I have addressed the issue of how to view mathematics textbooks in K-12 several times before (
"Changing order of topics: an example from practice", and "
The Book Gods" amongst others, but it seems to rear its unlovely head periodically amongst real teachers and mathematics educators. A recent post to a math teacher list I read posed the following:
I know this question might sound silly, but I need to know what you think. Do you believe it is counterproductive to literally teach math by the book? Today's text books are very complete and I have seen teacher developing strict routines on the book activities. Do you fully rely on your textbooks to teach Math?
Given the relatively progressive orientation of the list in question, I was a bit surprised to see this particular notion raised there, but I figured that when I posted what follows, I was merely reflecting pretty much universally-held sentiments there:
I think textbooks provide resources for teachers and students. Treating them as bibles is a huge error, one that far too many teachers fall into making. The unstated assumption behind teaching a textbook as given is that the author(s) know(s) more about your students than does the teacher. This is an absurd notion, one that any honest author (or publisher) would have to reject.
I have been amazed and appalled by teachers who cannot believe it's not only possible to depart from the textbook but in fact necessary for effective mathematics teaching. The extreme opposite position from mine is expressed by the late John Saxon, who insisted in the introduction to his books that every problem must be covered, and in the order given. If one analyzes Saxon Math books, this claim on his part becomes even more glaringly ridiculous than I hope it sounds on
first hearing. His Algebra 1 text, for example, looks like he took the table of contents, cut it up, threw the topics in the air, and then inserted them back in whatever random order they fell.
But even the most logically and carefully constructed textbook cannot possibly meet the needs of ANY class or student. It is the job of the teacher to change things around, supplement, omit, re-order, pare, edit, and otherwise perform thoughtful experimentation upon textbooks, even those books which the teacher selects herself. I recommend strongly looking at a twelve minute TED talk by Dan Meyer in this regard, though there is so much he packs into that talk that the issue of dealing with textbooks is only one important idea in it.
Thus, I was caught off-guard when Tad Watanabe, a respected colleague wrote:
My answer is "it depends." If a textbook series is carefully and thoughtfully developed, I think it is a good idea to follow it very closely at least a few years. The changes we make should be based on the actual "data" of how students respond to the instruction recommended by the series. I think too many teachers make too many changes prematurely. It is particularly troublesome when teachers change lessons from an NSF curriculum because what it suggests just doesn't fit their conceptualization of mathematics teaching. Now, if the textbooks aren't well written, then that will be a different issue. But, even in that case, there must be some convincing evidence that the quality of the book just isn't there.
Having coached mathematics at every grade level from 3rd through 12th, I was well-aware of the propensity of teachers being asked to implement a progressive reform program such as EVERYDAY MATHEMATICS (EM) in K-5, CONNECTED MATHEMATICS (CMP) in 6-8, or CORE-PLUS/CONTEMPORARY MATHEMATICS IN CONTEXT (CMIC) in 9-12 to undermine the textbook authors' philosophy in a host of ways. In the case of EM, I saw teachers who taught the book as written but because they never understood or did not buy into the underlying pedagogy managed to weaken the potential effectiveness of the program: leaving out the games that were included to help students build and reinforce basic arithmetic facts; refusing to allow discussion of student errors and instead immediately correcting mistakes so that, as one third grade teacher explained to me, they wouldn't have a chance to take root in the minds of the rest of the students (why correct ideas and information never seemed to do quite so good a job of spreading themselves she did not say); and generally making a very traditional, teacher-centered class out of a non-traditional, student-centered textbook.
In a near-by district, I saw the department chair at the middle school simply refuse to teach from CMP at all except when she knew I was coming to observe, and even then using the texts in such a perfunctory and disconnected manner that it couldn't have been lost on the students that their teacher had little or no respect for the material she was presenting.
In the case of CMIC, while I supervised secondary math student teachers for the University of Michigan at two enormous comprehensive high schools in Ann Arbor, I discovered that the books were only being used for students deemed 'difficult,' in need of remediation, and likely to be non-college-bound. At the same time, the department chair at one of these schools misled parents by citing the PSAT scores of these same low-level students as 'proof' that CMIC was an inferior program that CAUSED standardized test scores to decline. Later, she bragged to me that she had "killed Core Plus Mathematics" in Ann Arbor.
Thus, I realize quite clearly that there are serious reasons to worry about how 'teacher choice' might impact the use of any textbook, no matter how good, and particularly those which challenge assumptions and habits of new and veteran teachers alike.
Nonetheless, I offer the following in response to the above-quoted suggestion that "it depends":
I think this needs to be looked at from the perspective of pedagogical content knowledge (PCK).
If we assume that the textbook author(s) as viewed through the lens of the textbook have consistently superior PCK and that the teacher in a given classroom has consistently superior PCK, there's no problem. The teacher will use her PCK to make appropriate adjustments to the textbook that arise, as do many such decisions, in the heat of the moment: see the quintessential example of Deborah Ball's reaction to a third grade student asserting that "I think some numbers can be both odd AND even," one that no textbook author on the planet could reasonably be expected to have anticipated, nor any classroom teacher, for that matter. But only one person, the classroom teacher, actually is in a position to make a decision based on PCK at that moment, and only that person MUST make that decision.
If she decides to bow to the wisdom of the textbook/author(s), then since this incredibly powerful moment was not anticipated, she must pay it little or no heed and move on with the lesson as written. Of course, such moments have the potential to bifurcate into a host of possibilities depending on more variables than anyone can conceivably enumerate, let alone take into account. Many such moments get lost, the less fruitful paths pursued (for reasons that may have to do with: 1) following the text as written; 2) weak PCK on the part of the teacher; 3) reasonable decisions that just don't pan out; and 4) 'merely' the vicissitudes of complex human interactions that occur in activities like classroom mathematics teaching).
But to immediately ensure that more will be lost than need be the case because reason #1 MUST be adhered to (in order to satisfy the needs of textbook authors, publishers, project developers, researchers, etc.) simply denies the fundamental importance of teacher PCK in the making of every teaching decision in the moment that it arises.
Now, of course, rarely, if ever, do textbook author(s) and/or classroom teachers possess superior PCK. In the lower grades, particularly in the post-Liping Ma era in which we all know that most (American) elementary teachers don't know enough mathematics content and hence almost assuredly lack superior PCK, we tend to assume that a randomly selected teacher will have inferior PCK, and will come up short in PCK when weighed against that of the textbook author(s). And so it SEEMS like a no-brainer to agree with Tad here (and generally, I agree with Tad on most things). But I think we would be wrong to do so. Not because it isn't probably true that the PCK (or at least the CK) of the author(s) is deeper and broader and grounded in more experience, thought, research, etc., but because even with all that, mistakes are made. Mistakes that for any given set of kids might not be so bad but for others will make a lesson sink like a stone and create more difficulties than are probably good for anyone.
This isn't to say that 'problematic' situations are to be avoided in math. On the contrary, I think certain kinds of problematizing mathematics are essential for effective teaching and learning. But knowing in advance that doing a given problem at given point in a sequence of problems is going to result in good rather than in destructive problematizing is a very difficult if not impossible task, and certainly knowing this in a way that will apply nearly universally to class upon class, year after year, regardless of circumstances, would require god-like omniscience. Indeed, I doubt it's possible even for a deity. There are simply too many variables and likelihood of combinations of them that will make a given problem in a given sequence a loser FOR A SPECIFIC KID OR GROUP.
And that's why ultimately it must be the combination of the PCK of the author tempered by that of the teacher that rules.
Now, of course, we can all agree that there are teachers who will undermine a well-constructed lesson or unit simply because they "don't get it." I've been in PD sessions where the refrain is, "My kids won't do this," or "My kids CAN'T do this," when of course what is meant is "I don't get this" and/or "I don't like this" and/or "I can't teach like this."
So what does one do as a supervisor, department chair, content coach, author, researcher, principal, publisher, project director, etc.?
First, accept that teachers will tend to follow the path of least resistance unless they are confident, flexible, curious, and secure. When things go wrong, most teachers return to their comfort zone, which is generally teacher-centered, direct instruction with WAY TOO MUCH EXPLAINING, over-scaffolding, etc. While some folks can resist that temptation, they are few and far between. Those listed above had best do everything possible to raise the teacher's believe that she is safe: that heads won't roll, jobs won't be lost, people won't die, salaries will not drop, etc., simply because something doesn't quite go as planned or predicted. Try doing that in today's education deform, "teacher accountability" atmosphere, with everyone now worrying about "racing to the top" for the $$ (Oh, you thought that meant that we're all speeding to get kids to reach some ideal peak of learning? My ass, it does.)
Second, to be crude, bad teachers can fuck up a wet dream. You can hand them the most perfectly constructed, well-thought out lesson and they can and will make it fail if it's simply not in them to make it succeed. They'll dilute everything, they'll change the balance of the lesson to center on them, they'll micro-manage all the empty space in the lesson that should be left for the students to think in ways that turn an investigation into mere practice, thinking into watching, inquiry into imitation, a problem into an exercise.
But then, good teachers can still fail with the same wonderful lessons if, in fact, those wonderful lessons are for the most part simply text-book centered rather than teacher-centered but STILL NOT STUDENT-CENTERED. If the burden of the work and thinking is taken over by the text rather than by the teacher, it's not an improvement. Only if the STUDENTS have to do the thinking and the real work is a book a truly helpful resource. All the little subsets of questions so thoughtfully provided by authors because they assume that the kids must have such things laid out for them in bite-sized pieces with everything resolving neatly at the end, a simplistic moral lesson learned by everyone (e.g., "follow order of operations or bad things will happen!")
Which of course brings me back to that 12 minute TEDxNYCE talk by the brilliant Dan Meyer. Didn't watch it yet? Shame on you! Stop what you're doing (reading my silly diatribe) and
watch it: At the risk of infantilizing my readers and trivializing the power of Dan's talk by reducing it to my notes, here is something I crafted after multiple viewings of what he presented in New York. And since I fear some folks just won't click on that link or take the time to watch the talk, I present my notes here:
Notes from Dan Meyer TEDxNYED talk 3/6/2010
Five symptoms that you're doing mathematical reasoning wrong in classes:
1. Lack of initiative (students don't self-start)
2. Lack of perseverance
3. Lack of retention
4. Aversion to word problems
5. Eagerness for formulas
An impatience with irresolution:
“[Contemporary drama] creates an impatience, for example, with irresolution. And I'm doing what I can to tell stories which engage those issues in ways that can engage the imagination so that people don't feel threatened by it.” - David Milch, creator of NYPD BLUE, DEADWOOD
Sit-com sized problems that wrap up in 22 minutes, three commercial breaks, and a laugh track. No problem worth solving is that simple.
Our textbooks create impatient problem solving attitudes.
What problems worth solving give you exactly the information you need?
Real problems either have a surplus of information that must be sorted through or a lack of adequate information, some of which must be searched for.
Ski Life/Slope problem with four separate layers of information: 1) Visual; 2) mathematical structure (grid, labels, measurements, axes, points); 3) subsets (of questions) all leading to the question we really want to talk about: 4) which section is the steepest?
The layers are presented at once, breeding impatience with real problems because everything is handed to the kids (#3, the subsets of questions, over-scaffolds the question so that #4, the real question of interest, becomes trivialized)
Instead, Meyer starts with the (stripped-down) visual and immediately asks the final question, starting conversations, argument, disagreement; the situation is problematized in part for students because there is no vocabulary, terminology, labeling, etc. to frame the conversation. It's the difficulty of talking meaningfully about the skiers and what steepness means that leads to the labeling, the mathematical structure, etc.: the math serves the conversation, the conversation doesn't serve the math.
“The formulation of a problem is often more essential than its solution, which may be merely a matter of mathematical or experimental skill.” - Albert Einstein
But we just give problems to students; we don't involve them in the formulation of problems.
Water tank: Eliminate all the sub-sets, all the specific, non-distracting information, and reduce this to an under-represented math question where the students have to decide what matters (or not):
A water tank. . . . how long will it take to fill it? Plus a real photograph (not line art or clip art) or, even better, a video of someone filling a tank, slowly, tediously, driving kids to look at their watches and asking, “How long is it going to take to fill up?”
We don't get our answers from an answer key at the back of the book: we just watch the end of the movie.
1. Use multimedia
2. Encourage student intuition
3. Ask the shortest question you can
4. Let students build the problem
5. Be less helpful
The textbook is helpful, but in all the wrong ways.
We have all these cheap or free tools available to do these things. This makes it a most exciting time to be a mathematics teacher.
