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Visions of Mathematics: What Content Should We Teach?

CJ Westerberg, July 29, 2014 7:59 PM

visionsof math.jpg

Most Recent Update: The Case Against Algebra 2

What Content Should We Teach?

". . . .So I am glad that there are educators out there
questioning the conventional wisdom . . ."

                               -- Ben Daley, CEO and CAO, High Tech High

by Ben Daley

At a meeting of math teachers last week, I heard the question, "What is our vision for mathematics at High Tech High?" It is my belief that there is quite a bit of agreement on this topic. Here, I outline the skills and dispositions we want our students to acquire and the approach to math that we are taking.

But first, a digression on why we don't teach math the way I learned it.

"Everyone" knows how to teach mathematics. First, you take the smart kids and put them together in a room, because they need to be challenged. Then, you take the dumb kids and put them in a different room, because they need more support. Then you put the average kids in another room because they don"t belong in the first two rooms.

Now that the students are sorted appropriately, everyone knows how the class goes. First, a short lecture on section 1.1. Now, work an example problem on the board. Next, assign problems one through thirty-five odd. Students start to work in class. Any they don't finish they can do for homework. Tomorrow go over the homework. Short lecture on section 1.2. Test on Friday.

Interestingly, how class goes is exactly the same for the smart kids, the dumb kids, and the average kids. Just more problems and faster pace for the smart kids. Fewer problems and slower pace for the dumb kids.

Since everyone knows how math is supposed to be taught, and since, indeed, this is how math is taught through the United States, what are the outcomes of our approach?

1964
"The first systematic cross-national assessment of mathematical competencies was conducted in 1964 and included 13- and 17-year-olds from 12 industrialized nations. The results of this study indicated that American adolescents were among the most poorly educated mathematics students in the industrialized world." (Geary, 1996)

2003
"In their most recent lackluster showing on the world stage, students in the United States scored below average in mathematics literacy and problem solving in an international comparison." (Cavanagh & Robelen, 2004)

2009
"Students in the United States show little distinction compared with most other countries in reading, mathematics, or science at any grade level or age - and, according to one international test, are near the bottom of the pack in math." (Zehr, 2009)

Of course, international comparisons should be taken with a grain of salt. Harvard professor Daniel Koretz offers a fascinating discussion of such comparisons in his book, Measuring Up: What Educational Testing Really Tells Us (2008). Nonetheless, there seems to be a disconnect between how everyone knows that math should be taught and the results of that teaching. So I am glad that there are educators out there questioning the conventional wisdom on how math should be taught, even as we know that we don't have all the answers.

Who Is Our Ideal Graduate?

In The Global Achievement Gap (2009), Tony Wagner, outlines seven survival skills for the new economy.  (Ed. links here and here)  These are:

Critical thinking and problem solving
Collaboration and leadership
Agility and adaptability
Initiative and entrepreneurialism
Effective oral and written communication
Accessing and analyzing information
Curiosity and imagination

These strike me as worthwhile goals both for our students in general and for our students in terms of their mathematics knowledge and skills. To read more about the seven survival skills, read Tony's article entitled "Rigor Redefined" (2008).

What Content Should We Teach?

HighTechHigh.Math.jpg

Analyzing data - 1/3

College physics math - 1/3
 
SAT prep math - 1/6

Math as Art - 1/6


The diagram represents a proposed framework for the types of math taught at High Tech High. Of course, at different grade levels, students are working on different levels of math, but these represent the overall goals for allocation of time.

Analyzing data: As one parent, who is a UCSD economist, pointed out to me a few years ago, learning how to analyze data well is one of the most practical skills we could teach our students. Consider how a politician can misrepresent reality by manipulating data. And think about all the people who analyze data for a living. To prepare our students well for citizenship and productive work in a democracy, we should probably be doing a better job of teaching them how to analyze data.

College physics math: We never want to hear that a student has gone off to college to take freshman physics or calculus and feels like they were not well prepared in high school. No matter what we do, we need to be sure that students are getting the targeted, specific math that is needed to do well in college physics (and math). This could involve dropping many things that are traditionally taught in high school math classes but are unrelated to this goal.

SAT prep math: The SAT is a reality for our students, more important to them than the California standards tests. We want to prepare our students to do well on the SAT, whatever we may think of the value of this test.

Math as Art: All the math that we learn is not only to prepare us for college, career, and citizenship. Math can be a beautiful thing just for its own sake. I know I always loved math in high school just because it had a simplicity and elegance about it, and when I got the problem right, no one could disagree with me.


F. Scott Fitzgerald wrote, "The test of a first-rate intelligence is the ability to hold two opposing ideas in mind at the same time and still function." When teaching the math that we think is most important, we are pulled in multiple directions. We can follow a vision such as the one outlined above. Or we can follow the state standards and attempt to prepare students for the end-of-year multiple-choice tests. Or somehow we have to attempt to do both as well as we can and still keep functioning.

###





Ben Daley is the chief operating officer and chief academic officer for High Tech High  He acts as an advisor to fifteen high school students and teaches and advises students in the HTH Graduate School of Education. Ben joined High Tech High to teach physics as a founding faculty member in fall 2000.  He was the second director of
High Tech High Original Recipe.


####

For the full version of this article, describing methods used in HTH math classrooms,
and other writings by Ben Daley, visit

http://dp.hightechhigh.org/~bdaley/writing or purchase his book, Screeds on Schooling, at the HTH bookstore: http://www.hightechhigh.org/books

References

Cavanagh, S. & E. Robelen (2004).
U.S. Students Fare Poorly in International Math
Comparison. Education Week, December 7, 2004.

Geary, D. (1996).
International differences in mathematical achievement: their nature,
causes, and consequences. American Psychological Society. Retrieved from
http://web.missouri.edu/~gearyd/International%20Differences%20Math.pdf

Koretz, D. (2008).
Measuring Up: What Educational Testing Really Tells Us. Cambridge:
Harvard University Press.  (Ed Note:  link and article here)

Wagner, T. (2008).
Rigor Redefined. Educational Leadership, 66, 2, 20-25.

Wagner, T. (2009).
The Global Achievement Gap. Basic Books: New York.

Zehr, M. (2009).
U.S. Running with Pack on International Tests. Education Week,
August 26, 2009.

###

Bio:  "A New Hampshire native, Ben Daley wisely moved to San Diego, CA at the first opportunity. As a student at Haverford College, Ben majored in physics and was credentialed in secondary physics and math. After graduation, he traveled to the Philippines and taught science and math at an international school in Manila. Upon his return to the U.S., he taught physics and AP physics at the Madeira School, a girls boarding school in suburban Washington, D.C. He then moved to California to coach basketball and to teach physics at Pomona and Pitzer Colleges. He earned an M.A. in science education at the University of California, Santa Barbara".

Published in The Daily Riff October 2010.  Originally published in Spring 2010 UnBoxed, A Journal of Adult Learning in Schools found on the High Tech High website

Other related stories:
 "Person of the Year in Education"  High Tech High's Larry Rosenstock

"Is This the Best High School in America?"

"Why Our Kids Don't Get Math"

"An Amazing Time For Math In This Country"


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