In a recent post on math-teach@mathforum.org in response to my comments about pedagogical content knowledge, Jonathan Groves wrote, in part:

Perhaps it is tricky to define constructivism precisely and to know exactly what it is. I am not even sure if I know exactly what it is though I do have at least an idea of what it is. I have learned some about constructivism in some education classes I took at Austin Peay State University when I originally was trying to become a high school math teacher (which I later abandoned so that I can graduate in four years and teach college instead).

But the constructivism I learned I remember vaguely, and I learned some more about it when I was completing my online faculty training for Kaplan University, where I will begin teaching math online on March 25. The faculty training consisted not only on

information on how to use their online system for their online classes and about Kaplan University itself and some of their expectations for faculty but also about some of the theories of online learning and how online learning differs from traditional higher education learning. Constructivism is used a lot in their classes since direct instruction can't be used well in online classes. Even with live seminars, instructors still can't use direct instruction as their primary teaching method since the seminar is only one hour once a week.

The constructivism I learned from the training is very general; it says nothing about constructivism in mathematics or in any particular subject for that matter. That is perhaps not a good idea, but the faculty training course is designed for all the new faculty, regardless of what subjects they are teaching.

It is extremely important to know how constructivism is relevant to mathematics learning and teaching not only for those discussing it but also for those teaching mathematics. This knowledge can help us fellow mathematics teachers discuss constructivism intelligently and without confusion to others and to teach mathematics more effectively. We first need to learn exactly what constructivism is and

how it is relevent to learning and teaching mathematics.

[F]lexibility and content knowledge of mathematics itself is extremely important for teachers. I myself would hate to lead a discussion and it sidetracks into another related interesting topic but then find myself not knowing how to handle this situation. Class would then become a disorganized mess, and no students would benefit from it in the least bit. Lesson plans don't have to be followed rigidly and shouldn't be; the teacher should allow for digressions if they are relevant. But if the teacher cannot handle these digressions, chaos will reign, and chaos never works in any classroom.

I would also hate it if this new related interesting topic is something I know little about. I doubt that will happen with the classes I will soon teach in the next few weeks. One such course is Basic Algebra (Developmental Mathematics) for the Spring 2 quarter at Florida Tech (starts March 9) and Math 103 College Mathematics for Kaplan University (starts March 25). Kaplan uses 10-week terms, and they have new terms starting about every 3 weeks or so. Thus, one of your colleagues' classes may start

at the beginning of one term and yours start at the beginning of another and yet your colleague hasn't finished teaching his class yet. If I am wrong and something does arise I know little about, but since the courses are online, getting around this problem will be easier since I will usually have time to think about such posts

before I reply to them.

It seems obvious that subject knowledge is required for the mathematics teacher, yet the lack of mathematics knowledge potential teachers have when they graduate and then get their teaching licenses later makes one wonder if it is as obvious as it seems. At least that is a major problem in states like Tennessee and Kentucky that are known

for providing education that is not so good.

That leads to an extremely important question: What content knowledge should math teachers have? Of course this depends on the grade level or level of college they are teaching.

If new topics are related to the curriculum but not part of the curriculum itself and there is little time to devote to them, then having students think about it as homework or to lead a short class discussion or as extra credit research assignment are wonderful options to pursue. The last one is good since it encourages students to become curious about mathematics and to learn to study it on their own, which is a great benefit for them. And it teaches them that mathematics studied do not have to be topics restricted to what is learned in the classroom; in other words, it encourages them to learn whatever mathematics interests them, even if it is not discussed in the classroom.

Being able to make these decisions effectively is vital for preventing classrooms from turning into chaos and into doing what is best for the students.

[W]hat is best depends on who the students are, what the required curriculum is for the class, what math classes the students will or are likely to take in the future, and too many other factors to list here. Depending on the experience of other colleagues, whether in your school or not, when facing a tough decision in

teaching is vital. Kaplan University encourages us instructors to ask our department chair and our mentor and other colleagues when we face difficult decisions and other problems in teaching. And reading widely about teaching and discussing with others may

help prepare us for these difficulties later and to make good decisions. That is why several colleges I have seen take professional development seriously. Kaplan University does.

We definitely need better teacher education and professional development programs in this country. This will help not only in mathematics teaching but in the teaching of any subject. Like mathematics, most other subjects aren't being taught very effectively

these days. For example, schools didn't teach me much world geography or world history (our high school world history class was really European history; our teacher skipped all the chapters on history in the Americas, in Africa, in Asia, etc. except when the history pertained directly to European history. Even then, that was really

history on European colonization). And our U.S. government and economics classes weren't very good either. I wouldn't be shocked if that is the case throughout the U.S. since somehow I get the impression that many young American adults these days don't know much about how economics (except maybe business and economics majors!)

and the government work. I could give more examples, but I think these suffice in illustrating my point.

Where we start these improvements, I am not sure. One idea is to make sure our mathematics teachers have enough subject knowledge and knowledge of teaching to get started. And we should make sure mathematics teachers at all levels appreciate and enjoy mathematics. I have heard this problem is most severe at the elementary school

level, and I am not surprised since these teachers are forced to teach math,

regardless if they want to. Higher up, they can choose whatever subject they want. And I suppose elementary school math is harder to get excited about than more advanced math. And a lot of these teachers who don't like math may too have had teachers in elementary school who didn't like math who turned off their interest permanently, which in turn hurt their ability to learn the mathematics they ought to know to teach elementary school math effectively. If students learn to dislike math early, say in elementary school, they may learn to dislike it the rest of their lives. Exactly how to fix these problems, I don't know, and I strongly wish I knew! Does anyone have any ideas here?

My reply

I think the relevant ideas from constructivism are not complex. From my point of view, constructivism simply asserts that people construct their own individual understanding of anything they learn (in or out of formal instructional settings) through an INDIVIDUAL interaction between past knowledge/experience and new information.

Why this is important for teachers, most particularly for teachers of subjects in which there is a propensity for many instructors to think that "it is obvious" that something is true, or that it suffices for them to state a logical sequence of definitions and reference to previously-taught axioms and theorems in order for students to learn a new theorem, is that very little is obvious to many students, and it's lousy pedagogical practice to assume otherwise. Further whipping through the carefully constructed steps of a proof, while it may be beautiful to those who understand and are capable of appreciating it aesthetically, is lost on most students seeing the ideas for the first time and just learning to grapple with proof-theoretic mathematics. This approach not only leaves most students in the intellectual dust, but gives them the VERY false impression that if they were mathematically competent, all would be clear, just as it is to the instructor and must have been to the mathematician who first constructed the proof.

But of course, that is precisely NOT the case. In all likelihood, the professor did NOT grasp the workings of the proof (if it was of any depth and complexity) at first blush. And it is certain that the mathematician who constructed the proof didn't have it spring from her/his head like Athena from that of Zeus. There were generally a series of fits and starts, a good deal of creativity, intelligent/inspired guess work, going up blind alleys, retracting steps, and much frustration requiring sustained effort on the part of the mathematician or mathematicians who ultimately succeeded in constructing the proof in question. (Yes, there are "simple" proofs at various points in the development of any mathematical subject, but it's not long before those are left behind, and those who are going to follow the development of the topic will need to be able to juggle a host of definitions, axioms, previously-proved theorems, and, of course, abstract concepts, in order to make sense of new material and proofs). We do an enormous disservice to many students by hiding ALL the rough edges as if they never existed and as if we "got it" first time and every time.

At a lower level (namely K-14, where there is a lot less formal or even informal proving and a lot more teaching of definitions and procedures, much of the same still holds except that it's conceivable that the teacher has at best a tenuous hold on what's being taught, making many lower-grade or poorly-qualified higher-grade teachers loath to go off the reservation of any lesson plan. Hence, they are wont to fail to offer alternative explanations for concepts and procedures, often holding to the dreaded "my way or the highway" philosophy, sometimes from fear, sometimes from laziness, sometimes from just being a rigid character, but in all cases failing to serve well many students.

Teachers who have recognized the usefulness of the constructivist approach to understanding how we learn are less likely to come at teaching as a "one-size fits all" process. Constructivism doesn't tell anyone how to teach, but it surely points towards the notion that we can't pour our own understanding of anything into anyone else's head. That's simply not how folks learn. There is NO unmediated learning. There is NO way to get through the filtering and processing EACH individual will do every time we teach and s/he tries to understand and learn. Any teacher who gives a "dynamite" lecture and leaves the classroom with a confident belief that the students must have learned (if they were prepared and paying attention) is likely deluding him/herself. It's not that no one learned: everyone who WAS paying attention learned. But just what each one learned is unknowable, and to have ANY idea at all requires doing more than simply lecturing, no matter how well.

That is not to say that all lectures are equally good (or bad) or that lecturing doesn't have its place. But it does suggest that there needs to be a lot more. And given how much lecturing and its variants (so-called direct instruction, teacher-centered instruction, and similar approaches in which teachers can say with straight faces, "Boy, I taught good, but they learned lousy") dominate math teaching in this country, we might expect that mathematics teachers at all levels would welcome ideas about how to be more effective by varying instructional strategies and styles. If a group of mathematics teachers and teacher educators called for a shift in emphasis towards more student-centered instruction and less of the sage-on-the-stage approach, folks could look at what has been going on (and not going on) for decades honestly, recognize their past successes and failures, take some ownership and responsibility for both, and respond fairly to the 'new' ideas (of course, many of them actually aren't new, but rather are a return to some things that fell out of fashion for various reasons, not all of them because the methods didn't work).

Thus, it would perhaps shock a neutral observer to note how some mathematicians and K-12 math teachers responded, and the vehement attacks that have been launched on various notions that were introduced in the 1989 NCTM standards volumes (though not necessarily solely there, as there were individual practitioners using a host of more student-centered practices in their classrooms long before the first of those volumes was published). Twenty years later, the same distortions, lies, misinformation, epithets, etc., about this call for a shift in emphasis continue to appear on the internet, in the media, and in public and private discourse. Reform is portrayed as utterly wrong-headed, extremist, throwing babies out with bath water, ad nauseum, with little sign that the original opponents and those allies they gained will ever lighten up at all.

We're a long way from seeing broad-based and meaningful reform practice in math classrooms in this country, but it is clear, too, that reform ideas aren't going to go away, despite the best (or worst) efforts of its hard-line foes. And so it becomes necessary to go over a lot of the same ground periodically to try to sweep away the enormous amount of dreck about reform ideas that spews from a few persistent sources.

The necessary content knowledge for teachers is a majr focus of the research and writing coming out of the University of Michigan's mathematics teaching program, particularly that of Hyman Bass and Deborah Ball and their colleagues. Indeed, it is the major research question they've been looking at for about a decade.

Of course, part of the issue here goes beyond simple concerns about curriculum and towards questions about "the system." Recent calls for a national curriculum, national standards that apply to every state, etc., are simply an extension of the idea that there's some blueprint we could devise that, if followed to the letter or nearly so, would produce a mathematically competent country. I think this idea is in fundamental conflict with constructivist theories of learning, and these calls are an attempt, well-meaning or not, to make it increasingly impossible for individual classroom teachers to make the sorts of pedagogical judgments which in fact should be a major part of what they do. I have argued here and elsewhere that no one is better-positioned than the individual classroom teacher to make such choices about his/her students. Rather than managing curriculum from the top down in some blindly monolithic fashion, the people in charge should be putting energy into ensuring as much as possible that teachers are well-equipped to make those decisions as they arise.

The phony "accountability "movement would appear to be all about finding out which practices work and which don't and rewarding the successful and "dealing with" those who are not, but the real agenda is to reduce teaching to formulas: a wacky stop-watch brandishing "management science" approach out of the worst nightmares of those of us who realize that there's no such formula. A genuine accountability effort would entail creating structures in which teachers are helped to make better choices while not hamstrung so that they can't dare do anything that isn't "by the book." A real approach to accountability would entail ensuring that teachers give "accounts" of some of the important choices they make, both to encourage reflective practice on the part of each individual teacher and to collect a body of knowledge that can be shared with other teachers. Of course, when education is viewed as a business and creativity, originality, and any sort of "risk-taking" behavior is likely to be punished, this sort of model can't happen (and it's hard not to see the irony in these management notions coming from various executives whose companies may have been burning through money like a '65 Buick badly in need of a ring job burns through oil, while the genius managers are collecting millions of bonuses and perks. Apparently accountability is only for others).

What pedagogical content knowledge is, exactly, would take books to contain. What it is in small part is some of the choices I pointed to previously when I looked at roads not taken by Bill and by each teacher every time s/he chooses to either stay on the prescribed path of the lesson (and of course in constructing the lesson in the first place) or not. It's unlikely that I or anyone could cover every possible choice (even just the reasonable ones and some examples of less reasonable ones) that could arise. However, there are books starting to appear that, if their titles are to be believed, focus on mathematical pedagogical content knowledge. I don't have any of them and can't comment on how effectively they accomplish any of this. If and when I get my hands on some, I will try to report on what I see.

I don't think there are any simple or exact fixes to the current state of mathematics teaching and learning, naturally, but being open to real reflection on the issues, one's practice, and much else makes for a good start. The closed-mindedness exhibited all-too-often in response to constructivism and its relation to mathematics education helps no one, not even the entrenched.

Why this is important for teachers, most particularly for teachers of subjects in which there is a propensity for many instructors to think that "it is obvious" that something is true, or that it suffices for them to state a logical sequence of definitions and reference to previously-taught axioms and theorems in order for students to learn a new theorem, is that very little is obvious to many students, and it's lousy pedagogical practice to assume otherwise. Further whipping through the carefully constructed steps of a proof, while it may be beautiful to those who understand and are capable of appreciating it aesthetically, is lost on most students seeing the ideas for the first time and just learning to grapple with proof-theoretic mathematics. This approach not only leaves most students in the intellectual dust, but gives them the VERY false impression that if they were mathematically competent, all would be clear, just as it is to the instructor and must have been to the mathematician who first constructed the proof.

But of course, that is precisely NOT the case. In all likelihood, the professor did NOT grasp the workings of the proof (if it was of any depth and complexity) at first blush. And it is certain that the mathematician who constructed the proof didn't have it spring from her/his head like Athena from that of Zeus. There were generally a series of fits and starts, a good deal of creativity, intelligent/inspired guess work, going up blind alleys, retracting steps, and much frustration requiring sustained effort on the part of the mathematician or mathematicians who ultimately succeeded in constructing the proof in question. (Yes, there are "simple" proofs at various points in the development of any mathematical subject, but it's not long before those are left behind, and those who are going to follow the development of the topic will need to be able to juggle a host of definitions, axioms, previously-proved theorems, and, of course, abstract concepts, in order to make sense of new material and proofs). We do an enormous disservice to many students by hiding ALL the rough edges as if they never existed and as if we "got it" first time and every time.

At a lower level (namely K-14, where there is a lot less formal or even informal proving and a lot more teaching of definitions and procedures, much of the same still holds except that it's conceivable that the teacher has at best a tenuous hold on what's being taught, making many lower-grade or poorly-qualified higher-grade teachers loath to go off the reservation of any lesson plan. Hence, they are wont to fail to offer alternative explanations for concepts and procedures, often holding to the dreaded "my way or the highway" philosophy, sometimes from fear, sometimes from laziness, sometimes from just being a rigid character, but in all cases failing to serve well many students.

Teachers who have recognized the usefulness of the constructivist approach to understanding how we learn are less likely to come at teaching as a "one-size fits all" process. Constructivism doesn't tell anyone how to teach, but it surely points towards the notion that we can't pour our own understanding of anything into anyone else's head. That's simply not how folks learn. There is NO unmediated learning. There is NO way to get through the filtering and processing EACH individual will do every time we teach and s/he tries to understand and learn. Any teacher who gives a "dynamite" lecture and leaves the classroom with a confident belief that the students must have learned (if they were prepared and paying attention) is likely deluding him/herself. It's not that no one learned: everyone who WAS paying attention learned. But just what each one learned is unknowable, and to have ANY idea at all requires doing more than simply lecturing, no matter how well.

That is not to say that all lectures are equally good (or bad) or that lecturing doesn't have its place. But it does suggest that there needs to be a lot more. And given how much lecturing and its variants (so-called direct instruction, teacher-centered instruction, and similar approaches in which teachers can say with straight faces, "Boy, I taught good, but they learned lousy") dominate math teaching in this country, we might expect that mathematics teachers at all levels would welcome ideas about how to be more effective by varying instructional strategies and styles. If a group of mathematics teachers and teacher educators called for a shift in emphasis towards more student-centered instruction and less of the sage-on-the-stage approach, folks could look at what has been going on (and not going on) for decades honestly, recognize their past successes and failures, take some ownership and responsibility for both, and respond fairly to the 'new' ideas (of course, many of them actually aren't new, but rather are a return to some things that fell out of fashion for various reasons, not all of them because the methods didn't work).

Thus, it would perhaps shock a neutral observer to note how some mathematicians and K-12 math teachers responded, and the vehement attacks that have been launched on various notions that were introduced in the 1989 NCTM standards volumes (though not necessarily solely there, as there were individual practitioners using a host of more student-centered practices in their classrooms long before the first of those volumes was published). Twenty years later, the same distortions, lies, misinformation, epithets, etc., about this call for a shift in emphasis continue to appear on the internet, in the media, and in public and private discourse. Reform is portrayed as utterly wrong-headed, extremist, throwing babies out with bath water, ad nauseum, with little sign that the original opponents and those allies they gained will ever lighten up at all.

We're a long way from seeing broad-based and meaningful reform practice in math classrooms in this country, but it is clear, too, that reform ideas aren't going to go away, despite the best (or worst) efforts of its hard-line foes. And so it becomes necessary to go over a lot of the same ground periodically to try to sweep away the enormous amount of dreck about reform ideas that spews from a few persistent sources.

The necessary content knowledge for teachers is a majr focus of the research and writing coming out of the University of Michigan's mathematics teaching program, particularly that of Hyman Bass and Deborah Ball and their colleagues. Indeed, it is the major research question they've been looking at for about a decade.

Of course, part of the issue here goes beyond simple concerns about curriculum and towards questions about "the system." Recent calls for a national curriculum, national standards that apply to every state, etc., are simply an extension of the idea that there's some blueprint we could devise that, if followed to the letter or nearly so, would produce a mathematically competent country. I think this idea is in fundamental conflict with constructivist theories of learning, and these calls are an attempt, well-meaning or not, to make it increasingly impossible for individual classroom teachers to make the sorts of pedagogical judgments which in fact should be a major part of what they do. I have argued here and elsewhere that no one is better-positioned than the individual classroom teacher to make such choices about his/her students. Rather than managing curriculum from the top down in some blindly monolithic fashion, the people in charge should be putting energy into ensuring as much as possible that teachers are well-equipped to make those decisions as they arise.

The phony "accountability "movement would appear to be all about finding out which practices work and which don't and rewarding the successful and "dealing with" those who are not, but the real agenda is to reduce teaching to formulas: a wacky stop-watch brandishing "management science" approach out of the worst nightmares of those of us who realize that there's no such formula. A genuine accountability effort would entail creating structures in which teachers are helped to make better choices while not hamstrung so that they can't dare do anything that isn't "by the book." A real approach to accountability would entail ensuring that teachers give "accounts" of some of the important choices they make, both to encourage reflective practice on the part of each individual teacher and to collect a body of knowledge that can be shared with other teachers. Of course, when education is viewed as a business and creativity, originality, and any sort of "risk-taking" behavior is likely to be punished, this sort of model can't happen (and it's hard not to see the irony in these management notions coming from various executives whose companies may have been burning through money like a '65 Buick badly in need of a ring job burns through oil, while the genius managers are collecting millions of bonuses and perks. Apparently accountability is only for others).

What pedagogical content knowledge is, exactly, would take books to contain. What it is in small part is some of the choices I pointed to previously when I looked at roads not taken by Bill and by each teacher every time s/he chooses to either stay on the prescribed path of the lesson (and of course in constructing the lesson in the first place) or not. It's unlikely that I or anyone could cover every possible choice (even just the reasonable ones and some examples of less reasonable ones) that could arise. However, there are books starting to appear that, if their titles are to be believed, focus on mathematical pedagogical content knowledge. I don't have any of them and can't comment on how effectively they accomplish any of this. If and when I get my hands on some, I will try to report on what I see.

I don't think there are any simple or exact fixes to the current state of mathematics teaching and learning, naturally, but being open to real reflection on the issues, one's practice, and much else makes for a good start. The closed-mindedness exhibited all-too-often in response to constructivism and its relation to mathematics education helps no one, not even the entrenched.

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