NCTM Denver 2013: Fennell and Wray's Math Specialists Get Ready Now: Common Core Assessments Are Coming

Annual Meeting - Friday, April 19, 3:30 pm

Francis (Skip) Fennell - NCTM Past President; McDaniel College, Westminster, Maryland
Jon Wray - NCTM Board of Directors; Howard County Public Schools, Ellicott City, Maryland

Skip Fennell, Jon Wray, and Beth Kobett (who was absent for this presentation) are the leads on ems&tl, the Elementary Mathematics Specialists & Teacher Leaders Project. As the name implies, the focus here is on supporting math specialists, such as district-level curriculum directors, instructional coaches, and anyone who is in a position to support mathematics teachers.

For this presentation, Fennell and Wray looked at the upcoming Common Core assessments, PARCC and Smarter Balanced (SB), and suggested ways math specialists can help teachers prepare for the tests.

Francis (Skip) Fennell

The challenge Fennell and Wray presented was essentially to focus on the upcoming assessments and respect the influence they will have on curriculum and instruction, without focusing too narrowly on the assessments and cause instruction and learning to suffer. This means, for example, not turning classroom practice into test prep, and using sample items from both PARCC and SB wisely.

Fennell and Wray used the concept of assessment literacy to describe the ability for teachers and specialists to understand a testing program. Many teachers have no formal training in assessment, so math specialists must be able to help them build their assessment literacy. Part of this is simply becoming more familiar with the schedules and formats of the upcoming PARCC and SB assessments. Both consortia offer more than just an end-of-year test, and teachers are going to need to help students interpret new kinds of technology-enabled assessment tasks.

Jon Wray

Fennell sees great potential in the CCSSM, but said, "If the Common Core becomes political, it's dead." Teachers and specialists need to work with the standards in ways that doesn't reduce them to a checklist of vaguely connected ideas. Using a number of items and task prototypes, Fennell and Wray showed examples of sample items from PARCC and SB and showed the many ways these could be used richely in lessons if the teacher provides the right support and instruction. "There are a lot of ways sample items can be used as instructional gems, " said Fennell. A list of potential questions and strategies for various tasks can be found in their slides.

The presentation wrapped up with an urging to better understand the role of formative assessment around these sample tasks. Also, encouragement was made to use materials from both PARCC and SB, regardless of the test your state has adopted. More task resources were linked to, including Illustrative Mathematics, the Institute for Mathematics and Education (especially the progressions documents), The Mathematics Common Core Toolbox, the PARCC Educator Leader Cadre Portal, and the Smarter Balanced Scientific Sample Pilot Test Portal.

The slides for this presentation are available here.

NCTM Denver 2013: Saxe et al's Engagement in Mathematical Discussion: Linking Practices and Outcomes

Research Pressession - Wednesday, April 17, 8:30 am

Geoffrey B. Saxe - University of California, Berkeley
Maryl Gearhart - University of California, Berkeley
Ronli Diakow - University of California, Berkeley
Nicole Leveille Buchanan - University of California, Berkeley
Jennifer Collett - University of California, Berkeley
Bona Kang - University of California, Berkeley
Kenton De Kirby - University of California, Berkeley
Marie Le - University of California, Berkeley
Discussant: Deborah Loewenberg Ball - University of Michigan

This group from Berkeley presented their findings from the use of Learning Mathematics through Representations, or LMR. LMR is a research-based curriculum unit for the teaching of integers and fractions in the elementary grades. Despite only being 19 lessons long (at the time of their study), it carefully attended to students' definitions of number, unit intervals, subintervals, and early fraction sense. When compared to similar coverage by Everyday Math, LMR produced significantly higher learning gains at all stages.

I'd have been hard-pressed to take a worse picture than this.

The group did find variability in LMR results. After checking for curriculum coverage differences and not finding anything significant, they developed measures for both content and participation. The group found that communication was key, especially for the lowest-achieving students.

This research group also did intensive classroom observation and video collection. They looked at interesting teacher moves designed to disrupt student thinking in ways that elicited student protest, where students became motivated to express their understandings of concepts in ways that corrected the teacher's intentional mistakes. The group paid specific attention to the trajectory of student understanding about unit intervals. Over six weeks, with pre-, interim, and post-assessments, they showed how students' understanding of the fraction 8/7 grew (for most students) over time.

Saxe's anthropological approach to studying shifts in understandings over time add some theoretical nuance to this work, examining the semi-durability of ideas as they are reproduced and altered. This kind of detail is often lacking in research, but can provide some key insights about teaching and learning.

The Discussant, Deborah Ball, began her comments with "Wow." I think that says a lot. She commented on the project at a meta-level, about the project itself, and "being greedy," she asked questions about what else we can get out of this body of work. Ball appreciated the connectedness of the project, both to other projects and across the history of Saxe's work. She also appreciated how the work was situated across all aspects of instruction, including teaching, curriculum, and class discussion. Ball called the work "programmatic" in the ways it carefully broke down the issues of the study and carefully applied the right methods, the care taken with definitions, and the depth with which instruction was analyzed. Ball also asked about "correcting the teacher," wondering more specifically what they perceive that move/strategy to be, and how it might not fit into either direct or dialogic instruction. For her "greedy" questions, Ball asked:

1. What are you learning about teaching?
2. What are you learning about the challenge of "drop-in" curriculum?
3. What are you learning about the assessment of student learning?
4. What did you learn about who was talking in class? What were the relationships with social or identity markers? What are you learning about the use of problems that weren't situated in the real-world of the students?
5. How can the rest of us learn how to do this kind of programmatic work?

To answer #4, the nature of the work and issues with Human Subjects precluded them from collecting demographic information about students. In the interest of time, the panel decided to take most of the other questions under considerations while allowing time for Q&A from the audience. The most interesting answer in the Q&A was in regards to the availability of the LMR curriculum. Saxe said they tried getting it published commercially, but commercial publishers want to sell K-5 series of textbooks, not a 19-lesson replacement unit. So instead, the group is planning to post all the materials online and make them free for teachers to use in their classrooms.

RYSK: Ball's Unlearning to Teach Mathematics (1988)

This is the 16th in a series describing "Research You Should Know" (RYSK) and part of my OpenComps. I also Storified this article as I read.

Dan Lortie's 1975 book Schoolteacher clarified an idea that teachers already know: how we teach is greatly influenced by the way we've been taught. Lortie called the idea apprenticeship of observation, and it specifically refers to how teachers, having spent 13,000+ hours in classrooms as students, take that experience as a lesson in how to be a teacher. What we often fail to deeply reflect on, however, is that we were only seeing the end product of teaching. We didn't see the lesson planning, go to summer conferences, attend professional development workshops, study the science of learning, or take part in the hundreds of decisions a teacher makes every day. Just observing isn't a proper apprenticeship, even after thousands of hours watching good teachers. I think of it this way: I watch a lot of baseball, and I can tell good baseball from bad. This hardly makes me ready to play, sadly, because I'm not spending hours taking batting practice, participating in fielding drills, studying video, digesting scouting reports, and working out in the offseason. Just as watching a lot of baseball doesn't really prepare me to play baseball, watching a lot of teaching doesn't really prepare someone to teach. Still, all those hours heavily influence our beliefs, both of teaching and of subject matter.

Deborah Ball (CC BY-NC-ND
House Committee on Education
and the Workforce Democrats)

In 1988, the year she earned her Ph.D at Michigan State, Deborah Ball was spending a lot of time thinking about math teachers' apprenticeship of observation. In an article called Unlearning to Teach Mathematics, she describes a project involving teaching permutations to her class of introductory preservice elementary teachers. The goal was not simply to teach her students about permutations, but also to learn more about their beliefs about the nature of mathematics and to develop strategies that might enlighten those beliefs and break the cycle of simply teaching how you were taught.

By selecting permutations as the topic, Ball hoped to expose these introductory teachers to a topic they'd never studied formally. By carefully observing how her students constructed their knowledge, Ball would be able to see how their prior understandings about mathematics influenced their learning. The unit lasted two weeks. In the first phase of the unit, Ball tried to engage the students in the sheer size and scope of permutations, like by thinking about how the 25 students could be sat in 1,551,121,000,000,000,000,000,000 different seating arrangements. Working back to the simplest cases, with 2, 3, and 4, students, students could think and talk about the patterns that emerge and understand how the permutation grows so quickly. For homework, Ball asked students to address two goals: increase their understanding of permutations, but also think about the role homework plays in their learning, including how they approach and feel about it and why. In the second phase of the unit, Ball has her students observe her teaching young children about permutations, paying attention to the teacher-student interactions, the selection of tasks, and what the child appears to be thinking. In the last phase of the unit, the students become teachers and try helping someone else explore the concept of permutations. After discussing this experience, students wrote a paper reflecting on the entire unit.

From other research, Ball knew that teacher educators often assumed their students had mastery of content knowledge. Even moreso, future elementary math teachers themselves assumed they had mastery over the mathematical content they'd be expected to teach. She knew, however, that there was something extra a teacher needed to teach that content. Citing Shulman's pedagogical content knowledge, along with numerous others, Ball describes some ways we can think about what that special content knowledge for teaching is, but admits that her permutations project was too narrow to explore how teachers construct and organzie that knowledge. The project would, however, give insight to her students' ideas about mathematics, and assumptions they make about what it means to know mathematics. For example, a student named Cindy wrote:

I have always been a good math student so not understanding this concept was very frustrating to me. One thing I realized was that in high school we never learned the theories behind our arithmetic. We just used the formulas and carried out the problem solving. For instance, the way I learned permutations was just to use the factorial of the number and carry out the multiplication ... We never had to learn the concepts, we just did the problems with a formula. If you are only multiplying to get the answer every time, permutations could appear to be very easy. If you ask yourself why do we multiply and really try to understand the concept, then it may be very confusing as it was to me. (p. 44)

Comments like this revealed that many of Ball's students relied on a procedural view of mathematics, one where the question "Why?" had been rarely asked. Ball also noticed a theme in her students' reflections about knowing math "for yourself" versus for teaching. Alison wrote:

I was trying to teach my mother permutations. But it turned out to be a disaster. I understood permutations enough for myself, but when it came time to teach it, I realized that I didn't understand it as well as I thought I did. Mom asked me questions I couldn't answer. Like the question about there being four times and four positions and why it wouldn't be 4 x 4 = 16. She threw me with that one and I think we lost it for good there.

From observing a young student learn about permutations in phase two, Ball noticed that some of her students started to challenge some of their assumptions they made about themselves as learners. Both from her experience and from the literature, Ball knew that elementary preservice teachers are often the most apprehensive about teaching mathematics. In some cases, these students choose to teach elementary in the hopes of avoiding any mathematical content they might find difficult. Changing these feelings about mathematics and about themselves is a difficult task for the teacher educator, but Ball did see progress. Christy, for example, said, "Most of all, I realized that I do have the ability to learn mathematics when it is taught in a thoughtful way" (p. 45). Unfortunately, not all shared this experience, as Mandy said she "did not enjoy the permutations activities because I was transported in time back to junior high school, where I remember mathematics as confusing and aggravating. Then as now, the explanations seemed to fly by me in a whirl of disassociated numbers and words" (p. 45).

In her conclusion, Ball says activities like the permutations project can be used by teacher educators to expose students' "knowledge, beliefs, and attitudes" (p. 46) about math and teaching math. By understanding the ideas prospective teachers bring with them, teacher educators can better develop preparation programs that address those beliefs in ways that strengthen the positive ones while changing some negative ones. Also, by including these kinds of activities with introductory preservice teachers, this can raise their expectations for what they will encounter later in methods classes. Summarizing, Ball concludes:

How can teacher educators productively challenge, change, and extend what teacher education students bring? Knowing more about what teachers bring and what they learn from different components of and approaches to professional preparation is one more critical piece to the puzzle of improving the impact of mathematics teacher education on what goes on in elementary mathematics classrooms. (p. 46)

References

Ball, D. L. (1988). Unlearning to teach mathematics. For the Learning of Mathematics, 8(1), 40–48. Retrieved from http://www.jstor.org/stable/40248141

RYSK: Gravemeijer's Local Instruction Theories as Means of Support for Teachers in Reform Mathematics Education (2004)

This is the 12th in a series describing "Research You Should Know" (RYSK) and part of my OpenComps.

Gravemeijer (from above) at the 2011 RME Conference

I began my recent reading of the literature on learning trajectories by reading Clements & Sarama's (2004) Learning Trajectories in Mathematics Education, and then went back to where the idea formally began, Simon's (1995) Reconstructing Mathematics Pedagogy from a Constructivist Perspective. Now I'm jumping to 2004 again with Koeno Gravemeijer's Local Instruction Theories as Means of Support for Teachers in Reform Mathematics Education. Koeno Gravemeijer (pronounced Koo-no Grav-meyer) has worked at multiple institutions in the Netherlands and spent time at Vanderbilt working with Paul Cobb, but he's best known for his long time association and leadership with the Freudenthal Institute and his advancements of Realistic Mathematics Education (RME).

When Martin Simon introduced the concept of hypothetical learning trajectories in his 1995 paper Reconstructing Mathematics Pedagogy from a Constructivist Perspective, he described them as part of a teaching cycle that was informed by the teacher's knowledge and then revised after assessment of student understanding. While much of the focus was placed on the idea of the trajectory, Simon made clear that no two trajectories will be alike, as each one is hypothesized for a unique group of students who are uniquely constructing knowledge. In other words, you can't just prescribe a trajectory and ask teachers to follow it to the letter. Instead, Simon suggested we needed to build an understanding of the knowledge teachers were using to inform and modify their trajectories:

A possible contribution that can be made by the analysis of data and the resulting model reported in this paper is to encourage other researchers to examine teachers' "theorems in action" and to make teachers' assumptions, beliefs, and emerging theories about teaching explicit. (p. 142)

This paper by Gravemeijer is, in part, a response to Simon's call to other researchers. Gravemeijer first states that in a constructivism-inspired reform mathematics, the traditional goals of instructional design must change:

What is needed for reform mathematics education is a form of instructional design supporting instruction that helps students to develop their current ways of reasoning into more sophisticated ways of mathematical reasoning. For the instructional designer this implies a change in perspective from decomposing ready-made expert knowledge as the starting point for design to imagining students elaborating, refining, and adjusting their current ways of knowing. (p. 106)

Next, Gravemeijer recognizes that while every teacher can use their knowledge to hypothesize a learning trajectory, we (researchers, teacher educators, curriculum designers) need to have some knowledge in common if we want to help teachers:

The example Simon (1995) worked out shows that designing hypothetical learning trajectories for reform mathematics is no easy task. We can, therefore, ask ourselves what kind of support can be given to teachers. It is clear that we cannot rely on fixed, ready-made, instructional sequences, because the teacher will continuously have to adapt to the actual thinking and learning of his or her students. Thus it seems more adequate to offer the teacher some framework of reference, and a set of exemplary instructional activities that can be used as a source of inspiration. (p. 107)

This is where Gravemeijer introduces the concept of a local instruction theory, which he describes as "the description of, and rationale for, the envisioned learning route as it relates to a set of instructional activities for a specific topic" (p. 107). I admit, it's difficult at first to discern this from a hypothetical learning trajectory, but I think the key is the relationship to instructional activities (which are more fixed/solid) instead of a trajectory's relationship to student understanding (which is more flexibile/fluid). By addressing the relationship of learning to the instructional activities, Gravemeijer uses local instruction theories to describe a common foundation teachers can use for building trajectories, saying that "Externally developed local instruction theories are indispensable for reform mathematics education" and that it is "unfair to expect teachers to invent hypothetical learning trajectories without any means of support" (p. 108). (If you're still confused, I think I can safely oversimplify it like this: Simon says trajectories are about student learning, not mathematical tasks. Gravemeijer agrees, but since trajectories are unique because student learning is unique, it helps if we have some agreed-upon ideas about how mathematical tasks should be designed.) Given Gravemeijer's long association with the Freudenthal Institute, he naturally describes how design principles from Realistic Mathematics Education (RME) provide the kind of instructional design framework for creating a local instruction theory.

Design Research and RME

Some curricula and instructional strategies are developed then subjected to treatment and control groups to test their effectiveness. That's not design research and not how RME has been developed. Instead, design research consists of cyclical iterations of thought experiments, teaching experiments, and retrospective analyses. It's similar to how teachers improve their instruction as they gain experience: they plan an activity for year one, then conduct that activity, then reflect on the activity so it will be better in year two. Of course, a team of researchers who are carefully theorizing, observing, collecting data, and analyzing the results across multiple classrooms can more quickly and effectively improve tasks and instruction than a teacher can alone.

Gravemeijer describes the design research he conducted with Paul Cobb and others around the development of mental computation strategies for addition and subtraction with elementary students. There are numerous papers and at least part of one dissertation all related to this work, so I won't describe it here. I will, however, describe the three RME design principles that Gravemeijer cites as helping form the local instruction theory that guided the design research process.

Guided Reinvention

Hans Freudenthal (1973) believed mathematics is best learned when students get to experience a process of learning that's similar to the way the mathematics was invented.

If mathematics is to be applied, applying mathematics should be taught and learned. Applying is often interpreted, as mentioned above, as substituting numerical values for parameters in general theorems and theories. This is a misleading terminology. Mathematics is applied by creating it anew each time -- I will expound this in more detail too. This activity can never be exercised by learning mathematics as a ready-made product. Drilling algorithms may be indispensable, but inventing problems to drill algorithms does not create opportunities to teach applying mathematics. This so-called applied mathematics lacks the flexibility of good mathematics. (Freudenthal, 1973, p. 118)

I've heard criticisms of this approach. "How in the world can a student reinvent mathematics that took mathematicians hundreds of years to understand?" That's a valid question, and the best answer is: "Through carefully designed curriculum and instruction." The goal is not to replicate the invention of the mathematics, but learn from history how a mathematical idea might be constructed in the mind of a student. Of course, this takes an extensive and special knowledge of the history of mathematics, and largely explains why Freudenthal's Mathematics as an Educational Task is almost 700 pages long.

Didactical Phenomenology

The concept of didactical phenomenology relates the mathematical "thought thing" and the phenomenon it describes. This is not a theory I know well but hope to study more in the future.

Mathematical concepts, structures, and ideas serve to organise phenomena -- phenomena from the concrete world as well as from mathematics -- and in the past I have illustrated this by many examples. By means of geometrical figures like triangle, parallelogram, rhombus, or square, one succeeds in organising the world of contour phenomena; numbers organise the phenomenon of quantity. On a higher level the phenomenon of geometrical figure is organised by means of geometrical constructions and proofs, the phenomenon "number" is organised by means of the decimal system. So it goes in mathematics up to the highest levels: continuing abstraction brings similar looking mathematical phenomena under one concept -- group, field, topological space, deduction, induction, and so on. (Freudenthal, 1983, p. 28)

Traditionally we teach an abstract mathematics and then find examples for students to make the mathematics concrete. With didactical phenomenology, we focus on progressive mathematization, suggesting "looking for phenomena that might create opportunities for the learner to constitute the mental object that is being mathematized" (Gravemeijer, p. 116). Yes, it's hard to understand without a lot of specific examples, and that's why Freudenthal wrote almost 600 pages on this topic. It's all in a book I have yet to read, so I'll forgive myself for giving a better description here.

Emergent Modeling

I can best describe emergent modeling with an example. Imagine an elementary class learning about fractions. Instead of giving students a formal model (like a numerator and denominator), the concept of emergent modeling says we should let students reach these models informally and progressively. If a task involves the sharing of parts of cookies with the students, students might begin with breaking apart actual cookies. Once realizing this isn't convenient, students might move to drawing cookies on paper. At some point they'll realize that drawing all the details of the cookie isn't necessary and just use a circle to represent a cookie. Up until this point, these are all models-of a cookie. The key step in this process is when students start using circles to model other contextual situations, like working with fractions of time, money, space, etc. Now the circle is a model-for a part-whole relationship, and not representing a specific object like a cookie. These models-for have the power to generalize to other contexts, and eventually students no longer need the circle and rely on formal mathematics to represent and work with fractions. Gravemeijer describes a similar process in this paper, except with how bead strings, unifix cubes, and rulers can lead to marked and empty number lines as students develop ideas of cardinality, ordinality, and distance as they learn mental strategies for addition and subtraction.

Conclusion

I hope by now you have some sense for a local instruction theory. The three RME principles above -- guided reinvention, didactical phenomenology, and emergent modeling -- do not describe a detailed instructional sequence of tasks and instructions for a teacher. They are, however, a way of theorizing how a particular instructional sequence should work, grounded in the design research conducted by Gravemeijer et al. This kind of local instruction theory is what allows teachers to design hypothetical learning trajectories that focus on the construction of student understanding, and provide some common ground for helping teachers become better at trajectory hypothesizing.

References

Freudenthal, H. (1973). Mathematics as an educational task (p. 680). Dordrecht, The Netherlands: D. Reidel.

Freudenthal, H. (1983). Didactical phenomenology of mathematical structures (p. 595). Dordrecht, The Netherlands: D. Reidel.

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

Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26(2), 114–145. doi:10.2307/749205

You can press "Enter," but think twice before pressing "="

I just had a epiphany tonight while reading an article by Alibali et al. (2007) about students' understanding of the equal sign. While some students see it properly as a relational symbol, the most common misunderstanding is that equals is operational -- a sign that indicates "get the answer" or "add them up." It is this operational conception that leads some students to believe x = 10 in a problem like 5 + 5 = x + 3. (Some students also incorrectly believe x = 13, figuring the three has to be added with the two fives somehow.)

So here's my surprise: I had never considered that students might be using a tool every day that is reinforcing that operational conception -- their calculator. Go ahead and search Google Images for calculators. Doesn't every one use an equals button to perform the "get the answer" function? Should that button be labeled with something else? Some say "Enter" but still have an "=" sign on the button.

This is what's fun about being a researcher -- I suddenly want to do an experiment with two sets of classrooms, one that gets traditional calculators and one that get modified calculators without "=" signs for the "Enter" button. Let them go about their business for a year without any other attention paid to the issue, and measure students' understandings of the equal sign at the end of the year and see if the treatment group has better understanding than the control. I know it sounds trivial, but it's often in these small steps where we make new knowledge.



Alibali, M. W., Knuth, E. J., Hattikudur, S., McNeil, N. M., & Stephens, A. C. (2007). A longitudinal examination of middle school students’ understanding of the equal sign and equivalent equations. Mathematical Thinking and Learning, 9(3), 221-247.

Modeling Dimensional Analysis

I generally ask myself two questions when I examine the design of a mathematical task:

1. What is the context?
2. How can we model the mathematics?

Mathematical concepts with tasks for which these two questions can be answered easily tend to be easier to learn, while teaching and learning generally becomes more difficult when one or both of those questions can't be answered. For dimensional analysis (sometimes called the unit factor method or the factor-label method), the first question is easy to answer. It doesn't take much of an imagination to design a measurement conversion task that is set in a real-world context. A model, however -- whether visual, mental, or a concrete manipulative -- is generally absent. Typical dimensional analysis problems look like this:

Q: What is 60 miles per hour in meters per second?

A: \( \frac{60 \mbox{mi}}{1 \mbox{hr}} \times \frac{5280 \mbox{ft}}{1 \mbox{mi}} \times \frac{12 \mbox{in}}{1 \mbox{ft}} \times \frac{2.54 \mbox{cm}}{1 \mbox{in}} \times \frac{1 \mbox{m}}{100 \mbox{cm}} \times \frac{1 \mbox{hr}}{60 \mbox{min}} \times \frac{1 \mbox{min}}{60 \mbox{sec}} = \frac{9656064 \mbox{m}}{360000 \mbox{sec}} = \frac{26.8224 \mbox{m}}{\mbox{sec}} \)

For those who successfully learn dimensional analysis this way, there's a certain beauty to how the units drive the problem and how the conversion factors are nothing more than cleverly written values of one, the multiplicative identity. Unfortunately, many students struggle with this method. Some are intimidated by the fractions, some can't get the labels in the right place, and some just can't get the problem started.

What we need is a model. Let's start with the most basic of unit conversion models, a ruler with both inches and centimeters:

(Yes, I'm still using the same ruler I got as a 7th grader in a regional MathCounts competition.)

With only simple visual inspection, students should be able to use a ruler to estimate conversions between inches and centimeters. This is an informal model, one students can literally get their hands on. We can assist the learning by making the models progressively more formal. Here we model a trivial conversion from one inch to centimeters with a double number line:

(Yes, you still have to know your conversion factors!)

Such a simple example looks almost too easy to be useful, but we can add number lines for more complex conversions. We can even abstract the model further and go beyond conversions of distance. Suppose we wanted to convert 3 gallons to liters. I could model that conversion with number lines this way:

(I could have used any number of transition units, but I knew 1 quart was roughly 946 milliliters.)

Filling in the question marks from top to bottom, I'll see that 3 gallons, 12 quarts, 11,352 milliliters, and 11.352 liters are all the same volume. It's easy to see they're the same because on each number line those values are the same distance from zero. Because we're only converting one kind of unit (volume), we only need one dimension.

In our initial example we were converting 60 miles per hour to meters per second. That's two kinds of units, distance and time, so our model needs two dimensions. Furthermore, it can help to think of 60 miles per hour as a line, not just a point. After all, we often travel at a speed of 60 miles per hour without actually traveling a distance of 60 miles in exactly one hour.

Can you guess where our double (or however many are necessary) number lines will go in this model? The following video will demonstrate what I would call the graphing model or two dimensional model for performing conversions.

With the work shown in the video, we haven't just done one conversion. In fact, we're prepared to write 60 miles per hour 15 different ways, not that we'll ever be asked to do that. If we needed 60 miles per hour in centimeters per minute or feet per second, all the work is done. Just choose the appropriate quantity from the vertical and divide by the appropriate quantity from the horizontal. Of course, if we're in a hurry, we won't find all those intermediate figures and instead just proceed from miles to meters and hours to seconds as quickly as possible. Will that be quicker than the traditional method shown above? Probably not, but the purpose of using a model is understanding, not speed. Once the understanding is established, students can move on to a formal method or use technology when appropriate.

RYSK: Erlwanger's Benny's Conception of Rules and Answers in IPI Mathematics (1973)

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

In 1973, Stanley Erlwanger was a doctoral student at the University of Illinois at Urbana studying under Robert Davis (who taught many of us math as an advisor for Sesame Street) and Jack Easley when he published his landmark "Benny" article in Davis's new Journal of Children's Mathematical Behavior. (Now simply the Journal of Mathematical Behavior.) This and other Erlwanger articles became known as disaster studies (Spieser & Walter, 2004, p. 33) because they painfully reveal learning gone wrong, and they continue to impact the way we think about learning math and how we do research in mathematics education.

During the back-to-basics movement of the 1970s there was a push for programs that supported individualized instruction. One such program was Individually Prescribed Instruction, or IPI. IPI was designed for students to "proceed through sequences of objectives that are arranged in a hierarchical order so that what a student studies in any given lesson is based on prerequisite abilities that he has mastered in preceding lessons" (Lindvall and Cox, as cited in Erlwanger, 1973, p. 51). To measure that mastery, IPI relied heavily on assessments that were checked by the teacher or an aide, who would then have the opportunity to conference with the student and check for understanding. Erlwanger, however, saw a conflict inherent in the program: while the goals of IPI were "pupil independence, self-direction, and self-study" (Erlwanger, 1973, p. 52), teachers were supposed to have "continuing day-by-day exposure to the study habits, the interests, the learning styles, and the relevant personal qualities of individual students" (Lindvall and Cox, as cited in Erlwanger, 1973, p. 52). So is a teacher, with a class of students each working at their own pace, supposed to continuously monitor each individual student? How? The logical way to do this is to monitor assessment results and focus attention on strugging students. After all, if a student is passing the assessments and "mastering" objectives, how much could go wrong?

Benny was a twelve-year-old boy with an IQ of 110-115 in a 6th grade IPI classroom. Benny had been in the IPI program since 2nd grade, and the teacher identified Benny as one of her best students. By sitting down and talking to Benny about the math he was learning, Erlwanger discovered that Benny's conception of math was not only very rule based, but in many cases Benny's rules yielded wrong answers. For example:

• Benny believed that the fraction \(\frac{5}{10} = 1.5\) and \(\frac{400}{400} = 8.00\) because he believed the rule was to add the numerator and denominator and then divide by the number represented by the highest place value. Benny was consistent and confident with this rule and it led him to believe things like \(\frac{4}{11} = \frac{11}{4} = 1.5\).
• Benny converted decimals to fractions with the inverse of his fraction-to-decimal rule. If he needed to write 0.5 as a fraction, "it will be like this ... \(\frac{3}{2}\) or \(\frac{2}{3}\) or anything as long as it comes out with the answer 5, because you're adding them" (Erlwanger, 1973, p. 50).
• When Benny adds decimals, he adds the number and moves the decimal point the total number of places he sees in the problem. So \(0.3 + 0.4 = 0.07\) and \(0.44 + 0.44 = 0.0088\). Benny's rule for multiplication is very similar: \(0.7 \times 0.5 = 0.35\), \(0.2 \times 0.3 \times 0.4 = 0.024\), and \(8 \times 0.4 = 3.2\). Because these are correct answers, that only served to reinforce Benny's rules about the addition of decimals.
• Benny thinks different kinds of numbers should yield different answers: "2 + 3, that's 5. If I did 2 + .3, that will give me a decimal; that will be .5. If I did it in pictures [i.e., physical models] that will give me 2.3. If I did it in fractions like this [i.e., \(2 + \frac{3}{10}\)] that will give me \(2\frac{3}{10}\)" (Erlwanger, 1973, p. 53).

As you might guess, Benny got a lot of wrong answers and sometimes failed to achieve the 80% mastery mark on his assessments. It's clear that Benny isn't simply guessing and getting wrong answers -- his methods are consistent and he can confidently explain his reasoning. When Benny is wrong, he tries to change his answers until he gets ones that match the answer key, a process he called a "wild goose chase" (Erlwanger, 1973, p. 53). Because Benny's teacher/aide is only looking for answers that match the key (and trying to do so quickly), the emphasis is on the answer, not the reasoning. It was only Benny's persistence that resulted in him mastering more objectives than most of his classmates.

This style of learning led Benny to believe that math is little more than a collection of arbitrary rules and singularly correct answers: "In fractions, we have 100 different kinds of rules" (Erlwanger, 1973, p. 54). Erlwanger asked Benny where he thought the rules came from. "By a man or someone who was very smart. ... It must have took this guy a long time ... about 50 years ... because to get the rules he had to work all of the problems out like that..." (Erlwanger, 1973, p. 54). For both reasons of scholarship and concern for Benny, Erlwanger returned to the school twice a week for 8 weeks to work with Benny one-on-one. Unfortunately, despite Benny's eagerness to learn, Erlwanger found this to be too little time to change Benny's firmly-established view of mathematics and little progress was made.

What Benny Means to Theory, Research, and to Khan Academy

(It might be helpful to read yesterday's post about constructivism and the Khan Academy before reading this section.)

Erlwanger summed up the theoretical aspect in his conclusion:

Benny's misconceptions indicate that the weakness of IPI stems from its behaviorist approach to mathematics, its mode of instruction, and its concept of individualization. The insistence in IPI that the objectives in mathematics be defined in precise behavioral terms has produced a narrowly prescribed mathematics program that rewards correct answers only regardless of how they were obtained, thus allowing undesirable concepts to develop. (1973, p. 57)

Looking back at Benny in 1994, Steffe and Kieren summarized that

Erlwanger was able to demonstrate how Benny's understanding of mathematics conflicted with any "common sense" understanding of what would be regarded as "good mathematics." This was a crucial part of Erlwanger's work, because by demonstrating what a "common sense" view of mathematics should not be, Erlwanger was able to falsify (naively) the behavioristic movement in mathematics education at that very place where behaviorism has its greatest appeal -- at the level of common sense. (p. 72)

Prior to Benny, the large majority of research in mathematics education depended on quantitative methods -- using statistics to summarize and compare the performance of treatment and control groups. Erlwanger had opened the door to qualitative research, which essentially meant that researchers could now see the value of interviews, case studies, and similar methods. In other words, Benny showed researchers that they can, and should, talk to children.

Although we're approaching the 40th anniversary of the Benny study, anyone who has been paying attention to the debates regarding Khan Academy should be able to draw parallels between it and IPI and realize we're retreading a lot of the same water. In a recent Wired Magazine article about Khan, stories are told of students working individually, at their own pace, with their progress measured by a computer that judges answers right or wrong. The article highlights Matthew Carpenter, a fifth grader who has completed "an insane 642 inverse trig problems" (Thompson, para. 2). Carpenter has earned many Khan Academy badges, a sign of progress that pleases his teacher and amazes his classmates. Unfortunately, the article provides no evidence that Matthew Carpenter is not Benny. I, and hopefully everyone, sincerely hope he is not Benny. I hope he's developing a proper view of the nature of mathematics and developing solid mathematical reasoning and understanding. But I can't be sure, and maybe Carpenter's teacher can't be sure, either. While we sometimes can and do use behaviorist programs of instruction to learn, we can't rely on them to be sure that learning is happening the right way. That's Benny's lesson, and that's why we need to be critical (but not necessarily dismissive) of Khan Academy. People who fail to do so might be surprised with the results they get for all the wrong reasons.

References

Erlwanger, S. H. (1973/2004). Bennyʼs conception of rules and answers in IPI Mathematics. In T. P. Carpenter, J.A. Dossey, & J. L. Koehler (Eds.), Classics in mathematics education research (pp. 48-58). Reston, VA: NCTM.

Speiser, B., & Walter, C. (2004). Remembering Stanley Erlwanger. For the Learning of Mathematics, 24(3), 33-39. Retrieved from http://www.jstor.org/stable/40248471.



Steffe, L. P., & Kieren, T. (1994/2004). Radical constructivism and mathematics education. In T. P. Carpenter, J. A. Dossey, & J. L. Koehler (Eds.), Classics in Mathematics Education Research (pp. 68-82). Reston, VA: NCTM.



Thompson, C. (2011, July). How Khan Academy is changing the rules of education. Wired. Retrieved from http://www.wired.com/magazine/2011/07/ff_khan/all/1.

RYSK: Brownell's The Place of Meaning in the Teaching of Arithmetic

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

Prior to the emergence of mathematics education as its own field of research in the 1960s and 1970s, those who chose to study the learning and teaching of mathematics were generally either mathematicians (George Polya, for example) or educational psychologists. William A. Brownell was one of the latter, having earned his doctorate in educational psychology in 1926 at the University of Chicago. Brownell worked at a number of universities but is best known for his work at Duke University from 1930 to 1949 and eleven years as dean of the School of Education at the University of California, Berkeley. Brownell died in May of 1977.

Brownell spent most of his career studying how children learn mathematics and authored a textbook series for the teaching of arithmetic. When the NCTM asked experts for nominations for articles to be included in their Classics in Mathematics Education Research, nine different articles were nominated. As the project leaders explained, "There was never a question whether to include an article by Brownell; the only issue to which of the many classic articles that he wrote to include" (Carpenter, Dossey, & Koehler, 2004).

In this article, Brownell defines the terms meaning of a thing and the meaning of a thing for something else. Or, in short, meaning of and meaning for. Using an example of the times, Brownell says that while he knows little about the workings of an atomic bomb (meaning of), he knows plenty about what having and using atomic bombs means for other things, like our culture (meaning for). In arithmetic, Brownell sees meanings of to be mathematical understandings, usually developed in classroom settings; meanings for are the connections to arithmetic we generally experience in every-day life outside of the classroom. Brownell insists that the distinction between the two is clear, although each type of meaning is relative, with varying degrees of complexity and depth.

Brownell saw, in 1947, too little teaching of arithmetic understanding. Common practice looked like this (you need only watch 20 seconds or so):

Through repetition, many students learned arithmetic. Many didn't. Again, learning was relative, not absolute, and Brownell never claimed that any students received zero understanding. What Brownell wanted was for students to see more meaning, which he organized into four categories:

1. Meanings of whole numbers, fractions, decimals, percent, ratio, and proportion.
2. Meanings of operations, including when to add, subtract, multiply, and divide.
3. Meanings of principles and relationships of arithmetic, like the zero property, communitive property, etc.
4. Meanings of place value that go beyond procedural "borrowing" and "carrying."

Brownell cited three sources for lack of meaning: (a) anecdotal evidence from adults who were poor at arithmetic, (b) poor military test results in arithmetic, and (c) the experience of secondary math teachers who receive students without arithmetical understanding. Brownell reasoned:

School personnel and, to some extent, the public at large are beginning to awaken to the fallacy of treating arithmetic as a tool subject. To classify arithmetic as a tool subject, or as a skill subject, or as a drill subject is to court disaster. Such characterizations virtually set mechanical skills and isolated facts as the major learning outcomes, prescribe drill as the method of teaching, and encourage memorization through repetitive practice as the chief or sole learning process.

Sound familiar? Of course. This kind of message has been echoed in NCTM materials at least since the Agenda for Action (1980) and common thinking can be found in articles such as Keith Devlin's Why We Should Reduce Skills Teaching in the Math Class.

So why didn't we (and don't we) teach arithmetic for meaning? Brownell anticipated four objections:

1. Is it really necessary to understand meanings to learn arithmetic?
2. Are these meanings too difficult for children to learn?
3. Does it take too much time to teach meanings?
4. Does learning meanings of and for arithmetic really improve ability?

To this, Brownell begins by asking why we teach arithmetic at all. If it is simply for students to be able to compute quickly and correctly, and nothing else, then the objections stand. However, if you believe that one of the goals of teaching arithmetic is to introduce to students a logical system of thinking, then meaning is necessary. Without it, computational efficiency -- which we should all still want -- will deteriorate once the drills are over. In this early day of research, Brownell apologizes for not citing "an impressively large body of competent research," but claims a strong case for meanings can still be made without it. Now, more than 60 years later, the research exists and the teaching of arithmetic looks very different. Students receive much better instruction about place value and use manipulatives like base-10 blocks, five-frames, and rekenreks to help their understanding.

References



Brownell, W. A. (1947). The place of meaning in the teaching of arithmetic. Elementary School Journal, 47, 256-265.

Carpenter, T. P., Dossey, J. A., & Koehler, J. L. (2004). Introduction. In T. P. Carpenter, J. A. Dossey, & J. L. Koehler (Eds.), Classics in Mathematics Education Research (pp. 1-6). Reston, VA: National Council of Teachers of Mathematics.



National Council of Teachers of Mathematics [NCTM]. (1980). An Agenda for Action: Recommendations for School Mathematics of the 1980s. Reston, VA. Retrieved from http://www.nctm.org/standards/content.aspx?id=17278.