AERA Philadelphia 2014: Supporting Common Core-Driven Curriculum Adaptations for High School Algebra

Annual Meeting - Sunday, April 6, 12:25 pm

+Raymond Johnson (@MathEdnet) - School of Education, University of Colorado Boulder and Freudenthal Institute US
+Heather Leary (@kolorkid) - Institute of Cognitive Science, University of Colorado Boulder
+Bill Penuel (@bpenuel) - School of Education, University of Colorado Boulder

Abstract: The adoption of the Common Core State Standards for Mathematics (CCSSM) has created a need for many teachers and school districts to adapt their current curricular materials. Using the methods of design-based implementation research (Penuel et al., 2011), this project partnered with high school algebra teachers, district curriculum staff, and university researchers to support teachers in the selection and use of high-quality mathematical tasks. This participatory design process yielded a set of principles for task analysis that considered qualities of CCSSM alignment, cognitive demand, language, and technology. Results indicate a need for careful guidance in task rating, attention to teachers’ desires to further modify tasks, and possible benefits for task implementation.

This poster broadly summarizes the first year of an ongoing project that brings together education researchers, web developers, school district curriculum supervisors, and a team of district algebra teachers to support Algebra 1 teachers' curriculum adaptations to meet the new demands of the Common Core State Standards. To give a little context to the poster, you can see our research questions address not only the what of curriculum adaptation, but the how - ideally, we want to theorize ways these kinds of collaborations can work and a process other districts can follow to bring these kinds of curriculum changes to scale.

The perspective we take on our research is design-based implementation research, or DBIR. You can learn more about DBIR at learndbir.org. Design research has typically been carried out at the classroom level, and DBIR takes much of that thinking to a larger educational system. Too often professional development is only applied to teachers who struggle to implement new ideas and tools without broader institutional support. DBIR tries to change that by engaging stakeholders throughout the system to tackle persistent problems of practice in a way that is scalable and sustainable.

Design work with curriculum has been theorized by McKenney, Nieveen, and van den Akker (2006) to focus on iterative cycles of work towards the development of three products: design principles, curricular products, and professional development. These are in order of importance and reflect our values in this project. Our first priority is not a higher quality curriculum, although that is a strongly desired outcome. But to reach that outcome, our highest priority is giving teachers a set of principles for evaluating the quality of the materials they use and choose from every day. With these principles in place, quality resources to choose from, and a professional development program to tie the two together, we aim to create curriculum reform in a way that's sustainable. Two web-based products have supported this work: first, the Task Rating site, where teachers evaluate tasks given our co-designed principles. The rating data collected in this site is used in professional development for reflection and debate. The data also informs our second web tool, the Curriculum Customization Service. This site catalogs the high-quality mathematical tasks identified in the rating process alongside digital versions of the district curriculum as well as resources from NSDL, the National Science Digital Library. Teachers with accounts on the service can save resources into playlists for lesson and unit plans and use an uploading and sharing tool to share their curriculum with colleagues.

Our findings are preliminary. First, we found that mathematical tasks acted as a boundary object (Star & Griesemer, 1989; Star, 2010), an object around which we could organize our work despite each community (researchers, district supervisors, teachers, and web developers) having somewhat different perspectives on the role of mathematical tasks.

We found some tensions in our first year of work. A persistent tension related to the modification of mathematical tasks. Although the goal of the project was to adapt the curriculum, there was an effort taken to have teachers consider tasks as written. This was difficult for teachers due to another tension, their consideration of tasks given the unique contexts of their classrooms. While we value those contexts, rating tasks using the design principles was difficult to do consistently when teachers saw not the task as written, but a different version of the task that they would likely enact. Lastly, the most difficult design principles to develop were those that evaluated the language used in tasks. Through multiple revisions, rubrics for rating task language incorporated more structure from both research and an understanding of ongoing district effort to support language learning.

At year's end, we had collectively rated 40 tasks, most of which were cataloged in the Curriculum Customization Service. We also looked for consistency in ratings. Rater agreement varied depending on the principle applied to the task and the task itself. Preliminary measures indicated raters were generally in 60% agreement in their rating of cognitive demand, and alignment to the Common Core varied between 50%-80%. This work to measure agreement is ongoing and do not support any conclusions as yet.

The second year of the project has focused on supporting the implementation of tasks, using ideas like the task launch (Jackson et al., 2012) and a focus on promoting quality classroom discussions. We expect our work next year will be the difficult work of scaling, within the district to other Algebra 1 teachers and possibly to those in other partner districts.

Acknowledgement: This work was supported by a grant from the National Science Foundation (Award #1147590). The opinions expressed herein are those of the authors and do not necessarily reflect those of the NSF.

References

Jackson, K. J., Shahan, E. C., Gibbons, L. K., & Cobb, P. (2012). Launching complex tasks. Mathematics Teaching in the Middle School, 18(1), 24–29.

McKenney, S. E., Nieveen, N., & van den Akker, J. (2006). Design research from a curriculum perspective. In J. van den Akker, K. Gravemeijer, S. E. McKenney, & N. Nieveen (Eds.), Educational design research (pp. 67–90). New York, NY: Routledge.

Penuel, W. R., Fishman, B. J., Cheng, B. H., & Sabelli, N. (2011). Organizing research and development at the intersection of learning, implementation, and design. Educational Researcher, 40(7), 331–337. doi:10.3102/0013189X11421826

Star, S. L. (2010). This is not a boundary object: Reflections on the origin of a concept. Science, Technology, & Human Values, 35(5), 601–617. doi:10.1177/0162243910377624

Star, S. L., & Griesemer, J. R. (1989). Institutional ecology, “translations” and boundary objects: Amateurs and professionals in Berkeley’s Museum of Vertebrate Zoology, 1907-39. Social Studies of Science, 19(3), 387–420. doi:10.1177/030631289019003001

NCTM Denver 2013: Danielson's They'll Need it for Calculus

Annual Meeting - Saturday, April 20, 8:00 am

Christopher Danielson - Normandale Community College, Bloomington, Minnesota

As Steve Leinwand noted in his Thursday talk, math teachers are a relatively conservative, risk-averse bunch. Perhaps our conservatism comes from the perceived slow but steady progress of math over millennia where it's easy to take comfort in the old because the new can seem so difficult to obtain. Some of this rubs off in the way we teach, the activities we choose for students, and our judgement about what's important for students to know.

Chris Danielson's session kicked off by calling out some mathematics that gets taught in the name of "needing it for calculus," despite no widespread need for it anymore. Simplifying radicals. Rationalizing the denominator. Simplifying rational expressions. Factoring quadratics. Composition of functions. The binomial theorem. It's not that someone, somewhere doesn't have a use for these things, but what is increasingly becoming the exception should not prove the curriculum rule. Mediocre proficiency with these topics is not what leads students to be successful in calculus. What students really need for calculus is a deep understanding of slope as a rate of change and accumulation.

Christopher Danielson

This is a familiar story for some of us. We cringe when we ask students "What's slope?" and they parrot back, "rise over run" without knowing much beyond that. Yes, that might be one way to describe slope, but there are other, and arguably more important ways to describe slope. Danielson's focus on slope as a rate of change not only is most fundamental for calculus, but it is in alignment with the research on teaching slope (Lobato & Thanheiser, 2002; Peck & Matassa, 2012; Stump, 1999, 2001).

Danielson led the well-attended workshop through a number of middle-school appropriate tasks involving rates of change. Because the tasks were set in informal contexts, students would be most likely to work in terms of "dollars per bicycle rental" or "enjoyment per piece of candy," depending on the context of the problem. Time was spent not just looking at rates, but doing simple calculations to compare changes in rates over time, a fundamental conception needed for calculus. The problems in the workshop were adapted from tasks found in Connected Mathematics, a popular NSF-funded curriculum for the middle grades.

For Christopher's presentation, related tweets, and participant notes, see his post at http://christopherdanielson.wordpress.com/2013/04/21/the-goods-nctmdenver/.

References

Lobato, J., & Thanheiser, E. (2002). Developing understanding of ratio-as-measure as a foundation for slope. In B. H. Litwiller (Ed.), Making sense of fractions, ratios, and proportions (pp. 162–175). Reston, VA: NCTM.

Peck, F., & Matassa, M. (2012). Beyond “rise over run”. RME in the classroom. Workshop at ICME-12, Seoul, South Korea. Retrieved from http://rmeintheclassroom.blogspot.com/2012/07/icme-12-workshop-and-sharing-group-on.html

Stump, S. L. (1999). Secondary mathematics teachers’ knowledge of slope. Mathematics Education Research Journal, 11(2), 124–144. Retrieved from http://www.springerlink.com/index/R422558466765681.pdf

Stump, S. L. (2001). High school precalculus students’ understanding of slope as measure. School Science and Mathematics, 101(2), 81–89. doi:10.1111/j.1949-8594.2001.tb18009.x

NCTM Denver 2013: Hirsch's Mathematical Modeling: The Core of the Common Core State Standards

Annual Meeting - Friday, April 19, 2:00 pm

Christian R. Hirsch - Western Michigan University

Hirsch might claim that modeling is at the core of the Common Core, but at a glance it looks like a standard without standards. Yes, the fourth Standard for Mathematical Practice is "Model with mathematics," but the high school content standards chooses to mark standards in other domains as related to modeling instead of grouping the modeling standards together. This makes it more difficult to see the modeling connections across the high school standards, but that shouldn't reduce their importance.

Christian Hirsch has been at Western Michigan for 40 years and he is probably best recognized as the principal investigator for the Core-Plus Mathematics Project. Along with IMP, Core-Plus is one of the most recognized secondary, integrated, NSF-funded curricula to come out of the post-Standards curriculum development period in the 1990s.

Christian Hirsch

Hirsch opened his talk by detailing how all of the mathematical practices can be addressed with a modeling-focused framework of curriculum and instruction. "Real world problems, if even solvable, take a lot of time and perseverance." To Hirsch, Standards for Mathematical Practice 1 and 4 are the focal points of the entire process, at least in classrooms with good instruction. "I'm talking about classrooms where classes begin with problems. I'm not talking about classrooms where the problems are saved until the end."

The key to modeling and making mathematics problematic, says Hirsch, is to identify problems in context, study those problems through active engagement, and reach conclusions as the problems are at least partially solved. The learning lies not only in the solutions to the problems, but the new mathematical relationships that are discovered along the way.

Hirsch used several examples of problems involving modeling in this presentation. The first dealt with the business prospects of a climbing gym. Assuming a survey had been conducted that found the number of expected climbers is related to price \(x\) by the equation \(n(x) = 100 - 4x\), how many daily climbing wall customers should the gym expect? I didn't catch all the details of this problem, but the next question involved finding the optimal and break-even revenue points for the gym, which is nicely modeled by a quadratic. Hirsch advocated using a computer algebra system to assist with the calculations, and advised to help students realize that rounding to the nearest cent, if necessary, also slightly moves answers away from their true zeroes or maximums.

Hirsch's next problem dealt with finding the optimal location for an oil refinery with wells 5 km and 9 km from shore. I sense that this and the previous problem are in Core-Plus, but unfortunately that wasn't made clear and no handouts or downloads for this talk have been provided. While I don't like leaving presentations early, at this point I had a pretty good sense for this one and left to catch an overlapping presentation starting at 2:45. The problems Hirsch chose and the approaches to solve them were pretty solid 30 or more years ago and are still pretty solid today, and I wasn't feeling like the presentation was suddenly going to break new ground. (For me, at least. I totally understand that problems and approaches like this might be new ground in many classrooms.)

NCTM Denver 2013: Abels, Matassa, & Johnson's Making Sense of Algebra with Realistic Mathematics Education

Annual Meeting - Thursday, April 18, 2:45 pm

Mieke Abels - Freudenthal Institute for Science and Mathematics Education, University of Utrecht
+Michael Matassa Jr. - Freudenthal Institute US, University of Colorado Boulder
+Raymond Johnson - Freudenthal Institute US, University of Colorado Boulder

When it came time to propose session for the 2013 NCTM Annual Meeting in nearby Denver, we at the Freudenthal Institute US at CU-Boulder knew we should have some kind of "Intro to RME" workshop. Because I was already proposing to be a lead speaker on another session, I needed to find someone else to take the lead. Michael Matassa said he would do it, but then +David Webb had a better idea: Why not ask Mieke Abels from the Freudenthal Institute to do it? Mieke would be a perfect choice - she's been involved in FIUS from the beginning and she continues to be involved in curriculum development for Mathematics in Context and curriculum in the Netherlands. Happily, Mieke agreed and Michael and I were happy to back her up as co-presenters.

The picture at the top is Nederland, CO, which is amusing to our Dutch colleagues

Our goal in this presentation was to bring out the curriculum design features and give attendees a sense for informal and preformal approaches to algebra for the middle grades. Too often it seems "early algebra" gets interpreted as "algebra early," as if a school could just box up their high school algebra textbooks and ship them down to the middle school. Making big jumps to formal mathematics is risky, and that's one reason Realistic Mathematics Education (RME) adheres to a principle called progressive formalization. To illustrate, we started with a task you could give to 6th graders, or perhaps even younger students.

Tug-of-war, taken from Mathematics in Context

Those of us who have mastered formal algebra tend to want to write equations for this and solve. But for young students, RME design principles suggest we support students by relying on a "realistic" context. While "real-world" contexts are certainly realistic, RME's use of "realistic" means it can be imagined by the learner. The power of the context is not necessarily its authenticity, but its capacity to be mathematized.

On the tug-of-war task students will inevitably find different ways to substitute different animals for each other until it becomes clear which side would win the tug-of-war. Some students will likely try to redraw the animals, while others might use letters ("E" for elephant, etc.) as a substitute. Even though a formal equation might use "E" to represent the pulling strength of an elephant, it's fully expected at this stage for students to try writing things like "E = O + 2H" to represent the animals in the middle of the above slide, and interpret "2H" simply as the abbreviation "2 horses."

The Iceberg Metaphor

The concept of progressive formalization is often represented with the iceberg metaphor (Boswinkel & Moerlands, 2003; Webb, Boswinkel, & Dekker, 2008), which places formal mathematics above the water line. The tip of the iceberg is only supported because of the iceberg's "floating capacity, which is where informal and preformal mathematics is placed. Examples like the tug-of-war problem are informal because they rely almost entirely on the realistic context with little or no mathematical abstraction.

Here's another example of an informal task:

Three Frogs, taken from Mathematics in the City

Again, the frog jumping doesn't have to be something replicable in the real world. It need only exist in the imagination of the student, and to solve it students need to find ways to represent the jumps and steps in their work. At this point students will have worked often with number lines (including open number lines) and easier problems involving frog jumping, making number lines a natural model for this problem, like this:

Using an open number line to represent Sunny's jumps

The nature of this problem and the need to draw double number lines that end in a particular place helps students consider what it means to be variable in this problem, versus what quantities remain constant. Other types of problems with other contexts use other kinds of models. For example, the familiar model of a balance is used in RME-based curricula (the 1 and 5 represent weights):

The balance model, found in the Digital Mathematics Environment

Student work for these kinds of tasks can be an indicator of where in the formalization process students might be. If students are redrawing pineapples and lemons, they are still working at an informal level. For convenience they might replace pineapples and lemons with letters, suggesting a small amount of formalization, and eventually they'll be using those letters to write equations and not need to think in terms of the fruit and the balance. Just like tug-of-war, the balance model suggests an understanding of equals that is relational, which helps students who tend to interpret equals as operational.

The use of models is at the heart of the preformal level of the iceberg. Nearer the bottom we would place models of informal contexts. For example, a student who draws sectors of a circle to represent a fraction of a pizza is using the circle as a model of the pizza. Nearer the top of the preformal level we would place models for mathematical abstraction. The student who uses sectors of a circle to represent a fraction of seats occupied in a bus is using the sectors of the circle as a generalized representation of a part and whole, and not a specific representation of a bus. In RME students become familiar with many models, such as number lines, open number lines, double number lines, ratio tables, five frames, rekenreks, area models, and balance models.

Models are key at the preformal level of the iceberg

Another preformal model useful for systems of equations is notebook notation. Here a problem shows two combinations of long and short candles. Working informally, students would find some combination of candles that makes the problem solvable, such as doubling the second combination to make two long candles and two short candles for $6.80. Since the top arrangement has one more short candle and is a dollar more, then short candles must cost $1.00. A preformal way of working with these combinations is notebook notation:

Notebook Notation, taken from Mathematics in Context

From the notebook it becomes easier to see how students will learn to write and manipulate formal systems of equations. The column headings become the variables, and equal signs are placed in front of the total. Formal matrix notation is reachable from notebook notation as well.

Although progressive formalization is often presented as a direct informal-to-preformal-to-formal process, it is not expected that students will learn this way. Students who can work formally or preformally with easier problems are likely to reach to a lower level when problems become more difficult. Because they have achieved the formalization with easier problems, they become more likely to formalize more difficult problems when they can reason with less formal strategies when necessary.

Another way to view progressive formalization is with a learning trajectory, which connects specific contexts and representations along a path towards formal mathematics. Creation of both iceberg models and learning trajectories can be a productive activity for professional development and curriculum planning and alignment.

A learning trajectory for equations and systems of equations, with connecting links

RME isn't meant to be deeply complex, but contexts, models, and the connections between them need to be carefully chosen. Curriculum developers at the Freudenthal Institute take a design research approach to this work, testing and revising in iterative cycles to improve the curriculum over time. FI (formerly IOWO) was founded by Hans Freudenthal in 1971, giving the Netherlands over 40 years to gradually improve their mathematics curriculum and teaching. This type of adherence to a core philosophy for so long is generally unknown in education in the U.S., but schools in the Netherlands have used it to score near the top of international rankings on the mathematics portion of the PISA assessment.

If you'd like more information about RME, the following might be of interest:

• Mathematics in Context (MiC) - units used here were "Comparing Quantities" and "Graphing Equations"
• Mathematics in the City (MiTC) - unit used here was "The California Frog-Jumping Contest"
• Contexts for Learning Mathematics
• Digital Mathematics Environment (DME) - for balance problems, login as guest then Math Secondary Education -> Algebra -> Lesson - System of Equations (Netherlands-based website, more information)
• 4th International RME Conference, September 27-29, 2013, in Boulder, CO - see fius.org
• Utrecht Summer School on Mathematics and Science Education, August 19-30, 2013
• Paper: From Informal to Formal, Progressive Formalization an Example on "Solving Systems of Equations" by Martin van Reeuwijk (citation below)
• Link to slides (pdf, pptx) and handouts

References

Boswinkel, N., & Moerlands, F. (2003). Het topje van de ijsberg [The top of the iceberg]. De Nationale Rekendagen, een praktische terugblik [National conference on arithmetic, a practical view] (pp. 103–114). Utrecht, The Netherlands: Freudenthal Institute. Retrieved from http://www.fisme.science.uu.nl/publicaties/literatuur/5467.pdf

Van Reeuwijk, M. (2001). From informal to formal, progressive formalization an example on “solving systems of equations”. In H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.), The future of teaching and learning of algebra: The 12th ICMI study conference (pp. 613–620). Melbourne, Australia. Retrieved from http://repository.unimelb.edu.au/10187/2812

Webb, D. C., Boswinkel, N., & Dekker, T. (2008). Beneath the tip of the iceberg: Using representations to support student understanding. Mathematics Teaching in the Middle School, 14(2), 110–113. Retrieved from http://www.nctm.org/publications/article.aspx?id=20793.

NCTM Denver 2013: Herbst et al's Methods to Study Decisions in Mathematics Teaching

Research Pressession - Tuesday, April 16, 3:00 pm

Pat Herbst - University of Michigan
Daniel Chazan - University of Maryland
+Karl Kosko - Kent State University
+wendy rose aaron - Oregon State University
Justin Dimmel - University of Michigan
Orly Buchbinder - University of Maryland
Ander W. Erickson - University of Michigan

There are many things about teaching that can be described, but some very important things can be very difficult to measure. This group is working on creating an infrastructure and set of methods to make quantitative claims about phenomena in teaching, such as teacher decisions, recognition of norms and obligations  mathematical knowledge for teaching, and teacher beliefs. Developing these methods should help get the field beyond "what works" to a place of "how things work," ideally resulting in a model where we can better justify certain teaching practices and predict the results of their use. In addition to exploring interventions in classrooms, or working with teachers watching videos of classrooms, the group is using LessonSketch, a cartoon-like technology that allows users to quickly create and/or watch classroom scenarios.

There are several aspects to this research. First is looking at the breaching of norms. Using LessonSketch, teachers watched animations of classroom practice and "pinned" instances of norms being breached. The group found that participants were more likely to pin and remark on actions that breached norms, with more experienced teachers being more adept at identifying the breaches.

Wendy Aaron, Pat Herbst, and Justin Dimmel (right to left)

Using similar tools, the group also looked at teachers' preferences toward different student strategies, evidence for hypothesized instructional norms, and different kinds of professional obligations. The group found that when a norm was breached, teachers' justifications for the breach could be categorized as: (a) disciplinary, (b) institutional, (c) individual, or (d) interpersonal. The group considered these categories as part of the collective knowledge of the teaching profession, even though teachers' obligations may not be indicative of personal acceptance.

+Karl Kosko dug into some of the measurement of decision making. Teachers were provided scenarios of teaching geometry that ended at key points, followed by four potential actions designed to assess the likelihood of norm compliance. Using multinomial regression, the group found that teachers' pedagogical content knowledge for geometry didn't have any measurable effect on decision making in this study, but years of experience and perceived autonomy did have an effect.

In the conclusion, Dan Chazan hoped that work such as this will move teaching practice away from being perceived as an independent variable that determines student achievement. Instead, teaching will be an "achievement verb," something that is dependent on a host of input variables that shape teaching practice.

NCTM 2013 Denver: Superfine's et al's Supporting Underprepared Algebra Students: Results from a Design-Based Research Program

Research Presession - Tuesday, April 16, 1:00 pm

Alison Castro Superfine - Learning Sciences Research Institute, University of Illinois at Chicago
James Lynn - Learning Sciences Research Institute, University of Illinois at Chicago
Timothy Stoelinga - Learning Sciences Research Institute, University of Illinois at Chicago
Mara Martinez - Learning Sciences Research Institute, University of Illinois at Chicago
Cynthia Schneider - Charles A. Dana Center, University of Texas at Austin
Diane Briars - Pittsburgh, Pennsylvania
Discussant: Phil Daro, Public Forum on School Accountability, San Francisco

Algebra 1 continues to be a make-or-break-it course for many high school students. Some 75% of students who fail Algebra 1 will also fail on a 2nd attempt, or to get them through they will be placed into a pseudo-algebra course that grants credit without the rigor and support.

This project arose from the need to find a different way to deal with struggling algebra students. With NSF, Chicago Community Trust, and Gates Foundation funding, this group worked as a design team consisting of 25 researchers, curriculum developers, and practitioners to develop and oversee an intensified algebra program where students take algebra two periods each day. The key partner was the Dana Center at Texas at Austin, and you can find the project website at http://www.utdanacenter.org/intensifiedalgebra/index.php. Much of what was described in the presentation can be found in the key design features and course and lesson structure.

Themes from the literature

The model for curriculum selection is one that I believe is quickly becoming familiar. Tasks are rated by cognitive demand (Stein, Smith, Henningsen, & SIlver, 2009) and teachers are evaluated on how they implement the task, including the "launch" (I'm guessing they're using the work of Jackson, Shahan, Gibbons, & Cobb, 2012) and investigation and debrief phases of the lesson.

As design research, revisions are frequently made and evaluated. This group placed a focus on fidelity, or the extent to which practice follows the intent of the curriculum. Making a total of 58 observations of five teachers in three schools, the group looked for ways they could better support teachers' enactment of tasks with high cognitive demand.

Alison Castro Superfine presenting

Two trends stood out to the researchers. First, they were far more likely to see teachers explain ideas instead of students. Also, use of the AgileMind technology (which was part of the curriculum) varied significantly across the teachers.

While the project and revision process is ongoing, there are signs of positive results. On problems related to linear functions, students in the project showed significant growth, although the percentage correct on these problems still hovered around 50%.

As for how the research has informed the design process, the group observed three challenges: (a) the needs and time cycles of curriculum developers and researchers, (b) working in settings of implementation, and (c) contributing to theories of practice. This group appeared to be applying Gravemeijer's (2004) concept of a local instruction theory to inform their design process, although no author was cited in the presentation and they used local instruction theory in a somewhat different context.

The discussant, Phil Daro, praised the group for their decision to take a design approach to this problem. He questioned a statement in the presentation about not being able to expect high quality enactment of a low quality task, saying this has been documented from successful classrooms in Japan. Daro also liked the group's not-to-strict view of fidelity: "Implementation research is cursed by the idea of fidelity. I'd like to see fidelity go away." Lastly, Daro said this kind of work and partnership is the kind of work that should go on for 20-30 years, creating quality from within. "Don't design something and tell teachers that it's good for them. Develop empathy for teachers first and then give them something they'll think is cool."

References

Gravemeijer, K. (2004). Local instruction theories as means of support for teachers in reform mathematics education. Mathematical Thinking and Learning, 6(2), 105–128. doi:10.1207/s15327833mtl0602_3

Jackson, K. J., Shahan, E. C., Gibbons, L. K., & Cobb, P. (2012). Launching complex tasks. Mathematics Teaching in the Middle School, 18(1), 24–29.

Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2009). Implementing standards-based mathematics instruction: A casebook for professional development (2nd ed., p. 182). New York, NY: Teachers College Press.

Common Core Conundrum: Absolute Value

I should begin by first disclaiming that while I'm generally pro-standards, I'm also somewhat agnostic about standards. How can that be? Without going into much detail, I believe (a) what we "count" as mathematics is socially determined and standards documents are just part of that determination, (b) I will never perfectly agree with a standards document, but the amount of agreement should not be underestimated, and (c) if there's any power to standards, it's in how they're implemented -- and "good implementation" is very likely to be seen as just "good teaching" under a different set of standards, or no standards at all. It is with this mindset that I watch with some amusement (and sometimes, disappointment) arguments against the Common Core State Standards because of some poorly designed accountability measure. I'm pretty sure poorly designed accountability measures would be a concern right now regardless of the standards in place.

Still, I occasionally see things in the Common Core math standards (CCSSM) that make me stop and wonder, "How are teachers going to deal with this?" One recent instance of that was with how the CCSSM addresses the topic of absolute value. As I looked through Discovering Algebra: An Investigative Approach, I saw the typical "distance from zero" definition followed by an investigation that included this rather unhelpful picture and caption:

(Clearly portrays Elvis? The middle guy looks like Matthew Perry joined a Vegas lion-taming act.)

From there, Discovering Algebra presents students with equations to solve, such as:

$$ | x - 2| + 7 = 12 $$

But do the CCSSM call for solving these kinds of equations? I went searching through the CCSSM for all mentions I could find of absolute value, and here's what I found.

Grade 6

CCSS.Math.Content.6.NS.C.7 Understand ordering and absolute value of rational numbers.
CCSS.Math.Content.6.NS.C.7a Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. For example, interpret –3 > –7 as a statement that –3 is located to the right of –7 on a number line oriented from left to right.
CCSS.Math.Content.6.NS.C.7b Write, interpret, and explain statements of order for rational numbers in real-world contexts. For example, write –3 oC > –7 oC to express the fact that –3oC is warmer than –7 oC.
CCSS.Math.Content.6.NS.C.7c Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. For example, for an account balance of –30 dollars, write |–30| = 30 to describe the size of the debt in dollars.
CCSS.Math.Content.6.NS.C.7d Distinguish comparisons of absolute value from statements about order. For example, recognize that an account balance less than –30 dollars represents a debt greater than 30 dollars.
CCSS.Math.Content.6.NS.C.8 Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.

and

CCSS.Math.Content.6.SP.B.5c Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.

If all goes according to plan, there shouldn't be a need to define absolute value in 8th or 9th grade algebra, as it will already have been introduced in 6th grade. I know there are concerns about content being "developmentally inappropriate" for lower grade levels in the CCSSM, but without any personal data to the contrary, I imagine students could come to understand absolute value along with understanding positive and negative numbers. The last standard above, from the moment I first saw it, has been a point of fascination for me. How will 6th grade teachers help students learn topics like interquartile range and mean absolute deviation? You need absolute value for mean absolute deviation, but I don't think that will be the hangup for those who struggle with that standard.

So where else is absolute value mentioned?

Grade 7

CCSS.Math.Content.7.NS.A.1c Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.

and

CCSS.Math.Content.7.SP.B.3 Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable.

Now we have absolute value as it relates to subtraction, and we get another mention of mean absolute deviation in the 7th grade statistics and probability standards. There's nothing yet about solving equations with absolute value or graphing absolute value functions.

Grade 8

Nothing. No mention of absolute value or mean absolute deviation.

High School

The CCSSM doesn't specify what specifically belongs in 9th grade versus other grades, but here's what I found across the whole of the HS CCSSM standards.

CCSS.Math.Content.HSN-VM.C.12 (+) Work with 2 × 2 matrices as a transformations of the plane, and interpret the absolute value of the determinant in terms of area.

CCSS.Math.Content.HSA-REI.D.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

CCSS.Math.Content.HSF-IF.C.7b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

The first of the three standards above, I think we can agree, goes well beyond what we'd expect to find in a traditional Algebra 1 class. The other two standards address the graphing of various kinds of functions, and depending on the activity might belong in either a traditional Algebra 1 or Algebra 2 course, or more likely, both. But do you notice what's not there? Solving equations with absolute value functions. Sorry, \( | x - 2 | + 7 = 12 \), but it looks like you didn't get invited to the CCSSM party. Even if you interpret the second standard to include solving absolute value equations, it only asks for an approximation or to use technology.

Conundrum

The researcher in me wonders what Algebra 1 teachers will do when they get to a lesson on absolute value. Will some just teach it out of habit? Will some think it was excluded due to a CCSSM oversight? Will some modify the lesson to emphasize the graphing of absolute value, but not solving absolute value equations? How many will not even notice it went missing from the standards? Will anyone be willing to kill the little darling?

For years we've lamented math curricula in the United States that was "a mile wide and an inch deep." Given a fixed amount of time, going deeper means going narrower, and there is evidence that the CCSSM supports that (Porter, McMaken, Hwang, & Yang, 2011). For reasons I perceive to be mostly political, the Common Core State Standards don't go out of their way to specifically address content omitted compared to previous standards documents. In 1989, NCTM made a list of areas of emphasis and de-emphasis and many an argument in the math wars was waged using false dichotomies rooted in those lists. I'm sure those who wrote the CCSSM didn't want a repeat of those arguments, so now math teachers and those who support them are left to dig through the standards and find these omissions on their own.

(By the way, if you're looking for an interesting lesson for understanding and graphing absolute value, NCTM is offering one from a recent Mathematics Teacher for complimentary download: Angela Wade's "Teaching Absolute Value Meaningfully.")

References

Porter, A. C., McMaken, J., Hwang, J., & Yang, R. (2011). Common Core standards: The new U.S. intended curriculum. Educational Researcher, 40(3), 103–116. doi:10.3102/0013189X11405038

RYSK: Ball, Thames, & Phelps's Content Knowledge for Teaching: What Makes It Special? (2008)

This is the 17th in a series describing "Research You Should Know" (RYSK) and part of my OpenComps. I also Storified this article as I read.

My last two posts summarized the underpinnings of Shulman's pedagogical content knowledge and Deborah Ball's early work building upon and extending Shulman's theories. Now we jump from Ball's 1988 article to one she co-authored in 2008 with University of Michigan colleagues Mark Thames and Geoffrey Phelps, titled Content Knowledge for Teaching: What Makes It Special?

This article starts by looking at the 20+ years we've had to further develop Shulman's theories of pedagogical content knowledge (PCK). Despite the theory's widespread use, Ball and colleagues claim it "has lacked definition and empirical foundation, limiting its usefulness" (p. 389). (See also Bud Talbot's 2010 blog post and related efforts.) In fact, the authors found that a third of the more than 1200 articles citing Shulman's PCK

do so without direct attention to a specific content area, instead making general claims about teacher knowledge, teacher education, or policy. Scholars have used the concept of pedagogical content knowledge as though its theoretical founcations, conceptual distinctions, and empirical testing were already well defined and universally understood. (p. 394)

To build the empirical foundation that PCK needs, Ball and her research team did a careful qualitative analysis of data that documented an entire year of teaching (including video, student work, lesson plans, notes, and reflections) for several third grade teachers. Combined with their own expertise and experience, and other tools for examining mathematical and pedagogical perspectives, the authors set out to bolster PCK from the ground up:

Hence, we decided to focus on the work of teaching. What do teachers need to do in teaching mathematics -- by virtue of being responsible for the teaching and learning of content -- and how does this work demand mathematical reasoning, insight, understanding, and skill? Instead of starting with the curriculum, or with standards for student learning, we study the work that teaching entails. In other words, although we examine particular teachers and students at given moments in time, our focus is on what this actual instruction suggests for a detailed job description. (p. 395)

For Ball et al., this includes everything from lesson planning, grading, communicating with parents, and dealing with administration. With all this information, the authors are able to sharpen Shulman's PCK into more clearly defined (and in some cases, new) "Domains of Mathematical Knowledge for Teaching." Under subject matter knowledge, the authors identify three domains:

• Common content knowledge (CCK)
• Specialized content knowledge (SCK)
• Horizon content knowledge

And under pedagogical content knowledge, the authors identify three more domains:

• Knowledge of content and students (KCS)
• Knowledge of content and teaching (KCT)
• Knowledge of content and curriculum

Ball describes each domain and uses some examples to illustrate, mostly from arithmetic. For my explanation, I'll instead use something from high school algebra and describe how each domain applied to my growth of knowledge over my teaching career.

Common Content Knowledge (CCK)

Ball et al. describe CCK as the subject-specific knowledge needed to solve mathematics problems. The reason it's called "common" is because this knowledge is not specific to teaching -- non-teachers are likely to have it and use it. Obviously, this knowledge is critical for a teacher, because it's awfully difficult and inefficient to try to teach what you don't know yourself. As an example of CCK, my knowledge includes the understanding that \((x + y)^2 = x^2 + 2xy + y^2\). I've known this since high school, and I would have known it whether or not I became a math teacher.

Specialized Content Knowledge (SCK)

SCK is described by Ball et al. as "mathematical knowledge and skill unique to teaching" (p. 400). Not only do teachers need this knowledge to teach effectively, but it's probably not needed for any other purpose. For my example, I need to have a specialized understanding of how \((x+y)^2\) can be expanded using FOIL or modeled geometricaly with a square. It may not be all that important for students to understand both the algebraic and geometric ways of representing this problem, but I need to know both so I can better understand student strategies and sources of error. Namely, the error that \((x + y)^2 = x^2 + y^2\).

Horizon Content Knowledge

This domain was provisionally included by the authors and described as, "an awareness of how mathematical topics are related over the span of mathematics included in the curriculum" (p. 403). For my example of \((x + y)^2 = x^2 + 2xy + y^2\), I need to understand how previous topics like order of operations, exponents, and the distributive property relate to this problem. Looking forward, I need to understand how this problem relates to factoring polynomials and working with rational expressions.

Knowledge of Content and Students (KCS)

This is "knowledge that combines knowing about students and knowing about mathematics" (p. 401) and helps teachers predict student thinking. KCS is what allows me to expect students to incorrectly think \((x + y)^2 = x^2 + y^2\), and to tie that to misconceptions about the distributive property and exponents. I'm not sure I had this knowledge for this example when I started teaching, but it didn't take me long to figure out that it was a very common student mistake.

Knowledge of Content and Teaching (KCT)

Ball et al. say KCT "combines knowing about teaching and knowing about mathematics" (p. 401). While KCS gave me insight about why students mistakingly think \((x + y)^2 = x^2 + y^2\), KCT is the knowledge that allows me to decide what to do about it. For me, this meant choosing a geometric representation for instruction over using FOIL, which lacks the geometric representation and does little to address the problem if students never recognize that \((x + y)^2 = (x + y)(x + y)\).

Knowledge of Content and Curriculum

For some reason, Ball et al. include this domain in a figure in their paper but never describe it explicitly. They do, however, scatter enough comments about knowledge of content and curriculum to imply that teachers need a knowledge of the available materials they can use to support student learning. For my example, I know that CPM uses a geometric model for multiplying binomials, Algebra Tiles/Models can be used to support that model, virtual tiles are available at the National Library of Virtual Manipulatives (NLVM), and the Freudenthal Institute has an applet that allows students to interact with different combinations of constants and variables when multiplying polynomials.

Some of the above can be hard to distinguish, but thankfully Ball and colleagues clarify by saying:

In other words, recognizing a wrong answer is common content knowledge (CCK), whereas sizing up the nature of an error, especially an unfamiliar error, typically requires nimbleness in thinking about numbers, attention to patterns, and flexible thinking about meaning in ways that are distinctive of specialized content knowledge (SCK). In contrast, familiarity with common errors and deciding which of several errors students are most likely to make are examples of knowledge of content and students (KCS). (p. 401)

In their conclusion, the authors hope that this theory can better fill the gap that teachers know is important, but isn't purely about content and isn't purely about teaching. We can hope to better understand how each type of knowledge above impacts student achievement, and optimize our teacher preparation programs to reflect that understanding. Furthermore, that understanding could be used to create new and improved teaching materials and professional development, and better understand what it takes to be an effective teacher. With this in mind, you can gain some insight to what Ball was thinking when she gave this congressional testimony:

References

Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389–407. doi:10.1177/0022487108324554

A First Day Statistics Activity

I have the honor of again teaching our undergraduate statistics course in the School of Education, better known here as EDUC 4716 Basic Statistical Methods. Perhaps the most interesting thing about the course is that it's not required for any education programs, minors, or certificates. Instead, the course attracts students largely from the Department of Speech, Language, and Hearing Sciences (who don't need it to graduate, but do need it to apply to grad school or, more recently, to get certified) and sociology majors. So how does this course end up in the School of Ed? Probably due to the legacy we have in quantitative methods, thanks to people like Robert Linn, Gene Glass, Lorrie Shepard, and now faculty like Derek Briggs and Greg Camilli. Somehow all of their hard work and success filters down and gives a relative stats-hack like me a chance to teach undergrads.

Many of my students are upperclassmen and have spent much of their college experience avoiding math courses. In fact, on last year's FCQ (Faculty Course Questionnaire) my students' average rating for the item "Personal interest in this subject prior to enrollment" was a 1.8 out of 6 -- a response the university tells me is at the 0th percentile across campus. I like to think of this as a great opportunity in a "nowhere to go but up" kind of way, a chance for me to change the way students think of mathematics and see themselves as mathematical beings. Then again, it's hard to make big changes in only 15 class meetings of 2.5 hours each. If I'm going to make a difference, class has to get off to a solid start.

My opening activity this year started with the preparation of four simple index cards with different distribution shapes:

Four common distributions, clockwise from top left: normal, left skewed, right skewed, and normal.

I have the benefit of a small class of 14 students. So I cut my graphs into a total of 14 pieces:

14 pieces for 14 students. Note on the bottom I've provided the hints A, B, C, and D.

When class started, I mixed up the graph pieces and handed one to each student. Then I told the class to find the other people in class who had the graph pieces that aligned with theirs. Once they had a completed graph, form a group at one of the tables and discuss which of the following they thought their group's graph might describe:

• People born each month of the year
• Student GPAs at this university
• Student heights at this university
• Starting salaries of new graduates from this university

It took my class less than 3-4 minutes to find their groups and then I gave them another 3-4 minutes to discuss what their graph shape might describe. As a class, I had each group share their ideas and then we discussed them. Not everybody agreed initially about which shape matched which description, which led into important comments about how we might think about unbiased sampling of students and imagining different scales and labels along the horizontal axes.

So in less than 15 minutes I combined group-making, statistics, and active, student-centered problem solving into one activity. This activity also gets students thinking about distribution shapes, which I sometimes worry we ignore in the rush to calculate centers and spreads. If you're wondering how to adapt this for your classroom, I offer these suggestions:

• If you have a few more students, cut more slices.
• If you have twice as many students, consider making two of each distribution shape and scaling the x-axis to match one of 8 potential descriptions. (i.e., a normal distribution scaled for heights in inches could be distinguished from one scaled for SAT scores.)
• If you want to use this for Algebra 1, you can make graphs that describe things like, "Toni walked to the bus stop at 2 mph, rode the bus at 30 mph to the bike shop, then rode a bike back home at 12 mph." Such an activity begins CPM's Algebra Connections and was the inspiration for my activity.
• If you want to use this for Algebra 2 or higher, you can use graphs of functions that students will become familiar with (parabolas, cubics, hyperbolas, etc.). I don't think it's worth fretting over vocabulary at this point -- just give students an opportunity to think about how the functions behave and what phenomena they could possibly model.

Settling slope and constructive Khan criticism

This was co-written with Frederick Peck, a fellow Ph.D. student in mathematics education at the University of Colorado at Boulder and the Freudenthal Institute US. We each have six years of experience teaching Algebra 1 and are engaged in research on how students understand slope and linear functions. Fred shares his research and curriculum at RMEintheClassroom.com.

---

Sal Khan (CC BY-NC-ND Elvin Wong)

The Answer Sheet has recently been the focus of a lively debate pitting teacher and guest blogger Karim Kai Ani against the Khan Academy's Salman Khan. While Karim's initial post focused mainly on Sal Khan's pedagogical approach, Karim also took issue with the accuracy of Khan Academy videos. As an example, he pointed to the video on slope. Specifically, Karim claimed Sal's definition of slope as "rise over run" was a way to calculate slope, but wasn't, itself, a definition of slope. Rather, Karim argued, slope should be defined as "a rate that describes how two variables change in relation to one another." Sal promptly responded, saying Karim was incorrect, and that "slope actually is defined as change in y over change in x (or rise over run)." To bolster his case Sal referenced Wolfram Mathworld, and he encouraged Valerie Strauss to "seek out an impartial math professor" to help settle the debate. We believe that a better way to settle this would be to consult the published work of experts on slope.

Working on her dissertation in the mid-1990s, Sheryl Stump (now the Department Chairperson and a Professor of Mathematical Sciences at Ball State University) did some of the best work to date about how we define and conceive of slope. Stump (1999) found seven ways to interpret slope, including: (1) Geometric ratio, such as "rise over run" on a graph; (2) Algebraic ratio, such as "change in y over change in x"; (3) Physical property, referring to steepness; (4) Functional property, referring to the rate of change between two variables; (5) Parametric coefficient, referring to the "m" in the common equation for a line y=mx+b; (6) Trigonometric, as in the tangent of the angle of inclination; and finally (7) a Calculus conception, as in a derivative.

(CC BY-NC-SA Raymond Johnson)

If you compare Karim and Sal's definitions to Stump's list, you'll likely judge that while both have been correct, neither have been complete. We could stop here and declare this duel a draw, but to do so would foolishly ignore that there is much more to teaching and learning mathematics than knowing what belongs in a textbook glossary. Indeed, research suggests that a robust understanding of slope requires (a) the versatility of knowing all seven interpretations (although only the first five would be appropriate for a beginning algebra student); (b) the flexibility that comes from understanding the logical connections between the interpretations; and (c) the adaptability of knowing which interpretation best applies to a particular problem.

All seven slope interpretations are closely related and together create a cohesive whole. The problem is, it's not immediately obvious why this should be so, especially to a student who is learning about slope. For example, if slope is steepness, then why would we multiply it by x and add the y-intercept to find a y-value (i.e., as in the equation y=mx+b)? And why does "rise over run" give us steepness anyway? Indeed, is "rise over run" even a number? Students with a robust understanding of slope can answer these questions. However, Stump and others have shown that many students -- even those who have memorized definitions and algorithms -- cannot.

(CC BY Amber Rae)

This returns us to Karim's original point: There exists better mathematics education than what we currently find in the Khan Academy. Such an education would teach slope through guided problem solving and be focused on the key concept of rate of change. These practices are recommended by researchers and organizations such as the NCTM, and lend credence to Karim's argument for conceptualizing slope primarily as a rate. However, even within this best practice, there is nuance. For instance, researchers have devoted considerable effort to understanding how students construct the concept of rate of change, and they have found, for example, that certain problem contexts elicit this understanding better than others.

Despite all we know from research, we should not be surprised that there's still no clear "right way" to teach slope. Mathematics is complicated. Teaching and learning is complicated. We should never think there will ever be a "one-size-fits-all" approach. Instead, educators should learn from research and adapt it to fit their own unique situations. When Karim described teachers on Twitter debating "whether slope should always have units," we see the kind of incremental learning and adapting that moves math education forward. These conversations become difficult when Sal declares in his rebuttal video that "it's actually ridiculous to say that slope always requires units*" and Karim's math to be "very, very, very wrong." We absolutely believe that being correct (when possible) is important, but we need to focus less on trying to win a mathematical debate and focus more on the kinds of thoughtful, challenging, and nuanced conversations that help educators understand a concept well enough to develop better curriculum and pedagogy for their students.

Khan Academy (CC BY-NC-ND Juan Tan Kwon)

This kind of hard work requires careful consideration and an open conversation, even for a seemingly simple concept like slope. We encourage Sal to foster this conversation and build upon what appears to be a growing effort to make Khan Academy better. Doing so will require more than rebuttal videos that re-focus on algorithms and definitions. It will require more than teachers' snarky critiques of such videos. Let's find and encourage more ways to include people with expertise in the practice and theories of teaching mathematics, including everyone from researchers who devote their lives to understanding the nuance in learning to the "Twitter teachers" from Karim's post who engage this research and put it into practice. This is how good curriculum and pedagogy is developed, and it's the sort of work that we hope to see Sal Khan embrace in the future.

---

*Sal's point is that if two quantities are both measured in the same units, then the units "cancel" when the quantities are divided to find slope. As an example, he uses the case of vertical and horizontal distance, both measured in meters. The slope then has units of meters/meters, which "cancel". However, the situation is not so cut and dry, and indeed, has been considered by math educators before. For example, Judith Schwartz (1988) describes how units of lb/lb might still be a meaningful unit. Our point is not to say that one side is correct. Rather, we believe that the act of engaging in and understanding the debate is what is important, and that such a debate is cut short by declarative statements of "the right answer."

References

Schwartz, J. (1988). Intensive quantity and referent transforming arithmetic operations. In J. Heibert & M. J. Behr (Eds.), Number Concepts and Operations in the Middle Grades (Vol. 2, pp. 41–52). Reston, VA: National Council of Teachers of Mathematics.

Stump, S. L. (1999). Secondary mathematics teachers' knowledge of slope. Mathematics Education Research Journal, 11(2), 124–144. Retrieved from http://www.springerlink.com/index/R422558466765681.pdf

RYSK: Stump's Secondary Mathematics Teachers' Knowledge of Slope (1999)

This is the ninth in a series of posts describing "Research You Should Know" (RYSK).

I think just about every Algebra 1 student I ever taught came to me from Prealgebra knowing what slope was. At least they thought they knew what slope was. They could usually echo the words "rise over run," and I admit that very early in my career I probably would have found that somewhat satisfactory. But with each new Algebra 1 class (I taught 14 sections in 6 years), my students' limited understandings of slope became more frustrating. Honestly, it wasn't until my last year of teaching that I really felt I had the kinds of problems, activities, and explanations to help students construct an understanding of slope that I was happy with.

In discussions with my graduate school colleagues Fred Peck and Michael Matassa, I found that my experience wasn't unique. We were interested in exploring slope further, and that led us to an article by Sheryl Stump. Her name seemed familiar, and as soon as I saw her picture I realized that I'd had lunch with her at a conference just a few weeks before. I suppose if I'd found the article earlier I could have talked to her about slope instead of swapping stories about our common Midwestern roots, but she's been kind enough to reply to my emails when we've wanted to know more.

I think one of the reasons I liked Stump's article was that it focused on teachers instead of students. After all, the biggest reason I became dissatisfied with my students' understanding of slope was because my own understanding had grown more sophisticated with each trip through the curriculum. In her article, Secondary Mathematics Teachers' Knowledge of Slope (1999), Stump investigated the definitions, understandings, and pedagogical content knowledge of 18 preservice and 21 inservice teachers. Nearly all the inservice teachers had degrees in math or math education, including eight with math or math ed masters degrees, and their teaching experience ranged from 1 to 32 years.

Stump's review of previous literature on slope revealed a number of descriptions, including ratios, tangent, and -- because of its applications in physics, calculus, and other real-world applications -- the key concept of linear functions and rates of change. Stump also found everyday ways of thinking about slope, such as the downward slant of a hill from top to bottom. These different, yet related, descriptions of slope had led to misunderstandings in previous studies with students. No one had yet tackled this kind of research with teachers, so Stump designed and administered a survey and conducted interviews to understand what her study participants understood about slope. Her questions, "What is slope?" and "What does slope represent?" elicited responses that were sorted into seven categories:

• The category of geometric ratio included representations such as "\(\frac{\mbox{rise}}{\mbox{run}}\)" and "vertical change over horizontal change" and focused on slope as a geometric property.
• The category of algebraic ratio included representations such as "\(\frac{y_2 - y_1}{x_2 - x_1}\)" and "the change in y over the change in x, in which slope was defined by an algebraic formula.
• The words "slant", "steepness", "incline", "pitch", and "angle" were categorized as involving a physical property.
• Responses referring to slope as the rate of change between two variables were categorized as involving a functional property.
• The parametric coefficient category included references to m in the equation y = mx + b.
• A trigonometric conception of slope referred to the tangent of the angle of inclination.
• A calculus conception included mention of the concept of derivative. (p.129 )

Both the preservice and the inservice teachers in Stump's study averaged about 2.5 representations per teacher in their definitions. The geometric ratio representation of slope was easily the most common for both groups (83% of preservice, 86% of inservice). but preservice teachers most commonly (61%) used algebraic ratio as a second representation, while inservice teachers commonly (81%) described a physical property. Descriptions of slope using the parametric, trigonometric, and calculus conceptions were rare or nonexistent.

Stump then gave the two groups six math questions, each designed to test different understandings of slope. Both the first question, about rate of growth, and the second question, finding a linear equation given its parameters, were answered correctly by 100% of the teachers in both groups. Questions about slope as speed, read from a graph, were answered correctly by about two-thirds to three-fourths of teachers in each group. The most dissimilar performance came on a question about angle of inclination, answered correctly by 33% of preservice teachers and 67% of inservice teachers.

Next Stump asked the teachers, "What mathematical concepts must students have experience with before they can truly understand slope?" (p. 132). By a wide margin, both groups said a geometric representation was most important, but only three teachers in each group mentioned experiences with functional relationships. Similarly, when asked for real-world contexts for understanding slope, both groups tended to choose a physical property instead of a functional property. About a quarter of the teachers in each group didn't mention either, naming algebraic or geometric representations instead (p. 133).

Stump's teacher interviews allowed her to dig more deeply into teachers' understandings about how students learn about slope. When asked about student difficulties, almost all the inservice teachers referred to a calculation procedure, saying "they put the x's over the y's" or "the order in which they subtract them" (p. 139). Preservice teachers predicted similar difficulties with symbol manipulation. One preservice teacher said:

My guess is that some might be frightened off as soon as you introduce a mathematical definition or a formula for a line, like the slope-intercept of the equation. As soon as some people see equations, they just go nuts, especially with symbols instead of numbers. ... Not because they don't understand what slope is, but because they are not making the connection between the intuitive and even the not-so-intuitive idea of taking the ratio of this to this. Not making the connection between that and the symbolic abstract equation on paper. That's just a guess. I haven't had experience with that. (pp. 139-140)

In her discussion section, Stump acknowledges teachers' tendency to think of slope first as a geometric ratio, with a smaller majority commonly thinking of it as a physical property. Very few teachers -- less than 20% -- had a functional conception of slope. Stump continues:

Considering the importance of the study of functions for high school students, it is especially troubling that functional situations involving slope were missing from so many teachers' descriptions of their instructional practices. Their students may thus miss opportunities to make this important connection while forming their conceptions of slope. Rizzuti (1991) found that instruction that included multiple representations of functions allowed students to develop comprehensive and multi-faceted conceptions of functions. Based on the results of the present investigation, it is questionable whether the participating teachers could assist their students in developing such a rich conception of slope. (p. 141)

Finally, Stump asks some important questions for further study, such as, "When textbooks connect various representations of slope, do teachers emphasise those connections for their students? Can teachers learn to make connections even if textbooks do not emphasise them?" (p. 141). I don't think we really know the answers to those questions, but I do absolutely agree with Stump's closing recommendation: "Both preservice and inservice mathematics teachers need opportunities to examine the concept of slope, to reflect on its definition, to construct connections among its various representations, and to investigate functional situations involving physical slope situations" (p. 142). It's good to see that kind of work being done, such as with Fred Peck and Michael Matassa's teaching experiment research and curriculum on slope they shared at ICME-12.

References

Stump, S. L. (1999). Secondary mathematics teachers’ knowledge of slope. Mathematics Education Research Journal, 11(2), 124–144. Retrieved from http://www.springerlink.com/index/R422558466765681.pdf