Read this university professor’s appeal to his academic math colleagues that elementary and secondary school teachers have their students learn arithmetic, algebra, geometry, and calculus through teachers organizing students to work collaboratively when doing math. The professor called it the “laboratory method.”

*The laboratory method has … the flexibility which permits’ students to be handled as individuals or in groups. The instructor utilizes all the experience and insight of the whole body of students. He arranges it so that the students consider that they are studying the subject itself, and not the words, either printed or oral, of any authority on the subject. And in this study they should be in the closest cooperation with one another and with their instructor, who is in a desirable sense one of them and their leader.*

* Instructors may fear that the brighter students will suffer if encouraged to spend time in cooperation with those not so bright. But experience shows that just as every teacher learns by teaching, so even the brightest students will find themselves much the gainers for this co-operation with their colleagues.*

* …[T]he student might be brought into vital relation with the fundamental elements of trigonometry, analytic geometry and the calculus, on condition that the whole treatment in its origin is and in its development remains closely associated with thoroughly concrete phenomena. With the momentum of such practical education in the methods of research in the secondary school, the college students would be ready to proceed rapidly and deeply in any direction in which their personal interests might lead them (1631286).*

E. H. Moore, exiting President of the American Mathematical Society, trained a generation of mathematics professors at the University of Chicago. He was an advocate of tying together both content and pedagogy. He urged that school math lessons cover two instructional periods (rather than one) so teachers and students would have sufficient time to both understand the beauty of math and apply it to their daily lives. He wrote this article in 1903.

Moore’s advocacy of applied math and “laboratory” methods was roundly dismissed by traditionalists of the day. They believed that the essential knowledge of math taught in sequence of separate math subjects algebra, geometry, etc. through lecture, textbook assignments, memorization, and practice was the best way students would learn, understand, and appreciate the beauty of the subject. Controversy over content and pedagogy go back well over a century.

Like the subject of science, math has gone through its curricular and pedagogical wars (see here, here and MathWars). It continues with the roll-out of Common Core math.*

Over the decades, for both advocates of different math content and their critics , curricular reforms in the early 20th century (separate or unified subjects), post-Sputnik reforms at mid-century (the “New Math”), and end-of-century (National Council of Teachers of Mathematics “standards”), cycles of change oscillated back and forth between “traditionalists” who wanted students to acquire essential content knowledge and skills through textbooks by memorizing and practicing algorithms and “reformers” who designed courses and textbooks that integrated math subjects and sought deeper student understanding of content through individual and collaborative inquiry.

What happened in the policy sphere in the last half-century when commissions and state boards published “new” curricular frameworks in math–especially following *A Nation at Risk* (1983)* *and U.S. students’ mediocre scores on international tests– is clear. There is a documentary trail of “traditionalists” fighting “reformers” that even Hansel and Gretel could find their way without resorting to bread crumbs.

Not so, however, for what happened in elementary and secondary classrooms after state officials publicized new math frameworks and approved textbooks. What occurred in classrooms varied greatly across California as well as the nation when the “New Math” was unveiled in the 1960s and when NCTM “standards” hit schools in the 1990s. I expect a similar pattern of wide variation when math teachers implement Common Core. For what happened in California policy and classroom practice between the 1960s and 1990s and elsewhere across the nation, see here, here and REVIEW OF EDUCATIONAL RESEARCH-2005-Remillard-211-46.

Resolving these tensions between “traditionalists’ and “reformers” of the math curriculum and their associated ways of teaching will not occur easily. Perhaps there is a middle ground where partisans for each side can agree on both content and pedagogy, where both sides give a little and join forces to construct a curriculum and pedagogical content knowledge that includes what both sides seek for students. I do not see it yet as the impending Common Core standards in math get implemented in classrooms. For that middle ground to be trod by both sides of the divide in content and teaching of math, there would have to be a coming to grips with the historic division among policymakers, math professors, researchers, parents, and teachers over the primary purpose of elementary and secondary school students studying arithmetic, algebra, geometry, and advanced math subjects.

Is the purpose mathematical literacy for all students because in a democratic and technological society, it is essential? Or is the purpose to prepare students for entering college and getting good jobs? Or is it to produce engineers, scientists, and mathematicians that will keep U.S. secure and economically competitive in a world where other nations can do harm to the country? Or is it because math has been part of Western culture for millennia and every student should know that heritage?

For generations, reformers and traditionalists have waved their flags emblazoned with one or more of these purposes to rally followers. The truth is that, because of limited resources and external events, these flag wavers have had to make hard choices. Choosing among purposes involves making value choices. And conflict erupted repeatedly because of those value-choices.

The struggle over purposes of math often degenerating into “math wars” over what content and which ways of teaching are best have occurred also in history content and pedagogy.

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*Even lyricist and long-time math teacher at the University of California, Santa Cruz, Tom Lehrer got into the New Math wars of the 1960 with his rhyming ditties.

Reblogged this on The Echo Chamber.

Thanks for re-blogging post on math education struggles in U.S.

ITYM “even mathematician and maths lecturer Tom Lehrer”…

Many thanks, Tom, for pointing out that Tom Lehrer was a mathematician and song writer who taught math for many years at University of California, Santa Cruz. I will add that to description.

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Thanks for referring folks to my post on math education.

It’s great stuff. I just wish I had found your blog sooner.

Thanks, Michael.

As a relatively new entrant to the teaching profession, I was surprised at the extent of historical bickering in the U.S. over mathematical content and pedagogy.

As I mentioned in your lead-in article to this series of posts, Larry, I believe the analogy of “labs” from science best unifies procedural skill and conceptual understanding in the teaching and learning of mathematics. While each of these may serve as the central tenet for learning mathematics as espoused by the “traditionalists” on one hand and the “reformers” on the other, why the supporters of one versus the other see the two as mutually exclusive escapes me.

While there certainly may exist different mixtures of each focus (procedural and conceptual) as offered in a rich array of courses, I cannot envision one or the other standing alone effectively, as the troubling persistence of stagnant student test scores over time attest.

Has any research been conducted as to the efficacy of various combinations of each of these pedagogical emphases?

While I need to re-read his paper, “Basic Skills versus Conceptual Understanding,” Hung-Hsi Wu’s subtitle to that paper: “A Bogus Dichotomy in Mathematics Education” best captures my sentiment.

I have not read the piece you mentioned, Dave. The historical dichotomies in math, as you say, are hardly mutually exclusive save for the rhetorical moves of policymakers and ideologues in one camp or the other. When one gets to the classroom, however, I often see mixes of both in practice. I also see that the degree to which math teachers know their math and know how to combine both content and skill into activities, lectures, group work, etc. –that is, convey concepts and algorithms in ways that student can understand–those are teachers who bridge both procedural and conceptual.

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

Excellent overview of of the ongoing struggle on how best to teach mathematics in primary and secondary schools…

Thanks for re-blogging post, Dave.

Here is a link to the article by Hung-Hsi Wu which was mentioned above.

Click to access wu.pdf

Here is a link to a more recent article of his, also in American Educator.

Click to access Wu.pdf

There are many other articles on his webpage.

Click to access Wu.pdf

Thanks for the links to Wu’s work. I did not read his work. Was my loss. Not now.