Math instruction took another big hit recently. “Big” because the* New York Times*, one of the top U.S. newspapers ran it as a cover story of its magazine section. So here again, amid the Common Core standards in math that ask teachers to go beyond the “right” answer and periodic efforts over the past century (yes, I mean “century”) to move math teaching away from learning the rules of arithmetic, algebraic equations, and geometry proofs, comes another blast at how teachers teach math.

Elizabeth Green’s well-written article (drawn from a forthcoming book) on persistent patterns (mostly ineffective) in teachers implementing the New Math of the 1960s, the New NEW math of the 1980s, and now the math Common Core standards shines yet another light on the puzzle of why teachers teach as they do. And why policy after policy adopted to change math instruction has failed time and again in practice leaving each generation innumerate. Green has her own answers which to my experience as a teacher, historian, and researcher make a great deal of sense.

Moreover, as Green braids many threads together to explain persistence in poor math teaching, she also identifies others that begin to capture the complexity of teaching. Her answers as to what to do are, however, largely unsatisfying because she excludes pieces necessary to complete the puzzle. Without the full puzzle picture on the jigsaw box, glomming onto a few pieces risks even yet another failure to remedy the puzzling persistence of poor math instruction.

Green does not blame teachers. She points to state and federal policies, teacher education institutions, and the taken-for-granted way that new teachers have learned about teaching from watching a few feet away how teachers have taught them for 16-plus years. All of this captures important threads in unraveling the puzzle of persistent failure in routine, teacher-centered math instruction focused less on understanding deeply and practically math concepts and more on knowing the rules to get the right answer. But not all of the threads.

Nowhere does Green mention the power of the age-graded school to influence how teachers teach.

The age-graded school (e.g., K-5, K-8, 6-8, 9-12), a 19^{th} century innovation, has become an unquestioned mainstay of school organization in the 21^{st} century. Today, most taxpayers and voters have gone to kindergarten at age 5, studied Egyptian mummies in the 6^{th} grade, took algebra in the 8th or 9^{th} grade and then left 12^{th} grade with a diploma.

If any school reform–in the sense of making fundamental changes in organization, curriculum, and instruction–can be considered a success it is the age-graded school. Consider longevity–the first age-graded structure of eight classrooms appeared in Quincy (MA) in the late 1840s. Or consider effectiveness. The age-graded school has processed efficiently millions of students over the past century and a half, sorted out achievers from non-achievers, and now graduates nearly three-quarters of those entering high school Or adaptability. The age-graded school exists in Europe, Asia, Africa, Latin America, and North America covering rural, urban, and suburban districts.

As an organization, the age-graded school allocates children and youth by their ages to school “grades”; it sends teachers into separate classrooms and prescribes a curriculum carved up into 36-week chunks for each grade. Teachers and students cover each chunk assuming that all children will move uniformly through the 36-weeks to be annually promoted.

The age-graded school is also an institution that has plans for those who work within its confines. The organization isolates and insulates teachers from one another, perpetuates teacher-centered pedagogy, and prevents a large fraction of students from achieving academically. It is the sea in which teachers, students, principals, and parents swim yet few contemporary reformers have asked about the water in which they share daily. To switch metaphors, the age-graded school is a one-size-fits-all structure.

Why have most school reformers and educational entrepreneurs been reluctant to examine an organization that influences daily behavior of nearly 4 million adults and well over 50 million children? Dominant social beliefs of parents and educators about a “real” school, that is, one where children learn to read in 1^{st} grade, receive report cards, and get promoted have politically narrowed reform options in transforming schools. For example, when a charter school applicant proposes a new school the chances of receiving official approval and parental acceptance increase if it is a familiar age-graded one, not one where most teachers team teach and groups of multi-age children (ages 5-8, 9-11) learn together. Sure, occasional reformers create non-graded schools, the School of One, and particular community schools but they are outliers.

These familiar age-graded schools–don’t ask fish to consider the water they swim in–are missing in unraveling the puzzle of persistent ways of teaching math that Elizabeth Green has so nicely laid before us.

Reblogged this on David R. Taylor-Thoughts on Texas Education.

Thanks for re-blogging post on math teaching patterns, David.

Larry, you could argue that getting rid of the age-graded system would improve education, but I disagree that eliminating it is necessary to change teaching patterns. For example, the TIMSS video study described in The Teaching Gap showed a clear difference in the pattern of teaching math in Japan, Germany, and the United States, all of whom had age-graded schools. A study by James Spillane found three different patterns of teaching math by American teachers who all believed that they were implementing reform as intended. Their analysis found differences in 1) the nature of the tasks given to students and in 2) the discourse norms that prevailed in the classroom.

http://www.jstor.org/discover/10.2307/1164544?uid=2&uid=4&sid=21104474596507

Green’s solutions would go a long way in improving the teaching of math. What is necessary for those solutions to work is building a culture of collaboration among teachers. Currently, teachers consider the classroom their private domain, and my sense it that the majority prefer to work alone. What it takes for real change in the pattern of teaching is for teachers to watch each other teach, and to discuss in detail all aspects of that teaching and of teacher decision-making as the lesson unfolds. Otherwise, they interpret the reform recommendations in terms of their current practices.

https://mathematicsteachingcommunity.math.uga.edu/index.php/885/reform-what-it-takes-to-improve-core-teaching-practices

Thanks, Burt, for taking the time to comment. You say:”Green’s solutions would go a long way in improving the teaching of math. What is necessary for those solutions to work is building a culture of collaboration among teachers.” I agree. What makes it hard to build that culture, however, is precisely the age-graded school, a structure that separates teachers and makes each classroom a small kingdom. Of course, there are age-graded elementary schools where a culture of cooperation does exist and in secondary school math departments. Too often, they tend to be transient, slowly dissolving as key people–teachers and principal– exit.There is nothing magical about a non-graded school’s structure, or teaming, or multi-age groupings. But a structure that makes cooperation harder such as the age-graded school can be changed into one that helps rather than hinders collaboration. It is creating the conditions for cooperation that I speak of. David K. Cohen’s research in California in the late 1980s and Jim Spillane’s work in Michigan which you cite get at the thorny problems of classroom implementation and teacher beliefs that complicate further the influence of the age-graded school.

Being schooled in the post Sputnik era, all of us headed for college took science and math classes every year. Senior year Physics and Trigonometry. Being a “word person” I never understood what mathematics was about. I took it because my counselor advised me to. And I could learn to do the problems “to get the right answer” but could never have explained “why”! Indeed it wasn’t until my third graduate school program where my Statitics teacher’s text included a set of dice which we rolled 100 times to create our very own set of numbers to use in the work that I made the connection I needed for decades: 2x + 2y = 2(xy) is a compound sentence without an exclamation point!

Thanks, Ann, for detailing your experiences in learning math and how things came together for you in grad school.

Hi Ann.

The math teacher in me could not overlook your compound sentence analogy, at least the algebraic portion, which should state:

2x + 2y = 2(x+y).

I omitted the exclamation point, too.

More importantly, thanks for sharing your excitement in making your connection. I love it when students see the light. It’s a shame it took so long for that concept to be understood, which underscores the shortcomings of the trade-offs we make in our public education system in educating millions upon millions of students each year.

BTW, feel free to point out any grammatical or other linguistic mistakes I may have made in my comment, here, or below.

Dave

Hi Larry,

I am in the process of reading Elizabeth’s book and was delighted to hear your thoughts on it.

Can you think of any learning situations (outside of the formal school structure), where students from different age groups learn together? I can think of the Samba schools in Brazil.

Also, you mention that the age-graded school “prevents a large fraction of students from achieving academically.” Would you mind highlighting how the age-graded schools prevent students from learning?

Kunal

Thanks, Kunal, for your comments and questions. Learning situations within formal school structures would be those scattered non-graded elementary schools–many of which began in the 1970s–that do exist now. Informal schools that you ask about also have multi-age groupings. Summer “Freedom Schools” occurring in 2014 come to mind quickly. As for how age-graded schools preventing many students from achieving academically, I refer to the structural practice within age-graded schools of promoting and retaining students at the end of the school year for not learning–as measured by test scores–what had to be learned, say, at the end of the first or third or sixth grade. Kids learn at different rates. You do not need to have a Ph.D. in psychology to see that fact in front of your eyes. All you need is to be a parent of more than one child. Thus to be labeled a failure–built into the age-graded school structure–when learning is more often than not a function of time, not I.Q has a harmful effect on children and youth. That is what I meant.

Reblogged this on Reflections of a Second-career Math Teacher and commented:

Ms. Green’s NYT article strikes several chords with me, Larry, geometry pun intended.

The first follows.

“In fact, efforts to introduce a better way of teaching math stretch back to the 1800s. The story is the same every time: a big, excited push, followed by mass confusion and then a return to conventional practices.”

Sadly, history tends to repeat itself in these efforts constrained in some sense by time-honored notions of a proper “set point” for developing mathematical proficiency. In this way, our national educational system, at least with respect to mathematics instruction, maintains a state of balance, or social equilibrium. As you’ve pointed out in earlier posts, implementation is a key element for successful, systemic change. Every earlier effort to change mathematics instruction across our nation appears to have failed miserably in this critical phase.

Dave

Thanks for your comments, Dave, for both my post and what Ann had to say. i appreciate your re-blogging the post for your readers.

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