is restricted to 1. This form helps us immediately identify the slope of the line (and the y-intercept). Also, it’s the form used when we talk about linear functions.

The point-slope form is also a special case of the standard form where b is restricted to 1. In addition, the point-slope form is, in principle, essentially the same as the slope-intercept form because the information given to figure out the equation is the same (i.e., in both cases, the slope is given, plus one point on the line: in the slope-intercept form the point given is where b is the y-intercept; in the point slope form the point given is. Note that, in the latter case, if, then is the point where the line intersects the y-axis. Therefore, it can be used to write the linear equation in slope-intercept form. Likewise, given the y-intercept, we can write the linear equation in point-slope form).

Note the respondent helps students to connect the general forms of equations (i.e., using symbols) with the concrete examples they have worked on. Furthermore, instead of asking students to mindlessly memorize different forms of linear equations, the respondent emphasizes the connection between different forms and between the algebraic and geometric representations of linear equations. Throughout, we see the respondent is deliberate at using definition as a basis for logical reasoning, emphasizing and connecting key mathematical ideas, and scaffolding the reasoning process in a coherent way from simple to complex and from specific to general. Overall, the response is complete and characterized with precision, coherence, and purposefulness.

Appendix B Quadratic Functions and Their Graphs

Quadratic functions are significantly more complex, mathematically, than linear functions. However, the key to helping students learn this topic is for students to have a firm grasp of the graph of the quadratic function and its algebraic expression. Several central mathematical ideas are behind the scenario questions (see 3.1 to 3.4 in Table 2) on quadratic function and its graph, including: (1) Enable students to have comfort and familiarity with the graph of the quadratic function, through knowing the graph of the unit quadratic function. This foundational understanding sets students on the path to understand, (a) how standard quadratic functions relate to;that is, the graphs of are dilations of the graph of. Or put it differently, the graphs of are similar to the graph of the unit quadratic function. In addition, it is important to help students understand, (b) that the graph of is congruent to the graph of some standard quadratic function through translation. And combined (i.e., a and b), these two ideas help students understand that all the graphs of quadratic functions are similar to each other; (2) Help students understand how to rewrite the standard form into the vertex and root forms, both graphically and algebraically (i.e., completing the square); and (3) Help students understand why the vertex and root forms are better than the standard polynomial form that is typically given. In other words, what information about the graph does each of these forms contain about the quadratic function? How does the quadratic formula fit in all of this? And what do we know about the connection between the roots of the quadratic function (if the discriminant is greater than or equal to zero) and the constants? Let’s see how these key ideas are scaffolded in the exemplary responses to the questions.

The Graph of Quadratic Functions. Question 3.1 focuses onenabling students to have comfort and familiarity with the graph of the quadratic function, through knowing the graph of the unit quadratic function.

As shown in the following response, the respondent is purposeful and emphasizes the key ideas underlying the mathematical activities:

The key idea I hope students would understand is that the graph of is similar to the graph of through dilation and translation (and reflection if a < 0). But how do we get there? What kind of translation will take us from the graph of to the graph of (after

transforming the graph of into the graph of through dilation by scale factor of and centered at the origin)?

In addition, the respondent is very deliberate at using definition as a basis for logical reasoning:

I’d first help students understand that, for a point to be on the graph of a function (whether linear or quadratic), it means that the x-coordinate and y-coordinate of the point are connected through the given function in the following way:. Furthermore, the graph of a function is the collection of all points of the form.

With this foundational understanding, the respondent shows a carefully thought-out guided practice to scaffold complex ideas through simple, concrete examples:

With this understanding, the first quadratic function we would consider is. To help students graph by hand, I would have them set up a table with the following x-values.

Then I would ask them: how do we find the points on the graph of g that have these x-values? I would expect students to recall the definition above and realize that they first need to calculate for each of the x-values listed.

Then each point would be a point on the graph of g. Therefore, their graph of g would include the points (0, 0), (−1, 1), (1, 1), (2, 4), (−2, 4), (3, 9), (−3, 9).

Once the students are familiarized and feel comfortable with this simple quadratic function, the respondent pushes their thinking further, again through a concrete example:

Then I’d say, now let’s look at how the graph changes when we change the coefficient of in the function. First of all, what is the coefficient of in the unit quadratic? [Answer = 1] That means, we want to look at graphs of where a is not 1 (or 0). Let’s first consider the case when a is positive, say 2. What will the graph of look like? I would expect students to suggest we use the same method as before and expand the table.

Then we would consider when a is negative, say −2. Again, we would expand the table.

As can be seen, the respondent is very deliberate in carefully scaffolding the complex ideas in a systematic and coherent way to the students. At the end of these activities, the respondent is very purposeful about bringing mathematical closure to the key concepts and ideas that students should acquire through these exercises:

After graphing and a few more cases, I’d have a class discussion along the line of: What happens to the graph when the coefficient of changes? And by extension (or generalization), what does the coefficient “a” do to the graph of g to get to the graph of? What happens when “a” is positive? What happens when “a” is negative? I would expect students to offer the following observations:

Ÿ When a is positive and greater than 1, the graph appears to keep the same shape as the graph of g but looks “thinner” than the graph of g.

Ÿ When a is positive and between 0 and 1, the graph appears to keep the same shape as the graph of g but looks “wider” than the graph of g.

Ÿ When a is negative, the graph appears to be the reflection across the y-axis of the cases above.

I would explain to students that describing the graph of as the “same shape” but “thinner” or “wider” than the graph of g means the graph of is similar to the graph of g. By definition of similarity, that means the graph of is a dilation of the graph of g. I would have students confirm this fact with the following exercise (assuming students have had ample practice with dilations before this): dilate the points of g found earlier

by a scale factor and center at the origin (0, 0). Students should find that the dilated points coincide with the

points found earlier for the graph of and, therefore, this dilation transforms the graph of g to the graph of.

Next, we see that the respondent extends students’ thinking by pushing them to work with general forms (i.e., use symbols) following the same logic they used with concrete examples. The respondent is very deliberate at demonstrating the interconnectedness of mathematical ideas and showing the logical progression of mathematical ideas:

Following the same logic as outlined above (i.e., how I’d help students understand the connection between g and when a ¹ 1 or 0), I’d systematically ask students to graph the following functions, by hand:

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First, set a = 2 and vary q (e.g., let q be 1, 2, −-1, −2). Next, set a = −2 and vary q in the same way. Then have a discussion on what “q” does to the graph: What happens when “q” is positive”? What happens when “q” is negative? I would want students to conclude that the q translates the graph of q units up when q > 0 and q units down when q < 0.

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In the same manner, I’d set a = 2, q = 1 but vary p and have students graph each case. Then, I’d have a discussion on what “p” does to the graph. I would want students to conclude that the p translates the graph of h p units to the left when p < 0 and p units to the right when p > 0.

Key idea: the graph of h is the translation of the graph of (i.e., every point in the graph of translates p units horizontally and q units vertically).

Finally, the respondent shows the purposefulness by emphasizing the key ideas and bringing mathematical closure to what students are doing:

Once students have graphed a few functions of the form, I’d ask students to compare how constants p, q might be related to the constants b, c if is expanded to the form. The purpose is to help students see that all quadratic functions of the form can be re-written as. And the key to accomplish this is via the so-called “completing the squares”.

Different forms of quadratic function. Question 3.2 focuses on helping students understand how to rewrite the standard form into the vertex and root forms, both graphically and algebraically (i.e., completing the square)

As can be seen below, the respondent is purposeful about the key mathematical ideas that students need to understand:

The key mathematical ideas for this question are for students to understand that any quadratic function of the form can be re-written into: (1) the vertex form through the technique of completing the square; and (2) the root form if we know the root(s) of the quadratic function (and the roots are given by the quadratic formula, assuming discriminant b2 − 4ac ≥ 0).

In addition, the respondent demonstrates a deliberate effort at scaffolding ideas in a systematic and coherent way: showing the interconnectedness of ideas both geometrically and algebraically and scaffolding the reasoning from simple to complex and from specific to general. These features are well demonstrated in the following response:

So how would I go about accomplishing this teaching and learning goal?

(1) Standard form to vertex form:

I would graphically show the logic of completing the square and then algebraically manipulate the standard form to vertex form. In other words, I would start with the diagram approach first. Once students understand what “completing the square” looks like graphically, I’d go over how we to do the general case through algebraic manipulation.

Step 1: Graphically

Consider the expression where k, l are positive numbers. Then we can think of as, which is equal to the area of the square with side length k, plus the areas of two rectangles each with side lengths k and, as shown below.

As the diagram shows, we could form a larger square with side length if we added in a square at the top right corner. What would be the area of such a square,?

Then the area of this larger square is. Therefore, if we add to our original expression, we “complete the square”. Symbolically:.

Step 2: Algebraically

I would start with concrete cases of (e.g.,) and ask students, what do we need to add to each of these expressions in order to be able to rewrite each one in the form? Building upon the graphical approach in Step 1, students should see the following pattern:. I would summarize their work on the concrete cases with this general case, which can be transformed into:

Then I would do concrete cases of (e.g.,) and ask students, how can we rewrite each of the expressions in the form, plus some constant? Based on students work, I would summarize with the general case:

Finally, I would have concrete cases of (e.g.,) and ask students, how can we rewrite the expressions so that each one has in it?

Finally, the respondent is purposeful in terms of bringing a mathematical closure to what students are learning, namely, converting quadratic function from its standard form to vertex form:

Once they understand the concrete cases, I’d work with them through the general case in the function form:

I would ask students to determine what ?and ?? should be so that

Our conclusion would be

I would then ask students, where have we seen quadratic functions look like this? Hopefully they would recall

the form from work in question 1. Then, as a final exercise, I would have the students find the relationship between the constants, which would get the following:.

When helping students to understand how to convert quadratic function from standard form to root form, the respondent is explicit about using definition as a basis for logical reasoning:

Step 1: Definitions of “zero of a quadratic function” and “root of a quadratic equation”

We define to be a zero of a quadratic function if. We define to be a root of a quadratic equation if is a solution to the equation. That means,

a root of a quadratic equation is a zero of the quadratic function (and vice versa).

Using this definition, the respondent carefully scaffolds the ideas through a systematic and coherent progression by helping students understand: (1) the relationship between roots of a quadratic equation and x-intercepts of a quadratic function; (2) using this relationship to locate the x-intercepts of a quadratic function; and (3) transform the quadratic function from standard form to root form.

Why do we need different forms of quadratic functions? Questions 3.3 and 3.4 focus on different forms of quadratic functions. The primary purpose of these two scenario questions is to examine ways in which teachers help students to understand that each form of the quadratic function provides some information about the quadratic function and to see the connection between algebraic and geometric representations of quadratic function. An exemplary response again exhibits characteristics of coherence and purposefulness. For example, the following response shows a deliberate effort at showing the logical progression from using concrete examples to using general cases:

First, I’d ask students to graph a specific quadratic function, say. I would expect them to do this by, first, rewriting the function into the form as explained in question 2, and then, using

the method outlined in question 1, they would translate the graph of to get the graph of. Then I’d ask them to work in groups or pairs to figure out the following:

1. At what value (s) of x does the graph intersect the x-axis? What does it mean when the graph of the function intersects the x-axis?

2. Where is the line of symmetry? Draw it.

At what value of x does the function achieve its maximum or minimum value? How do you know? When does a quadratic function have a maximum value? When does it have a minimum value?

Not only that, the respondent connects mathematical ideas students are learning now with what they were doing previously:

The first set of questions relates back to question 2, (2). There we showed how to find the roots of the quadratic equation algebraically (and by connecting Step 2 of (1) and Step 3 of (2) we can get the quadratic formula, which is a quick and mechanical way to get the roots directly from the quadratic function in standard form, assuming the discriminant is greater than or equal to zero). We can also find the roots of the quadratic equation (which are the x-intercepts or zeros of the quadratic function) by using the graph: we see that the graph of

intersects the x-axis k units above the vertex. Because the graph of is congruent to the graph of f (as discussed in question 1), we can find the x-intercepts by finding the x-values when. The result is, which means the x-intercepts of f are from the axis of symmetry (see graph below).

In addition to helping students connect mathematical ideas learned previously, the respondent also is deliberate at helping students to connect the algebraic and geometric representations:

By looking at the graph, students should also notice that the vertex is the point at which the function achieves

its maximum (when the constant a < 0) or minimum value (when the constant a > 0). The vertex is because, if a > 0 then and thus so f achieves its minimum when, i.e., x = h. The argument is similar when a < 0.

Finally, the respondent is purposeful at emphasizing key mathematical ideas in order to bring a mathematical closure to the concepts that students are learning:

At the conclusion of the exercise, I would ask, so what is the value of writing the quadratic function in the

form? I would expect students to say: it tells us the vertex, the axis of symmetry, and how to translate the graph of to the graph of f. I would then ask, so what is the value of the root form

(which they learned to derive in Question 2)? I would expect students to say: it tells us the roots of the quadratic equation, which are the x-intercepts, as well as the zeros, of the quadratic function.

In this way, students can see that the vertex form and root form provide them with all there is to know about the graph of a quadratic function. However, it’s the standard form that immediately tells us the quadratic function is a member of the broader family of polynomial functions. Therefore, each form tells us something about the function but not everything (the vertex form doesn’t immediately give us the roots, the root form doesn’t immediately give us the vertex, the standard form doesn’t tell us much about how the graph looks).

To summarize, the cases we show exemplify precision, coherence, and purposefulness, key attributes of a deep content understanding for teaching K-12 students. Throughout, these responses demonstrate a consistent effort at emphasizing key mathematical ideas, the logical progression of mathematical concepts, and the connectedness among different concepts, procedures, and ideas. In addition, responses show careful attention to scaffolding ideas to students in a systematic and coherent way: from simple to complex, from specific examples to general cases.