Welcome to Math for Teachers!

[picture of Niles]

NOTE: This is the syllabus for a PREVIOUS VERSION of the course. See nilesjohnson.net/teaching.html for current courses.

This is the course homepage and syllabus for Math 1135, Number and Operations for Teachers. Here you will find our course calendar, current assignments, basic course information, and links to additional content of interest. This syllabus may need to be updated as the course progresses, and you will always find the current version at http://www.nilesjohnson.net/teaching/AU14/1135.html.

[children writing numbers]

Excellent math teachers have been far too scarce in American education. This course and its sequel, Math 1136, give you the keys to how and why elementary mathematics works. We will develop a thorough understanding of the ideas, how they are related to eachother, and how they are related to more advanced ideas. This is the basic knowledge which underpins truly great teaching!

Course Objectives

This course covers the concepts of whole numbers (positive and negative), place value (base-ten and alternate bases), decimals, and fractions. Some content on irrational numbers appears at the end, and this is extended in Algebra and coordinate geometry for teachers (2137). The four arithmetic operations are covered both conceptually and algorithmically. Attention is given to ensuring that students can perform the algorithms correctly and explain why they give accurate answers. Lastly, the course covers the concepts of proportions and how they are related both to multiplication/division and to fractions. Factors, divisibility, and some elementary number theory complete the course.


  • Counting numbers, decimals
  • Meaning of fractions
  • Meaning of addition and subtraction
  • Meaning of multiplication
  • Multiplying fractions, decimals, integers
  • Meaning of division
  • Dividing fractions, decimals, integers
  • Meaning of ratios, rates, proportions
  • Greatest common divisor, least common multiple
  • Rational and irrational numbers

Section Information (#32509)

Math 1135 focuses on concepts of number systems and operations, including some algebra and number theory.

Our class meets at the following place and times following the OSU Registrar's academic calendar:

Hopewell 112
Mondays, Wednesdays 3:55—5:15
and Fridays 3:05 – 5:10.

Homework Assignments(+)

Assignment information will be posted here as the semester unfolds. A number such as A.B.c refers to problem or practice exercise c at the end of section A.B.

You only need to turn in the Problems; the Practice Exercises are suggestions for your study, and may be discussed in class. Remember to type your homework and bring a printed copy to class!

Past Homework

Homework 25 due Monday, 12/8

Problems: 8.4.5, 8.5.17, 8.6.13, 8.7.1
Additional problem: Explain the relationship between the numbers in each row of the table in Activity 8N in terms of how the Spirograph flowers are constructed. Use this to complete the table.

Homework 24 due Friday, 12/5

Problems: 8.4.4, 8.5.8, 8.5.9, 8.6.4, 8.6.5
Additional problem: Explain why the square-root of 6 must be irrational. (Note: this will require a new lemma, but the idea is similar to the one we used in class.)

Writing assignment due Monday, 12/1

Participate in our Carmen course discussion on history related to prime numbers. See details there for resources and guidelines.

Homework 23 due Monday, 11/24

Problems: 8.1.6, 8.3.7, 8.3.9

Homework 22 due Friday, 11/21

Problems: 8.1.1, 8.1.2, 8.2.3

Homework 21 due Wednesday, 11/19

Problems: 7.2.14, 7.3.4, 7.3.7, 7.4.2

Homework 20 due Friday, 11/14

Problems: 7.1.8, 7.1.9, 7.2.2, 7.2.12

Homework 19 due Monday, 11/10

Write up and turn in your answers to activities 7A and 7B.
Also do problem 7.1.3 and graph the distances and times in a coordinate plane.

Homework 18 due Friday, 11/7

Study the review sheet and write solutions for extra credit.

Homework 17 due Friday, 10/30

Study activities 6M, 6N, 6O, 6P and be ready to discuss in class.

Homework 16 due Monday, 10/27

Problems: 6.3.5, 6.3.9, 6.3.17
Additional Problem: Explain how to use the scaffold method in base four.

Homework 15 due Friday, 10/24

Problems: 6.1.5, 6.1.8, 6.2.3, 6.2.7

Homework 14 due Monday, 10/20

Problems: 5.2.6, 5.2.10, 5.3.1, 5.4.6, 5.4.8

Homework 13 due Friday, 10/17

Problems: 5.1.1, 5.1.8, 5.1.16

Writing assignment due Wednesday, 10/15

Participate in our Carmen course discussion on the history of place-value arithmetic. See details there for resources and guidelines.

Homework 12 due Wednesday, 10/15

Problems (turn in): 4.6.4, 4.6.5, 4.6.12
Additional: Give a multiplication algorithm for base-four place value, and work a couple of two and three digit examples.

Homework 11 due Friday, 10/10

Study review sheet and turn in written solutions for extra credit.

Homework 10 due Monday, 10/6

Problems (turn in): 4.4.16, 4.4.18, 4.5.5
Additional Problem: Reconsider problem 4.3.6 and explain your expressions in terms of the meaning of multiplication.

Homework 9 due Friday, 10/3

Problems (turn in): 4.3.6, 4.3.14, 4.4.6
Additional Problem: Represent the following numbers in base ten and base four: three; thirty; three hundred; twelve; fourty eight; one hundred ninety two.

Homework 8 due Monday, 9/29

Practice Exercises (do not turn in): 3.5.3, 4.2.1
Problems (turn in): 3.4.4, 3.5.4, 4.2.1, 4.2.2

Homework 7 due Friday, 9/26

Practice Exercises (do not turn in): 3.3.6, 3.4.10
Problems (turn in): 3.3.11, 3.3.12, 3.4.10, 3.4.26

Homework 6 due Monday, 9/22

Practice Exercises (do not turn in): 3.2.7, 3.2.11, 3.3.6
Problems (turn in): 3.2.8, 3.2.11, 3.3.5

Homework 5 due Wednesday, 9/17

Practice Exercises: Read about "math drawings" and problem solving in § 2.1
Problems: 2.4.3, 3.1.5, 3.2.3

Writing assignment due Monday, 9/15

Participate in our Carmen course discussion on the history of zero. See details there for resources and guidelines.

Homework 4 due Monday, 9/15

Practice Exercises (do not turn in): 2.3.8, 2.3.9, 2.4.2
Problems (turn in): 2.3.18, 2.4.8, 2.4.14, 2.5.13

Homework 3 due Friday, 9/12

Practice Exercises (do not turn in): 2.2.1, 2.2.6, 2.3.1
Problems (turn in): 2.2.10, 2.2.14, 2.3.1

Homework 2 due Monday 9/8

Practice Exercises (do not turn in): 1.3.6, 1.3.9
Problems (turn in): 1.2.5, 1.2.7, 1.3.3, 1.4.5
Additional Problem: In class we discussed two methods of expressing numbers in base four (Shelby's and Scotty's methods). Give another example of each, and draw pictures that illustrate the difference between the methods. Click here for a description of the two methods.

Homework 1 due Wednesday 9/3

Practice Exercise 1.1.4
Problems 1.1.1, 1.1.2, 1.1.7, 1.2.1
Additional Problem: Alice was born August 1, 2000, and her cousin Bob is fifteen years older. Write Alice and Bob's ages using base 4 and the digits 0, A, B, C.

[number elves 1-5]

Basic information


Niles Johnson
Office: Hopewell 189
Office Hours: Mondays and Wednesdays 10:00 – 11:30; Thursdays by appointment


Mathematics for Elementary Teachers with Activities, 4th Edition by Sybilla Beckmann. ISBN 0-321-82572-1. The 4th edition (Monkeys on the cover) is substantially improved over earlier editions. Moreover, it is available in electronic or loose-leaf formats which are cheaper and more portable than the bound version.

Exam Schedule

We will have three in-class midterms and a final exam. All students must take the exams at the scheduled times indicated here.

Midterms (3):

Friday Sept. 19, Friday Oct. 10, Friday Nov. 07.


Scheduled by the registrar. See campus schedule for exam times.


Active participation in class is an essential part of this course, and so attendance every day is required. Please let Niles know as soon as possible if an illness or other commitment will prevent you from attending. Homework is due at the beginning of each class, and late homework cannot be accepted unless prior arrangements have been made.

Math Learning Center

The Math Learning Center (Warner 214) is a great resource, and I encourage you make use of it. The staff there has experience with our course and would love to help you!

Calculators and Mathematical Software

There are a variety of modern tools which support mathematics learning and application. We'll use several of them in this course, but no technology will be used on the quizzes or exams. In particular, calculators and cell phones will not be permitted. The only materials you'll need are writing instruments and your mind.

Assessment (+)

Your final grade will be based on written homework, written and oral in-class participation, quizzes, midterms, and a final exam. The precise breakdown is as follows:

  • In-class participation: 12%
  • Homework: 5%
  • Writing Assignments: 4%
  • Quizzes: 14%
  • Midterm exams: 15% each, for a total of 45%
  • Final exam: 20%

The two lowest quiz and homework grades will be dropped. If your final exam score is higher than one of your midterms, then that grade will replace the lowest midterm grade.


Thoughout this semester we will be focused on the how and why of elementary mathematics. This means that you will be responsible both for knowing the content and for knowing how to explain the content. We will practice this in a variety of ways, and much of this practice will take place in class.

Teaching mathematics requires listening carefully to students, assessing their ideas, and responding in ways that make sense. Our class participation is designed to practice these essential skills. You will have opportunities on a daily basis to listen to your fellow classmates explain ideas and ask questions. You will be asked to respond with your ideas and with additional questions. Together we will see how and why mathematics works! Participating in this course includes all of the following:

  • Show interest in mathematical ideas.
  • Show interest in different ways of approaching mathematical ideas.
  • Listen carefully to different ways of solving a problem.
  • Carefully evaluate a proposed method of solution.
  • State whether you agree or (respectfully) disagree with a statement.
  • Show interest in learning with and from others.

Homework and Quizzes

Homework and Quizzes will be scored using the rubric below. Scores are based both on mathematical correctness and quality of explanations. Homework should be typed and presented as you would an essay. If necessary, you may draw diagrams by hand on separate pages. Quizzes will be hand-written during the first 10 or 15 minutes of class.

The homework is intended to give you time to develop your explanations and understanding of the content. You are encouraged to work with your classmates on this, but you must write your own explanations. Homework is designed to help you learn, and not as an assessment tool. Therefore grades for homework will be recorded on a complete/incomplete basis (scores of 3 and higher are complete). The numerical scores are given only for your information.

The weekly quizzes are opportunities to evaluate your current grasp of the material. They will be very short and based on previous homework problems or class activities. Quiz grades will be recored as scored.

Grading Rubric

The descriptions on the rubric below are meant as a guide to help answer the question What constitutes a good explanation of mathematics? This is a subtle and challenging question—one that is worthy of considerable time and energy. As you write your explanations, consider the distinction between Procedure and Conceptual Meaning. A complete explanation addresses both of these, but the conceptual meaning is essential. Explanations that address procedure only are a disservice to students and do not support their future learning. They are also much less interesting!

Quizzes and exams will follow this scale; Niles will use these for comments on homework too, although homework grades are recorded on a complete/incomplete basis. Homework will be considered complete if the average score is R (3) or higher.

Excellent work that exceeds the assignment guidelines.
Good plus (10)
Correct procedure with an explanation that effectively addresses relevant definitions and concepts.
Good (9)
Correct procedure with nearly complete explanation.
Emerging/Good (8)
Well-developed but incomplete explanation.
Procedural errors are minor or nonexistant.
Emerging plus (7)
Emerging explanation that shows understanding.
Procedural errors are minor or nonexistant.
Emerging (6)
Explanation that mentions core definitions or conceptual meaning relevant to the question.
Possibly some non-minor procedural errors.
Relevant/Emerging (5)
Work that has merit but also has significant shortcomings in the procedure and/or explanation.
Relevant effort (3)
Work that shows relevant effort but is seriously flawed.
No credit (0)
No work submitted, or no relevant effort shown.
[number elves 6-0]

Websites on math standards

Websites of elementary school texts from high performing countries

Other websites of interest

  • Project INTERMATH, which focuses on building teachers' mathematical content knowledge through mathematical investigations that are supported by technology.
  • Problems and tasks that Sybilla Beckmann wrote for her 6th graders during the 2004/2005 school year organized according to the grade 6 Georgia Performance Standards.
  • Report by the National Council on Teacher Quality on mathematics in the U.S.: No Common Denominator. The report includes a great appendix on content that mathematics teachers must know, and ranks Beckmann’s textbook highest overall among elementary mathematics content textbooks.

GEC Information

This Mathematics course can be used, depending on your degree program, to satisfy the Quantitative and Logical Skills category of the General Education Requirement (GEC). The goals and learning objectives for this category are:


Courses in quantitative and logical skills develop logical reasoning, including the ability to identify valid arguments, use mathematical models and draw conclusions based on quantitative data.

Learning objectives:

Students comprehend mathematical concepts and methods adequate to construct valid arguments and understand inductive and deductive reasoning, scientific inference and general problem solving.

Disability Statement

Students with disabilities that have been certified by the Office for Disability Services will be appropriately accommodated, and should inform the instructor as soon as possible of their needs. The Office for Disability Services is located in 150 Pomerene Hall, 1760 Neil Avenue; telephone (614) 292-3307 and VRS (614) 429-1334; webpage http://www.ods.ohio-state.edu/

Academic Misconduct Statement

It is the responsibility of the Committee on Academic Misconduct to investigate or establish procedures for the investigation of all reported cases of student academic misconduct. The term "academic misconduct" includes all forms of student academic misconduct wherever committed; illustrated by, but not limited to, cases of plagiarism and dishonest practices in connection with examinations. Instructors shall report all instances of alleged academic misconduct to the committee. For additional information, see the Code of Student Conduct:

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