On Major Problems and Grand Challenges, Part 1

Last month the NCTM Research Committee asked its members to help it identify the grand challenges for mathematics education. Grand challenges, said NCTM, (a) are hard yet doable, (b) affect millions of people, (c) need a comprehensive research program, (d) are goal-based with progress we can measure, and (e) capture the public's attention and support. I'm a month too late to contribute to NCTM's survey, and before blogging my thoughts into the wider conversation I thought I should look back at someone else's previous attempt. Maybe I'd gain some perspective on what grand challenges are and how persistent they might be.

Hans Freudenthal (Wikimedia Commons, CC-BY-SA)

In 1980, Hans Freudenthal gave a plenary address at ICME that later turned into an article in Educational Studies in Mathematics titled, Major Problems of Mathematics Education. I've briefly summarized the article on the MathEd Wiki and here I'll note the progress I think we've made on Freudenthal's 11 problems.

1. Freudenthal believed we "need[ed] more pardigmatic cases, paradigms of diagnosis and prescription, for the benefit of practitioners and as bricks for theory builders" (p. 135). In the case of arithmetic, which was Freudenthal's example, I think Cognitively Guided Instruction (CGI) is very much the kind of thing Hans was looking for.
2. Freudenthal wanted us to more carefully consider how people learn and observe their learning processes. I think several decades of teachers' awareness of constructivist theories of learning has changed how most people think of learning, and newer work in the area of teacher noticing puts fine points on what teachers notice and why.
3. How do we design curriculum and instruction around progressive formalization? There is always more to learn, but the Freudenthal Institute in the Netherlands has now worked on this for decades and the frameworks for curriculum design are well-established.
4. How do we retain and leverage mathematical insight? Freudenthal wrapped this into the conceptual vs. procedural debate, one that's still very much alive. However, I think we have better examples of productive approaches to this problem, and some research results (the BEAR project work at Berkeley comes to mind) showed that more focus on the conceptual didn't come at the expense of procedural facility. Still, this problem gets wrapped up in people's beliefs about mathematics and the teaching and learning of mathematics, and those beliefs sometimes aren't swayed by current evidence.
5. How do we reflect on our learning? This is another problem we now know much more about, particularly due to Schoenfeld and his work on metacognition.
6. How do we develop a mathematical attitude? This is still a challenge, and not just because some students say they don't like math. I think this problem might be closest to what Jo Boaler is currently trying to change with her focus on mindsets in learning mathematics.
7. How do we coordinate students working together when the are at different levels of learning? Many teachers and scholars have worked quite hard on this problem and I feel like most teachers now see the benefit of heterogeneous ability groups. For more, I'd suggest Ilana Horn's book, Strength in Numbers.
8. How do we create contexts for mathematizing? I think there's been a wealth of work in this area, from work based in Realistic Mathematics Education, work on word problems like that from Verschaffel, Greer, and de Corte, and, most recently, Dan Meyer's work. I could go on, as there are many more examples, and perhaps future work will give us a clearer picture about which contexts work best and why.
9. Can we teach geometry by having the learner reflect on spatial intuitions? Maybe it's my lack of expertise in geometry education research, but I really don't know where we stand on this problem. Freudenthal seemed to be reaching in his article on this problem, and maybe a more tangible articulation of the problem would have helped me better judge any solutions we might have.
10. How can technology increase mathematical understanding? Freudenthal admitted not being tech-savvy even in 1981 (he used "the ballpoint" as an example of technology that changed instruction, and not in an obviously historical way), but I think we now have numerous examples of tech that helps increase understanding. We also have a lot of examples of tech that doesn't, and I'm sure Freudenthal would have seen problems in our ability to judge the good from bad.
11. How do we use a holistic approach to educational development for change? In his native Netherlands, Freudenthal would likely be pleased today to see his colleagues' commitment to design-based, participatory approaches to research. We have some of that here in the U.S., too, but we also struggle for a "scientific" approach to finding "what works" based on experimental studies. We also have too much faith in how standards affect change; if Freudenthal thought curriculum development for change was a wrong perspective, surely he'd think the same about standards. Those things are just part of a much bigger picture.

Looking at this list, I think we have a lot to be proud of. Even though Freudenthal's article wasn't some sort of directive or command to fellow and future math education researchers and teachers, many people over many years worked so we'd have some answers to these questions. Still, there's a gap between ''what the field of math ed knows'' and ''what a teacher does with this knowledge, if they know it," which hints at what might be a grand challenge of its own. I'd like to get to that, but in a later post. Next, I'll look at some of the grand challenges that I've seen others post on the web in response to NCTM's call for input.

RYSK: Dewey's The Child and the Curriculum (1902)

This is the 20th in a series describing "Research You Should Know" (RYSK).

In my last RYSK post, I joined some other math teachers in discussing Richard Skemp's Relational Understanding and Instrumental Understanding (1976). Skemp's is a classic article that wrestles with a duality; in Skemp's case, the distinction between math for procedural skill versus a deeper mathematical understanding. For this meeting we turned the clock back further to Dewey's The Child and the Curriculum (1902), another classic article struggling with a duality in learning.

As D. C. Phillips (1998) noted in his review of The Child and the Curriculum, Dewey had a particular passion for dualisms, addressing more than three dozen of them in Democracy and Education (1916) alone. As Skemp and many others have shown, dualisms can be a starting point towards building a more nuanced understanding, as "neither is the world divided into a series of polar opposites, nor is it one" (Phillips, 1998, p. 404). Somewhere in between the opposites and the same lies the understanding many of us seek.

The Child and the Curriculum presents a particular dualism that very much persists to this day: should education be rooted in content, or in the needs and wants of the child? In Dewey's time, the push for a content focus was seen in the report by The Committee of Ten, not totally unlike how we currently push for content with documents like the Common Core State Standards. Dewey, like Skemp, also uses the metaphor of the map, using it to describe the logical versus psychological ordering of subject matter. Again, we still struggle with this duality today; last November Jere Confrey remarked at a conference, "There are some parts of the common core standards that I would express as mathematicians’ thought experiments," meaning we often guess how mathematical understanding is developed based on the structure of the mathematics instead of research on how children actually learn. These, of course, are not opposites, but they aren't the same, either.

Most of our discussion used Dewey as a prompt for thinking how Dewey's words more than a century ago frame modern challenges in education. (Reading Dewey seems particularly good for this kind of activity.) I was joined by +Nik Doran, +Bryan Meyer, +Nat Banting, and +Chris Robinson was feeding us ideas in the chat as we went along.

I plan to have more of these discussions, and hope we can get into some literature that really addresses research in math education versus some of these more theoretical or philosophical pieces. If you have suggestions for articles to read, please add them and vote them up or down in Google Moderator!

References

Phillips, D. C. (1998). John Dewey’s The Child and the Curriculum: A century later. The Elementary School Journal, 98(5), 403–414.

RYSK: Skemp's Relational Understanding and Instrumental Understanding (1976)

This is the 19th in a series describing "Research You Should Know" (RYSK).

This classic think piece from Richard Skemp, despite being now almost 40 years old, still gets a great deal of attention amongst mathematics educators. I'd somehow missed it in my own preparation, which I find surprising how much I've spent studying the math wars. Skemp describes two perspectives on understanding mathematics, one which he calls relational and the other he describes as instrumental. Relational understanding is related to what we might think of a "deeper" understanding, something that reflects how and why mathematics works and is applied. Instrumental understanding relates to those reliable and typically efficient procedures we apply to produce mathematically correct answers. The part of Skemp's article that really sticks out for me is his suggestion that "mathematics" might be used too broadly: "I used to think that maths teachers were all teaching the same subject, some doing it better than others. I now believe that there are two effectively different subjects being taught under the same name, 'mathematics'" (p. 91)

Today +Chris Robinson, +Joshua Fisher, +Nat Banting, +Nik Doran, and I (pictured left-to-right along the bottom) met via Google+ Hangout to discuss Skemp's article. We discussed examples of each kind of understanding, whether one is a subset or prerequisite to the other, and the various influences that lead us to emphasize one over the other, such as curriculum and assessment.

(Some research needs to be summarized, while some needs to be expanded on. That's my explanation for why it will likely take you longer to watch the video than to read the original article.)

I've been in discussions like this before and I always find them to be fascinating. Still, it seems difficult to really get at the root: What is mathematics, and how can and should our beliefs about mathematics change? There are also strong implications for curriculum design and learning theory, as finding the right balance in our approach to both kinds of understanding should lead to better student outcomes.

References

Skemp, R. R. (1976/2006). Relational understanding and instrumental understanding. Mathematics Teaching in the Middle School, 12(2), 88–95. Originally published in Mathematics Teaching. Retrieved from http://www.jstor.org/stable/41182357

Upcoming Hangout to discuss Skemp's Relational Understanding and Instrumental Understanding (1976)

Few things brighten my day more than seeing math teachers on social media seek out research:

@doingmath @natbanting Thanks! I think @nik_d_maths and I should include a classical education piece in our journal each month.
— Chris Robinson (@absvalteaching) March 29, 2013

Thanks to +Bryan Meyer, I quickly got pulled into this conversation. Together we decided to meet via Google+ Hangout this Sunday at 19:00 UTC (3 pm ET, noon PT) to talk about a classic article in mathematics education, Skemp's Relational Understanding and Instrumental Understanding, originally published in 1976 in Mathematics Teaching. If all goes according to schedule, I'll be joined by +Chris Robinson, +Nat Banting, +Nik Doran, +Bryan Meyer, and perhaps some others in our 10-person Hangout. If you'd like to join us (which means you need to have read the article), keep your eye on my Google+ posts for the event invitation. I'll probably reshare it to the Mathematics Education Research and Mathematics Education (K-12) communities. If more than 10 show an interest, I think we'll just organize multiple Hangouts...somehow. Stay tuned to the invite on Sunday and be prepared to keep your options open.

Getting the Article

In my ideal world, research would be published via open access, and I could simply link to a copy of any article I wished to share. We don't totally live in that world (yet), but thanks to JSTOR's "Register and Read" program, a lot more research is open to the public. If you follow this link to a 2006 reprint of Skemp's article, you should be able to register or log into your JSTOR account and select the article to be placed on your virtual shelf. There are restrictions -- you can't download the article, the article must remain on your shelf for at least 14 days, and you can only have three articles on your shelf at once.

If none of this works for you, you're on your own.

Update (2013-04-07): The video and wrap-up of the Hangout is here: http://blog.mathed.net/2013/04/rysk-skemps-relational-understanding.html.

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: Gutiérrez's (Re)Defining Equity: The Importance of a Critical Perspective (2007)

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

Do you ever find yourself talking about something, defending something, or promoting something when you suddenly realize you don't have a good definition of that thing?

In one way or another, I've been thinking about equity in math education ever since I was an undergraduate. I remember debating the value of "equality of opportunity" versus "equality of outcomes," and getting a sense for how the NCTM Standards prescribed a type of school mathematics for all students. Here at CU-Boulder, issues of equity and social justice are never far away. But what, exactly, do we mean by equity in math education? And why is it important?

Rochelle Gutiérrez focuses on issues of equity as an associate professor of mathematics education at the University of Illinois at Urbana-Champaign. Even though she's been publishing on issues of equity for well over a decade, in 2007 she wrote a book chapter titled, (Re)Defining Equity: The Importance of a Critical Perspective. In that chapter, she argues why we need a definition of equity that gives teachers and researchers a clear sense of purpose.

When equity is loosely defined, it comes under attack from several directions. First is a belief that not all students can learn, and that mathematical proficiency has more to do with natural ability than with effort. The second threat to equity is a "deficit theory" towards groups of students that haven't had much historical success in mathematics, whether that deficit is seen as biological or cultural. The third threat to equity, says Gutiérrez, comes from within the research community itself: so many issues get covered under the umbrella of "equity" that few of them get the kind of focused attention they need, even while many agree equity is important. As Gutiérrez puts it:

Perhaps the lack of a clear definition is what contributes to a general consensus that equity is worth striving for, everyone having his or her own vision of what it means. However, having a poorly defined target means we are only sure we are moving toward it when, in fact, we are very far away. (p. 38)

Gutiérrez argues that we should leave behind the traditional "excellence versus equity" and "traditional versus reform" debates in favor of a new perspective: dominant versus critical. Instead of teaching mathematics that "reflects the status quo in society, that gets valued in high-stakes testing and credentialing, that privileges a static formalism in mathematics," (p. 39), we should be favoring critical mathematics, that which "squarely acknowledges the positioning of students as members of a society rife with issues of power and domination" (p. 40). This includes using math to examine social and political issues, to highlight perspectives of different cultures, and to challenge the view that mathematics is a static entity. Gutiérrez does not wish to create a dichotomy here -- in fact, she argues the importance of learning dominant mathematics because it can help students better understand and criticize the world.

With that perspective in mind, Gutiérrez defines equity. First, she warns not to confuse it with equality; whereas equity implies "justice" or "fairness," equality implies "sameness." Gutiérrez is *not* arguing that all students should experience the same instruction using the same materials, or that we should expect all students to have the same outcomes. Gutiérrez fully recognizes that, within any group, experiences and outcomes will vary, and that some students will have interests that lead them away from mathematics. That's okay. Instead, she says equity in mathematics should consist of three main parts:

1. "Being unable to predict students' mathematics achievement and participation based solely upon characteristics such as race, class, ethnicity, gender, beliefs, and proficiency in the dominant language" (p. 41, emphasis original). To clarify, Gutiérrez says, "I contend that only when there is sufficient variation within groups and no clear patterns associated with power or status in society between groups can we conclude that this aspect of equity is being addressed" (p. 42, emphasis original). As for measuring achievement, Gutiérrez says we should use standardized tests (but not exclusively), because those are often the tools we use to grant power to individuals.
2. "Being unable to predict students' ability to analyze, reason about, and especially critique knowledge and events in the world as a result of mathematical practice, based solely upon characteristics such as race, class, ethnicity, gender, beliefs, and proficiency in the dominant language" (p. 45, emphasis original). It's this aspect of equity that Gutiérrez uses to stress the critical aspects described above.
3. "An erasure of inequities between people, mathematics, and the globe" (p. 48, emphasis original). Gutiérrez claims "This aspect of equity addresses the fact that having equal access to cultural capital and critical stances to society are necessary but insufficient conditions for change" (p. 48). While this aspect of equity is by far the most difficult to measure, and may not happen in our lifetimes, it should be the key goal of any long-term reform in mathematics education.

Gutiérrez closes her paper with this paragraph:

It might be the case that the first two aspects of equity must be addressed before we would see any changes in the third aspect. That is, students who gain both (1) dominant and (2) critical mathematics identities will lead to different kinds of mathematicians in the academy, thereby changing what counts as mathematics as well as how it is evaluated. The important thing to consider in this (admittedly simplistic) model is that neither the first nor the second aspects of equity are sufficient to redress injustices in the world. Students need to be able to do both -- be able to play the game of mathematics that is currently associated with power and intellectual potential, and be able to change the game of mathematics to serve a better society. (p. 49)

References



Gutiérrez, R. (2007). (Re)defining equity: The importance of a critical perspective. In N. S. Nasir & P. Cobb (Eds.), Improving access to mathematics: Diversity and equity in the classroom (pp. 37-50). New York, NY: Teachers College Press.

Reflections of Multicultural Education

It’s the day after the last day of what has been a very busy semester. Being busy is good, and being awash in new information every day is something I relish. But there comes a time when we must pause and reflect, and too often this semester I have not given myself that time. Admittedly, just keeping up with the flood of new information proved to be too much, and the student-to-student whispers of “You can’t read everything, you know” proved too regularly to be true. But finally, now, I can take a few hours and think about the last core course of my doctoral program: Multicultural Education (MCED), taught by Linda Mizell.

Assessing the value of this class has been difficult, as there were plenty of moments during the semester when I felt I wasn’t making much scholarly progress. One reason for that feeling – and a reason I appreciate – is that prior coursework had left me better prepared for MCED than I expected. (Or so I thought.) Rarely were the issues we explored in MCED not ones I’d considered in prior courses like Culture and Ethnography, Ethics in Education, Policy Issues, Education Research and Policy, and Perspectives on Classrooms, Teaching, and Learning. It is a credit to my institution that attention to multiculturalism and equity permeates into most corners of the school, although I admit there are times where I still sense it as artificially layered on to a lesson or, even worse, uncomfortably absent. A second reason for that lack-of-progress feeling stemmed from not being able to keep up with all the reading and assignments for the semester. As I finished the last of my papers last night, I thought back to what remained unfinished and one reading in particular stood out: Eduardo Bonilla-Silva’s Racism Without Racists.

So after submitting my last final paper, I pulled Bonilla-Silva back off the shelf and picked up where I’d left off. I had read all but the last two chapters, but it was in those last two chapters where things appeared to get most interesting. In this, the third edition of Racism Without Racists, Bonilla-Silva added a new chapter at the end addressing the “Obama Phenomenon.” I started reading and almost immediately I was taken back to what I thought made this book so interesting, engaging, and challenging to begin with: Bonilla-Silva’s outspoken criticism of a system that perpetuates racism and inequality. In general, I do not disagree with Bonilla-Silva’s message. But the style with which the message was delivered came as a bit of an uncomfortable shock.

In his detailed analysis of interviews with both white and minority students, Bonilla-Silva exposed the racism found in peoples’ language. For example, in an interview with a white girl named Jill who claimed, “One of my best friends is black” (p. 58), Bonilla-Silva asks her to go into more detail. Jill then describes her friend as “bright” but with “terrible GMAT scores,” and then says, “What he lacks in intellect he makes up for in…he works so hard and he’s always trying to improve himself.” In his analysis, Bonilla-Silva addresses the contradiction about intelligence and points out that Jill never mentions this friend by name. This example by itself might seem lacking in evidence, but it is far from an isolated incident in the text. The dissection of racism in peoples’ speech happens on page after page. Sometimes it’s subtle, sometimes less so, and I remember feeling during my first reading that I’m glad Bonilla-Silva wasn’t interviewing me, because he seemed to make everybody sound racist!

Now, reflecting exactly on that thought, I see how that thinking exposes how I largely missed Bonilla-Silva’s greater point (even though it’s the title of the book): the kind of racism we’re dealing with now is less about the individual and more about a system. Bonilla-Silva wasn’t after Jill to make her sound like a racist – at least not the kind of racist most people imagine when they hear that label. Bonilla-Silva was instead exposing how Jill, along with most of the other interviewees in the book, demonstrates the systems and structures of racism and how they exist in what we all say, do, and believe. In other words, it’s not about Jill. For the same reason, I shouldn’t have worried about Bonilla-Silva interviewing me, as the interview would have only helped me understand how my actions, behaviors, and attitudes are being affected by the subtle yet significant culture of racism that still exists in our society. And until we are forced to recognize it, there is very little we can or will do about it.

It’s also this same system that allowed much of the country to endorse President Obama, and how that endorsement gives us a false sense of accomplishment that we’ve somehow reached a “post-racial” society. (We haven’t.) As an educator I wonder how we can have policies like NCLB which are so bold to declare a school a failure when achievement gaps persist, yet our greater society and government doesn’t always extend that same failure judgment to the enormous gaps in achievement, income, wealth, health, etc. that we see in our society. Sure, the #Occupy protesters have their message, but it’s unfortunate that so few were shouting until the perils of inequity reached beyond minorities.

The system that Bonilla-Silva describes should not have been an “uncomfortable shock” to me. From where I now stand, I can see how other readings described much of the same system, yet somehow by using more academic or less forceful language I was led to think I understood when I didn’t. Perhaps the best example of this is Beverly Daniel Tatum’s book “Why Are All the Black Kids Sitting Together in the Cafeteria?” I remember thinking as I read it, “I really like Beverly Daniel Tatum because she’s making me feel comfortable about a difficult topic.” Where I feared an interview with Bonilla-Silva, I would have welcomed the opportunity to speak with Beverly Daniel Tatum.

But somehow, disguised by my initial affection for the authors, I didn’t immediately see how in many ways Bonilla-Silva and Tatum were largely describing the same system of racism. I’m glad I read Tatum first and then Bonilla-Silva, because now as I reflect I can see how Tatum’s message didn’t really sink in for me; if it had, I wouldn’t have been so challenged by Bonilla-Silva. The lesson for me is not that I need to keep reading more critical work (although that would certainly help), but that it’s going to take more effort to make myself feel uncomfortable about issues of culture, race, class, power, etc., before somebody else gets the chance to do it for me.

For me, the simple title to this post has a double meaning. First is the more obvious, that I’m finally taking some time to think about a class I experienced over the past semester. Second, and more importantly, is the idea that multicultural education has a reflective property like a mirror bouncing light around a corner. As an educator who had a relatively monocultural upbringing in the rural Midwest, and who apparently can still be surprised by the injustices in the world around me, I need to use what I’ve learned about multicultural education to shine some light not only around corners, but back on myself. There’s so much more for me to see, most of which is hidden by its largeness, not its smallness. As an educator this is what we do: we help students explore and understand the world around them, and our reward for doing so comes both in our students’ growth and our own.

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.

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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.

Download a PDF of the original article

A Quick-and-Dirty Guide to Fighting the Math Wars

I just posted this to a reply to a post by David Wees on Google+, but I thought it might be useful to some if it had some permanence here.

I've been in and out of "Math Wars" debates for 10+ years, and I find it's helpful to examine the issue at a more granular level. Here's a quick list of questions I jotted down:

What is your definition of mathematics? (Someone who answers, "It's a subject you learn in school" may have very different views from someone who answers, "It's a human activity we undertake to solve problems relating to number and shape.")

What is your philosophy of mathematics? (A Hardyist and a Mathematical Maoist have very different views, as do a Platonist and a Formalist. And for all the consistency in mathematics, this is not something with which we as individuals are necessarily consistent.)

What is our goal for students learning mathematics? (Is it to prepare them for work? For more school? To gain an appreciation of mathematics? For mental exercise?)

How should we assess mathematics? (Often when we claim that students do or do not perform well in mathematics, we are basing those claims on an assessment that may not embrace a balanced view of the issues above. Or, failing that, we make those claims without regard to the biases of the assessment.)

What learning theories do we use, and how do we use them? (A difficulty with learning theories is that in most all cases we can design curriculum and pedagogy around them that show they work -- at least to a degree. The workings of the human brain aren't easy to study, explain, or leverage in a classroom.)

How do we perceive "failure" or "success" of practices of the past? (I fear sometimes we stereotype certain historical movements, such as "New Math" and the "Back to Basics" movement, and we falsely assume that those movements were implemented in every classroom with high fidelity. We also sometimes forget that as time has passed, we are trying to teach higher and higher levels of mathematics to more and more students.)

How do we avoid false dichotomies? (False dichotomies were addressed in that article and Zwaagstra was wise to try to avoid them. But it's such an *easy* trap to fall into! [I've probably done it here without realizing it.] For example, he cited a paper by Alfieri, et al. (2011) that claimed through meta-analysis that "unassisted discovery does not benefit learners." But why would a well-trained constructivist teacher believe discovery should be unassisted? That's the same as assuming that a traditional teacher only has students listen to lectures and work problems in isolation. No teacher or student thrives exclusively on either. Interestingly, Zwaagstra in the next sentence says learners should be "scaffolded," an idea developed by Jerome Bruner in support of learning in a social constructivist environment.)

What skills, abilities, and philosophies do we believe teachers need to be successful? (I'm not sure we fully comprehend the effects on the received curriculum when it's taught by a teacher with skills, abilities, and philosophies that run counter to those supported by the curriculum. In such cases it's easy to misplace blame for poor outcomes.)

I'm sure there are more that I could add, but I strongly recommend that anyone who is serious about this debate to take on these issues one by one. Only if there is some agreement, or at least some sympathy and understanding, on these issues does it become truly productive to talk about "what works."

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.

Constructivism and the Khan Academy

Not long after sitting down at my computer this morning, there was this tweet from David Wees:

“How would you explain constructivism to someone not (well) versed in pedagogy? You have 140 characters. #edchat #BCed”

I took David’s challenge and what followed was a pretty good conversation with David Cox, Ira Socol, and Jennifer Borgioli. For the sake of clarity, yet with an attempt at brevity, I thought a follow-up post would be good here. My goal is to share the kind of knowledge that David asked for -- a short explanation for someone who might be new or unclear about these ideas -- so please excuse me if I don’t touch on some of the nuanced bits (and there are many, trust me!) of the theory.

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Before we talk about learning theory, we should take a step back and talk about epistemology - the branch of philosophy concerned with the nature of knowledge. There are multiple epistemologies, but two are important here.

Objectivism: An objectivist epistemology holds that knowledge and meaning exists independently of the learner. It is not just believing that “a rock would be a rock if we were here or not.” Instead, it is a belief that the rock carries some meaning of what it is to be a rock, and when we study rocks we are discovering that meaning. Objectivism is sometimes called empiricism or externalism.

Constructivism: A constructivist epistemology holds that there is no objective knowledge. This doesn’t mean that there aren’t objects, but that the knowledge and meaning we associate with objects is constructed by us as we engage and interact with the world. Constructivism can take several different forms, depending on the importance placed on social and historical interactions.

Hopefully you can already see how different epistemologies can affect a person’s view of teaching and learning. Now let’s compare three learning theories associated with these epistemologies.

Behaviorism: Behaviorist learning theory is often associated with an objectivist epistemology. Human actions, including exhibitions of our knowledge, are viewed as behaviors that respond to a stimulus. The process begins with a transmission of knowledge from the teacher (which can be a non-human source of knowledge) to the student. If the response is the expected behavior, the student is rewarded. If the response is not the expected behavior, the student is punished. By stimulating the student with rewards and punishments, the teacher encourages the student to receive the transmissions of knowledge.

Information Processing: IP theory still applies an objectivist epistemology, but differs from behaviorism in that learning is seen as an inner cognitive process and not just a response to a stimulus. The brain is seen roughly as analogous to a computer -- it has memory and processing systems that serve to store and analyze information, although the analogy doesn't extend to understanding exactly how those systems actually work. Application of this theory in teaching generally involves heavy doses of repetition to ensure that knowledge is retained in memory.

Constructivism: Not surprisingly, constructivist learning theory is associated with a constructivist epistemology. Because knowledge is constructed by the learner, the teaching/learning process focuses on creating conditions for that construction to happen. There is no "transmission" of knowledge. Depending on the form of constructivism, the teacher might facilitate the construction of knowledge through the inclusion of contexts and social interaction.

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The science of education would be much easier if we could prove that some theories never work and one works all the time. But we can’t. However, I don’t know of an educational psychologist that doesn’t think constructivist learning theory (in at least one of its variants) works better than those based on an objectivist epistemology. So why doesn’t every teacher do it? Or do it well? Teachers in classrooms have resource, time, and other constraints that makes constructivism more difficult than we all wished it was. Also, it’s not always clear cut which theory is being applied by a teacher. Suppose you were to peek into a classroom and see a teacher speaking to the entire class. Maybe the teacher is trying to transmit knowledge in a behaviorist/IP way. Maybe the teacher is trying to help students get into a certain frame of mind and is a constructivist. You can’t tell at a glance because the learning theories don’t always present themselves as extreme opposite ends of the spectrum. But when there’s controversy, we like to pretend that they do. Enter Salman Khan.

There’s been a lot written about Sal Khan and the Khan Academy over the past several months, including a recent article in Wired Magazine that became a large part of this morning’s discussion on Twitter. The idea of learning by watching videos isn’t necessarily behaviorist or solely an application of information processing theory, but it’s more easily seen as a medium for the transmission of knowledge, not construction, and the point-keeping for problems right and wrong also fits the stimulus/response model. Phrases such as "Khan and Gates both admit there’s no easy way to automate the teaching of writing" also point at behaviorism and IP. (There’s an underlying assumption here that if teaching can be automated, learning will be automated.) The Wired article quotes parents and teachers who are amazed at the progress their kids are making, measured by problems completed, modules finished, and badges earned. Are those students learning? Of course they are, but exactly what they are learning and how well they understand it is at the core of the debate.

Behaviorism and information processing aren't mentioned by name in the article. Perhaps they don't need to be; it’s a style of education that most all of us are familiar with and perhaps it doesn’t need much explaining. Constructivists are named as Khan’s critics, and the theory is described using terms such as "play around" and "fumbling around," the latter of which was probably an unfortunate choice of words by a constructivism supporter. Saying that "it’s better to give kids activities that let them discover the principles of math and physics on their own" doesn’t give enough credit to teachers in good constructivist learning environments. When done well, teachers don’t just "give" activities and students aren’t "on their own." Instead, there’s a careful orchestration going on and the teacher is with the students 100% of the way, asking questions, providing feedback, provoking the student to look at tasks in ways that help students construct deep understandings. Can a video do this? Obviously there are severe limitations -- not limitations that prevent all learning, but limitations that might be preventing the best kind of learning.

Look for an upcoming post about what happens when the instruction is based on objectivism but the student, a kid we'll call "Benny," constructs knowledge in his own, incorrect way. The "Benny" paper by Stanley Erlwanger in 1973 had huge ramifications for research and teaching in mathematics education, and has interesting parallels to learning via the Khan Academy.

I'd like to give great thanks to Jackie Hotchkiss for helping review a draft of this post. (Any final shortcomings are solely mine, of course.) When in doubt, talk to an educational psychologist!

Teaching as a "Moral Craft"

I recently read The Peculiar Problems of Preparing Educational Researchers by David F. Labaree, and was particularly struck by this paragraph:

The main reason for [teaching as a moral craft] is that, unlike most professionals, teachers do not apply their expertise toward ends that are set by the client. A lawyer, doctor, or accountant is a hired mind who helps clients pursue goals that they themselves establish, such as to gain a divorce, halt an infection, or minimize taxes. But teachers are in the business of instilling behaviors and skills and knowledge in students who do not ask for this intervention in their lives and who are considered too young to make that kind of choice anyway. By setting out to change people rather than to serve their wishes, teachers take on an enormous moral responsibility to make sure that they changes they introduce are truly in the best interest of the student and not merely a matter of individual whim or personal convenience. And this reponsibility is exacerbated by the fact that they student's presence in the teacher's classroom is compulsory. Not only are teachers imposing a particular curriculum on students, then, but they are also denying them the liberty to do something else. The moral implications are clear: If you are going to restrict student liberty, it has to be for very good reasons; you had better be able to show that the student ultimately benefits and that these benefits are large enough to justify the coercive means used to produce them (Cohen, 1988; Fenstermacher, 1990; Tom, 1984).

I've heard other people argue that the teaching profession doesn't compare well to other professions, like being a doctor or lawyer. After reading this, I'm happy they don't - teachers have good reason to feel like they're doing something special.

References
Labaree, D. F. (2003). The peculiar problems of preparing educational researchers. Educational Researcher, 32(4), 13-22. Retrieved from http://edr.sagepub.com/cgi/content/abstract/32/4/13.

A 2-for-1 "Soft Skills" Special: The Sit-Stand Paradox and Defective Girls

Written for The Virtual Conference on Soft Skills, July 3 - July 31, 2010

Of all my courses as a pre-service math education major, I think I enjoyed educational psychology the least. When you spend much of every day deciphering the infallibility of mathematics, the "theories" of social science don't hold up well against the scrutiny of a brain hardened by the concept of rigorous proof. I now realize I should have adjusted my perspective in whatever way necessary to ensure I got more out of the class, but even if I had I don't think anything would have fully prepared me for a classroom full of independently-minded students. You just have to jump in there, year after year, class after class.

In my six years of teaching high school math I developed some wonderful relationships with my students. Without overstepping the bounds of a teacher-student relationship, my students became my friends, something I seem to remember being told I should never let happen. But I would look forward to seeing my students each day; I would try to make the most of my time with them, and I would miss them when they were gone. If that doesn't describe "friends," then I apparently don't know what a friend is. I might be in the "ivory towers" of academia now, but I honestly think of my former students from those six years every single day.

That's not to say that there weren't MANY bumps and hiccups along the way, and I regret the lack of effort and deficits in my own character that prevented me from forming stronger relationships with ALL my students. But as I reflect back, two lessons learned (one a realization, the other a piece of advice) helped strengthen that special student-teacher bond.

The Sit-Stand Paradox

Ask any teacher what period of the day is likely to be their least favorite and most will answer, "last period." It doesn't matter if your school has four, six, seven, or eight periods -- there is always a last period. By far my toughest group of kids I ever attempted to teach was a last-period business math class. It's a bad sign when, on the very first day of class, a student who you've just met pulls you aside and tells you, "I don't know who decided to put this mix of students together in the same class, but it's a really, really bad idea." I'd like to think my chances with them would have been much better if I'd seen them before lunch.

So what makes last period so tough? I think the explanation is simple: after a long day at school, students are tired of sitting and teachers are tired of standing. Should you be trying to hide this fact from your students? No! They're people, not circus animals that might attack if they sense fear. If you want a class that works with you, not against you, share your motivations and frustrations. Establish common goals and understandings so you can move forward together. Maybe it's time for an out-of-seat activity, a lesson outside, or a trip to a less familiar room in the school. I know it sounds easier than it really is, but even something as simple as sending your students to the board while you sit at their desk can be just the change in perspective everybody needs. If you're worried that instruction might suffer with a little chaos, think of how much it's already suffering when all the students are watching the clock hoping to be somewhere else.

Don't Treat Boys Like They're Defective Girls

During my third year of teaching I had the privilege to share a classroom with Miss Sandra Miley, a 30+ year educator who took a distinct pleasure in teaching freshman boys' seminar and P.E. If you didn't already know, freshman boys are at that awkward age (which lasts from about 11 to 25, as far as I can tell) that can make them awfully hard to teach. Not so for Miss Miley, who passed on this hard-earned wisdom:

"Do you want to know the secret to teaching boys? It's simple. Don't treat them like they're defective girls."

Ever since I was given that Yoda-like advice, I've been trying to unravel the mysteries contained within. Certainly Miss Miley had a perspective from 30+ years in the classroom that I may never match, but I think I got the point. As teachers, our jobs are made easier (not necessarily more enjoyable or effective) when students sit at attention, take notes, raise their hands, follow rules and instructions, and hang on our every word. If you have students who fit that description, I'd bet dollars to doughnuts that the majority of them are girls. Should you want or expect every student to belong in that category? I sure hope not. So don't punish boys who don't happen to behave like those girls. Such behavior is probably not in their DNA.

If you're not convinced, here's a little anecdote to consider. A highly-respected education researcher shared this hypothesis at a conference last fall (identities have been hidden to protect the unpublished):

"I've never been brave enough to try to publish this, but I've long wondered if boys develop better problem-solving skills because they aren't paying attention in class. Girls who listen carefully to instructions and take notes always know exactly where to start because the teacher told them. Boys who goof off during instructions spend a lot more time and effort sorting out the aspects of a problem for themselves, and that practice pays off in the long run."

Peeping Tom: Finding Windows in the Ivory Tower

Today on Twitter, Tom Whitby posted:

I understand the intent of Tom's post: we have too many teachers who have become detached from some of the "big thinkers" in education. It's easy for a teacher, with all the pressures and responsibilities, to become isolated in their classroom with their students. Fortunately, it's easier than ever to traverse the branches of the internet and find leaders in education online, as well as other teachers who want to share, discuss, and debate big ideas in education.

While I'm sure Tom didn't intend for his list to be all-inclusive, all the names listed have something in common: none of those people are current professors of education. I'm not saying that professors have cornered the market of good ideas, but rarely do I see them mentioned on Twitter or elsewhere outside the ivory towers of academia. (Not suprisingly, several professors who are breaking down this wall are professors of educational technology, such as Alec Couros and Scott McLeod.) Trust me: ed school professors care just as deeply about students, schools, and the improvement of our educational system as anyone, and many have wonderfully big thoughts and ideas. In addition, they have a scholarly duty to promote ideas that have been tested and shown to have positive effects, not just ideas that sound like good ideas.

This might not be the place for a lame sports analogy, but I'm thinking of it this way. I love baseball, and I could happily spend hours listening to Bob Costas and Peter Gammons describe the nuances of the game. But if my job is to walk up to the plate and hit a major league fastball, do I want Costas or Gammons as my hitting coach? No! Give me Joe Varva or Rudy Jaramillo. Never heard of them, you say? Well, they're both major league hitting coaches, for the Twins and Cubs, respectively. Costas or Gammons could probably help me get a swing that looks like Joe Mauer's, but I'd need Varva or Jaramillo to help me develop my best swing, not one modeled after somebody else's. And neither Varva or Jaramillo themselves played in the majors. They know what they're doing because they tirelessly treat their jobs as a science. Alfie Kohn isn't Joe Varva. He's Peter Gammons -- an intelligent and thoughtful commentator who is making positive contributions to his profession and our enjoyment, but not necessarily a scientist.

So while you're asking your colleagues about Kohn, Marzano, and Sir Ken, try asking them if they have heard of Linda Darling-Hammond, Deborah Ball, Michael Apple, Truus Dekker, Alan Schoenfeld, or Lorrie Shepard. Don't know them? You should, but if you don't, don't be too hard on yourself. I was disappointed to the see that the list of Race to the Top scorers was heavily populated with educational consultants, institute founders, foundation advocates, and others who might profit from the results, instead of more ed school researchers. So maybe Arne Duncan doesn't know many of the names on my list, either. But it's not all his fault, and not all your fault, either. Our system of higher education and scholarly publishing is holding up those ivory walls, walls that work both ways. Stick to Alfie Kohn and let the wall crumble, or read Linda Darling-Hammond and try to knock it down.