RYSK: Cobb, Zhao, & Visnovska's Learning From and Adapting the Theory of Realistic Mathematics Education (2008)

This is the 21st in a series describing "Research You Should Know" (RYSK).

It's Open Access Week (#OAweek) so I thought it would be fitting to use this "research you should know" post to highlight one of my favorite open access articles in mathematics education, Learning From and Adapting the Theory of Realistic Mathematics Education by Paul Cobb, Qing Zhao, and Jana Visnovska. Because the article is open access, I get to be less interested in summarizing it and more interested in giving you a reason to read it.

Realistic Mathematics Education (RME) is a theory for the design and development of mathematics curriculum. It is still deeply rooted in the Netherlands, where Hans Freudenthal greatly influenced mathematics instruction there with his belief that mathematics was a human activity, and that activity was characterized by mathematizing the real or readily imagined world. ("Realistic" comes from the Dutch phrase "zich realiseren," which in English means "to imagine.") This mathematization can be thought of in two ways, horizontal and vertical: "horizontal mathematization involves going from the world of life into the world of symbols, while vertical mathematization means moving within the world of symbols" (Freudenthal, 1991). Hans Freudenthal died in 1990 but his work continues, primarily at the Freudenthal Institute for Science and Mathematics Education at the University of Utrecht in the Netherlands.

There have been four primary avenues where RME has established itself in the United States. The first is with the middle school curriculum series Mathematics in Context, which grew from a partnership between mathematics education researchers at the University of Wisconsin (primarily Thomas Romberg) and researchers at the Freudenthal Institute. The second is the K-8-focused work of Mathematics in the City, which primarily brought together Cathy Fosnot from the City College of New York and Maarten Dolk of the Freudenthal Institute. The pair also wrote most of the Young Mathematicians at Work book series. The third place where RME is established in the U.S. is here at CU-Boulder, home of Freudenthal Institute US and its director, David Webb. David worked on the Mathematics in Context project at Wisconsin, and brought FI-US with him to CU-Boulder. The fourth place I recognize RME having a significant influence in the United States is in the work of Paul Cobb, particularly in his long research partnership with Koeno Gravemeijer, a researcher from the Freudenthal Institute. Cobb and Gravemeijer spent more than a decade working and publishing together, and that work did a lot to strengthen ties between RME as a design theory and theories in the learning sciences.

Like any idea or theory, RME has limitations. Over its 40+ years of existence it's proven to not be a static thing (van den Heuvel-Panhuizen, 2002), and this article by Cobb, Zhao, & Visnovska describes some of the important ways their work has both informed and been influenced by RME. They describe three adaptations: the first involves accounting for classroom activity and discourse in RME, the second acknowledges the mediating role of the teacher in making curriculum modifications and adaptations, and the third looks at how RME can focus on teacher learning, not just student learning. For details, I'll let you read the article for yourself at http://educationdidactique.revues.org/276. If you have any questions about the article or RME, leave a comment, find me on social media, or email me. We RME folks want to spread the word!

References

Freudenthal, H. (1991). Revisiting Mathematics Education: China Lectures. Dordrecht: Kluwer.

van den Heuvel-Panhuizen, M. (2002). Realistic Mathematics Education as work in progress. In F. L. Lin (Ed.), Common Sense in Mathematics Education: Proceedings of 2001 The Netherlands and Taiwan Conference on Mathematics Education (pp. 1–39). Taipei, Taiwan.

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

RYSK: Simon's Reconstructing Mathematics Pedagogy from a Constructivist Perspective (1995)

This is the 11th in a series describing "Research You Should Know" (RYSK) and part of my OpenComps.

After reading Clements & Sarama's (2004) Learning Trajectories in Mathematics Education a few days ago, I wanted to go back to the origins of learning trajectories: a 1995 paper from Martin Simon that explored how mathematics can and should be taught differently with a constructivist mindset. Simon is a professor of math education at NYU and has a history of researching how students and their teachers come to understand their mathematical knowledge.

From the start, I immediately appreciated two strengths of this article: Simon's clear writing and the relatively straightforward description of constructivism he offers. When you're still trying to sort out what constructivism is and is not (like me and many classroom teachers), it's a whole lot easier to parse this article from 1995 than, say, a heavily theoretical, mid-2000s piece by Jim Greeno. Recognizing that there are multiple (and often subtly different) ways to describe constructivism, Simon lays out his interpretation like this:

Constructivism derives from a philosophical position that we as human beings have no access to an objective reality, that is, a reality independent of our way of knowing it. Rather, we construct our knowledge of our world from our perceptions and experiences, which are themselves mediated through our previous knowlege. Learning is the process by which human beings adapt to their experiential world. (p. 115)

So when we have an idea that "works," meaning it does what we need it to do to make sense of our experiences, then we've constructed knowledge. This can be a tough sell to the mathematically-minded who say things like, "I didn't construct two plus two equals four. That's an objective fact." In response, the constructivist would disagree about having access to objective reality, but acknowledge that we (almost?) universally construct the knowledge that 2+2=4 because we experience no evidence suggesting otherwise (what Simon and others refer to as disequilibrium).

There's also a theoretical debate about the construction of knowledge as an individual, cognitive process versus a social process. This is an interesting debate, to be sure, and it has been pushing the leading edges of theories for learning mathematics for about the past twenty years. If you're wearing a theoretician hat, then you care deeply about how this debate might be won. But if you're wearing a researcher hat (like Simon is here), then you use both theories to help you gain whatever insights they might afford you. Simon (crediting work by Cobb, Yackel, and Wood) calls this coordination of psychological and sociological approaches "social constructivism," and compares it to how a physicist can better explain the nature of light by considering it both a particle and a wave.

Simon takes pains in this article to separate the theory of constructivism from a notion of "constructivist teaching." It's a mistake that I've seen and heard many times, and it's important to understand the differences and nuances. Simon states:

As I stated above, constructivism, as an epistemological theory, does not define a particular way of teaching. It describes knowledge development whether or not there is a teacher present or teaching is going on. ... There is no simple function that maps teaching methodology onto constructivist principles. A constructivist epistemology does not determine the appropriateness or inappropriateness of teaching strategies. ... The commonly used misnomer, "constructivist teaching," [suggests that] constructivism offers one set notion of how to teach. The question of whether teaching is "constructivist" is not a useful one and diverts attention from the more important question of how effective it is. From a theoretical perspective, the question that needs attention is, In what ways can constructivism contribute to the development of useful theoretical frameworks for mathematics pedagogy? (p. 117)

Using this perspective and a lot of theoretical support from work done in the early 1990s, Simon sets out to explore "the ongoing and inherent challenge to integrate the teacher's goals and direction for learning with the trajectory of students' mathematical thinking and learning" (p. 121, emphasis in original). Unlike a traditional perspective, where the pedagogical focus tended towards chopping mathematical content into manageable pieces to be demonstrated and practiced, Simon wished to focus on student understanding and a plan for mathematical tasks that improved that understanding.

I won't describe Simon's teaching experiment in great detail (after all, it was data-rich enough for Simon to publish multiple papers), but it involved a group of preservice elementary teachers and a set of tasks designed to elicit understandings about how multiplication related to the simple area formula A = l x w. Simon knew that his students had no trouble multiplying or using the formula. That wasn't the problem. Instead, he gave them this task:

Determine how many rectangles, of the size and shape of the rectangle that you were given, could fit on the top surface of your table. Rectangles cannot be overlapped, cannot be cut, nor can they overlap the edges of the table. Be prepared to describe to the class how you solved this problem. (p. 123)

Students used their rectangle (I'm imagining an index card) to measure the length and width of their table. A few groups questioned whether the rectangle should maintain its orientation, or if the long edge should always align with the edge of the table. This launched a class discussion and Simon pushed students to explain how they found the area without defaulting to "I used the formula." Some students talked about rows and columns, some talked about counting rectangles, but comments about "overlapping" rectangles suggested misunderstanding was still apparent. Compounding the problem was that these students were not accustomed to provide this level of justification.

Simon tried varying the task to elicit better student explanations. Some students seemed to get it while others still struggled or remained silent. (The transcript excerpts in Simon's paper are very valuable here, if you can get a copy of it.) Simon began to worry that students were actually misunderstanding things about area, not just how multiplication relates to rectangle area, so he assigned a second task about finding the area of an irregular shape. This was less of a problem for students, so Simon returned to the "turned rectangle" problem and tried another activity with measuring tables, both with rectangular cards and also with sticks. Some students were stuck in their thinking that the area unit must be the size and shape of the card, while others began to see how using the long edge of the card for length and width of the table created new, square units not shaped like the card.

All these classes were observed by researchers who took notes and videotaped the classroom activity. Simon also kept a journal of his reflections after each lesson and planning session. Following the teaching experiment, Simon analyzed his role as the decision maker in the classroom activities. First, he had hypothesized that his students would otherwise be satisfied with knowing and using a formula for area, but had probably never explored why the formula worked. This hypothesis was based on prior experience with similar students, prior research, and pretesting. Simon carefully thought out what he thought would happen in his initial activity, saying this thinking

provides an example of the reflexive relationship between the teacher's design of activities and consideration of the thinking that students might engage in as they participate in those activities. The consideration of the learning goal, the learning activities, and the thinking and learning in which students might engage make up the hypothetical learning trajectory, a key part of the Mathematical Learning Cycle described in the next section. (p. 133)

The "Mathematical Learning Cycle" Simon, in a simplified way, suggests how a teacher's knowledge can be used to create a hypothetical learning trajectory (containing a learning goal, a plan for learning activities, and a hypothesis about the learning process), and how assessment of student knowledge gives the teacher new and better knowledge upon which to refine the hypothetical learning trajectory. (I can't help but wonder if Simon once thought he'd be remembered for "learning cycles," not "learning trajectories.") The trajectory as planned is always hypothetical because it is just the teacher's prediction and the true trajectory cannot be known in advance. Modification of the trajectory happens as the teacher increases his/her knowledge about what students understand, which can be during a planning session between lessons or on-the-fly during a classroom activity. Of course, the more knowledge a teacher has in advance -- about their students, about the mathematical content, and about theories of learning that content -- the better the hypothetical learning trajectory can be. Sometimes we don't have all the information we want, says Simon:

As a teacher, I often do not have a well-developed map of the mathematical conceptual area in which I am engaging my students; that is, I may not have fully articulated for myself (or found in the literature) the specific connections that constitute understanding or the nature of development of understanding in that area. ... Thus, in such cases, my operational definition of understanding is the ability to overcome these particular difficulties; I may not have unpacked the difficulties in order to understand the conceptual issues that are implicated. Thus, even if I do not have a thorough knowledge of what constitutes mathematical understanding in a particular domain, having a rich set of problem situations that challenge students and having knowledge of conceptual difficulties that they typically encounter provide me with an approximation that lets me be reasonably effective in promoting learning in the absence of more elaborated knowledge. (This is not to suggest that the more elaborated understanding would not be more powerful.) (p. 139)

In his summary, Simon reiterates some major themes:

1. Student understanding is prioritized in the design of instruction
2. Teachers learn as students learn
3. Planning instruction includes the creation of a hypothetical learning trajectory
4. Because of #2, teachers need to constantly revise #3

Lastly, Simon emphasizes the challenge of teaching using the methods and example he's described. "Teachers will need access to relevant research on children's mathematical thinking, innovative curriculum materials, and ongoing professional support in order to meet the demands of this role" (pp. 142-143). I plan on summarizing more work on learning trajectories, so hopefully I can provide a little bit of that needed support.

References

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

Nielsen's Reinventing Discovery (2011) in the Context of Education Research

As a Ph.D. student I've taken my share of methods courses, giving me skills in everything from ethnography to ANOVA. But as important as those things are, I've sensed that there are new research methods emerging thanks to technological advancements and online communities. Our lives are too data-rich and our means of communication are too plentiful to limit ourselves to the same methods for research -- and learning -- that we used just 10 years ago.

Even though I feel like I live in the thick of this revolution, engaging with teachers and researchers on Google+ and Twiter, I wanted a broader perspective on how researchers use networks to make new discoveries. For this I turned to Michael Nielsen's book Reinventing Discovery: The New Era of Networked Science. Although Nielsen is a pioneer in quantum computing, I hoped to find some ideas that I could apply to a social science like education research.

Nielsen uses a variety of examples and concepts to describe what works and what doesn't (or hasn't) in networked science. Instead of listing them here, watch this TEDx talk by Nielsen:


If that talk wasn't long enough for you, Neilsen held a longer talk at Google that is worth checking out.

As much as I like Neilsen's example of Tim Gowers's Polymath Project, I can't imagine a direct translation to education research. One of the beautiful aspects of mathematics is that it usually doesn't require conducting an experiment, interviewing subjects, sampling a population, or agreeing on a conceptual framework -- the kinds of things that make social science untidy and difficult. Frankly, if solving problems in education were structured like proving mathematical theorems, I think we'd be solving more problems and finding better solutions than we are currently.

Neilsen's story about Qwiki hits home for me. For some time now I've imagined creating and maintaining a wiki that essentially translates the contents of the NCTM's Second Handbook of Research on Mathematics Teaching and Learning into knowledge that teachers could access and use. Just like Qwiki, it's easy to get math teachers and educators to agree that this would be a great resource to have. Unfortunately, I'm not sure how a math education wiki like the one I've imagined would avoid Qwiki's fate. Without incentives for experts to contribute and maintain the site, I'd probably spend more time fighting spam than helping teachers.

Neither the Polymath Project or Qwiki offer a blueprint for a new kind of mathematics education research. Thankfully, Nielsen describes some general characteristics for successful networked science. First, in his chapter titled "Restructuring Expert Attention," Nielsen suggests networked science has these attributes:

• Harnessing Latent Microexpertise -- The project must allow even the narrowest of expertise. A 3rd-year algebra teacher might not have the broad expertise of an experienced math education researcher, but that 3rd year teacher might have small elements of expertise that exceed that of the recognized experts.
• Designed Serendipity -- The project needs to be easy to follow and encourage participation from a variety of experts. You want problems to be seen by many in the hopes that just a few will think they have a solution they wish to contribute.
• Conversation Critical Mass -- One person's ideas need to be seen by others so they create more ideas, and the conversation around all the contributions keeps the project going.
• Amplifying Collective Intelligence -- The project should showcase the fact that collectively we are smarter than any one individual.

Those are all great characteristics of any project. But what makes this any different than any traditional, offline project? Nielsen offers several suggestions. Unlike a large group project with clear divisions of labor, technology allows us to divide labor dynamically. Wikipedia certainly would not have grown the way it did if labor had been divided statically between a set of contributors. Also, networked science uses market forces to direct the most attention to the problems of greatest interest. Lastly, contributing to an online project rarely feels like committee work, and participants can more easily ignore poor contributions or disruptive members.

Projects like Wikipedia and Linux exhibit the above attributes, but Nielsen explains that such projects needed something extra in order to scale to thousands of participants. Nielsen describes these in a chapter called "Patterns of Online Collaboration," and they are: (1) being modular, (2) encouraging small contributions, (3) easy reuse of earlier work, and (4) signaling to what needs attention. When I look at this list and think of Wikipedia, I can see how well a wiki or open source software project fosters these patterns. But how do we build such a project in education? Given Nielsen's framework above, a project that would interest me needs three key aspects:

• The content of the project has to be something that both teachers and researchers can contribute, such as a collection of math tasks, curriculum plans, or perhaps pedagogical techniques.
• Teachers need to be able to easily use and modify each other's content.
• (This one's the crux!) When teachers use content, there needs to be a way to collect and submit feedback about the use of that content, and that feedback becomes data that researchers can use not only to improve the content of the site, but to produce new and traditional reports of research.

It's that last bullet that's the hardest but most intriguing. There are so many places to get lesson ideas on the internet, but I don't know of any that collect data about the effectiveness of the lesson in a format suitable for research. Khan Academy claims to do this this kind of data collection internally, but KA is a closed project that lacks nearly all of the attributes Nielsen has described in his book. The project I want needs to be an open one, with all of its moving parts exposed and no more owned or identified with a single participant as Jimmy Wales is identified with Wikipedia. If you have ideas for what such a project could/should look like, leave them in the comments!

RYSK: Freudenthal's Why To Teach Mathematics So As To Be Useful (1968)

This is the fourth in a series of posts describing "Research You Should Know" (RYSK). While the article is not actually the report of research findings, it is part of a foundation upon which a generation of mathematics education research has been based.

Starting in the late 1960s, Dutch mathematician Hans Freudenthal saw the trend of "new math" spreading from the U.S. to the world. He pushed back with a philosophy of mathematics education now known as Realistic Mathematics Education (RME). The following article by Freudenthal, Why To Teach Mathematics So As To Be Useful, provides early insight to the core principles of RME: mathematics as a human activity, mathematization from contexts, and mathematics for all students. This article is also the first article in the first ever issue of the journal Educational Studies in Mathematics. Thankfully, instead of simply summarizing the article, I've been granted permission to reprint it here so you can read Freudenthal's words for yourself.

---

HANS FREUDENTHAL

WHY TO TEACH MATHEMATICS

SO AS TO BE USEFUL

My first task at this moment is to welcome you who have come here from various countries to sacrifice one week of your holidays for the benefit of mathematical education all over the world. I trust this meeting will be as useful as according to the general theme of this conference mathematical education should be held to be. I trust we all will learn as much from each other's experiences and arguments as we like to do and often have done at such opportunities. With great satisfaction I remember the meeting of December 1964 at Utrecht and I hope the few among you who have participated in that conference will share my feelings of gratitude. But whenever I shall remember those pleasant days and evenings, and lively discussions, I will never forget the man whom I met first and last on that occasion, the liveliest among all of us, the much regretted Wittenberg, this fiery nature who died much too early as though he had burnt himself in his own fire. Though I admit there was none among us who shared his opinions, I am sure everybody was impressed by his honest search into the truth of our educational philosophy. To my mind, his definitive absence overclouds the bright sky of this day.

The present colloquium is an activity of the ICMI sponsored by the government of the Netherlands and by IMU. It is not the first in this new period of ICMI and in this year. In January we met in Lausanne with the physicists, in a meeting sponsored by UNESCO, which was attended by some among you. In my opinion the resolutions adopted at Lausanne are a mile-stone in the philosophy of mathematical education. If I substitute my wishes and hopes for my opinion, I would say they should be so. It is evident that the use of mathematics has been a key criterion in all arguments on mathematics at that meeting.

In this introductory address I feel I have to justify the general theme of the present conference rather than to tell you about techniques of teaching useful mathematics. This means that I will not speak about how to teach mathematics so as to be useful but about why we should teach mathematics so as to be useful, or rather about why we should teach mathematics so as to be more useful.

Of course this is a question of educational philosophy, and as such it will be answered in a different way according to which philosophy we adhere to. Yet educational philosophy is not an abstract system. It depends on the real educational system in which we live, and on our, positive or negative, attitude with respect to that system. Is the variety of national educational philosophies really a drawback to international talks on mathematical education or should I say that there is no better opportunity to test them than to have them bump against each other? Are not we too often and too readily inclined, when reading or hearing about the educational experiences in another country, under another educational system, to sigh: it is just a pity, but this does not apply to our situation? I would say whenever this happens, then something is wrong either in the one system or the other, or, most likely, in both.

It is generally admitted that there is a wide gap between the educational philosophy of the U.S.A. and the Socialist European countries on one hand and the continental Western European countries on the other hand, though this gap has been narrowing to a considerable extent. On the one side one has for long times pursued the ideal of one kind of education for all youth, on the other side one has always overstressed that part of the educational system which provides educational facilities for a small group of students selected more on social than on intellectual grounds. I have to admit, and I do it with shame and distress, that in the Western countries of continental Europe, if we speak about mathematical education, we more often than not, mean the gymnasiums and lycées, and tacitly forget the about more than 90% who do not attend this type of schools. I agree that a more balanced educational system can be as bad if its highest level is too low to do justice to the most gifted students. But instead of discussing the question which kind of justice is the least evil, I would rather try to do the most justice to all people and to the society they belong to.

I need not explain to you why mathematics can be useful though the fact itself is one of the most recent and most astonishing features of the history of civilization. It would be more difficult to tell how mathematics can be useful provided that we do not limit ourselves to counting up instances of the all-pervading influence of mathematics in our culture, but ask what happens in the individual if he applies mathematics or if he tries to. Much has been done to investigate the learning process, though it is a fact that most of this research has been rather laboratory than classroom-oriented. Very little, if anything, is known about how the individual manages to apply what he has learned, though such a knowledge would be the key to understanding why most people never succeed in putting their theoretical knowledge to practical use.

Since mathematics has proved indispensable for the understanding and the technological control not only of the physical world but also of the social structure, we can no longer keep silent about teaching mathematics so as to be useful. In educational philosophies of the past, mathematics often figures as the paragon of a disinterested science. No doubt it still is, but we can no longer afford to stress this point if this keeps our attention off the widespread use of mathematics and the fact that mathematics is needed not by a few people, but virtually by everybody.

Mathematics is distinguished from other teaching subjects by the fact that, even in its actual totality, it is a comparatively small body of knowledge, of such a generality that it applies to a richer variety of situations than any other teaching subject. Modern mathematics can be seen as an effort to reduce this body of knowledge even more and to enhance the flexibility of what remains to be taught. At the same time this fact about mathematics is the source of the principal dilemma in teaching mathematics so as to be useful. In an objective sense the most abstract mathematics is without a doubt the most flexible. In an objective sense, but not subjectively, since it is wasted on individuals who are not able to avail themselves of this flexibility. On the other hand, teaching applied mathematics is as bad, if it means mathematics in a specialized context, which does not account for the greatest virtue of mathematics, its flexibility.

Though it might look different, I am still busy with the question why mathematics has to be taught so as to be useful, after we had agreed that it is useful and that students are expected to use it. There are two extreme attitudes: to teach mathematics with no other relation to its use than the hope that students will be able to apply it whenever they need to. If anything, this hope has proved idle. The huge majority of students are not able to apply their mathematical classroom experiences, neither in the physics or chemistry school laboratory nor in the most trivial situations of daily life. The opposite attitude would be to teach useful mathematics. It has not been tried too often, and you understand that this is not what I mean when speaking about mathematics being taught to be useful. The disadvantage of useful mathematics is that it may prove useful as long as the context does not change, and not a bit longer, and this is just the contrary of what true mathematics should be. Indeed it is the marvellous power of mathematics to eliminate the context, and to put the remainder into a mathematical form in which it can be used time and again.

Between two extreme attitudes one may be inclined to try compromising. If this means teaching pure mathematics and afterwards to show how to apply it, I am afraid we are no better off. I think this is just the wrong order. I have always considered it a remarkable fact that people are able to apply simple arithmetic, but not quadratic equations or even linear functions. Do not object that arithmetic is so easy. It is not. Take such problems as:

   If I have got ten marbles and I give three away, how many are left?
   If I have got ten marbles, and John has three less, how many does he have?
   If there are ten students in the room and three are girls, how many are boys?
   If I am ten years old now, how old was I three years ago?
   If B is between A and C, B is at a distance of 7 miles from A, and C is at a distance of 10 miles from A, how far is B from C?

It is not so easy to learn that in all these and a hundred other situations the same arithmetical operation applies. It takes some time, but finally everybody succeeds in understanding it. Why? I daresay, because arithmetic starts in a concrete context and patiently returns to concrete contexts as often as needed. The counterexample is fractions. In its traditional teaching the concrete context is no more than a ceremony which is hurried through in a jiffy. If afterwards the abstract theory of fractions has to be applied, its comes too late, on too high a level, and is not connected to any previous experience on a level where fractions should have been introduced. What is the reason for this change of attitude of the teacher? Is the patience of the schoolmaster exhausted when fractions turn up? I believe the answer is rather that the schoolmaster himself does not know fractions in a concrete context, and that for this reason he is not able to teach them in a more responsible way than he is used to do.

I am afraid this answer applies to the greater part of our mathematics teaching. Even the fact that a teacher applies mathematics himself, does not necessarily imply that he knows how he is able to do so and to use such a knowledge in his teaching.

The problem, however, is still much more serious. In the past, and mostly even now, textbook writing has been dominated by quite other aims than by the goal of a mathematics that could be useful. Mathematics is a peculiar subject. Arithmetic and geometry have sprung from mathematizing part of reality. But soon, at least from the Greek antiquity onwards, mathematics itself has become the object of mathematizing. Arranging and rearranging the subject matter, turning definitions into theorems and theorems into definitions, looking for more general approaches from which all can be derived by specialization, unifying several theories into one -- this has been a most fruitful activity of the mathematician, and no doubt our students are entitled to enjoy these fruits. No doubt modern mathematics is both much more flexible and much simpler than the mathematics of fifty years ago. No doubt our students have to learn the most modern mathematics. Teachers are more and more prepared and more and more inclined to bridge the gap between school mathematics and grown-up mathematics which had become wider from year to year.

However, this is not the whole story. The problem is not what kind of mathematics, but how mathematics has to be taught. In its first principles mathematics means mathematizing reality, and for most of its users this is the final aspect of mathematics, too. For a few ones this activity extends to mathematizing mathematics itself. The result can be a paper, a treatise, a textbook. A systematic textbook is a thing of beauty, a joy for its author, who knows the secret of its architecture and who has the right to be proud of it. Look how such an author would justify his construction: Why have you defined addition on page 10 in such a circumstantial way? -- because this more general definition will prove useful on p. 110. Why have you proved this geometrical theorem in such an unnatural manner? -- because at this stage I restrict myself to affine notions which have to precede metric notions. Why do not you mention forces as an instance of vectors? -- because mechanics has to be based upon vector algebra and not the other way round.

Systematization is a great virtue of mathematics, and if possible, the student has to learn this virtue, too. But then I mean the activity of systematizing, not its result. Its result is a system, a beautiful closed system, closed, with no entrance and no exit. In its highest perfection it can even be handled by a machine. But for what can be performed by machines, we need no humans. What humans have to learn is not mathematics as a closed system, but rather as an activity, the process of mathematizing reality and if possible even that of mathematizing mathematics.

New mathematics has been met with criticism. People who apply mathematics often feel uneasy when observing that the mathematics they have been used to apply is replaced by something they judge less suited for applications. It is a fact that biologists, economists, sociologists are better prepared to apply modern mathematics than physicists who carry the burden of a longer tradition. In the universities the gap between the mathematics of mathematicians and that of physicists has become terrifying. It is a habit of physicists to treat any particular subject with that kind of mathematics which prevailed at the time when that subject turned up in the history of physics. For instance, though physicists know eigenvalues of symmetric matrices because Laplace introduced them in a physical context, they still deal with orthogonal matrices with such oddities as Eulerian angles, because Euler was not yet acquainted with eigenvalues.

It would be a disaster if this lag would become permanent, though I hope it will not. Time ago I eavesdropped on a talk between a physics professor and his assistants, criticizing his course and particularly such a subject as Lagrange multipliers: this is not physics, one of them said, this is plain linear algebra.

Probably we will have to wait for the next generation to have physicists reconciled with modern mathematics teaching.

It is a pity that most of the criticism against modern mathematics is made with no knowledge about what modern mathematics really is. It is a pity, because there is ample reason for such criticism as long as mathematicians care so little about how people can use mathematics. We are not entitled to reproach physicists for identifying modern mathematics with a preposterous educational philosophy, since this identification is of our own making. I am convinced that, if we do not succeed in teaching mathematics so as to be useful, users of mathematics will decide that mathematics is too important a teaching matter to be taught by the mathematics teacher. Of course this would be the end of all mathematical education.

Mathematisch Instituut der
Rijksuniversiteit, Utrecht

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Obligatory disclaimer: Reprinted/republished with kind permission from Springer Science+Business Media:


Freudenthal, H. (1968). Why to teach mathematics so as to be useful. Educational Studies in Mathematics, 1, 3-8. Original copyright © D. Reidel, Dordrecht-Holland.

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3rd International Realistic Mathematics Education Conference (#RME11)

Starting tomorrow I'll be attending the 3rd International Realistic Mathematics Education Conference (#RME11), hosted by the Freudenthal Institute USA (FIUS) here at the University of Colorado at Boulder. The three-day conference features four keynotes, three plenaries, and only 18 breakout sessions, one of which I have the privilege of leading. I attended the previous RME Conference in 2009 before I really had a chance to become familiar with the theory and those who develop and promote it. RME is a theory of mathematics education worth knowing, but for this post I'd rather focus on some of the people who will be presenting. They're well-known in the field of math education research, even if they might not be in the math education blogosphere. I'm hoping this post helps change that.

David C. Webb is the Executive Director for the Freudenthal Institute USA and an assistant professor of mathematics education at the University of Colorado at Boulder. (He's also my advisor.) His involvement with the Freudenthal Institute goes back to his graduate school days at the University of Wisconsin, where we was advised by Tom Romberg and worked on the Mathematics in Context project, an NSF-funded curriculum that combined the goals of the NCTM Standards with the philosophies and design theory of RME. Following Romberg's retirement and David's move to the CU, FIUS came to Boulder in 2005. For RME11, David will lead Friday's plenary titled, "Informed Classroom Practice: Progress and Challenges" and will co-lead Sunday's closing plenary titled, "Design, Research, and Practice: Building a Community of Designers and Practitioners."

Henk van der Kooij (pronounced "koy") is a senior staff member at the Fruedenthal Institute, University of Utrecht, the Netherlands, where he conducts research and trains mathematics teachers. I had the pleasure last summer of taking a class co-taught by David and Henk, and even after ten days of going 8am to 3pm, it was not unusual to catch Henk sitting to the side of the room, molding a not-so-great mathematical task into a much better one. For this year's conference, Henk will co-lead the closing plenary with David Webb, participate in a Q & A Friday afternoon with Keono Gravemeijer and Mieke Abels, and conduct one session called, "What Mathematics is Important for (Future) Work?"

Koeno Gravemeijer gets the honor of the opening keynote at this year's conference, titled, "Helping Students Construct More Formal Mathematics." I've read a number of his articles and book chapters (see here for a sample), and seen many more referenced, so I'm quite excited to see and hear him in person. I'm not sure what areas of math ed research Gravemeijer hasn't tackled, from design research to statistics education, and the list of articles returned in Google Scholar makes me want to just stop what I'm doing and read for about a month.

Doug Clements will deliver Saturday morning's keynote, "Learning Trajectories -- The Core of Standards, Teaching, and Learning." My introduction to Doug Clements came last spring when my advisor asked me to read Clements's chapter in the Second Handbook of Research on Mathematics Teaching and Learning. The chapter, "Early Childhood Mathematics Learning," written with Julie Sarama, is perhaps the most thorough, dense, yet well-organized and enlightening (in a near-overwhelming kind of way) reading I've done yet as a graduate student. Some suggest we know (or we're close to knowing) all there is to know about early childhood mathematics, so summarizing that knowledge is no easy task. If you ever cross this chapter, take the advice my advisor gave me: "Take your time."

There are so many more excellent people presenting at this conference. Mieke Abels. Debra Johanning. Meg Meyer. The point of this post wasn't so much to drop names, or to think you'll be star-struck by this lineup (in a nerdy math ed researcher way), but to let you put a few names and faces together of people who share a common interest -- they can't stop thinking about how we can better teach and learn mathematics. And if you can think of them that way, then the walls of the ivory tower seem to crumble.

I plan on blogging, tweeting (with hashtag #RME11), and posting to Google+ throughout the weekend, although I still have to find enough spare moments to keep up with my other classwork due next week. (And finish putting together my own presentation!) So far I don't know of any other bloggers or members of the math ed Twitter community who will be attending, but be sure to make your presence felt if you're lucky enough to attend the conference.

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.