Barry Garelick, via a kind email:
An important part of the job of teaching math in K-12 is to stretch students–to teach them creative and personal engagement with the material. At some point this must involve expecting students to come up with previously unfamiliar steps on their own for new problems that do not lend themselves to known algorithms, prescribed methods, and predictable approaches. An effective way of doing this is to extend routine problems that students know how to solve into nonroutine problems.
Over the past two decades, however, disagreements between advocates of traditional or conventional math teaching and the math reform movement have resulted in a fragmented approach to teaching math. A key area of disagreement centers on the distinction between “exercises” and “problems”. Math reformers generally believe that conventional math teaching consists mainly of routine problems that are nonthinking, repetitive, tedious and do not lead to students learning to solve nonroutine problems.
Barry Garelick, via a kind email:
I am a mathematics teacher. I majored in math and, prior to going into teaching, used it throughout my career.
My facility with math is due to good teaching and good textbooks. I fully expected the same for my daughter, but after seeing what passed for mathematics in her elementary school, I became increasingly distressed over how math is currently taught in many schools.
Optimistically believing that I could make a difference in at least a few students’ lives, I decided that after I retired, I would teach high school math. To obtain the necessary credential, I enrolled in George Mason University Graduate School for Education in the fall of 2005.
The ed school experience did have some redeeming features. Most of my teachers had taught in K-12, and had valuable advice about classroom management problems and some good common-sense approaches to teaching that didn’t rely on nausea-inducing theories.
Those theories are inescapable, unfortunately.
Specifically, many education theorists hold that when students discover material for themselves, they learn it more deeply than when it is taught directly. In this vein, the prevailing belief in the education establishment is that although direct instruction is effective in helping students learn and use algorithms and mathematical procedures, it is ineffective in helping students develop mathematical thinking.
Barry Garelick, via a kind email
The New York Times recently published a piece called “The Faulty Logic of the Math Wars” by W. Stephen Wilson (a math professor at Johns Hopkins University) and Alice Crary. While the article itself is worth reading, I found the reaction of the readers to be equally fascinating. They revealed the ideological divide that defines this “war”. I was reminded of Tom Wolfe’s famous description of the reaction of the New Yorker literati to his 1965 article in the New York Herald Tribune that criticized the culture of The New Yorker magazine: “They screamed like weenies over a wood fire.”
Barry Garelick, via a kind email
The tendency to interpret the standards along the ideological lines of the reform movement can be seen most readily in how the Standards for Mathematical Practice are interpreted. The SMP themselves are sensible and few mathematicians or teachers would disagree with their principles. Their interpretation and implementation is another matter, however.
Standards for Mathematical Practice: The Cheshire Cat’s Grin.
Standards for Mathematical Practice: Cheshire Cat’s Grin, Part Two
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Standards for Mathematical Practice: Cheshire Cat’s Grin, Part Three
Barry Garelick:
Is it my imagination, or have you noticed that some public high school courses that are now called “honors” are equivalent to the regular “college prep” curriculum of earlier eras? And have you also noticed that what is now called “college prep” is aimed largely at students who are deemed low achievers or of low cognitive ability?
In fact, this trend is nobody’s imagination. Over the past generation, public schools have done away with “tracking” — a practice that began in the early 1900′s. By the 20′s and 30′s, curricula in high schools had evolved into four different types: college-preparatory, vocational (e.g., plumbing, metal work, electrical, auto), trade-oriented (e.g., accounting, secretarial), and general. Students were tracked into the various curricula based largely on IQ but sometimes other factors such as race and skin color. Children of immigrants, and children who came from farms rather than cities, were often assumed to be inferior in cognitive ability and treated accordingly.
During the 60’s and 70’s, radical education critics such as Jonathan Kozol brought accusations against a system they found racist and sadistic. They argued that public schools were hostile to children and lacked innovation in pedagogy. Their goal — which became the goal of the larger education establishment — was to restore equity to students, erasing the lines that divided them by social class and race. The desire to eliminate inequity translated to the goal of preparing every student for college. The goal was laudable, but as college prep merged with the general education track, it became student-centered and needs-based, with lower standards and less homework assigned.
Some of the previous standards returned during the early 80’s, when the “Back to Basics” movement reacted against the fads of the late 60’s and the 70’s by reinstituting traditional curricula. But the underlying ideas of Kozol and others did not go away, and the progressive watchword in education has continued to be “equality.”
Related: English 10.
Barry Garelick, via a kind email
In The Atlantic’s ongoing debate about how to teach writing in schools, Robert Pondiscio wrote an eye-opening piece called “How Self-Expression Damaged My Students.” In it, he tells of how he used modern-day techniques for teaching writing–not teaching rules of grammar or correcting errors but treating the students as little writers and having them write. He notes, however that “good writers don’t just do stuff. They know stuff. … And if this is not explicitly taught, it will rarely develop by osmosis among children who do not grow up in language-rich homes.”
What Pondiscio describes on the writing front has also been happening with mathematics education in K-6 for the past two decades. I first became aware of it over 10 years ago when I saw what passed for math instruction in my daughter’s second grade class. I was concerned that she was not learning her addition and subtraction facts. Other parents we knew had the same concerns. Teachers told them not to worry because kids eventually “get it.”
One teacher tried to explain the new method. “It used to be that if you missed a concept or method in math, then you were lost for the rest of the year. But the way we do it now, kids have a lot of ways to do things, like adding and subtracting, so that math topics from day to day aren’t dependent on kids’ mastering a previous lesson.”
This was my initiation into the world of reform math. It is a world where understanding takes precedence over procedure and process trumps content. In this world, memorization is looked down upon as “rote learning” and thus addition and subtraction facts are not drilled in the classroom–it’s something for students to learn at home. Inefficient methods for adding, subtracting, multiplying, and dividing are taught in the belief that such methods expose the conceptual underpinning of what is happening during these operations. The standard (and efficient) methods for these operations are delayed sometimes until 4th and 5th grades, when students are deemed ready to learn procedural fluency.
Barry Garelick, via a kind email:
Mathematics education in the United States is at a pivotal moment. At this time, forty-five states and the District of Columbia have adopted the Common Core standards, a set of uniform benchmarks for math and reading. Thirty-two states and the district have been granted waivers from important parts of the Bush-era No Child Left Behind law. As part of the agreement in being granted a waiver, those states have agreed to implement Common Core. States have been led to believe that adoption of such standards will improve mathematics and English-language education in our public schools.
My fear (as well as that of many of my colleagues) is that implementation of the Common Core math standards may actually make things worse. The final math standards released in June, 2010 appear to some as if they are thorough and rigorous. Although they have the “look and feel” of math standards, their adoption in my opinion will not only continue the status quo in this country, but will be a mandate for reform math — a method of teaching math that eschews memorization, favors group work and student-centered learning, puts the teacher in the role of “guide” rather than “teacher” and insists on students being able to explain the reasons why procedures and methods work for procedures and methods that they may not be able to perform.
Barry Garelick:
In the never-ending dialogue about math education that has come to be known as the “math wars”, proponents of reform-based math tend to characterize math as it was taught in the 60’s (and prior) as “skills-based”. The term connotes a teaching of math that focused almost exclusively on procedures and facts in isolation to the conceptual underpinning that holds math together. The “skills-based” appellation also suggests that those students who may have mastered their math courses in K-12 were missing the conceptual basis of mathematics and were taught the subject as a means to do computation, rather than explore the wonders of mathematics for its own sake.
Without delving too far into the math wars, I and others have written that while traditional math may sometimes have been taught poorly, it also was taught properly. In fact, a view of the textbooks in use at that time reveal that they provided both procedures and concept. Missing perhaps were more challenging problems, but also missing from the reformers’ arguments is the fact that not only are procedures and concepts taught in tandem but that computational fluency leads to conceptual understanding. (See http://www.psy.cmu.edu/~siegler/r-jhnsn-etal-01.pdf )
Barry Garelick:
In a well-publicized paper that addressed why some students were not learning to read, Reid Lyon (2001) concluded that children from disadvantaged backgrounds where early childhood education was not available failed to read because they did not receive effective instruction in the early grades. Many of these children then required special education services to make up for this early failure in reading instruction, which were by and large instruction in phonics as the means of decoding. Some of these students had no specific learning disability other than lack of access to effective instruction. These findings are significant because a similar dynamic is at play in math education: the effective treatment for many students who would otherwise be labeled learning disabled is also the effective preventative measure.
In 2010 approximately 2.4 million students were identified with learning disabilities — about three times as many as were identified in 1976-1977. (See http://nces.ed.gov/programs/digest/d10/tables/xls/tabn045.xls and http://www.ideadata.org/arc_toc12.asp#partbEX). This increase raises the question of whether the shift in instructional emphasis over the past several decades has increased the number of low achieving children because of poor or ineffective instruction who would have swum with the rest of the pack when traditional math teaching prevailed. I believe that what is offered as treatment for math learning disabilities is what we could have done–and need to be doing–in the first place. While there has been a good amount of research and effort into early interventions in reading and decoding instruction, extremely little research of equivalent quality on the learning of mathematics exists. Given the education establishment’s resistance to the idea that traditional math teaching methods are effective, this research is very much needed to draw such a definitive conclusion about the effect of instruction on the diagnosis of learning disabilities.1
Barry Garelick, via email:
Fran, by Way of Introduction
My high school algebra 2 class which I had in the fall of 1964, was notable for a number of things. One was learning how to solve word problems. Another was a theory that most problems we encountered in algebra class could be solved with arithmetic. Yet another was a girl named Fran who I had a crush on.
Fran professed to not like algebra or the class we were in, and found word problems difficult. On a day I had occasion to talk to her, I tried to explain my theory that algebra was like arithmetic but easier. Admittedly, my theory had a bit more to go. She appeared to show some interest, but she wasn’t interested. On another occasion I asked her to a football game, but she said she was washing her hair that day. Although Fran had long and beautiful black hair, and I wanted to believe that she had a careful and unrelenting schedule for washing it, I resigned myself to the fact that she would remain uninterested in me, algebra, and any theories about the subject.
My theory of arithmetic vs. algebra grew from a realization I had during that the problems that were difficult for me years ago when I was in elementary school were now incredibly easy using algebra. For example: $24 is 30% of what amount? In arithmetic this involved setting up a proportion while in algebra, it translated directly to 24 = 0.3x, thus skipping the set up of the ratio 24/30 = x/100. Similarly, it was now much easier to understand that an increase in cost by 25% of some amount could be represented as 1.25x. What had been problems before were now exercises; being able to express quantities algebraically made it obvious what was going on. It seemed I was on to something, but I wasn’t quite sure what.
Barry Garelick, via email:
The education establishment commits to fads like group and collaborative learning, but Garelick says they shouldn’t ignore and misinterpret traditional math.
Most discussions about mathematics and how best to teach it in the K-12 arena break down to the inevitable bromides about how math was traditionally taught and that such methods were ineffective. The conventional wisdom on the “traditional method” of teaching math is often heard as an opening statement at school board meetings during which parents are protesting the adoption of a questionable math program: “The traditional method of teaching math has failed thousands of students.” A recent criticism I read expanded on this notion and said that it wasn’t so much the content or the textbooks (though he states that they were indeed limited) but the teaching was “too rigid, too inflexible, too limited, and thus failed to adequately address the realities of educating a large, diverse, and rapidly changing population during decades of technological innovation and social upheaval.”
There is some confusion when talking about “traditional methods” since traditional methods vary over time. Textbooks considered traditional for the last ten years, for example, are quite different than textbooks in earlier eras. For purposes of this discussion, I would like to confine “traditional” to methods and textbooks in use in the 40′s, 50′s and 60′s. And before we get to the question about teaching methods, I want to first talk about the textbooks in use during this time period. A glance at the textbooks that were in use over these years shows that mathematical algorithms and procedures were not taught in isolation in a rote manner as is frequently alleged. In fact, concepts and understanding were an important part of the texts. Below is an excerpt from a fifth grade text of the “Study Arithmetic” series (Knight, et. al. 1940):
Barry Garelick, via email:
I attended Mumford High School in Detroit, from the fall of 1964 through June of 1967, the end of a period known to some as the golden age of education, and to others as an utter failure.
Raymond
I attended Mumford High School in Detroit, from the fall of 1964 through June of 1967, the end of a period known to some as the golden age of education, and to others as an utter failure. For the record I am in the former camp, a product of an era which in my opinion well-prepared me to major in mathematics. I am soon retiring from a career in environmental protection and will be entering the teaching profession where I will teach math in a manner that has served many others well over many years and which I hope will be tolerated by the people who hire me.
I was in 10th grade, taking Algebra 2. In the study hall period that followed my algebra class I worked the 20 or so homework problems at a double desk which I shared with Raymond, a black student. He would watch me do the day’s homework problems which I worked with the ease and alacrity of an expert pinball player.
While I worked, he would ask questions about what I was doing, and I would explain as best I could, after which he would always say “Pretty good, pretty good”–which served both as an expression of appreciation and a signal that he didn’t really know much about algebra but wanted to find out more. He said he had taken a class in it. In one assignment the page of my book was open to a diagram entitled “Four ways to express a function”. The first was a box with a statement: “To find average blood pressure, add 10 to your age and divide by 2.” The second was an equation P = (A+10)/2. The third was a table of values, and the last was a graph. Raymond asked me why you needed different ways to say what was in the box. I wasn’t entirely sure myself, but explained that the different ways enabled you to see the how things like blood pressure changed with respect to age. Sometimes a graph was better than a table to see this; sometimes it wasn’t. Not a very good explanation, I realized, and over the years I would come back to that question–and Raymond’s curiosity about it–as I would analyze equations, graphs, and tables of values.
Barry Garelick:
In Jay Greene’s recent blog post, “The Dead End of Scientific Progressivism,” he points out that Vicki Phillips, head of education at the Gates Foundation misread her Foundation’s own report. Jay’s point was that Vicki continued to see what she and others wanted to see: “‘Teaching to the test makes your students do worse on the tests.’ Science had produced its answer — teachers should stop teaching to the test, stop drill and kill, and stop test prep (which the Gates officials and reporters used as interchangeable terms).”
I was intrigued by the education establishment’s long-held view as Jay paraphrased it. This view has become one of the “enduring truths” of education and I have heard it expressed in the various classes I have been taking in education school the last few years. (I plan to teach high school math when I retire later this year). In terms of math education, ed school professors distinguish between “exercises” and “problems”. “Exercises” are what students do when applying algorithms or routines they know and can apply even to word problems. Problem solving, which is preferred, occurs when students are not able to apply a mechanical, memorized response, but rather have to apply prior knowledge to solve a non-routine problem. Moreover, we future teachers are told that students’ difficulty in solving problems in new contexts is evidence that the use of “mere exercises” or “procedures” is ineffective and they are overused in classrooms. One teacher summed up this philosophy with the following questions: “What happens when students are placed in a totally unfamiliar situation that requires a more complex solution? Do they know how to generate a procedure? How do we teach students to apply mathematical thinking in creative ways to solve complex, novel problems? What happens when we get off the ‘script’?”
Barry Garelick
The New York Times ran a story on September 30 about Singapore Math being used in some schools in the New York City area. Like many newspaper stories about Singapore Math, this one was no different. It described a program that strangely sounded like the math programs being promoted by reformers of math education, relying on the cherished staples of reform: manipulatives, open-ended problems, and classroom discussion of problems. The only thing the article didn’t mention was that the students worked in small groups.
Those of us familiar with Singapore Math from having used it with our children are wondering just what program the article was describing. Spending a week on the numbers 1 and 2 in Kindergarten? Spending an entire 4th grade classroom period discussing the place value ramifications of the number 82,566? Well, maybe that did happen, but not because the Singapore Math books are structured that way. In fact, the books are noticeably short on explicit narrative instruction. The books provide pictures and worked out examples and excellent problems; the topics are ordered in a logical sequence so that material mastered in the various lessons builds upon itself and is used to advance to more complex applications. But what is assumed in Singapore is that teachers know how to teach the material–the teacher’s manuals contain very little guidance. Thus, the decision to spend a week on the numbers 1 and 2 in kindergarten, or a whole class period discussing a single number is coming from the teachers, not the books.
The mistaken idea that gets repeated in many such articles is that Singapore Math differs from other programs by requiring or imparting a “deep understanding” and that such understanding comes about through a) manipulatives, b) pictures, and c) open-ended discussions. In fact, what the articles represent is what the schools are telling the reporters. What newspapers frequently do not realize when reporting on Singapore Math, is that when a school takes on such a program, it means going against what many teachers believe math education to be about; it is definitely not how they are trained in ed schools. The success of Singapore’s programs relies in many ways on more traditional approaches to math education, such as explicit instruction and giving students many problems to solve, in some ways its very success represented a slap in the face to American math reformers, many of whom have worked hard to eliminate such techniques being used.
Katharine Beals, Trumpeter Books, 2009 Reviewed by Barry Garelick, via email
Many school parents question the value of today’s homework assignments. They rightly wonder whether their children are getting the education they need in order to succeed in college. For the most part, they are well-meaning parents who were educated from the 1950’s through the 1970’s in a different style–a style derided by the current power elite in graduate schools of education and school administration. They describe the schoolroom remembered by today’s parents as: sitting in rows, facing front, listening passively to a teacher who talked to the blackboard, “memorizing by rote”, and thinking uncritically. In today’s classrooms, students are given a minimal amount of instruction, and instead are presented with a question–say a math problem–told to form groups and work out an approach to solving the problem. Or if not a math problem, they are told to discuss an aspect of a book they are reading. Homework assignments are often art projects, in which students must construct dioramas of the climactic event of a story they read, or decorate a tissue box with German phrases to help them learn the language, or put together a family tree with photographs and label each with the Spanish term for their place in the family.
In Raising a Left-brain Child in a Right-brain World, Katharine Beals explores today’s classrooms and describes in detail why this approach is particularly destructive and ineffective for students who are shy, awkward, introspective, linear and analytic thinkers. She is careful to explain that her use of the term “left brained” is her way of categorizing students who are linear thinkers–who process information by learning one thing at a time thoroughly before moving on to the next. (I use the term in the same fashion in this review.)
A particularly powerful passage at the beginning of the book describes the difficulties that left-brained children face and provides a stark and disturbing contrast with the traditional classrooms that the parents of these children remember:
Making matters worse is how today’s informal discussions favor multiple solutions, personal opinions, and personal connections over single correct answers. In previous generations the best answer, exerting an absolute veto power, favored the studious over the merely charismatic; how that there is no best answer, extroversion is king. … To fully appreciate the degree to which today’s classrooms challenge our children, we should consider how they might have fared in more traditional schools. Imagine how much more at ease they might be in general, and how their attitudes toward school might improve, if they enjoyed the privacy of quietly listening to teachers lecture instead of having to talk to classmates. …Imagine if they could read to themselves instead of to a group, do math problems on their own, and find, in the classroom, a safe haven from school yard dynamics. (p. 23)
by Carol Ann Tomlinson and Jay McTighe, Association for Supervision and Curriculum Development, 2006; Reviewed by Barry Garelick, via email:
The premise of this book is enticingly simple . It presents two solutions to two prevalent problems in education . The first is the vast amount of content required to be taught because of various state standards, and how one can thread that maze and “teach for understanding .” That is, how can educators get students to apply what they’ve learned to new and unfamiliar problems? The second is the diverse nature of today’s classrooms, the result of heterogeneous grouping of students of different abilities . How does an educator differentiate instruction to accommodate such diversity in a single classroom?
I read this book in a math teaching methods class a few years ago . One event in that class stands out regarding this textbook . In a chapter on assessing understanding, a chart presents examples of “Inauthentic versus Authentic Work” (p . 68) . For example, “Solve contrived problems” is listed as inauthentic; “Solve ‘real world’ prob- lems” is listed as authentic . The black-and-white nature of the dis- tinctions on the chart bothered me, so when the teacher asked if we had any comments, I said that calling certain practices “inauthentic” is not only pejorative but misleading . Since the chart listed “Practice decontextualized skills” as inauthentic and “Interpret literature” as authentic, I asked the teacher, “Do you really think that learning to read is an inauthentic skill?”
She replied that she didn’t really know about issues related to reading . Keeping it on the math level, I then asked why the authors automatically assumed that a word problem that might be contrived didn’t involve “authentic” mathematical concepts . She answered with a blank stare and the words “Let’s move on .”
That incident remains in my mind because it is emblematic of the educational doctrine that pervades schools of education as well as this book . The doctrine holds that mastery of facts and attaining procedural fluency in subjects like mathematics amounts to mind- numbing “drill and kill” exercises that ultimately stifle creativity and critical thinking . It also embodies the belief that critical thinking skills can be taught .
In a discussion of what constitutes “understanding,” the authors state that a student’s ability to apply what he or she has learned does not necessarily represent understanding . “When we call for an appli- cation we do not mean a mechanical response or mindless ‘plug-in’ of a memorized formula . Rather, we ask students to transfer–to use what they know in a new situation” (p . 67) . In terms of math and other subjects that involve attaining procedural fluency, employing worked examples as scaffolding for tackling more-complex prob- lems is not something that these authors see as leading to any kind of understanding . That a mastery of fundamentals provides the foun- dation for the creativity they seek is lost in their quest to get stu- dents performing authentic work from the start
Barry Garelick, via email:
Earlier this month, the Common Core State Standards Initiative (CCSSI)–a state-led effort coordinated by the National Governors Association Center for Best Practices (NGA Center) and the Council of Chief State School Officers (CCSSO)–issued the final version of its math standards for K-12.
The draft standards were released in March and CCSSI allowed the public to submit comments on the draft via their website. Over 10,000 comments were received. The U.S. Coalition for World Class Math was one of the commenter’s and I had a hand in drafting comments. We were concerned with the draft standards’ use of the word “understand” and pointed out that the use of this verb results in an interpretation by different people for different purposes. I am pleased to see that the final version of the standards has greatly reduced the use of the word “understand”, but I remain concerned that 1) it still is used for some standards, resulting in the same problems we raised in our comments, and 2) the word “understand” in some instances has been replaced with “explain”.
I am not against teaching students the conceptual underpinnings of procedures. I do not believe, however, that it is necessary to require students to then be able to recite the reasons why a particular procedure or algorithm works; i.e., to provide justification. At lower grade levels, some students will understand such explanations, but many will not. And even those who do may have trouble articulating the reasons. The key is whether they understand how such procedure is to be applied, and what the particular procedure represents. For example, does a student know how to figure out how many 2/3 ounce servings of yogurt are in a ¾ ounce container? If the student knows that the solution is to divide ¾ by 2/3, that should provide evidence that the student understands what fractional division means, without having to ask them to explain what the relationship is between multiplication and division and to show why the “invert and multiply” rule works each and every time.
Barry Garelick:
In the fall of 1970, I dropped out of the University of Michigan during my senior year with the intention of never re turning. I was a math major and I convinced myself that I would have a better chance being a writer than a mathematician
In the fall of 1970, I dropped out of the University of Michigan during my senior year with the intention of never re turning. I was a math major and I convinced myself that I would have a better chance being a writer than a mathematician. I figured I would work at any job I could get to support myself. The only job I could get was unloading telephone books from a truck into the cars of people who were to deliver them. The job was to last three days–I quit after the first. During that first day, around the time when my arms became like rubber and I could hardly even lift one phone book, I had a flash of insight and decided to return to school and get my degree. Then I would become a writer. In the summer of 1971, I got my degree, and vowed to never again set foot in another math classroom in my life, and told myself that if I ever did I would puke.
Barry Garelick:
It explores two different approaches to math; one is representative of the fuzzy math side of things, and the other is in the traditionalist camp. I make it clear what side I’m on. I talk about how the fuzzy side uses what I call a “separate path” in which students are given open ended and ill posed problems as a means to teach them how to apply prior knowledge in new situations. I present two different problems, one representing each camp.
The math may prove challenging for some readers, though high school math teachers should have no problems with it.
Much has been written about the debate on how best to teach math to students in K-12–a debate often referred to as the “math wars”. I have written much about it myself, and since the debate shows no signs of easing, I continue to have reasons to keep writing about it. While the debate is complex, the following two math problems provide a glimpse of two opposing sides:
Problem 1: How many boxes would be needed to pack and ship one million books collected in a school-based book drive? In this problem the size of the books is unknown and varied, and the size of the boxes is not stated.
Problem 2: Two boys canoeing on a lake hit a rock where the lake joins a river. One boy is injured and it is critical to get a doctor to him as quickly as possible. Two doctors live nearby: one up-river and the other across the lake, both equidistant from the boys. The unhurt boy has to fetch a doctor and return to the spot. Is it quicker for him to row up the river and back, or go across the lake and back, assuming he rows at the same constant rate of speed in both cases?
The first problem is representative of a thought-world inhabited by education schools and much of the education establishment. The second problem is held in disdain by the same, but favored by a group of educators and math oriented people who for lack of a better term are called “traditionalists”.
Barry Garelick:
“What’s a court doing making a decision on math textbooks and curriculum?” This question and its associated harrumphs on various education blogs and online newspapers came in reaction to the February 4, 2010 ruling from the Superior court of King County that the Seattle school board’s adoption of a discovery type math curriculum for high school was “arbitrary and capricious”.
In fact, the court did not rule on the textbook or curriculum. Rather, it ruled on the school board’s process of decision making–more accurately, the lack thereof. The court ordered the school board to revisit the decision. Judge Julie Spector found that the school board ignored key evidence–like the declaration from the state’s Board of Education that the discovery math series under consideration was “mathematically unsound”, the state Office of the Superintendent of Public Instruction not recommending the curriculum and last but not least, information given to the board by citizens in public testimony.
The decision is an important one because it highlights what parents have known for a long time: School boards generally do what they want to do, evidence be damned. Discovery type math programs are adopted despite parent protests, despite evidence of experts and–judging by the case in Seattle–despite findings from the State Board of Education and the Superintendent of Public Instruction.
Barry Garelick, via email:
By way of introduction, I am neither mathematician nor mathematics teacher, but I majored in math and have used it throughout my career, especially in the last 17 years as an analyst for the U.S. Environmental Protection Agency. My love of and facility with math is due to good teaching and good textbooks. The teachers I had in primary and secondary school provided explicit instruction and answered students’ questions; they also posed challenging problems that required us to apply what we had learned. The textbooks I used also contained explanations of the material with examples that showed every step of the problem solving process.
I fully expected the same for my daughter, but after seeing what passed for mathematics in her elementary school, I became increasingly distressed over how math is currently taught in many schools.
Optimistically believing that I could make a difference in at least a few students’ lives, I decided to teach math when I retire. I enrolled in education school about two years ago, and have only a 15-week student teaching requirement to go. Although I had a fairly good idea of what I was in for with respect to educational theories, I was still dismayed at what I found in my mathematics education courses.
In class after class, I have heard that when students discover material for themselves, they supposedly learn it more deeply than when it is taught directly. Similarly, I have heard that although direct instruction is effective in helping students learn and use algorithms, it is allegedly ineffective in helping students develop mathematical thinking. Throughout these courses, a general belief has prevailed that answering students’ questions and providing explicit instruction are “handing it to the student” and preventing them from “constructing their own knowledge”–to use the appropriate terminology. Overall, however, I have found that there is general confusion about what “discovery learning” actually means. I hope to make clear in this article what it means, and to identify effective and ineffective methods to foster learning through discovery.
Garelick’s part ii on Discovery learning can be found here.
Related: The Madison School District purchases Singapore Math workbooks with no textbooks or teacher guides. Much more on math here.
Via a Barry Garelick email:
“The article describes my experience tutoring my daughter and her friend when they were in sixth grade, using Singapore Math in order to make up for the train wreck known as Everyday Math that she was getting in school. I doubt that the article will change the minds of the administrators who believe Everyday Math has merit, but it wasn’t written for that purpose. It was written for and dedicated to parents to let them know they are not alone, that they aren’t the only ones who have shouted at their children, that there are others who have experienced the tears and the confusion and the frustration. Lastly it offers some hope and guidance in how to go about teaching their kids what they are not learning at school.”
Barry Garelick, via email:
By way of introduction, I am neither mathematician nor mathematics teacher, but I majored in math and have used it throughout my career, especially in the last 17 years as an analyst for the U.S. Environmental Protection Agency. My love of and facility with math is due to good teaching and good textbooks. The teachers I had in primary and secondary school provided explicit instruction and answered students’ questions; they also posed challenging problems that required us to apply what we had learned. The textbooks I used also contained explanations of the material with examples that showed every step of the problem solving process.
I fully expected the same for my daughter, but after seeing what passed for mathematics in her elementary school, I became increasingly distressed over how math is currently taught in many schools.
Optimistically believing that I could make a difference in at least a few students’ lives, I decided to teach math when I retire. I enrolled in education school about two years ago, and have one class and a 15-week student teaching requirement to go. Although I had a fairly good idea of what I was in for with respect to educational theories, I was still dismayed at what I found in my mathematics education courses.
In class after class, I have heard that when students discover material for themselves, they supposedly learn it more deeply than when it is taught directly. Similarly, I have heard that although direct instruction is effective in helping students learn and use algorithms, it is allegedly ineffective in helping students develop mathematical thinking. Throughout these courses, a general belief has prevailed that answering students’ questions and providing explicit instruction are “handing it to the student” and preventing them from “constructing their own knowledge”–to use the appropriate terminology. Overall, however, I have found that there is general confusion about what “discovery learning” actually means. I hope to make clear in this article what it means, and to identify effective and ineffective methods to foster learning through discovery.
Daniel de Vise:
The most noticeable change is a dramatic increase in students taking accelerated math classes in the middle years, an initiative that seems to have spread to every school system in the region. Educators view math acceleration as a gateway to advanced study in high school and, in turn, to college. Higher-level math classes have helped middle schools cultivate a community of students similar to those in honors and Advanced Placement high school classes.
At Samuel Ogle Middle School in Bowie, the number of students taking Algebra I, a high-school-level course, has doubled from 60 to 120 in the past two years.
Barry Garelick references Montgomery County’s experiment with Singapore Math. About Singapore Math. More here.
Barry Garelick:
The British mathematician J. E. Littlewood once began a math class for freshmen with the following statement: “I’ve been giving this lecture to first-year classes for over twenty-five years. You’d think they would begin to understand it by now.”
People involved in the debate about how math is best taught in grades K-12, must feel a bit like Littlewood in front of yet another first year class. Every year as objectionable math programs are introduced into schools, parents are alarmed at what isn’t being taught. The new “first-year class” of parents is then indoctrinated into what has come to be known as the math wars as the veterans – mathematicians, frustrated teachers, experienced parents, and pundits – start the laborious process of explanation once more.
It was therefore a watershed event when the President’s National Mathematics Advisory Panel (NMP) held its final meeting on March 13, 2008 and voted unanimously to approve its report: Foundations for Success: The Final Report of the National Mathematics Advisory Panel.
National Math Panel.
Frederick Hess: Math is fundamental. This observation is a groan-inducing cliché, but it’s also true. Math matters for employment, financial literacy, and even for navigating evidentiary claims about things like Covid-19 and climate change. Yet math education seems to have gotten sidelined amid broader debates about school culture, civics, and the rest. Lately, when math … Continue reading “Full of the kinds of things that teachers say privately but hesitate to speak aloud” →
Barbara Oakley: Out on Good Behavior: Teaching Math while Looking Over Your Shoulder, by Barry Garelick. We greatly enjoyed and got a lot out of this brief, sardonic memoir of an outstanding math teacher in an era when teaching math in public schools is becoming increasingly divorced from what neuroscience has revealed about how students … Continue reading Out on Good Behavior: Teaching Math while Looking Over Your Shoulder →
Barry Garelick, via a kind email: “Hell hath no fury like a mathematician whose child has been scorned by an education system that refuses to know better,” Barry Garelick wrote in his first published article on math education in 2005. He has been at it ever since, and his focus has remained the same: why … Continue reading Math Education in the U.S.: Still Crazy After All These Years →
Barry Garelick, via a kind email: I attended an “Ed Camp” recently. This is one of many types of non-professional development and informal gatherings where teachers talk about various education-related topics. The camp I attended was free of charge and took place at a charter school that prided itself on a student-centered approach to learning. … Continue reading A Math Teacher’s Day at Ed Camp →
Katherine Beals & Barry Garelick: “In general, there is no more evidence of “understanding” in the explained solution, even with pictures, than there would be in mathematical solutions presented in a clear and organized way. How do we know, for example, that a student isn’t simply repeating an explanation provided by the teacher or the … Continue reading Explaining Your Math: Unnecessary at Best, Encumbering at Worst →
Barry Garelick, via a kind email: “I am not an outright proponent of the philosophy that ‘If you want something done right, you have to live in the past’, but when it comes to how to teach math there are worse philosophies to embrace,” Barry Garelick explains as he continues from where he left off … Continue reading Teaching Math in the 21st Century →
Barry Garelick, via a kind email: I am currently working at a middle school in a neighboring school district. I do not have my own classes; I assist the math teachers there by identifying and working with students who are struggling. Like most schools these days, it has fallen under the spell of Common Core, … Continue reading Conversations on the Rifle Range 26: Moving On and the Sedimentation of Students →
Barry Garelick: t took me about three weeks to learn all the names of my students. Identifiable patterns of behavior took me a little longer. For example, Cindy, who is in one of the two algebra classes, tended to stop me in the midst of explaining a new procedure and say: “Wait, wait, I’m confused, … Continue reading Conversations on the Rifle Range 23: The Quadratic Formula Ultimatum, and the Substrate of Understanding →
Barry Garelick, via a kind email: There are a variety of methods one can use to discipline students: detentions, referrals, sending the student outside of class, contacting the parents. I was confused about most of them and resisted using them. Lunch-time detentions were especially tricky because of a dual lunch schedule at my school. Because … Continue reading Boundaries of Behavior, Parallelograms, and the Art of Forgiveness →
Barry Garelick: I currently am on a second career after retirement—I teach math in middle school. During my last few years of work, I started taking courses in ed school at night. The first course I took was taught by a professor who had what seemed to me to be a unique gift. He managed … Continue reading Who’s to Say Teachers Can’t Modify Common Core? No One →
Barry Garelick, via a kind email: I had come to the point in the chapter on systems of linear equations in my algebra 1 class where the book presented mixture, rate and current, and number problems. To prep them for the onslaught, I included a word problem into one of the warm-up problems I had … Continue reading Word Problems, No Guess and Check, and a Sound Bite for an Interview →
Barry Garelick, via a kind email: Back in September, when I was doing my sub-assignment for the high school, I attended a math department meeting the day before school began. Sally from the District office presided, and among the many things she told us at that session was that this year the students in the … Continue reading Teaching to the Authentic Assessment →
Barry Garelick, via a kind email: Mrs. Halloran, the teacher for whom I was subbing, was known for her strictness. On the day I met with her, she had me observe some of her classes and she introduced me. She told the students “I expect you all to behave well with Mr. Garelick. He and … Continue reading Classroom Rules and Procedures Meet Understanding →
Barry Garelick, via a kind email: After my assignment at the high school I took on various short-term sub assignments. These seemed relatively straightforward compared to the difficulties I had been through. In fact, I was reluctant to start subbing at first, but after the first few times, I started to regain my confidence. I … Continue reading On Math Education: JFK, the Beatles and Getting Back in the Swing of Things →
Barry Garelick, via a kind email: A video about how the Common Core is teaching young students how to do addition problems is making the rounds on the internet: http://rare.us/story/watch-common-core-take-56-seconds-to-solve-96/ Much ballyhoo is being made of this. Given the prevailing interpretation of Common Core math standards, the furor is understandable. The purveyors of these standards … Continue reading Undoing the ‘Rote Understanding’ Approach to Common Core Math Standards →
Barry Garelick, via a kind email: All my classes were getting ready to take their first quiz later in the week. My second period class was the second-year Algebra 1 class. We were working on systems of linear equations covering the various ways of solving two equations with two unknowns. I was preparing for my … Continue reading Teaching Math-Conversations on the Rifle Range 7: Winds and Currents, Formative Assessments, and the Eternal Gratitude of Dudes →
Barry Garelick: Barry Garelick, who wrote various letters under the name Huck Finn and which were published here is at work writing what will become “Conversations on the Rifle Range”. This will be a documentation of his experiences teaching math as a long-term substitute. OILF proudly presents episode number two: My back-to-school night was held … Continue reading Conversations on the Rifle Range 2: Negative Numbers, Back-to-School Night, a Tattooed Man and a Mysterious Stranger →
Out in Left Field, via a kind Barry Garelick email: Barry Garelick, who wrote various letters under the name Huck Finn and which were published here is at work writing what will become “Conversations on the Rifle Range”. This will be a documentation of his experiences teaching math as a long-term substitute. OILF proudly presents … Continue reading Conversations on the Rifle Range, I: Not Your Mother’s Algebra 1 and the Guy Who Really Knows →
The news last week that Shanghai students achieved the top scores in math on the international PISA exam was for some of us not exactly a wake-up call (as Secretary of Education Arne Duncan characterized it) or a Sputnik moment (as President Obama called it).
We’ve seen this result before. We’ve seen the reactions and the theories and the excuses that purport to explain why the US does so poorly in math. In fact, there are three main variations used to explain why Chinese/Asian students do so well in international exams:
- Version 1: They are taught using rote learning and then regurgitate the results on exams that test how well they memorize the procedures of how to solve specific problems.
- Version 2: They are taught using the reform methods of a “problem based approach” that doesn’t rely on drills, and instills critical thinking and higher order thinking skills
- Version 3: The teacher or the culture produces the proper conditions for learning.
Barry Garelick: If one could make a case against the perpetrators of reform math—complete with arrests and jail time—showing that such programs are a form of child abuse, the math wars would cease in a matter of days. As it is, however, reasoned arguments from those who oppose the reform programs haven’t seemed to carry … Continue reading On Math Reform →
The Economist: Look around the business world and two things stand out: the modern economy places an enormous premium on brainpower; and there is not enough to go round. But education inevitably matters most. How can India talk about its IT economy lifting the country out of poverty when 40% of its population cannot read? … Continue reading The Politics of K-12 Math and Academic Rigor →
Typical of many math textbooks in the U.S., this one is thick, multicolored,and full of games,puzzles,and activities,to help teachers pass the time, but rarely challenge students. Singapore Math’s textbook is thin, and contains only mathematics — no games. Students are given briefexplanations, then confronted with problems which become more complex as the unit progresses. Barry … Continue reading Miracle Math →
Barry Garelick: Education Secretary Spellings recently announced the formation of a presidentially appointed panel that was formed to address math teaching. According to the charter of this panel, one of its purposes is “to foster greater knowledge of and improved performance in mathematics among American students.” The panel is charged with producing a report in … Continue reading Polite Agreement or Something We Can Use? →
Barry Garelick on our “National crisis in mathematics education”. Fat Link on Barry Garelick.
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