Expert Interviews

	Jim Kaput Biography

1.  Interview with Jim Kaput

Jim Kaput is Chancellor Professor of Mathematics at University of Massachusetts, Darthmouth. He is the principal investigator of the SimCalc Project, funded by the National Science foundation, director of the Early Algebra Project, funded by the US Department of Education, and the founder and CEO of SimCalc Technologies, LLC.

Listen

"It’s my sense that technology can support algebra learning by helping build meaning...”

"We’re going to be moving ahead on two fronts…”
 

"This vision integrates the technology much more richly into the daily practice of the classroom…”

Read

INT: It’s now federal policy that school leaders at the state and district levels have to take a close look at the research literature, as they make decisions about where and how to apply technology to teaching and learning. So, focusing in on middle school math, Jim, here is the question: Based on your experience, what are the most promising ways that technology is supporting the current curriculum goals in middle school mathematics?

JK:   (Hear audio) It’s my sense that technology can support algebra learning by helping build meaning, because meaning -- building meaning is the essential problem of teaching and learning algebra, and technology provides some means for doing that, in particular, by helping students link representations, they can use their knowledge of number and arithmetic as a basis -- building tables, and then graphing those tabular data, and then finding formulas that fit that data, and checking those -- all of that is facilitated by varieties of technology. Of course, the dominant technology in algebra teaching and learning is the graphing calculator, and there’s rather a lot of writing about the use of graphing calculators in teaching and learning algebra; however, there’s not a lot of rigorous research on the impacts of that. There’s some, but not a lot.

INT: Right. And the general gist of that research, would you say that it’s positive?

JK:   Well, generally yes. There are some meta studies, and some reviews of literature, all of which indicate positive impacts of the use of technology in algebra teaching and learning, and pre-algebra work, as well.

There are lots of descriptive accounts of technology use, but not a lot of work rises to the level of rigorous research, particularly in terms of randomized assignments of students. We’ve taken the cue from those who are calling for more rigorous research and have built into our latest round of funded work a series of quite rigorous studies to test claims we make about student engagement and learning in a technology-rich context.

It is important to note, however, that the underlying technological base is evolving fairly rapidly, and there tends to be more attention to the innovation itself than attention to detailed research that evaluates outcomes in statistically rigorous ways. Most of the folks involved in the work are driven by innovation rather than by evaluation and consolidation, and that’s been a long-term problem, I think, in the field overall.

It’s important to note that the middle school mathematics curriculum, at least the algebra strand, is centered on building understanding of rate, ratio, proportion, and linearity -- linear change; that’s been at the heart of the current NCTM Principles and Standards, and even the earlier 1989 version. And that continues to be, I think, the core mathematics that the algebra needs to be built around, that notion of rate of change, particularly constant rate of change. However, you can’t appreciate the constant rate of change unless you see what it’s not, and so students need to be brought into contact with quantities that, in fact, change in a non-constant way. And there are, I think, some really good opportunities for use of technology to help with that.

INT: And now we’re talking about more than graphing calculators, I assume?

JK:   Well, graphing calculators augmented by data collection devices.

INT: Probeware?

JK:   Probeware, exactly. And, in fact, I’ve been using CBL [calculator-based lab] stuff, in conjunction with some software called “SimCalc MathWorlds,” that allows you to import physical motion. While most probeware allows you to import the motion and graph it -- position versus time graphs, for example, and in many cases velocity versus time graphs -- our stuff enables you to replay the motion after you import it, and so you can review the motion again and again, against its graph; you can compare how the graph changes with how the motion changes, and it’s very important for students to make that solid connection between the graph and the motion.

INT: Can you share some of your experiences working in the schools?

JK:   Yes. We’ve had an interesting evolution over the past 10 or 12 years. We started out building software and curriculum for computers, and did a lot of stuff that assumed that students would be learning either alone or in pairs in front of a computer, and discovered that as we went into schools, computers were actually quite rare in mathematics classes, doing a lot of this at the eighth- and ninth-grade level in algebra. What we did find were graphing calculators, so we had to port our software and our curriculum to the TI-83+ graphing calculator, because that’s what people had, and furthermore, we came to realize that this notion of technology, as represented by computers in a computer lab, and kids working on computers in the computer lab, really was an outlier in terms of daily practice. We needed to work with a kind of technology that is in the classroom on a daily basis, and where the teacher has a significant role to play. And so now, our approach has evolved to the point where the teacher plays a fairly central role in orchestrating the use of the technology, and using, for example, the CBL stuff that I was just describing. It is often done in a whole class demo-type situation with a common projection display that all of the students are watching. Or the students may work in small groups, and then show their motions in front of the class; that sort of thing. So, it’s been in evolution -- in the early ‘90s, where we were thinking about computers and computer labs and kids working individually or in pairs on computers, to a technology that is much more integrated into daily classroom practice in the hands of both the teacher and the students, and where the teacher plays a fairly prominent role in orchestrating the interaction with the technology and the curriculum.

INT: With the increasing availability of laptops in schools, do you think this work is going to get ported back onto laptops? Or do you think the hand-held calculating device is going to be powerful enough?

JK:   My sense of what is happening is that, first of all, the laptop initiatives get far more press and attention than the actual numbers in schools would represent; the numbers of kids out there in schools where they have access to laptops is tiny. It’s in the 1 percent or under range, and those are particularly wealthy communities, or under special circumstances, like the Maine situation. So, I don’t see laptops as a likely prospect in the next five years. I do see increasing sophistication of handhelds to more tablet-like, PDA style. I believe it’s going to be important to take advantage of that kind of technology, because it fits in a much more integratable way daily practice; the key is making sure the technology fits with what teachers and kids typically expect to do in the classroom, and to take advantage of the physical structures and patterns of work that already exist, and be affordable, obviously. You shouldn’t need to build another room in order to use powerful technology.

INT: So, in the next three to five years with the kinds of changes that you anticipate in what technology can do, do you think curriculum in middle school mathematics will advance, or will this simply be a better way of realizing the current curriculum in middle school math?

JK:  (Hear audio) Yes. We’re going to be moving ahead on two fronts, and we’ve been doing that for some time. One has to do with doing old things better - teaching the existing curriculum more effectively with greater student engagement and improved student achievement on standard measures. And then there’s the other side, which is using technology to do better stuff. On that side I see tremendous promise in the ability to have students create their own individual functions on their own portable devices, and then bring those functions together on the teacher’s device, typically a computer with a projection display in front of the room, which enables students’ work to be aggregated and so that the students can make individual functions that are aggregated into families of functions. The attention then shifts from the individual function to the family of functions, and I think that over the next three to five years, we’re going to begin to see a strong move on the part of the curriculum to take advantage of these technological affordances to shift attention upward from the individual object to the family of objects, and that’s a deeper understanding of algebra than is currently existing, and that’s the kind of better stuff that I foresee beginning to happen in the curriculum.

INT: Excellent.

JK:   I think it’s in the nature of things that the base mathematical object in the twenty-first century is a parameterized family of things, rather than the individual thing. I think there’s a shift that’s taking place, so that we don’t think about a particular quadratic function, but we think in terms of what the family of quadratic functions looks like, or the family of linear functions -- what does that look like? -- particularly if you say vary the b in y = mx + b, or vary the m in y = mx + b, or vary them both, as we do in some of our new network SimCalc activities. That’s a fundamental shift towards, I think, doing better stuff. You might ask why -- why is there a shift towards the parameterized family of things being a base mathematical object? That has to do with the fact that mathematics is moving in the computer medium, and it’s just as easy, in fact, probably easier, to make a family of things on a computer, than it is to make an individual one. And so, as the medium has shifted, the base mathematical object shifts as well -- and I think that’s where the curriculum is likely to head as the technology matures.

There’s one other shift that I’d like to identify. There was a time, beginning in the ‘80s and continuing through the 1990s, where folks thought that linking representations was what it was all about, and that we could get students to learn algebra better if we could get them to link tables, graphs, and formulas. Alan Schoenfeld wrote a paper called “Learning: The Microgenetic Analysis of One Student’s Understanding of a Complex Subject Matter Domain,” in which he studied a particular student’s work with linear functions in a nicely-linked environment. The students understanding was incredibly unstable, despite extended experience in this environment, where she had tables, graphs, and formulas that she could manipulate. It turns out, in retrospect, that this student’s understanding was not grounded in any of her other knowledge, so that the linear functions were about certain tables, and the tables were about certain graphs and vice versa. None of the whole thing actually meant much to her, and as a result, her understanding was very flimsy and unstable. The big change that technology can afford us to deal with that problem is that probewaretype devices, for example, the Calculator-Based Ranger (CBR), allow you to make the mathematics be about something other than just mathematical representations. The mathematics can be about some physical phenomenon that you, the student, are intimately, and kinesthetically involved in. And that, then, grounds these representations as meaningfully related to one another and as different ways of representing a concrete experience. And that’s, I think, a very deep change that the technology makes available to us, that we need to do a better job of taking advantage of. In SimCalc MathWorlds we also hot-link most any function to an animation, so the functions DO something for you. They don’t just sit there.

INT: Can you comment on how technology emerges as a tool in this classroom that you’re imagining?

JK:  (Hear audio) Yes. This vision integrates the technology much more richly into the daily practice of the classroom, and to the extent that that daily practice can be moved towards getting students to ground their learning in real physical activity that may be socially meaningful as well, then that daily practice can take advantage of the technology. If the daily practice continues to be stand and deliver -- let’s graph this function, now, let’s graph this next function, and let’s make a table for this function, and let’s graph that table, we’re not going to make too much progress. The teachers need to be able to take advantage of the technology, and that in turn means changing their practice often in fundamental ways. There’s another complication in that the teacher’s understanding of the mathematics needs to be fairly substantial to take advantage of these kinds of technologies. You’re not handing off understanding of the mathematics to the technology, where the technology then delivers it to the student. That’s not how I see the technology being used by the teacher; rather, the technology is a tool in the hands of the teacher.

INT: And in your experience in the projects that you’re involved with, Jim, are you seeing teachers come up to this level of sophistication in the understanding of mathematics, so they can use the technology?

JK:   I think that this is, in fact, the real bottleneck at this point. Certain teachers with whom we engage have both a knowledge of mathematics and the flexibility of pedagogy that enables them to take advantage of this kind of technology relatively quickly and powerfully; other teachers have the mathematics, but a rigid pedagogy that makes it quite difficult; yet other teachers have a tentative understanding of the mathematics, but are willing to try to improve student learning, and are flexible, and those teachers seem to be able to learn the mathematics as they teach it, using the technology, because they become substantially engaged with the mathematics in the same ways that we want the students to be substantially engaged in the mathematics. And so, we see strong progress among those teachers who are willing to change their practice in order take advantage of the technology. But I think there’s a lot of work ahead of us in terms of designing appropriate experiences for teacher professional development, and also for building this kind of experience into pre-service teacher education programs around the country.

INT: Jim, do you think we’ve covered the issues?

JK:   There’s one other area that I didn’t talk about, having to do with, graphically-editable functions that is to my mind directly applicable to algebra. Let me see if I can set the stage. Most functions that are dealt with in algebra are globallydefined, typically by a formula, but in fact, most functions that exist in the world, based on the data of the world, like economic data, or data based on the physical world, do not fit simple formulas. The temperature of the day, for example, plotted against time, is a function, but you’re not going to find an easy formula that describes it. The same applies to the velocity of the school bus that took you to school on a particular day; again, it’s a highly irregular function, but nonetheless, it’s a function of time. It turns out that if you define software that enables people to make functions in a piecewise way, and then graphically edit those with hot spots that you just drag around that embody the kinds of irregularity that real phenomena embodies, then you’ve got a whole new way of building functions in the classroom that is quite a lot more flexible than what you can build using formulas. And in particular, then, you can attack issues of slope as rate of change, and the idea of linear change, by providing examples of functions that are not quite linear. I tend to think of y = mx, and y = mx + b as a kind of degenerate case, and if you can put that kind of function alongside a function that consists, say, of two or three pieces, each of which is a straight line segment, but which have bends in them (a polygonal shape), then you’re looking at a contrast that makes the linear case more meaningful. Furthermore, you can look at the slope of each of those pieces and see how the slope changes; if you have, in particular, an interpretation of those, as say position-versus-time functions that actually drive an animated object on the screen, then you’ve got a chance to build a deeper meaning of slope as rate of change and a contrast with linear change, right before you on the computer or calculator screen. Some of the work that we do on the SimCalc project is based on exactly that, where students make piecewisedefined position functions that contrast as linear single piece functions where each function drives its own motion animation. Then students can compare the differences in the graphs and see how their respective motions differ, side by side. This both a deep mathematical experience and an exciting one, as, for example, when students build “races” with Y=2X.

INT: And are able to, therefore, grasp more of the variability of the real physical object?

JK:   Right. And furthermore, in this kind of approach, while you’re building the notion of slope as rate of change, which of course, is the reason why anybody would study slope in the first place; you’re doing it in a way that lays the basis for understanding of calculus that comes later. And that’s, I think, the really important aspect of this kind of approach, where we build in enough variability inside the function, using these piecewise-defined segments, that enable you to contrast with the degenerate case -- the linear function case -- where the change is constant.

INT: Great. Well, I think that wraps us up, Jim. This has been wonderful. You’ve really covered a lot of ground.

Reference: Schoenfeld, A., Smith, J., & Arcavi, A. (1993). Learning: The microgenetic analysis of one student’s understanding of a complex subject matter domain. In R. Glaser (Ed.), Advances in Instructional Psychology (Volume 4). Hillsdale, NJ: Erlbaum. This interview was conducted on 1/20/04. An edited version was reviewed and approved by Jim Kaput on 5/4/04.

Biography

Originally trained in mathematics, Kaput became interested in teaching teachers and reforming undergraduate education and, in the representational side of student learning - a continuing interest. His other continuing interest is finding ways to bring educational & economic opportunity to those for whom it is too frequently denied. In the 80’s he became interested in the empowering potential of newly available technologies, including concretely-based learning of ratio, proportion and elementary functions (at the Harvard Educational Technology Center), and new visual databases (e.g., as co-PI on the TableTop Project). In the late 80’s he began working with Tom Romberg on the question of how to bring empowering technology into the service of math-ed reform. As an Associate Director of the National Center for Research in Mathematical Sciences Education (NCRMSE) at Wisconsin, he began efforts to understand how the core math curriculum might be fundamentally reorganized to democratize access to big ideas, ideas such as calculus. Kaput is on many R&D project advisory boards, a consultant to the NSFconsultant tothe NSF systemic initiative programs, and is a frequent speaker at national and international meetings.

For more information about the work of Dr. Kaput, see:

• Dr. Kaput’s curriculum vita
• SimCalc project
• SimCalc Technologies LLC

		Printer-friendly version