22M:006 Logic of Arithmetic
Expanded Syllabus
This document contains an expanded form of the syllabus with additional details filled in
concerning the various topics to be covered.
The goal of this course is to ensure that students in it work on developing a deep
understanding of the mathematics of number systems, place value, arithmetic operations in a
variety of algorithms, and the mathematical and pedagogical relations between these algorithms.
Designed with prospective elementary school teachers in mind, wherever possible the
mathematics will focus on mathematics underlying elementary school contexts and applied
settings.
The class will meet twice per week in lecture format, and twice per week in discussion
section format. It is a three credit-hour course.
The intent of covering multiple algorithms for the various arithmetic operations in the
natural numbers, and the examination of those operations in bases besides base 10, is to have
students think about and articulate the differences and similarities in mathematically valid but
different approaches to mathematical problems. Discussing the pedagogical differences and/or
learning issues between those methods is also important and should be undertaken.
Some of the motivation for undertaking these kinds of explorations comes from
documents such as Adding It Up: Helping Children Learn Mathematics by the National Research
Council http://www.nap.edu/books/0309069955/html/ and The Mathematical Education of
Teachers by The Conference Board of Mathematical Sciences
http://www.cbmsweb.org/MET_Document/index.htm. Those documents argue that students
learn and use differing mathematical thinking processes and algorithms for computation, and that
teachers should be flexible and able enough to evaluate the mathematical soundness of varying
approaches (not necessarily ‘on the spot’, but with some exploration which may afford an
opportunity to learn for the teacher and the class).
Student errors in using standard and other algorithms also make for rich discussion and
exploration materials. Examples of such errors are cited in the sources below and include for
example misalignment of numbers and/or columns, subtraction or division in an inappropriate
order, mistaken notions of magnitude, etc.
Many other algorithmic approaches are undoubtedly available. We list some as examples
for possible use in the course, but instructors and their students are encouraged to find and
investigate additional ones.
The exploration of the various algorithms might take place as in-class and/or discussion
section activities with continuations as homework problems, or as lecture format material. For
example, an in-class activity might consist of having groups of students investigate two different
algorithms, with the goal of writing or explaining how those two compare.
The extension of the arithmetic operations and algorithms to other bases and to the
positive rationals and integers provides additional rich sources for students to undertake
mathematical investigations. Additionally work on understanding the arithmetic operations in
other bases can simulate for prospective teachers the type of learning processes their future
students (i.e. grade school children) may undergo when learning those operations in the natural
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numbers. Consequently exploration of extensions to other bases is a valuable activity for
prospective teachers.
Course Text
Mathematics For Elementary Teachers: A
Contemporary Approach, Sixth Edition
Gary L. Musser, Oregon State University
William F. Burger
Blake E. Peterson, Brigham Young University
ISBN: 0-471-16425-9
©2003
Course Grading:
Graded homework and/or in-class assignments, projects, activities
Mid-term tests or quizzes- up to 3 tests/assessments planned
Discussion section grade-participation and graded work
Final Exam given during finals week
25%
25%
25%
25%
Concise outline of syllabus:
Weeks 1-2: Introductions, pattern recognition, place value, models of natural numbers N.
Text chapters 1, 2
Weeks 3-7: Conceptual and multiple algorithmic study of arithmetic operations in N
Text chapters 3, 4
Weeks 8-12: Conceptual models of the positive rational numbers and extensions of operations,
decimals, irrationals, percents and proportions
Text chapters 6, 7 and 8
Weeks 13-14: Conceptual models and extensions of arithmetic operations to integers Z
Text chapter 8
Week 15: Prime numbers, LCM, GCD, applications Text chapter 5
Detailed outline of syllabus:
We use the following abbreviations for references below:
AIU for Adding It Up: Helping Children Learn Mathematics
MBP for Mathematics For Elementary Teachers: A Contemporary Approach, Sixth Edition by
Musser, Burger and Peterson (also referred to as “Text” below)
LDT for Mathematical Reasoning for Elementary Teachers (Second Edition)
by Long and DeTemple
Week #1
Introduction to goals and methods of course, expectations for student work and assessments.
Problem solving and patterns recognition. Patterns in natural numbers and numerical calculations, figurate numbers
(e.g. triangular numbers, pentagonal, Fibonacci sequences etc.). Text chapter 1.
Week #2
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Continuation-numerical patterns. Natural/whole numbers as counting numbers, the number line model, counting
number name ideas- linguistic differences and learning issues (cf. AIU, pp. 163-168, 236, 411-412). Place value
systems, other bases e.g. computers for base 2, grocery items for base 6 or 12 (e.g. six packs and twelve packs of
soda), abacus-type and clock arithmetic. K-6 devices and manipulatives: Cuisenaire rods, base ten number cubes,
mats, strips, units, packing blocks, abacus, calculator, etc. Text chapters 1 and 2.
Week #3
Place value systems continued. Conceptual and contextual models of addition. Multiple algorithms and their
mathematical and pedagogical relations-extensions to other bases. Text chapters 3, 4 and additional materials.
Some Addition algorithms:
1. The ‘common’ U.S. algorithm (MBP p. 154)
2. Regrouping (“carrying”) on the bottom as taught in Chinese schools and some American schools (AIU page 202,
method B)
3. Expanded notation (AIU p. 202, method C, MBP p. 154)
4. ‘Scratch mark’ method (LDT, p. 198 #20, MBP p. 164 #4)
5. Abacus types of methods (LDT pp. 169-170, 175-176 # 23 -29)
6. Lattice method (MBP p. 155).
Week #4
Conceptual and contextual models of subtraction in natural numbers. Multiple algorithms and their mathematical
and pedagogical relations-extensions to other bases. Student errors may include subtraction of a minuend digit from
the subtrahend digit below it when the latter digit is larger than the former. Text chapters 3, 4 and additional
materials.
Some Subtraction algorithms:
1. The ‘common’ U.S. algorithm (MBP p. 156)
2. Various methods and notations for regrouping/exchanging (“borrowing”), (LDT, p. 193, AIU p. 205)
3. Equal-additions algorithm, adding a number to both the minuend and subtrahend to make the column by column
subtractions possible (MBP pp. 157, 168 # 12).
4. Cashier’s algorithm (MBP p. 168 #11)
Week #5
Review and assessment on patterns, place value, natural numbers, conceptual models and
multiple algorithmic approaches to addition and subtraction
Conceptual and contextual models of multiplication in natural numbers. Multiple algorithms and their mathematical
and pedagogical relations-extensions to other bases. Student errors for investigation may include incorrect alignment
of digits when standard algorithm is used (leaving out most of the zeros).
Text Chapters 3, 4 and additional materials.
Some Multiplication algorithms:
1. The ‘common’ U.S. algorithm (MBP pp. 159)
2. Expanded notation (LDT p. 202, MBP p. 168)
3. Egyptian or duplation system (LDT p. 208 #16, MBP p. 169 # 19)
4. Russian peasant method (LDT p. 208 #17, MBP p. 169 # 18)
5. The ‘lattice’ algorithm (LDT p. 209 #18, MBP p. 159)
6. The ‘area’ or array method (LDT p. 202, MBP 110).
Week #6
Completion of multiplication algorithms and related questions.
Conceptual and contextual models of division in natural numbers. The division algorithm, multiple algorithms and
their mathematical and pedagogical relations-extensions to other bases. Questions about division by zero. Text
Chapters 3, 4 and additional materials.
Some Division algorithms:
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1. The ‘common’ U.S. algorithm (MBP p. 162)
2. Expanded or repeated subtraction algorithm method (LDT p. 204, MBP p. 161, AIU p. 211)
3. Scaffold method (LDT p. 204, MBP pp. 161-162)
4. A ‘short’ algorithm (LDT p. 206).
Week #7
Completion of natural number division algorithms and related questions. Summary and review of various arithmetic
operations and algorithm techniques, their uses and relations. Text Chapters 3, 4 and additional materials.
Week #8
Review and assessment on conceptual models and multiple arithmetic algorithms for
natural numbers
Conceptual and contextual development of fractions (rational numbers). Extensions of the arithmetic operations and
algorithms to fractions (rational numbers) multiple representations. Orderings and comparisons. Text Chapter 6.
Week #9
Extensions of the arithmetic operations and algorithms to rationals and multiple representations.
Decimals as place values systems, orderings and comparisons. Interpretations and meanings of repeating decimals as
rational numbers. Text Chapters 6 and 7.
Week #10
Decimals, percents and proportions, the finite decimal number system ‘FD’, i.e. the subset of the real numbers
consisting of those reals with finite decimal expansions; this is the ‘home’ of calculator arithmetic. FD is a dense
subset of the rational numbers and the real numbers, and is closed under addition, subtraction, multiplication but not
division, cf. AIU pp. 88-90, this is a logical place to discuss interpretations of calculator answers to arithmetic
problems and the potential rounding errors. Text Chapter 7.
Week #11
Completion of rationals , decimals, irrationals (as numbers with non-terminating, non-repeating decimal expansions
and in contexts such as diagonal length of a unit square), percents and proportions. Applications to
probability/statistics types of problems.
Week #12
Review and assessment on rational numbers, decimals and comparisons, arithmetic
operations and algorithms
Completion of decimals, percents and proportions. Applications to probability/statistics types of problems. These
may include basic problems on probabilities of outcomes from repeated or multiple coin flipping (Bernoulli trials),
games of chance, and “Monty Hall” type problems (conditional probability).
Text Chapters 7.
Week #13
Conceptual and contextual development of integers in multiple representations e.g. number line models, positive and
negative valued counters or chips, money problems – bills and checks, below zero temperatures, basement floors of
buildings, etc. Extensions of the arithmetic operations and algorithms to integers. Text chapter 8
Week #14
Extensions of the arithmetic operations and algorithms to integers. Text Chapters 5, 4
Week #15
Time-permitting: Prime numbers, prime factorization, GCD, LCM and applications to periodicity problems such as
calendar dates, rotation numbers for wheels, gears etc. of differing sizes, etc. Additional topics may include multiple
methods of estimation and mental math. Text chapter 5.
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Late homeworks will not be accepted unless there is a documented medical or religious excuse.
Make-up exams will be given only if there is a documented medical or religious excuse.
Disability?
I need to hear from anyone who has a disability which may require some modification of seating,
testing or other class requirements so that appropriate arrangements may be made. Please see me
after class or during my office hours.
Complaints about Faculty or T.A.s
Please see the additional documents available from the Schedule of Courses. In summary, first
see the person you wish to complain about, then see his or her immediate supervisor. The chain
of command is: graduate or undergraduate assistants, then Prof. Seaman, then the Math
Department's Executive Officer, David Manderscheid (phone: 335-0714).
Student Academic Misconduct (cheating)
Please see the information available in the Schedule of Courses.
This course is given by the College of Liberal Arts. This means that class
policies on matters such as requirements, grading, and sanctions for
academic dishonesty are governed by the College of Liberal Arts. Students
wishing to add or drop this course after the official deadline must
receive the approval of the Dean of the College of Liberal Arts. Details of the
University policy of cross enrollments may be found at:
http://www.uiowa.edu/~provost/deos/crossenroll.doc
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