**JIANFA CAI**

OK, my name is Jianfa Cai. I’m from the University of Delaware and this is academic year, which is 2000-01. I’m visiting the Harvard Graduate School of Education. I’m from China originally. I was a faculty man at the Beijing University before I came.

Well, conceptual understanding—this is certainly a very big question. But we usually say a—a student conceptually understands something. It really means that a student know the meaning, the notion of an idea or a concept. Not only that, but also be able to connect to other concepts, ideas, or elated facts. Let me give you specific example here. Let’s say take example of uh percentage concept—percent. The percent—when a student tries to understand, they have to understand what the percent really means. But also need to understand the symbol of percent. Uh but that’s not—not to—yeah. For conceptual understanding this is—uh present a concept, they also need to be able to relate to that concept, present a concept to other numbers, for example, uh fractions, uh decimals, or even whole numbers. But also will be able to relate to ratio, proportion, proportion of reasoning and so forth. So it’s conceptual, understanding something. Not only to know the concept or idea itself, but also be able to relate to that concept or idea to others.

Oh, so you are really talk—talking about what is the impact of teachers’ conceptions of mathematics on their teaching. Well, I want to say that the impact is—is huge—is huge. Well, one thing we pretty sure that uh—one thing we are not sure at this moment, there is no universal agreement about the what constitutes good teaching. We really don’t know. The—we could have uh different way of teaching but to view it as similar as ours. But one thing we do know from the research that teachers conception of mathematics has great impact on student’s learning and their teaching. Well, let me take it to an extreme case here. Suppose that—two different views here for the mathematics. One teacher view—again that’s extreme case here—extreme cases. The one teacher would view mathematics as a set of fixed facts. Another teacher views mathematics as a dynamic body of knowledge, a way of thinking. If a teacher view mathematics as a fixed set of knowledge or facts then that teacher more then likely will teach to assure the results, going through step-by-step, then try to show how to solve a problem. Students do the—lots of practice, repeat, repeat, uh until students know some procedures or algorism, then stop. That’s their teaching. They take that as a view. If you—a teacher viewed mathematics as a dynamic body of knowledge and a way of thinking, they more likely—that teacher will try to engage their students. Try to assure the students the process of solving a problem and where that come from, how to approach a problem. So really try to work together to reach the results. So that’s, I want to say, is two extreme uh cases here but you—most of the teachers will be in the middle.

OK. Well, uh if uh teachers view the mathematics as a dynamic body of knowledge, a way of thinking, and uh what the classroom would look like, I would say that classroom would be students uh pretty—pretty much engaged in the—the whole mathematical discourse. Uh, it’s not just a sit and listening to what teacher has to say, take the notes, and try to memorize what teacher has to say. So students will be a part of that discourse or be engaged. And teachers will try to uh make sure that students understand some of the things then move on and not just—teacher is not just an authority for that knowledge. And the students will try to—they themselves will try to construct their own meanings to understand it and then move on.

Well, the—what’s the importance of the mathematical communication in classroom? That’s uh the question, I guess. (interruption) Well, the communication I would say is certainly very important. Now when we talk about a mathematical communication there’s a little bit of difference between the—the normal kind of conversation or communication. The mathematical communication we are talking about communicate mathematically. Although they’re quite a bit of commonality between the normal communication or regular communication versus mathematical communication, the reason it’s important because the—in—to learn mathematics, the students have to know the concepts, the notions, and affects, but also need to know the thinking process. Through communication students are shown those things. Students show if they are—a teacher will know if a student knows the concept, the notion they are talking about, if they know the process to yield to—to collect answers. OK? So the reason that uh—that mathematical communicate is very important because that emphasizes on uh thinking. We want students not just to be able to memorize something, but also be able to think—think—most importantly will be able to think mathematically. That’s why I think that the mathematical communication is very important.

OK. Well, let’s—I would just say to learn mathematics well and to what mathematics is important in the middle school and a secondary school just—several things come—quickly come into my mind. But before I go into the—what mathematics uh skills would be important for them to have, I would like to talk about the affective aspects first. To me, to learn mathematics is not just to gain something. A teacher will put something into a student’s head. So students need to take active roles. To that end, I would say students pay very important role and uh students have to be persistent. Uh in this day if we go into the classroom, lots of students uh quickly want to give up what they are trying. I think it’s OK for a student to struggle a little bit when they try to learn mathematics or try to solve mathematical problems. That’s the one thing I want to say. Secondly, I want—I truly believe that everyone is able to learn mathematics. Everyone. And I have lots of evidence for showing you that. Even some of the students uh viewed as uh slow learners in classrooms, they could produce wonderful, very creative, uh ways to solve a problem. I could show you tons of evidence for that. OK. So these uh two things I would uh say would be very, very important. So we need to encourage every student to try to success. Now come back to the what mathematics in the schools would be important for the middle school and the secondary uh school students, the one thing that will quickly come into my mind, to be—the way to think abstractly. Now from elementary school move up to mid—middle school, the mathematics became a little, little abstract. Well, there’s quite a bit of evidence showing that students when they come to the 5th or 6th grade students, they suddenly got a difficulty. Although they will—doing perfectly fine in elementary school years. It’s one reason I believe, and there is some research evidence that shows that, because students were unable to think abstractly. So hard to transform—help students to transform from a mathematic to more abstract or algebraic kind of way of thinking is one of the important uh—uh, important things we need to focus on as teachers and the researchers. Yeah.

Sure. Well, a mathematic average is a very in—interesting concept. I—I said it’s interesting concept, let me just give you a few—few examples, a few aspects of the concept of why that’s very interesting and important. It’s important because we use these concepts almost every day. We are talking about uh an average number of people doing something. Average—averages always come up (?), but no the averages concept is—is also interesting because it involves uh computation algorithms. We all know we—in order to calculate the averages, we just add the numbers up and divide the total number of the numbers. Right? So usually there is uh algorithms we call add all of them up and divide. That’s the algorithm part. But the algorithm part is only just the small part of the average concept. But now why we could use this concept to represent the—the whole set of data? Let me give you quick example. For example we got a two classrooms. Each classroom, student got a—every student got a—a score in mathematics. Now we want to say now which class got a—performed the better on a cert—on a test. What do we look at? Do we compare the top scorers or the low scorers? No, we usually look at the average. So the average is representative for the scores in that—for a particular class. Is the tool to describe that set of data. All right. So this I call the statistical aspects of the concept. And before I talk about the computational algorithm aspect of the concept. So to understand this concept we need to know both. Even the—related to the computational algorithm, we could uh—a student could directly use that algorithm to calculate average. But also sometimes students need to use that algorithm very flexibly. So my research on that concept is that try to—to understand what students know about this concept. The ‘what’ of the finding I got from my study is that almost uh the majority of the students—the majority of the students don’t have difficulties to directly apply the algorithm to calculate an average or a mean. I’m talking about the middle school students. The subjects I used in my—in my study were the middle school students. So they didn’t have difficulties. But the difficulty will come from when a situation arises, they have to use the algorithm in—in a flexible way. Then students got lots of different ways. Like a half of—half of the samples I use in my study couldn’t do those kinds of problems. Although majority of them are able to directly apply the algorithm to calculate the mean. So what does this study mean? This study means that—what we in—in the states in schools—oh, by the way, I—this study has been extended to cross-nationally. Now, I also used the same tasks to examine the Chinese 6th grade students. The one thing that’s also interesting—although overall, Chinese students perform better, but uh Chinese students experience similar difficulties when they try to apply an al—a computational algorithm or average algorithm to a new situation. They exper—experience similar difficulties for that. So what it really means is that, you know, our teaching, we need to focus more on student’s conceptual understanding. That’s just one of the topic could uh show us those kind of uh points (?). So we need to focus on conceptual understanding a little bit more to help students understand those kinds of concepts better.

OK. Well, uh in the past several years I’ve conducted uh several cross-national studies mainly involving the US 6th grade students and the Chinese 6th grade students. My main goal of the studies is trying to understand what can we learn from those studies to improve student’s learning. So the focus of all of my studies is not just uh to compare—to do the comparison. That’s not my goal. Uh, it’s the mean. For me it’s a mean. It’s a means to try to learn the ways we could improve student’s learning. So one of the studies I did is that I tried to look at US and Chinese students uh performance on four different kinds of tasks—four different kinds of tasks. The first type I call computation tasks—purely computation tasks. It’s like uh, “What is 1/5 + 2/7? What is it?” OK, purely computation items. The second type of problems I call the simple problem solving tasks. It’s the—let me give you an example. Those kinds of tasks would be “John has 5 apples but you have 10 more then John. How much marbles—or how much apples do you have?” OK. Those kind of simple uh problems. Third type of problem I call complex but uh in-term thinking and the process is a little bit of constraint kind of problems. For those problems students usually are able to locate an algorithm an applied algorithm into the situation to solve the problem. And the fourth type of the problem I call uh complex but process relatively open kind of problem. Now for those kinds of problems, students don’t have an existing algorithm they can apply to solve the problem. They need to devise their own algorithm, figure out own ways to solve the problem. Now here’s my finding. For the first three types of tasks, Chinese 6th grade students performed significantly better then the US 6th grade students. Now those US and Chinese students are a similar age. OK. But for the last one, the US students performed significantly better then the Chinese students. That’s very interesting results because the last type of problem I would say somehow is—assess students understanding and their creativity, those kinds of aspects uh of those students uh mathematical learning. Well, if we look at it just within the country, for the US students—for the US students, they performed better on the uh process—they were open to the complex kind of problems, then also the process relatively constraint kind of problems. It’s vice versa for the Chinese students. So that—so this study—what this study tells me, I would say there’s two things we could learn. One is that maybe or it’s possible in US classroom and in Chinese classroom they’re all doing something good but on different aspects. Maybe we need—needed to uh pay our attention to identify those good things happening in the US classroom and the good things uh happening in Chinese classroom. If we could combine those good part—good parts, which happen in both classrooms, I think we could all our students more—help all our students more. That’s the one thing I think we learned. Secondly, the things we learned is that uh students have a computation ability. All those kind of routine problem solving skills not necessarily means that they are a—they also have ability to deal with non-routine kinds of—way of solving problems. So in teaching we need to focus on both—focus on both. I want to share with you one more finding from my study. In my study, also look at—looking at how students—what kind of strategies US and the Chinese students use to solve problems, especially for those complex problems. The one thing I find that—is that the US students tend to use what I call the visual representation or visual approach to solve it and the Chinese students more likely try to use symbolic sometimes even algebraic ways—uh approach to solve problems. Now this finding got an instructional implication here. Well, it’s important to help students to understand mathematics uh using visual approach or manipulative, but at a certain point—at a certain point, especially in middle and high school level, we need to push a step further to help students understand mathematics beyond the concrete level. That’s probably the way in the United States, especially in classrooms, we need to focus on.

OK. Well, we’re talking about apparent roles of our students learning—or our children’s learning. Well that’s the—that study comes from uh my experience. Uh in my early childhood, I think my parents—although themselves don’t have that much education. Actually uh my parents don’t have any former education but they were very, very supportive for my education. The—but they themselves—they couldn’t support me or help me for my schoolwork but they emotionally supported me. So given that uh experience there and the one thing we certainly uh try to look at is uh what can we do to improve students’ learning? To me the improve students learning is not solely teacher responsibility, school responsibility. And the parents, the community should be part of it. Myself as a—as a father, certainly I’m doing my part—doing my part. It’s important to do that part. So that—to that end I tried to look at what exact role the parents were playing in students learning of mathematics. Now I looked into the research literature, some studies look at uh where parents could be—get involved in the families and—and the children go to school to do kind of volunteer kind of work and so forth. Of course, those kinds of things are important. But uh my study would be in the focus—is—is a little of the micro lab to actually how they impact our students learning of mathematics. So through extensive library of literature I identified five different laws parents possibly could apply. Let me mention them. The first one I called “Parents as Motivator.” Motivate students in their learning. Second is “Parents as a Monitor.” Try to monitor students’ behaviors including their learning. Third is “Parents as a Resource Provider.” A fourth is “Parents as a Learning Counselor.” The parents try to help their children to find uh their learning difficulties, how to overcome, or how to find a—a good learning approach or form a learning habit. That’s the other law we’ll apply. The last one would be “Parents as a Contact Adviser,” which means that actually help students to learn mathematics. Now in these five laws, I look at these five laws and try to relate it to students’ achievement. The one thing I find is that these five parental laws will significantly contribute to the prediction of their achievement. So it’s—it’s a—a highly correlated—it’s related to their—their achievement—the student’s achievement. I—I also look at which—which laws are more—more important then others. So use uh technical words along the regression analyzers. Try to tease uh--tease out what laws would be more important then others. So what I found is that the parents as a motivator and the parents as a monitor are the most important predictors for their achievement. So to that end, this study means that—no, for the parents, for themselves, don’t have that much formal education, still could play important role to try to monitor and a motivator their student’s learning in their home. All right?

Well, uh I would focus on the instruction and assessment. OK. Well, the instruction—teacher’s instruction and assessment uh really help each other and assessment I would say as a way to help teachers to know students progress. It could help them make their instructional decisions. To provide them feedback about what students know and what students are doing. And instruction then could adjust to focus on students learning. But on the other hand, the assessment could be a means to exam the instruct—the—the effectiveness of instruction. So could—these two things would be related to each other and uh that’s why we usually say assessment and instruction should be—align up. Assessment should align the ways of instruction. OK? So one—one is—will help another. Because we certainly want to know—know the results of students’ learning. That’s part of the thing that the teachers need to know to adjust their instruction.

Well, that’s a very good question. Well, if you—we focus on the lot learning we possibly—the assessment we just mainly focus on if a student uh is able to perform the procedure, if they’re able to get the correct answers. But if we are focused on the conceptual understandings, the assessment should be different. Now for—to me, the traditionally used standardized assessment using multiple-choice items sometimes are not appropriate to assessing students’ conceptual understanding. Uh, but the—of course, some of the multiple-choice items, if we dev—develop them well, would be appropriate to assess a student’s conceptual understanding. But lots of the items wouldn’t be appropriate. OK. So in order to assess students’ conceptual understanding, they’re reasoning, or even mathematical communication, we’ve talked about it before, I think we need to look into the alternative ways to assess students. The one way I really like is that—so-called, using open-ended tasks, which means that students not only needed to provide the correct answer, but also they needed to provide their thinking and the reasoning process, how they got their answer, to show their strategies. And those can open-ended problems possibly could have multiple correct answers. Oh—some of them—most of them will have multiple collective ways to yield to the correct an—to yield the correct answers. So those kinds of ways would be appropriate to assess students’ conceptual understanding and communication. Of course, for assessment we also needed to use other means. So like uh in the states, teachers will use portfolios, journals, even informal observations because they’re also the means to assess students.

To that question I think I want to alter—alter a little bit. To me uh for the—the most important thing for teachers to have would be their attitude or the willingness to improve their teaching. Uh, I want to uh give you a story here. In 1997, I went to my wife’s graduation ceremony uh in Pittsburgh—my wife and I both graduated from the University of Pittsburgh. Uh, in 1997, the University of Pittsburgh invited the president of Purdue University to be the keynote speaker in that uh ceremony. And uh I forgot his name, but the one thing—actually only one thing I remember that’s the sentence—was the sentence that the president of the Purdue University said. He pointed to those who received the PhDs at the front of the room so that all—that’s not exactly his words, but that’s the—the—something like that. He said, “Well, congratulations, you got your PhD, but do you think you know a lot in your field? Do you think you are an expert in your field?” And the answer would be “No. You need to know that much in your field, not an expert in your field, although you got your PhD already. Then what does that mean since you got your PhD? You got your PhD means that you are able to learn. You are capable of learning.” So the reason I want to bring up this story here is that—for the teachers who got a certification, got their degree in universities or colleges. That doesn’t mean they are experts in teaching already. That means that they have the potential to be effective teachers, to be good teachers. That means that they can learn to be effective teachers. That’s why I said one of the important things would be that teachers should have a willingness, (?), a disposition waiting to improve their teaching through their teaching. Uh, myself as a—as a university professor, also as—it’s really a—a teacher—just a teacher at the university, I learn so much uh from my teaching. Even some of the classes I have taught for several times. So I think the one the—the—the thing I have passion on would be we need to encourage—help teachers to learn through their daily teaching—through their daily teaching. But uh one thing I really hope, something could be done to help teachers to have enough time to prepare, to learn, to improve their teaching. For these days I see little—little bit of difficulties because some teachers or most of teachers have to teach five or six lessons a day. I don’t know how much time they are going to have to prepare, to learn, to improve their teaching.

Well, before I came to the United States I can read—I could read English. So because the—the education of English in China uh is to help one be able to read—be able to read. So I—before I came to the United States I already had my Master’s Degree in Mathematics Education. So in college—even in high school, in college, and in—in graduate school, I spent some time to study English. Uh, so I can read. But the focus is not on speaking and then listening. So you can see I still have a—quite a heavy accent here in my speaking. OK? Well, that—for the speaking, for the listening, those kinds of aspects of English are literally learned uh start to—start to—when I—when I arrived in the English country. But at the first I have experienced so many difficulties. The difficulty will come from two. One is that—that in China we try to study so-called British English and in the United States it’s American English. The American English is sometimes in the structures and the—the phrase we use will be a little bit different from the British English. So that’s the one difficulty I have. OK. Secondly, is the—the culture. Some of the phrase—if the—my reading is mainly the scientific kind of reading, reading mathematics or educational lit—literatures and there’s not so many slangs there, the—the daily—daily phrases there. But when I come to the daily conversation, lots of those—those phrases or—or slangs come up. That would be difficult to learn. But I—eventually I believe I overcome that through a couple different ways. The one is taking the courage to try—to try—is willing to speak. I remember the first uh class—in the first class I was—I was taking, I wanted to express my views but I couldn’t because of my language barrier. So what I did, I simply uh took uh chalk and just wrote all my thoughts on the blackboard to show it to—to the rest of the class. Now that really takes the courage to—to do things. And the second thing is that—is waiting to ask people. Uh, I always said you have to have success is waiting to ask. I just keep asking people. If I do not—don’t know, I don’t assume. I’m just going to ask to—to learn. That’s the way to learn. You can’t learn mathematics, language, or anything else if we don’t know—don’t know. And just ask and try to—we need to learn. We need to learn. Then when we learn it—when we—then we get it.

OK. Well, being Chinese—although right now I’m an American citizen, but I still view myself as a Chinese. OK. Well, for cultural tradition in a society, uh China in the United States, I would say are two—are in the two opposite directions in many aspects. In the political system, tradition, the way we think are very different—are very different. Uh, let me just uh share with you a couple of examples here. Uh, to me in the United States from a Chinese perspective the Americans got too much freedom. The freedom is a wonderful thing, but to—sometimes because of freedom get out of control for some of the things. I just can’t believe that children, little children, uh 10 or 13-years-old children is able—or was able to carry a gun to school. That’s simply not acceptable in Chinese culture. But on the other hand, in Chinese society, to me I also viewed its just too much control—too much control. Because of much control, sometimes limited our thinking, even limited students thinking. Because of the overall culture—in the controlling of those kind of culture, the students were—were bringing it to the classroom. There are also those kinds of teachers in China sort of have those kind of controlling kinds of ways in their teaching. That—although I don’t have the empirical evidence to show, I hope someday I will, that will infect our—impact our students learning, limit students thinking—limit the students thinking. That’s the one thing. Secondly, I will say in terms of the family values. In the family values in China, the family—family is very, very important for everyone. OK. But in the United States I feel—I have to say, I feel a little bit sad that something’s happening. For the parents, if something happening, even not crucial, couldn’t get it right, quickly divorce—quickly divorce. But the—the divorce uh to me, personally, although I don’t have experience for that, I think has a tremendous impact on their children—their children. That’s always my belief. And as parents, myself as a father, probably I may not be able to give my children that much uh money when I pass away. They couldn’t uh inherit that much property from me, but one thing I do hope they will get from me would be that I want to assure them I love their mother. I love my wife. Those family values I think—although we’re talking about those families not really directly related to school, but it should be very, very important for students learning. So that’s it.