Construction delays

Prospective math teacher “John Dewey” got an “A” in math education from his professor, Mr. NCTM, despite their disagreements on constructivism.
Dewey thinks students could figure out a formula for the sum of interior angles of a convex polygon in five to 10 minutes with guidance from the teacher or 45 minutes with the professor’s approach.

Given a choice between giving students 45 minutes to reach an “aha” experience, or 5 to 10 minutes, I and others like me opt for the latter.

Dewey thinks his classmates — including the diehard constructivist — will shift to his views over time.

See Dave Marain’s newish blog, Math Notations, for more.

About Joanne


  1. Dewey thinks his classmates — including the diehard constructivist — will shift to his views over time.

    Not if all he has to rely on is sweet reason.

  2. Imagine the following activity, Joanne. You guess the grade where this was actually demonstrated. Also, how would you characterize this lesson? Traditional? Reform? Constructivist? Discovery-based? Active lesrning? Ah, labels!

    Children working in pairs draw, with a ruler: a triangle, quadrilateral, pentagon and hexagon.
    The instructor tells each group to take their scissors and cut the three corners of the triangle and rearrange the pieces to form a straight angle. The teacher is demonstratng this on the overhead with color transparency film or with an opaque projector. This activity is repeated for the other figures, except the teacher doesn’t tell them how many straight angles can be formed. She tells htem to make as many as possible with the pieces, but pieces cannot be used more htan once. The children are told to record their findings in a data table, which is also demosntrated by the teacher.
    # of sides # of vertices (corners) # of straight angles formed
    3 3 1
    4 4 ??
    5 5 ??
    6 6 ??

    The teacher is circulating, assessing, guiding, asking many questions… She asks the students to predict how many straight angles could be formed from cutting the vertices of a decagon. She doesn’t tell them what a decagon is — someone in the class will make an educated guess and she’ll guide them by relating the prefix deca- to common words.
    She will then ask each group to formulate a rule in words for the relationship between the numbers of sides and the number of straight angles. After 10-15 minutes, groups volunteer to discuss and display their findings. The teacher is asking many questions of varied levels of taxonomy.

    Later she has hte children combine pairs of straight angles to form ‘circles’ and begins to formulate hte numerical version of this important rule.

    So, what grade level? Could it be introduced as early as 4th or 5th? Should all children have had similar experiecnces BEFORE taking high school geometry? Is this lesson far too time-consuming just to get at a simple algebraic formula 180(n – 2)? Personally, I have seen lessons like these at the middle school level, but, in other countries, younger children (4th grade) have these kinds of experiences. My Korean students told me so! If you would like to read further views like this, pls visit my blog ( I will repeat this comment there and expand on it…
    Dave Marain