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Guidelines for fast teaching of pre-academic mathematics

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  • Shlomi Fish
    This is the first draft of this document. Those that are interested in this matter (and do not think pupils today are taught in a manner that is efficient
    Message 1 of 1 , Jan 7, 2001
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      This is the first draft of this document. Those that are interested in
      this matter (and do not think pupils today are taught in a manner that is
      efficient enough) are welcome to read it, comment or add their own
      material. I hope to post it to Slashdot or something like that one day,
      but meanwhile this forum will do.

      I'll establish a web-page for it soon enough.

      Regards,

      Shlomi Fish


      Algebra
      -------

      Linear Equations should be taught from as early as the first or second grade.
      Throughout their studies, children learn to solve equations such as
      [Square] + 5 = 6, [Square] - 10 = 5, 6 - [Square] = 2, but are not introduced
      to the general solving mechanism presented by algebra. Thus, they are unable
      to solve such equations as 2 * [Square] + 9 = 3 * [Square] + 6. Showing them
      the algebraic way of adding, substracting, multiplying or dividing both sides
      by a certain value, will both ease their lives, and give them important
      mathematical intuition.

      A balance can be used to demonstrate these concepts, with various unmarked
      weights as variables.

      Later on, the students should be introduced to a set of linear equations:
      x + y = 6 and x - y = 4, and further. The methods of adding or subtracting two
      equations should be taught, as well as the method of isolating a variable
      and substituting for it in the other equations. A discussion should be held on
      how to solve an equation set most quickly assuming a certain variable is
      the one requested.

      Second-grade equations should be shown a small time after the power
      and the root are introduced. Students should be explained that the key to
      solving x^2+A*x+B is finding two numbers whose product is B and whose sum is
      A. It should be practiced a little, but the general solution for quadratic
      equation should be presented with its derivation, soon afterwards.

      Geometry
      --------

      Geometry should be taught from the beginning from the Euclidean basics
      while teaching everything that is essentially nowadays taught starting at
      Junior High and High school. The concept of a proof should be emphasized, and
      used. The children should be given problems to prove as homework.

      I believe a geometry proof is more of a directed acyclic graph than a
      sequence. Thus, I think kids may find it easier to write each sequence of
      conclusions on one place on a big sheet (say A3), and connect those sequences
      with arrows. Later on in their education, they can be taught that it should be
      presented as a sequence. But I think it's important to show that mathematical
      thinking is in no way completely sequential.

      Perhaps it's possible to write a computer program that will help students
      arrange their proofs that way, and maybe even verify that the deduction is
      correct. The students should not depend on it though.

      Arithmetics
      -----------

      From early on, children should be introduced to the concept of the zero number
      as indicating "nothing", and negative numbers as indicating numbers on its
      opposite side. A line can be shown with one number written every 10
      centimeters, the zero at the middle and the negative numbers at its left side.

      One half can be introduced as the solution to the equation 2*[Square] = 1.
      Fractions are simply the result of dividing two integers by each other. As is
      the common practice nowadays, splitted circles can be shown in order to
      visualize fractions, but the students should be reminded that a fraction can
      be a part of any quantity. Sometimes it doesn't make sense (like 1/5th of a person, or
      a table) but it always exist.

      I believe children of tender age are able to grasp multi-digit arithmetics, so
      the move from single-digit numbers to two-digit, three-digit and beyond should
      be made as swift as possible. Al-Khuarizmi's algorithms' and the multiplication
      board should be taught of course, but I believe they can be done with by the
      second or third grade.

      The numbers at the right side of the floating digit can be introduced as an
      extension similar in spirit to the negative numbers. Children can be shown that
      100 = 1000/10, 10 = 100/10, 1 = 10/1, and that 0.1 = 1/10 (and 0.01 = 0.1/10
      etc.). Floating point arithmetic is a convenience more than a necessity and should
      be treated as such.

      The connection between floats such as 1/3 or 1/7 and their floating-point
      representation should be shown as soon as students have grasped the concept of floating-point arithmetics. Children should understand that every rational
      number has a repetetive part at the end, but it may not necessarily constitute
      of the full non-full number.





      ----------------------------------------------------------------------
      Shlomi Fish shlomif@...
      Home Page: http://t2.technion.ac.il/~shlomif/
      Home E-mail: shlomif@...

      The prefix "God Said" has the extraordinary logical property of
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