Harel Barzilai on...

Calculus Reform at Cornell:

Why Reform?

(I) Why Reform?

[student comments from course evaluations:]
"The course had too [many] topics for one semester. It was rushed and too hard to keep up. ... Having [the third prelim] a week before final is ridiculous." (Spring '93)

"There is too much that we are supposed to learn. There is no way the TA can be expected to teach the material effectively, considering the amount." (Spring '93)

"I think that the course is a bit overloaded. The course focuses on a lot of material in _very_ little depth. I would much prefer less material but in more depth." (Fall 92)

"[The] course went much too fast for a second semester calculus course" (Fall 1992)

"The material was extremely rushed" (Fall 1992)

Putting thing positively rather than negatively:
"It would be nice to step back from the syllabus and really see what we were doing. Need a broader overview" (Fall 1992)


  • Too many topics; too little depth, both (necessarily) depth we (instructors) are able to cover given number of topics, and depth students are able to absorb/comprehend (no matter how much depth we lecture on) given number of topics.


    (112, but more generally, problems with current teaching in mathematics, and more generally still, problems with current teaching:)

    Students focus on "how do you do number 3?", not concepts. (it's a race/sprint to the ?!?finish?!?)

    Students retain little at the end of the course. Remember very little the day (or week) after the Final

    Topics are isolated -- mathematics not seen as integrated

    Mathematics seems "impossible to understand" for many

    Mathematics seems "artificial" ("seems real" ==> "depth") not having to do with the 'Real World'

    Mathematics not seen a *creative* human endeavor ("creative exploration" ==> spending time on "depth")

    ("human" ==> communicating ideas to, learning from, one's peers)

    Bottom Line:

    They grow up with mistrust of, fear of,
    lack of:
    understanding of the importance (and real-world relevance) of
    lack of feeling for fun/creativity of, mathematics;

    bad associations overall with mathematics.

    Our Bottom Line: Grown-up citizen ex-calculus students:
    "why should I fund mathematics research (or even mathematics teaching (disillusionment)) anyway???"

    (II) Where do we go from here?

     /"in-depth"       \
    |                   |        =         
    |"natural"          |        ===       Projects
    |                   | ============
    |                   | =============
    |"fun"/"creative"   | ============      Cooperative Learning
    |                   |        ===      ("Collaborative Learning") 
    |"a human activity  |        =
    Projects:		New Mexico State; others
    Cooperative Learning:	Myrtle Lewin - Visiting Prof
    Here at Cornell, I tried group-learning (also "oral exams")


    Surface area of a pond (used in a STD calc text) within a project:

  • God does not come along and hand you a function

  • The mathematics "abstractions" *are* useful

  • Numerical integration as *useful* and *natural* technique to use.

    Medical dosage:

  • The mathematics "abstractions" *are* useful

  • Infinite Series as *useful* and *natural* technique to use

    Vectors & Sailing:

  • The mathematics "abstractions" *are* useful

  • Vectors are *natural* not "just made up by mathematicians"

  • Mathematics: helps us understand the natural processes around us

  • Fun!

    Quotes on Course Evaluations on Cooperative Learning:

    "I really thought that the collaborative learning ?projects? was a beneficial experience and was extremely useful. The 112 czar in the spring should try to incorporate this tool into the curriculum as best he can"(Fall '92)

    "The group work was very helpful but there was much too much material covered [in the non-projects-based version of Math 112] to properly comprehend it. Very rushed" (Fall '92)

    "Group work was very helpful and made class interesting" (Fall '92)

    "Working in groups was very helpful because we were able to practice our skills"(Spring '93)

    "Group work with other students was helpful" (Spring '93)

    "If we had more time, the group sessions would have benefited more" (Spring '93)

    To me personally:

    "I really learned a lot by explaining to others"

    "I learned so much by working at the black-board with my group as a whole"


    "You learn by teaching"

    Passive Listening << Active Doing << Communicating-->Teaching

    "You learn by communicating" "You learn by communicating" You "OWN IT" (versus are given it from above, "garnish & all" (TV dinner))


    CREATIVE (versus fixed laws handed down)

    Creative ---> "*I* explore" ---> "Active engagement"

    "Let's go back to nature!" -- or at least learn from nature:

    How do human beings naturally learn? (This philosophy used in teaching-reform for language-classes, e.g.: the "Rassias Method" at Dartmouth College)

    Projects <--> Cooperative Learning:

  • In depth ==> often too big to handle alone within time limits
  • Integrating several parts into cohesive whole

    "Oral Exams" (my excuse to personally tutor everyone):

    "Oral exams was a great help...working one on one made a definite difference ___ing my knowledge of the material." (Spring '93)

    I believe the most helpful source of learning for me was the oral exams...they were an excellent idea" (Spring '93)

    "oral exams were very helpful" (Spring '93)

    What can we learned from "oral exams" that we can use for calculus reform? ...

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