I recently read the 2016 book Math Education in the U.S.: Still Crazy After All These Years by Barry Garelick. It's only 190 pages and a few bucks on Amazon. I strongly encourage anyone to read it who is interested in the history of "new math" programs in the U.S.Barry, who was a math major himself, recounts his own experiences when his daughter was suddenly struggling with a curriculum called "Everyday Math". He proceeded to advocate for her and others by understanding how and why the math curriculum had changed to the point where his daughter was no longer learning. He began to tutor her, and others, and ultimately became a secondary math teacher himself.
When I read this, it was like Barry was a prophet foretelling the types of illogical math education ideas that would ultimately visit us here in Wake County, NC.
I am an avid reader and often judge a book by the extent to which I "can't put it down" or in the case of this one, the extent to which I highlight salient passages or insights contained in the book.
My best endorsement of this book is to just share a sampling of the contents. I hope you will check it out in full. It's eye opening and will leave you more aware that there are bigger themes at play when a school system chooses such curricula.
Excerpts from Math Education in the U.S.: Still Crazy After All These Years
by Barry Garelick:
"Discovery learning has always been a powerful teaching tool. But constructivists take it a step beyond mere tool, believing that only knowledge that one discovers for oneself is truly learned."
"There is little argument that learning is ultimately a discovery. Traditionalists also believe that information transfer via direct instruction is necessary, so constructivism taken to extremes can result in students’ not knowing what they have discovered, not knowing how to apply it, or, in the worst case, discovering–and taking ownership of–the wrong answer. Additionally, by working in groups and talking with other students(which is promoted by the educationists), one student may indeed discover something, while the others come along for the ride."
The knowledge base of the world is changing so fast, the theory goes, that learning how to learn is ultimately more important than learning facts.
The knowledge base of the world is changing so fast, the theory goes, that learning how to learn is ultimately more important than learning facts.
That critical thinking cannot occur without something to think critically about—namely facts—is of little concern to ed school gurus. And given the importance of critical thinking over facts, learning becomes a laissez faire type of thing in which information is presented in a never-ending spiral fashion in which topics are revisited and reviewed in the belief that “if they don’t learn it now, they’ll learn it later”.
If a school does poorly on standardized tests, the teachers are held accountable, not any textbook that the school board had a hand in adopting.
More likely it will be about how to facilitate classes to work in small groups (the mainstay of education school theory: students teach other students better than teachers can.
Sue White, director of mathematics for DC Public Schools stated as a witness that: “We have, in the District, six schools that are presently using EM and have topped out on mathematics scores.” They also topped out on reading scores, which wasn’t mentioned. Also not mentioned was the fact that Bunker Hill, which had a higher level of free/reduced price lunches, did not top out on math scores.
Newer teachers thought that the program had merit, probably because it matched the theories they had been hearing about in ed school, which promote things like child-centered curriculum, discovery learning, working in groups, and spiral process.
The teachers clearly bear an unfair burden in all this. So do the students learning centers or parents who can teach them what they’re not being taught. In the meantime, the school boards go on with their business: buying into the next educational fad that comes along, ignoring parents and expert opinions and continuing their longstanding tradition of adopting programs that are short
Over the past several decades, math education in the United States has shifted from the traditional model of math instruction to “reform math”. Although the shift has not been a uniform one, evidence of such transition is indicated by perennial articles in newspapers and the internet featuring parents who question and protest the methods being used to teach their children math.[iii] The traditional model has been criticized for relying on rote memorization rather than conceptual understanding. Calling the traditional approach “skills based”, math reformers deride it and claim that it teaches students only how to follow the teacher’s direction in solving routine problems, but does not teach students how to think critically or to solve non-routine problems. Traditional/skills-based teaching, the argument goes, doesn’t meet the demands of our 21st century world.
Reformers dismiss the possibility that understanding and discovery can be achieved by students working on sets of math problems individually and that procedural fluency is a prerequisite to understanding. Much of the education establishment now believes it is the other way around; if students have the understanding, then the need to work many problems (which they term “drill and kill”) can be avoided.
The treatment for low achieving, learning disabled and otherwise struggling students in math thus includes math memorization and the other traditional methods for teaching the subject that have been decried by reformers as having failed millions of students.
The criticism of traditional methods may have merit for those occasions when it has been taught poorly. But the fact that traditional math has been taught badly doesn’t mean we should give up on teaching it properly. Without sufficient skills, critical thinking doesn’t amount to much more than a sound bite. If in fact there is an increasing trend toward effective math instruction, it will have to be stealth enough to fly underneath the radar of the dominant edu-reformers. Unless and until this happens, the group-think of the well-intentioned educational establishment will prevail. Parents and professionals who benefitted from traditional teaching techniques and environments will remain on the outside — and the methods that can do the most good will continue to hide in plain sight.
For experts, struggle is suitable; e.g., an expert swimmer may struggle to perfect a swim stroke whereas a novice may struggle to keep from drowning—a struggle that doesn’t teach them how to swim.
learn at home by watching videos on the internet — videos consisting of direct instruction on mathematical procedures. The direct instruction of the classroom is often replaced with “stimulating and engaging activities”. This puts the onus on children to (1) have access; (2) be in a good home environment; and (3) self-motivate to pick up the lessons. But if a student does not understand something in the video, the rest of the lesson is not going to make sense.
Group work can be a healthy supplement to teacher-driven lessons or for highly social kids. But it is an inefficient way to get through a lesson in which new technical skills are to be learned. Here are four groups for which this approach is a particularly bad idea: (1) very poor performers—who shrink from participating and can panic at exposure among peers; (2) very high performers—who resent that others in the group look to them to carry the burden, (3) students with social handicaps—for obvious reasons; and (4) students with communication deficits—such as, but not limited to, having a different native tongue as classmates.
Parents confronting school administrators are patronized and placated. School officials will agree and say something like, “Yes, students should learn math facts and procedures (and we do this!). Yes, teachers ought to actually teach, (and we do this!). And yes, students should do drills (and we do this!)” This is all followed with: “We use a balanced approach,” which is often followed with: “We’re saying the same things; we’re in agreement”
Whether understanding or procedure comes first ought to be driven by subject matter and student need — not by educational ideology.
One problem I was having was that EM does not use a textbook. Students do worksheets every day from their “math journal” a paperbound book that they bring home. Without a textbook, however, it is not always apparent what was taught—particularly when the student doesn’t remember. Any explanation that a student has received about how to solve such problems is done in class.
Thus, there is no textbook a student (or parent) can refer to go over a worked example of the type of problem being worked. Worse, sometimes problems are given for which students have no prior knowledge or preparation. They appear to be reasonable problems—it is just not evident to the parent who steps in to help the struggling child that they have had little or no preparation for such problems. Then there is the issue of sequencing, or lack thereof—which I will discuss later.
I told him once in an email that I was not happy with EM and asked him his opinion. I’ve asked other teachers this question and they usually chose not to answer—perhaps out of fear for their jobs.
In EM, however, students are exposed to topics repeatedly, but mastery does not necessarily occur.
So teachers are left with a three ring circus of kids getting it, kids not getting it, and are expected to “adjust the activity” as needed.
Many teachers do not realize that they have been given an unenviable and impossible task. In fact, I have spoken with new teachers who speak of EM and other poorly conceived programs in glowing terms, speaking of them as leading to “deeper understandings of math.” Some have said “I never understood math until I had this program.” But it is their adult insight and experience that is talking and creating the illusion that the math is deep. Children cannot make the connections the adults are making who already have the experience and knowledge of mathematics.
According to the establishment, students should be “led” to their discovery of the answers. Providing explicit instruction is considered to be “handing it to the student” and prevents them from “constructing their own knowledge.”
The problem is that the reigning education theory focuses mostly on discovery, with only a nod to direct instruction. That’s mistaken.
Educators who promote “authentic learning” mistakenly believe that novices learn the same way that experts do. They believe that students construct their own knowledge by being forced to make connections with skills and concepts that they may not have mastered. The theory is that they learn what is needed in a “just in time” manner, thus providing the motivation for learning, which they assume would otherwise be a tedious and soul killing exercise.
In general, reform math promotes a teaching approach in which understanding and process dominate. As discussed above, teaching standard algorithms are delayed in the belief that learning those first will eclipse any understanding of what is going on when such procedures are followed. The result, reformers believe, will be students “doing but not knowing math”. By understanding how the tool works before being given the tool, reformers believe that when students get to more difficult and higher level math problems, they will be “thinking like mathematicians” and that conceptual understanding — more than procedural understanding and fluency — will guide their mathematical proficiency.
Schools and districts are quick to tell parents—both suspicious and unsuspecting—that such circumvention strategies are part of a deeper understanding of math facts as opposed to the “mind-numbing” and “interest-killing” approach to math which in the past “failed thousands of students.”
Also frequently overlooked is the fact that students in low income families who make up the “changing demographic” cited in such arguments do not have access to tutoring or learning centers, while students in more affluent areas are not held hostage — dare I say “tracked”? — to poor curricula and dubious pedagogical practices.
I also do not think that I am alone in drawing a distinction between reform and traditional modes of math teaching. While traditional math can be taught properly as well as badly, I believe that poor teaching is inherent in most if not all reform math programs. I base this on having seen good teachers required to follow programs that present content poorly, lack a coherent logical sequence and rely on questionable pedagogies. I would like to see studies conducted to document how U.S. students who do well in math and science and pursue STEM majors and careers are learning math. The chances are fairly good that such investigations would show that in K-8, many students are getting support at home, from tutors, or from the many learning centers that are springing up all over the U.S. at rapid rates. Since tutors and learning centers (and parents) tend to use traditional methods for teaching math, I somehow doubt that the clientele are exceptions to some ill-defined rule. In my view, as well as the view of many parents and teachers I’ve met, there are few exceptions to the educational damage reform math programs have caused, even when such programs are taught “well.”
The conversation turned to “student outcomes” and “growth-mindset.” This last phrase, a concept made popular by Carol Dweck, is the theory that students can develop their abilities by believing that they can do so. The term has taken hold as its own motivational poster in classrooms, professional development seminars and Ed Camps across America. Someone remarked that the idea of growth mindset itself is a student-centered concept. I suppose it is, if you combine belief in yourself with hard work, instruction, and practice—things I don’t hear much about when I hear about growth-mindset.
Having been in the position of a parent raising a daughter subjected to student-centered classrooms, I think what that parent meant was not so much, “Why should I be involved in my child’s education?” but rather: “I’m doing a lot of teaching at home that should be going on in the school.” Many parents have complained that students are not being taught grammar, math facts, and other necessities of education, but which teachers of student-centered classrooms consider “drill and kill” and “drudge work.” That may account for the popularity of learning centers like Sylvan, Huntington and Kumon, which all focus on these things.
It is as if the purveyors of these practices are saying: “If we can just get them to do things that look like what we imagine a mathematician does, then they will be real mathematicians.” That may be behaviorally interesting, but it is not mathematical development and it leaves them behind in the development of their fundamental skills.
Consider students whose verbal skills lag far behind their mathematical skills—non-native English speakers or students with specific language delays or language disorders, for example. These groups include children who can easily do math in their heads and solve complex problems, but often will be unable to explain—whether orally or in written words—how they arrived at their answers.
Most exemplary are children on the autism spectrum. As the autism researcher Tony Attwood has observed, mathematics has special appeal to individuals with autism: It is, often, the school subject that best matches their cognitive strengths.
And yet, Attwood added, many children on the autism spectrum, even those who are mathematically gifted, struggle when asked to explain their answers. “The child can provide the correct answer to a mathematical problem,” he observes, “but not easily translate into speech the mental processes used to solve the problem.”
What testing does is measure “markers” of learning and understanding. Explaining answers is but one possible marker.
Is it really the case that the non-linguistically inclined student who progresses through math with correct but unexplained answers—from multi-digit arithmetic through to multi-variable calculus—doesn’t understand the underlying math? Or that the mathematician with the Asperger’s personality, doing things headily but not orally, is advancing the frontiers of his field in a zombie-like stupor?
At best, verbal explanations beyond “showing the work” may be superfluous; at worst, they shortchange certain students and encumber the mathematics for everyone.
As Alfred North Whitehead famously put it about a century before the Common Core standards took hold: It is a profoundly erroneous truism … that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them.
Problems such as the one above are not necessarily bad, but when presented without a sufficient learning base of prior knowledge and procedural skills, they do little to promote problem solving skills. It is as if the advocates of open-ended and investigative problems are saying that presenting students with non-routine, and open-ended problems on a constant basis form a “problem solving” schema. Furthermore, they view such problem solving schemes as independent of the mastery of basic types of problems that are learned by example and scaffolded to present more challenge. The thinking is that by giving students a constant does of challenging problems, not only are problem solving “schemas” being developed—so the theory goes —but also all the students in the class are in the same boat. That is, all students will be struggling and there won’t be those few who get it while others are left feeling inadequate. The danger in such thinking is that there is a converse to this theory that is usually ignored: that is, a steady diet of problems held beyond everyone’s reach may well result in students being in the same boat–one in which all are feeling lost and inadequate.
The invert and multiply example has for years served as the poster child for the reform math movement. It is used as evidence that traditionally taught math is math taught wrong because it is presented as a bunch of tricks, relying on rote memorization with no conceptual understanding or connections to other concepts—students should see that math makes sense. Before I get too far into this, let me say that I believe that students should be taught why the invert and multiply rule for fractional division works, and I have done so in classes that I have taught. I will also say that the accusations about traditionally taught math are in large part based on mischaracterizations. I have talked about this in Chapter 8 so I will not go into further detail on it except to say that when I and many others I know were educated in the 50’s and 60’s, math was taught with understanding, and connected with prior concepts, and was not taught as merely rote memorization..
In today’s math teaching methods, students must demonstrate an “understanding” of computational procedures before they are allowed to use standard algorithms. Such topsy turvy approaches to math education have been around for more than two decades, but the interpretation and implementation of Common Core have made them more popular. To compensate for what reformers believe is a lack of understanding, the teaching of mathematics has been structured to drag work out far longer than necessary with multiple procedures, diagrams, and awkward, bulky explanations. In so doing, students are forced to show what passes for understanding at every point of even the simplest computations. Instead, they should be learning procedures and working effectively with sufficient procedural understanding. The approaches to math teaching in the lower grades in schools is a product of many years of mischaracterizing and maligning traditional teaching methods. The math reform movement touts many poster children of math education . Their views and philosophies are taken as faith by school administrations, school districts as well as many teachers—teachers who have been indoctrinated in schools of education that teach these methods. The reform movement has succeeded in foisting its beliefs upon ever growing populations of new teachers who believe this is the only way. In so doing, they math reformers have unwittingly created a poster child in which “understanding” foundational math is not even “doing” math.
