The learner will:
1. write the eleven (basic) axioms of our number system.
2. describe the key mathematical concept (s) of each axiom .
3. select two axioms and describe why they must be taught to young children.
4. describe the process of "inventing" a deductive system (math) versus
"describing" an
inductive system (science) from the real world .
Components of our place value system
5. name and describe the five components of our place value system.
6. describe and use the system for naming numerals of all sizes.
7. construct the place value system for numerals from one million to one ten-
thousandth,using:
a. actual numerical values;
b. powers of our base 10; and
c. powers of any base.
Set theory
8. define "sets," "elements of a set," "number" and "numeral" as used with young
children.
9. use elements of a set to describe:
a. set equality and inequality;
b. "ordering" of sets by the counting principle;
c. "counting" sets using the one-to-one correspondence;
d. cardinal principle of number; and
e. ordinal principle of number.
Computation with Addition and Subtraction
The learner will:
10. name and describe each of the eleven steps in a recommended model for
teaching
all computational operations .
11. identify and/or create sample lessons for each of the recommended eleven
steps
for teaching computation.
12. state a basic definition for both addition and subtraction, and then
translate that
definition into models which young children can understand and use.
13. discuss activities and techniques for making computational practice more
effective,
including the identification of recommended activities and then completing each
activity
as a child must complete it.
14. construct lessons for explaining regrouping in both addition and
subtraction,
including both manipulative explanations and expanded notation explanations.
15. identify and explain standard errors made by children when performing the
standard algorithm for either addition or subtraction.
16. discuss the following ideas or concepts as they apply to instruction in
addition/subtraction.
a. "guessing" to obtain an answer.
b. "error analysis" as used by the teacher.
c. mathematical definition versus classroom models for subtraction.
d. classroom games used as a drill and practice activity.
e. "sequential teaching" procedures.
f. the "terminal level skill" for any given operation.
Multiplication/Division Objectives
I. The learner will apply the "eleven step" teaching procedure to
multiplication, giving
special attention to the following skills:
17. state two basic models for illustrating multiplication.
18. illustrate each definition with concrete objects.
19. demonstrate symbolic methods for obtaining multiplication answers .
20. discuss procedures for helping children memorize basic multiplication facts.
21. illustrate the process of regrouping in a multiplication problem, using base
ten
blocks and expanded notation.
22. identify, describe, and discuss procedures for teaching the computational
algorithm
to young children.
23. use two alternate algorithms for solving multiplication problems.
II. The learner will apply the " eleven step" teaching procedure to division,
giving special
attention to the following skills:
24. state two different "story problem definitions" for division.
25. write story problems appropriate for both division definitions.
26. illustrate, with base ten blocks, the logic behind the division
computational
algorithm.
27. demonstrate how division can be sequenced for instruction, using the idea of
learning "one new step" in each lesson.
28. demonstrate the use of "estimation rules" in teaching long division.
29. identify the "check points" a child can use in a long division problem to
see if the
work is done correctly.
30. write correct estimation sentences for division problems.
31. The learner will identify "standard errors" made by children while doing
multiplication or division problems.
Fraction Concepts and Computation With Fractions
The learner will:
32. solve fraction computational problems, in computation or story problem
format, for
all four operations.
33. demonstrate an understanding of the value of fractions by:
a. arranging fractions in size relationships
b. using mathematical procedures to compare fraction size.
34. name and illustrate three basic procedures for presenting fractions to
elementary
school children.
35. demonstrate, with either numerals or concrete examples, the following,
two-way
fraction manipulations.
a. converting improper fractions to mixed numbers and back.
b. converting whole numbers to unit fractions and back.
c. expanding and reducing fractions in families.
36. name and illustrate four ways to find common denominators.
37. demonstrate the "regrouping" step in addition or subtraction of mixed
numbers.
38. show how fractional regions can be used to explain multiplication.
39. demonstrate two ways of dividing fractions, namely:
a. the "complete" invert and multiply procedure.
b. the "common denominator" procedure.
40. demonstrate and explain the use of "canceling" in a fraction multiplication
problem.
41. identify the proper order for teaching fraction computation procedures using
the
"sequential teaching" idea.
42. demonstrate ways of showing children the "common-sense logic" of fraction
multiplication and division.
Decimal Objectives
The learner will:
43. solve computation and story problems for each of the four basic operations
involving decimal numerals.
44. demonstrate the extension of the place value system into fractional
numerals.
45. correctly read and write decimal numerals.
46. illustrate the use of place value in arranging decimal numerals for addition
and
subtraction problems.
47. illustrate a multiplication of decimals problem, using fractions, in order
to explain the
movement of the decimal point.
48. illustrate the movement of the decimal point for each of the three possible
division
of decimals problems, and identify the teaching order for the three possible
types.
49. demonstrate the procedure for "rounding off" a decimal to any given place
value
position.
50. describe, in words and with illustrations, the movement of the decimal point
in each
computational situation involving decimals.
Per Cent Objectives
51. state the basic ratio used in all percentage problems.
52. solve each of the basic per cent problems, using both the ratio and an
appropriate
"short-cut" when possible.
53. demonstrate how the short form solution for Type I and II problems can be
obtained
from the basic ratio.
54. solve challenge story problems for each problem type, involving the
following:
a. percentages over 100%.
b. two-step percentage problems. (When the answer to the problem is not the
answer to the question.)
