OK, I’m John Dossey uh and uh, you know, I taught for ‘x’ years—what 33 years at Illinois State University uh in the math department.
Well, I—I think when you talk about the history of mathematics, uh especially as it plays out today with respect to reform and—and programs at the undergraduate level for people entering teaching or really entering the work place. Uh, I—I would probably go back and uh—and—until around the turn of the century uh David Eugene Smith at uh Columbia Teacher’s College um was involved with the International Mathematics Union, which was the professional mathematics group internationally, and uh in the period around I believe 1908, they decided that it was probably important to start to think about a professional group of mathematics educators. And uh I think that’s one important time in—in the history of—of really the teaching of mathematics that we want to look at because that’s at the time when we started to take a look at “What is it to train a teacher? What—What’s the um content—pedagogical knowledge long before Shaman?” Uh, speech at ? in 1986. Uh, but also it—it started to take a look at uh the differences between mathematics itself as a discipline, which was uh, you know, the classical study at that time of analysis, algebra, geometry, etc, uh to begin to look at a new uh discipline within the mathematical sciences broadly, and that was math education. Uh how—how do people come to know mathematics? Uh what are—what are curricular elements, the articulation of school programs with the university programs, etc? Uh throughout then uh I think the ensuing years, but especially in the 1930’s with the work of Brunell, um the work of Beagle and others around the school—mathematics study group uh in the late 50’s or early 60’s. Um I think we saw mathematics education begin to grow and—and research in math education begin to blossom. But I think when we talk about the history of math and changes of leading to reform we also have to look at the mid 70’s. Uh and in fact, actually, around 1980 the publication of the committee on under graduate program in mathematics uh guidelines that came out for the Mathematical Association of America—and it spoke uh I think in broad terms about uh the mathematical sciences as a word uh for the first time. Where we start to take a look at mathematics as an encompassing—not only kind of mathematics and math education, but statistics, operations research, uh new forms of applied mathematics other then just those that went into physics and chemistry, but those that go into the mathematics of finance, the statistics applied to sociology, and started to take a look at uh a quantitative literacy for those in mathematics in a much broader sense. I—I would—I would kind of pick those out as—as important dates. Uh, clearly today mathematics education itself has—has evolved into a discipline in and of itself uh but within, I think, I—I—I’d hope within that broad umbrella of things called the mathematical sciences.
Well, I think in—when you think about mathematics as ideas, I think back to a quote from the British mathematician uh, uh Pardy Mone. He talked about mathematics having permanence because uh unlike the work of other artists it’s uh—it’s made of ideas and it lasts long. And that’s—in mathematics we tend to—we tend to think of um fundamental beliefs that we have. Uh things uh—your—your—I think everybody’s familiar with two points determine a line as a—as a basic idea, an ? In mathematics uh we—we tend to, in—in the end at least, get down to what—what are the kind of fundamental beliefs and how do we build forward with those. And so our building blocks are ideas. They’re concepts, they’re relationships and once one takes on these uh—these basic understandings and ascribes to them, uh the building blocks are the same today as they were 3000 B.C. Uh or—in—the—the ideas are ways in which we—we—we begin to model things in the real world. Uh they’re—they’re—they give us tools to begin to talk about relationships, etc. Um, so that—and the basic idea of—I mean the basic building materials of mathematics are ideas. They’re not uh—they’re not experiments in a lab, they’re not uh, uh things that have a physical tangibility to them.
Uh, I think—I think quantitative liter—a person’s quantitatively literate when they uh, first of all, have the tools to understand a situation that involves number, or shape, form, uh chance, uh change. They’re able to quantify. They’re able to look at the patterns and bring out of it uh things that they can begin to assign number to or name or be able to compare and contrast to the elements in the situation. I think, first of all, it’s—it’s being able to understand such situations. I think at a second level that they’re able to predict that they’ve developed that understanding of quantitied, understand what’s regularity or what outcomes may be. And I’d say third that they move to a—a position where they can maybe begin to control situations. Now I—I—I want to quantify control because my wife always says that that’s a male thing uh in—or a military thing. But I think about in a production situation, about maintaining quality in a factory or—or beginning, for example, a classroom to predict the—the misconceptions that students might form in a situation in being able to think through first of—of how—how to deal with a situation to minimize the—the unwanted aspects. And I think mathematics gives us the tools to do that in many—many times to be able to look at a structure and look at—at what—what occurrences can happen and what are the actions that one can take either strategically in the big—big sense, or tactically in the small sense to uh—to alleviate the unwanted elements.
Uh, the development of the NCTM standards I think uh—I’m actually going to start over. (interruption) The uh—the NCTM standards uh—and I think more broadly standards that have been set at state levels uh that more directly affect teachers then—then kind of global professional recommendations that come from something like the NCTM or the Math Association of America. I think they play a vital role in defining in a public way what we think is important as professionals in a field uh for students to both know and be able to do. Uh in the—in the 1980’s—late 1980’s there was uh I think paranoia in our country about education. There may still be. Uh the deals don’t—don’t—basically with the ideas around how do our students uh compare internationally. Uh how—what—what do our students know and—relative to what the public think they should know. And—and how—how to deal with the—the discrepancies between these views. When you think about the United States and I think in the last year having something like 14,855 school districts uh and that each of those districts by and large have their own ability to—to once the doors are closed and even the classrooms within those districts, to enact a mathematics curricula of their own, that if we want to move students in a highly mobile nation of—of—of our own in a creative, rapid change to a—a level of competence that the public’s asking for and to a large degree that the profession has illish, strong ideas of—of how to reach, we have to have some guidelines. And those guidelines, I think, uh address the issues of—of common expectations because when students move from one district to another, they don’t have common expectations uh there’s—there’s a problem. Secondly, I think to—to begin to give some commonality in materials, commonality in assessments, uh opportunities for all students to learn a—a common core of material. If we don’t move to at least some publicly ascribable lists of objectives, uh the chances of changing where we’ve been all these years are pretty small. On the other hand—and you may find it strange to hear me say this, I don’t this that the—the NCTM standards or another set of standards are—are like uh, you know, the—the ten commandments that everyone has to—to do in exactly this form. Uh, in fact, that would be the worst thing to have a template uh to put down and—and—and have cookie cutter pieces around. But they are at least a way of beginning a conversation of—of what common goals should be and—and helping refine that thinking over time. As you know, the NCTM this year released its principles and standards for school mathematics uh, you know, 11 years after the original standards. And, you know, it shows a—a—a movement in—in terms of—of cutting down the number of content areas uh to five, which matches what many states are doing. And so again, it’s a way of—of beginning to move that to, again, common understandings. So I think that’s been the big power of—of the standards. I—there’s the danger of people who want to read it literally and—and use it in arguments rather then opening discussions. And I—I think that’s the downside you always have to work around.
Well I think—uh to me the—the biggest impact to the NCTM standards probably has been a policy impact. Uh it’s opened a—a—a avenue, a—a basis for discussion between legislators, between teacher educators, between classroom teachers, parents. It’s formed—it’s formed the table that people can come to to talk about uh mathematics education, what’s good, what’s bad. Uh it—it’s allowed uh, I think, the focus to be brought upon the role both of processes and content and how they interact with one another. In fact, that’s I think also a thing that—that showed up to a large degree in the—the revision of the standards is the uh—the enlargement of that—that kind of set of processes uh to include representation which is a thing that evolved over that 10 to 11 year period in many programs. So they talked about doing things with graphical models, concrete models, verbal models, uh doing it symbolically, doing it with tables of data, becoming to understand mathematics uh in a broad set of venues and being able to move between those uh representations. Uh so I—I think it has a living document and also is a—is a basis for policy discussions uh which clearly has, I think, shown through. In 46 or 47 states have changed their—their state uh outcomes uh as a result of these standards. Not only peril, though, but at least as a result of that opening discussion.
I think the role of communication in—in mathematics education uh takes on many meanings. Uh one of the meanings is the—the teacher’s ability to hear—hear what students are saying, to react to it uh, to be able to phrase uh what the student’s saying in a different way. I ask the student to—to think deeper about a topic. Uh another, I think, one of the roles of communication in—in mathematics program is that mathematics is of itself a social activity. It’s a coming together of saying, “We believe these things is—are fundamental in what things follow.” If—if you’re looking at mathematics—the doing of mathematics and the solving of problems uh by and large they’re not done by people uh stacked away in a closet somewhere working under a light with a green shade on their forehead but it’s really by people who are out uh tackling significant problems in society and trying to say, “What are the fundamental uh relationships here? What do we know about this from—from other areas? How do we put it together? How do we reach a solution? Or in the case where there are several solutions, which of these may play out better in this situation because some of them may not be uh—one may bring—bring to reality given the technology in that particular setting?” So it takes a lot of uh talking and shifting between representations, modeling things in different ways. A lot of communication can be visual communication. Uh part of it’s verbal uh communication and being able also to go to uh text and to—to read with meaning to learn. Uh a lot of mathematics is already embedded in—in printed or—or technologically uh reachable material but the thing is, is how does one bring meaning from that material? So I think from a—from a speaking standpoint, a reading standpoint, a working in groups and interpersonal communication standpoint um we can’t do without communication. Um the—the emphasis on communication I might mention in the standards, I would trace back to—to the work of the Equity Project of the College Board uh in the early 1980’s which was the first one I think to really talk about reading and writing as part of a way of coming to understand uh, uh mathematics. Not only to—to understand other things uh that, you know, society’s developed.
I—I think that the alignment of uh assessment and more importantly evaluation uh with standards is important uh and that when one comes to interpret the results uh one has to know what framework items or—or other forms of assessment uh, uh both how they were developed and for what purposes they were developed. Uh and I—I like to draw distinction between assessment and evaluation. Uh in—in particular, when I’m looking at assessment, I’m looking at gathering information to improve either the student’s learning or—or the teaching act that would lead to structuring uh opportunities for that students to learn. And when I’m looking at evaluation, I’m really looking at—at uh kind of a summative uh activity that—that results in some ranking or mark uh for the students or group of students. And when we, in either case, gather information that uh is not aligned with the instructional program and I—I’ll take the standards and move it to an instructional program, then—then we have data uh either whether it’s to inform teachers or to inform the public or some other group uh which probably misrepresents uh the situation in which it’s being interpreted. And uh I think there’s a very difficult task to do is to align assessment uh with standards or—or with instructional program uh because one has to make sure that one is—is measuring the—the same type of content or process intent uh that was either called for in an instructional design or was delivered in a classroom. Uh and—and I’m not saying that we should only teach things that were taught, but that when we’re measuring problem solving, we’re measuring, you know, higher-order processes, we know what were the inputs in terms of—of tools that the student had to work even though their coming to a novel situation. Uh otherwise one tends to look at the numbers and make judgments that—that are, I think, grossly uh unfounded in uh—in terms of—of generally what comes out. Because one can always add, subtract, do all kinds of things with the outcome of assessment, but if—if they don’t represent the—the—the basic type of content then—then uh they’re meaningless.
Uh, to me meaning—meaningful assessment is an assessment that’s—that’s getting at where the student’s mind is relative to uh a particular concept or—or principle if then type of statement, or a procedure that the—I mean an algorithmic procedure. Does the student under know—understand why the procedure works? Why that procedure’s better then another procedure? Uh, why it’s—it’s an appropriate procedure for this particular situation? They’re able to compare and contrast ideas. So I want the teacher to uh in meaningful—and when I talk about meaningful assessment, to be able to step both inside the student’s head and ask the kind of question about “Why are you thinking this way? What is this related to? Where—where does it lead to?” And then on a broader sense where one’s looking more at—at kind of uh summative rather than formative ideas, I’m looking for assessment that really talks to the—to the in goals of—of the uh particular disciplinary that not—not at—at—at small minute pieces but at things that show the student’s ability to—to bring processes together. To—to coalesce ideas and—and to apply them in—in—in situations that show some generality. That shows that the student is able to function as a problem solver in society or within the disciplinary area. And too often, I think, with standardized tests and other things we tend to look at the atoms and we tend to break uh learning down into little pieces, but we have no idea of the student’s ability to put them together like—what we’re beginning to do, I think, with student constructed responses and especially extended ones uh that causes other problems in assessment with rubric design, etc., but—but those, I think, are more meaningful. Apopum (?) would talk about them being—being closer to the end product. Uh, but then I think what the Scans Report and a lot of other things are calling for.
I—I think a lot of educational reform, at least the things that happened from 1985 forward, uh to a large degree came from the Nation At Risk Report in 1983. I think that’s almost all—all of the summaries point back to it, but in math education, in particular, I think the results of the 1986 national assessment and—and the release of the data from the second international math study are two areas where assessment played very large roles in—in the development of the standards. Uh, when we develop the original standards uh that came out—the Curriculum Evaluation Standards in 1989, we didn’t focus on these negative results as part of the document because uh—I shouldn’t say negative results, but these uh less—less then satisfactory results of student performance. Because they were a status report of where our students were relative to those items. But uh assessment I think uh gave us an idea of where students were, what was the status, where did we need to move out? Uh, since that time I think uh the Temes Study or the Continuing Nape Studies and actually the forthcoming Peces Studies all will provide us with continuing information about uh in a long-term trend to what degree are we moving in certain areas. Uh secondly, they—they give us targets uh, you know, to look at. And they—they give us a lot of I think raw data when people go in as the National Council of Teachers and Math, their series of kind of in depth analysis of—of the Nape assessments. When you start to look item after item and—and really what—what are our students misconceptions and how do these play out, I think they give us a lot of information for—for teacher education and I think a lot of our—our work in math education now has grown from Nape uh items and—and misconceptions. And we could give these items to students to talk about and then show them what did students do when they got those items. So helping teachers to be a little more prepared to deal with it. I think at state levels they’re providing us with—with uh markers about “Are the programs moving in—in the right direction?” Now that’s assuming that we have assessments that are—are meaningful in asking the right questions which isn’t always the case. So I think they—they both have fueled reform but also they continue to inform both at—at a policy level, at teacher education and parent, uh and—and right down to the student level uh the kinds of progress that we’re making.
Yeah, I—I think the—the development of teacher’s understanding of uh the uses of standardized test is I think a difficult task. Uh standardized test by in large talk us—inform us about groups because teachers rarely see the individual results of student’s work on individual items. And—and that’s even true of things like the national assessment. I think what these tests do is they—they provide with income when—when the tests are—are matched up. I mean they’re meaningful assessments and they’re aligned with uh one’s curriculum. They provide us with some uh information about movement uh over time. We have to be careful because there are differences between successive years classes and—and—and—and it’s a ques—it’s a question also about educational significance verse—versus statistical significance. In many cases we talk about growth as a result of standardized tests but we’re really only looking at—at statistical growth which may be less then one more item correct on the test. And so I—I—you know, I find it hard to really defend standardized tests other then they—they do give us a broad marker uh against a base line because to some degree the standardized test is a baseline that—that doesn’t change a great deal over years. And if our curriculum is moving in directions and the test does provide us meaningful markers relative to goals we have, then we can begin to see some trend. I would rather spend with teachers the time of—of looking at—at how we can—we can ask good questions that students give us artifacts in the classroom that we can look at and interpret, we can share with parents, we can actually share with the students and ask the students even how they would evaluate it. Uh, I think that’s a much more defensible uh, uh, uh time on—on assessment uh then—then I would probably would talk about with power timed test.
I think when I look back over Nape from the—the 19—mid-1980’s uh and even all the way back to the beginning of Nape math assessment in 1973, we went through a period of—of moving from really asking research questions in the beginning to becoming more of a—of um a baseline marker uh type of score when it shifted to the nation’s report card in the late 1980’s. In the early Nape, we asked questions that looked about differences in representation, differences in the way things were worded that really informed teaching and then, you know, in the mid—mid-1980’s it—it became getting a number and a marker in classifying children as basic. And I think that direction has been uh—been counterproductive for education. I—I’ve seen a shift in uh Nape from really the early 1990’s to the present where we’ve gone back to asking a lot more student constructed response items and focusing not only on interpreting those items, but also releasing them. Now over the Internet where you can see student work, the Rubrics, the items, uh and—and we’re—we’re gathering information that both informs the policy issues and clearly that’s—that’s a big purpose for Nape, but—but in addition we’re also beginning to inform teachers about goals and giving them exemplars that they can take and tailor them to their classroom. I think that to me has been one of the biggest impacts and changes from large scale assessment is—is the movement of it up to—to kind of marker numbers and then back to informing more about the teaching/learning process. Uh and I think most of us who come from content areas like math education, reading, uh this is really bringing information back, that it changes it to an assessment rather then an evaluation. Uh that it gives us ways of thinking about improving teaching and improving uh learning.
Well, if you ask me what uh my—my crystal ball look is I think I have to answer in two ways. One is as a math educator what I would wish for and secondly, as a realist in the world, what new may come. I’d like to see assessment continue as a math educator to focusing more on student thought, uh student’s ability to attack realistic and achievable uh, uh problems in ways that we—we can reliably—and I think that’s important—uh score. And uh—and whenever I say score I’m talking about a grade. I like to keep—keep Rubrics and grades separated because the items are only a part of a total assessment of a student. But I’d—I’d—I’d like to see us extend those to group problem solving situations uh where we would perhaps collect uh information of student’s working in groups and then have them uh have the opportunity to discuss an issue but then individually uh respond to it. So we’re seeing a broader range of student product, uh almost an assessment portfolio for each student. I’d like to see that as a—large-scale assessment moving in those directions. I think from another standpoint when it cost a—a—a hundredth of a thousandth of a cent to grade pages running through in an opt scan form versus dollars to grade a single item on a—a—an extended student constructive response, uh kind of a novel where student’s have to write a paragraph or more, uh I think the—the economics of assessment are—are not going to allow us to expand much beyond where we are today in terms of large scale assessment. In fact, there’s talk of creating market baskets of items just like we have a—a market basket of uh—of products from the grocery store to talk about the consumer price index. Uh I see us moving to much smaller sets of items to make judgments about how our students are—are actually doing in mathematics for uh public reporting which I—I think again is counterproductive given the—the focus that we are talking about a quantitative literacy and student’s ability to—to do processes. When we narrow it down we lose the reliability of our judgments and—and I—I think we really probably uh see on the horizon more of a return to objective uh items which involve less student production. On the other hand, I think that there’s the—the uh—there’s the other side of being able to quickly re—re—release sets of items of that—that type to talk to the public about what students can and cannot do because one can build equivalent market baskets of items. But on the other hand, I think we—we tend to find that teachers teach in—in the same direction to what is assessed. And as we move away from, I think, student-constructed responses, we’ll see also a movement away from processes in the classroom once again.
When—when one starts to look, I think, at results of international comparative studies in—in a mathematics education or most any area, uh one—one has to move beyond just looking at—at sums, totals, and ranks because what we have here are—we have artifacts that are not only of the discipline that’s being talked about but of the culture in which that uh—that discipline’s learning took place. Um, we have artifacts of the assessments systems in those countries. I’m currently working on some reanalysis of the Temes data uh at—at all three of the population levels—actually four if you count the two levels that at grade 12 the literacy in the advanced study students. And uh we—we see such things as difference in student response rates. Uh, for example, in Hungary, students tend not to answer a multiple choice question or a student constructed response question unless they know the answer whereas in the United States you’ll find that 98 percent of the kids will answer whether they know the answer or not, but they’ll get something down on paper. But when you start to take a look at percentages, then there are—there—you’ve got a—you’ve got a question of what those percentages really represent. So I think from a—from an international uh, uh study standpoint the strongest thing that comes from those is actually having information on items from different cultures where one can start to look at the curricula, one can start to take a look at the social no you in which that discipline both ranks within the country’s view of needs of a student and—and how those needs and goals play out in—in the student’s work. Um and as you begin to then look at clusters of countries that—that have similar expectations and do similar things over the same age ranges with students, one can begin then to make some comparisons and contrasts. But when one looks at it as a huge horse race, I think it’s a total misrepresentation of the data and can actually be quite harmful. Um, one also I think from the international studies uh, uh we’re beginning to branch out. I think with the Temes study had—had uh context studies that actually went in and took a look at—at how the school functioned in—in the different countries, uh what are the expectations of parents, uh what are teacher’s views of students, and I think we also need to move to what are students views of teaching in the subject, perhaps in—in a broader way. And so we had that plus the videotape studies. And I think those are beginning to give us a fuller, uh broader picture but one that begins to talk about what’s meaningful information in a country and how to interpret in different cultures uh both the outcomes of education uh from a test standpoint but also from a student process and thinking standpoint.
Well, if—if one were to ask me what—what uh I would say my soapbox is for math education for, you know, the next decade. I—I would say at least in the United States I would look for the—the question of bringing quality math education to all students. Uh, if we look back at the 70’s we saw uh some—some distinct differences between male and female performance along gender lines. And there were concentrated efforts uh to erase those uh—erase the barriers that were there and also to address uh I think assessment issues that sometimes disadvantaged one gender versus another by asking questions about sports or something that only affluent families, children would know about so that we made assessment fair uh or relatively fair for students of—of the different genders. But—but today we—we know—while we’ve eradicated to a large degree those differences uh in—in the classroom, we haven’t done very much to begin to erase the differences between the mathematics performance of white students, African-American students, Hispanic students, Asian-American students, uh and—and other groups. And I think addressing this issue given the growing uh diversity of—of our population uh has a lot to say about the future economy, work force. I mean if you’re looking at—from an economic standpoint—but I think you have to look at it actually from—from just the social/cohesive standpoint—is that in many ways mathematics is becoming the currency of—of—or a currency in a technological world and people who don’t have access to it are being more and more disenfranchised. And uh I don’t think we can allow this to happen as a—as a country, as an educational system. Uh, very clearly that—that would—that’s a—a topic that just deserves up most effort, is to make all children to have the opportunities in mathematics. I think we can do it, but it takes a lot of work and I think it’s going to take—take a lot of effort uh in—in directions that uh—uh, you know, here to for, kids just haven’t had equal opportunities to learn and addressing that is threatening to a lot of people. And uh it will be a tough—tough up hill battle, but I—I think it’s one we can make progress on.