Math 195/196 – Mathematics for Elementary Education I/II

Course Coordinator

Susan Johnson

Catalog Description

MATH 195: Mathematics for Elementary Education I

(2-2) Cr. 3. F.S.

Prereq: Satisfactory performance on placement assessment, 2 years high school algebra, 1 year of high school geometry, enrollment in elementary education or early childhood education
Whole number operations through analysis of properties, theoretical and hands-on models, mathematical analysis of elementary students’ thinking; standard and non-standard algorithms; structure of the decimal system; linear measurement; two- and three-dimensional measurement, shapes and spatial sense; number theory; algebra as it relates to elementary curricula/teaching profession. Students in the College of Liberal Arts and Sciences may not count MATH 195 toward General Education Requirements.

MATH 196: Mathematics for Elementary Education II

(2-2) Cr. 3. F.S.

Prereq: Minimum of C- in MATH 195 and enrollment in elementary education or early childhood education.
Integer, fraction and decimal operations through analysis of properties, theoretical and hands-on models, mathematical analysis of elementary students’ thinking; standard and non-standard algorithms; continuation of two- and three-dimensional measurement, shapes and spatial sense; probability and statistics; proportional reasoning; algebra as it relates to elementary curricula/teaching profession.


Math 195 Course Packet (for Math 195)


Math 196 Course Packet (for Math 196)


Mathematics for Elementary Teachers, 5th ed. (for both Math 195/196)


ISBN: 9780134423401 (e-book, 180 day subscription)

ISBN: 1323785280 (custom book)


Mathematics for Elementary Teachers, 5th ed. (for both Math 195/196)

Activity Manual




ISBN: 9780986615115

Syllabus for Math 195

The course focuses on the whole number system, numeration, algorithms and interpretations for whole number computation, topics from number theory, algebra, geometric shapes, and measurement. We will cover portions of chapters 1, 3, 4, 6, 8, 9, 10, 11, & 13 of the text and accompanying activities from the Activity Manual.

Syllabus for Math 196

The course focuses on systems of rational numbers (fractions and decimals) and integers, as well as physical representations, theoretical models and computations, algebraic reasoning, percent and proportional reasoning, surface area and volume, data analysis, and probability. We will cover portions of Chapters 2, 3, 5, 6, 7, 8, 12, 13, 15, 16 and additional material.

Objectives for Math 195

This course targets the mathematics subject matter specialization standard of the Iowa State Teacher Education Standards. It is designed to help you understand the central concepts, tools of inquiry, and structure of mathematics and prepare you to create learning experiences that make these elements meaningful for elementary students. At the end of this course, you will have both content and process knowledge. You will have experienced what it means to think mathematically, understand the value of conceptual insight, and appreciate how mathematical knowledge is constructed in an exploratory manner.

Unit 1

  • Use manipulatives with a non-base 10 place value numeration system
  • Convert symbols and pictures in the Alphabitia numeration system
  • Explore the Alphabitia numeration system in an effort to uncover the key features of our own numeration system
    • role of a zero symbol
    • how place value changes as we move right or left one place
    • how the number of symbols used is related to the base of the system
  • Become familiar with the Iowa Common Core’s Standards of Mathematical Practice
  • Discover more about our base 10 system by looking at other base systems
  • Examine 100s charts, think about patterns and counting by 10s
  • Count in base 5 & 6 (what comes next? what comes before?)
  • Understand 1:1 correspondence, cardinality and subitizing concepts
  • Determine what comes next, what comes before, in a variety of bases
  • Convert between base b and base 10
  • Investigate challenges and solutions to difficulties that arise from bases >10
  • Convert non-base 10 numbers to base 10
  • Use exponential expanded form to represent numbers
  • Use a number line (that reflects the structure of the decimal system) to model and explain rounding decisions
  • Categorize +, – story problems
    • Part-part-whole, add to, take away, or compare
    • Result unknown, start unknown or change unknown (for add to & take away types only)
    • Discrete or continuous quantities
  • Write /critique exemplars of each +/- story problem type
    • Clear cut, simple and age appropriate
    • When appropriate, use tenses and/or adverbs to clearly indicate order of actions (via words like “then”, “next”, “more”, “now”, “yesterday”, “this morning”, etc).
  • Recognize/demonstrate /use associative, commutative prop. of addition
  • Recognize/demonstrate/use learning paths for addition
  • Add/subtract in different bases, staying in base
  • Know/demonstrate the relationship between + and –
  • Draw strip diagrams for addition/subtraction problems
  • Know/demonstrate ways to lighten the load for learning single digit addition facts (addition chart)
  • Use the equal sign correctly to write an equation or series of equations
  • Know/recognize/demonstrate, and for Level 3, write equations for, procedures children use to perform single-digit subtraction.
  • Subtract in different bases, staying in base
  • Write equations that correspond to a method of calculation for addition and subtraction problems
  • Use number lines to find the difference between two numbers
  • Analyze methods of addition and subtraction other than the common algorithms
  • Add/subtract multi-digit numbers in base b
  • Use the lattice algorithm to add in base b or 10
  • Use the partial sums algorithm to add in base b or 10
  • Explain the common (standard) addition and subtraction algorithms in terms of bundled objects/place values, paying special attention to regrouping.
  • Make sense of and explain why alternative addition and subtraction algorithms give correct answers.
  • Experience solving unfamiliar problems
  • Become familiar with a variety of solution methods
  • Explain a solution to a given problem using two different strategies

Unit 2

  • Perform and understand notation for set operations
  • Draw, interpret and use Venn Diagrams to represent information, perform set operations and solve word problems
  • Know and understand the basic meaning of multiplication
  • Write and solve repeated addition, array, area, ordered pair, and multiplicative comparison situations for story problems.
  • Use arrays, organized lists, tree diagrams, strip diagrams and number lines to exhibit multiplicative structure
  • Interpret an expression for a product in terms of number of groups and number of items in a group
  • Use and identify the commutative and associative properties of multiplication
  • Write an expression that uses the operations and properties to describe a given picture/diagram.
  • Draw a picture/diagram to represent a given expression.
  • Know and apply order of operations to evaluate expressions
  • Identify and use the distributive property in a context and/or a set of equations
  • Write a story problem and draw an area diagram/array to correspond to a given expression involving the distributive property.
  • Use the distributive property to lighten the load for learning single-digit multiplication facts
  • Write equations and draw an area diagram/array corresponding to a method of calculation
  • Write/identify “how many groups” and “how many in each group” types of division problems
  • Write the corresponding multiplication (and addition) equations for division problems with and without remainders
  • Write or identify story problems in which the quotient is rounded up or down.
  • Explain the validity (or lack thereof) of 0/a, a/0 and 0/0 types of division problems.
  • Write the corresponding multiplication (and addition) equations for division problems with and without remainders
  • Understand and write equations (including generalized division algorithm form) that correspond to student-generated methods of division.
  • Explain scaffold division algorithm using “how many groups” analogy, including context situations
  • Identify, explain, and correct student errors in standard and non standard division algorithms
  • Explain standard division algorithm using “how many in each group” analogy, including how interpretation of number changes within the method
  • Identify, explain, and correct student errors in standard and non standard division algorithms
  • Use a geometry (area) argument to describe factors and multiples.
  • Write, identify and solve word problems involving multiples and factors
  • Use and understand terms: factor, multiples, prime, composite.
  • Use the Sieve of Eratosthenes to find prime numbers, and explain why this works
  • Use and understand rationale for an efficient method for determining if a number is prime

Unit 3

  • Use a factor tree to find the prime factorization of a number
  • Explain and use Number of Factors Theorem and observations from factor table activity
  • Find the GCF and LCM of two or more numbers, using both the list method and the prime factorization method.
  • Identify, solve and write story problems involving GCF and LCM
  • Define even and odd numbers
  • Use algebra and pictures to prove conjectures about even or odd numbers (numeric examples are not enough to prove something is true)
  • Know and explain the rationale behind divisibility tests for 2, 3, 4, 5, 9, and 10 using:
    • manipulative diagrams,
    • place value arguments, and
    • division algorithm proofs
  • Understand that a specific example is not a proof of all cases
  • Know, use and explain rationale for divisibility tests for 4, 5, 9 and 10
  • Construct combined divisibility tests
  • Know vocabulary: variable, expression, equation, formula
  • Write expressions that correspond to a design or pattern, and reflect the meaning of multiplication
  • Write numeric and algebraic equations that correspond to sums of different types of numbers
  • Use algebraic, geometric and numerical methods to justify formulas for adding sequences of numbers.
  • Flexibly use the formulas for sums of consecutive numbers and the sums of odd numbers to solve a variety of problems.
  • Formulate and flexibly use equations and/or geometric diagrams arising from a scenario.
  • Write equations to represent related quantities; write story problems to represent such equations
  • Distinguish between situations such as “three times as much as” and “three times more than”
  • Solve equations using number sense: annotated pictures
  • Solve equations algebraically and show correspondence to pan balance.
  • Connect features of a linear function with features of corresponding graph (intercept, slope) and table of values
  • Determine if a given sequence is arithmetic, geometric, or neither
  • Write successive terms and formula for arithmetic sequences
  • Given a series of figures or table of values, write the formula that arises from an arithmetic sequence
  • Know geometry terms from text 10.1 and 10.2
  • Know and use the parallel postulate
  • Understand why the formula for the sum of the measures of the angles in a polygon makes sense
  • Compute the sum of the measures of the angles in a polygon, as well as the individual measures of angles in a regular polygon
  • Solve angle problems involving parallel lines and/or polygons

Unit 4

  • Understand/use the definition of circle and sphere and solve related distance problems
  • Know definitions of quadrilaterals: rhombus, parallelogram, trapezoid, rectangle, square
  • Compare and contrast characteristics of quadrilaterals and show these relationships with Venn diagrams
  • Know, justify and use formula for number of diagonals in a polygon
  • Find the midpoint and slope of specific and general (using algebra) line segments
  • Prove a figure is a parallelogram by using general coordinates and computing slopes
  • Describe why measuring length with a ruler can be challenging for some students.
  • Explain why we multiply or divide when we convert units
  • Know assigned units in both the US customary & the metric systems
  • Convert among/between US customary and metric system units using dimensional analysis
  • Know terms related to different types of polyhedral and their parts
  • Visualize a named or general polyhedron to determine its characteristics, such as number of vertices, faces and edges
  • Know how Euler’s formula relates the number of vertices, faces and edges of any convex polyhedron
  • Draw a pattern for a named polyhedron, cylinder or cone
  • Name the polyhedron, cylinder or cone associated with a given pattern or net
  • Know definition of a platonic solid
  • Know names & characteristics of all platonic solids
  • Understand why only a certain number of platonic solids exist

Objectives for Math 196

Unit 1

  • Understand the definition of fraction and the role of numerator and denominator
  • Given the fractional amount of a quantity, represent the unit/ whole or another fractional amount for:
    • Area models (pattern blocks, rectangular models)
    • Discrete models
    • Length models (number line, bar models)
  • From the wording of a problem, identify the “whole” or “unit” for each given fraction, and understand how the “whole” may change in a problem
  • Use fractions to compare quantities
  • Draw pictures to depict equivalent fractions and explain their relationship to the calculation used to find them.
  • Connect procedures for finding common denominators and simplifying fractions to the meaning of fractions
  • Solve problems involving fractions with unlike denominators
  • Use multiple methods to find a fraction between two numbers
  • Find decimal equivalency of a fraction
  • Flexibly understand meaning of numerator, denominator in a way that broadens the definition of a fraction
  • Use and explain the reasoning behind several methods of comparing fractions
  • Know/use the following methods to solve percent problems (finding %, part (or portion), and whole):
    • pictures
    • percent equations
      • equivalent fractions
    • percent tables
  • Know/use the following methods to solve percent problems (finding %, part, or whole):
    • pictures
    • percent equations (using decimal equivalents or % notation)
      • equivalent fractions
    • percent tables
  • Solve percent increase or decrease problems using
    • equations,
    • percent tables*, and
    • strip diagrams*
  • Recognize the “whole” when solving percent increase or percent decrease problems
  • Understand nuances of language used in percent problems
    • When comparing two quantities to each other
      • % of
      • % as much/many as
    • When comparing a difference in two quantities to one of the quantities
      • % increase/% more than/% profit,
      • % decrease/% less than/% loss/% discount
    • When comparing percentages directly:
      • percentage point increase
  • Explain the logic behind the procedure for converting between mixed numbers and improper fractions and draw a picture to motivate that understanding.
  • Use several methods to add/subtract mixed numbers and identify student errors involved in these operations.
  • Use expanded form to express decimals as a sum of fractions
  • Analyze story problems involving fractions to determine whether or not they can be solved by adding the given fractions directly.
  • Identify/write fraction story problems of compare, add-to, take-away and part-part-whole problem types.
  • Understand and model the addition /subtraction of fractions as combining/separating like parts that refer to the same whole.
  • Model integer addition as “combining sets” with +/- manipulatives and corresponding equations
  • Model integer subtraction as “take away” with +/- manipulatives and corresponding equations
  • Use number lines to model integer addition (as combining sets) and integer subtraction (as take away)
  • Explain and model subtraction as comparison using a vertical number line.

Unit 2

  • Apply the meaning of multiplication (as defined by author) to write and analyze fraction story problems
  • Model proper fraction multiplication using:
    • Area models
      • fraction circles
      • rectangular area model (make the connection between it and the standard algorithm for multiplying fractions)
    • Length model/number line (if time)
    • Discrete model (if time)
  • Use the following models to show improper fraction multiplication
    • fraction circles
    • rectangular area (showing connection to std alg)
    • rectangular area (showing connection to FOIL)
    • number line
    • discrete (if time)
  • Model decimal multiplication with base 10 manipulatives and/or decimal squares
  • Explain why the procedure for multiplying decimals makes sense mathematically
  • Know and justify exponent rules for
    • multiplication
    • division
    • raising to a power
  • Represent numbers with powers and scientific notation
  • Use manipulatives to model multiplying negative numbers, when possible
  • Understand and justify the rules for multiplying negative numbers:
    • For (-) (-)=(+), explain with:
      • Pattern method
      • Substitution for Zero method (using the distributive property)
  • Interpret remainders from “How many groups?” fraction division problems correctly
  • Write “How many groups?” fraction division story problems and see connection to whole number division problems
  • Draw pictures to solve “How many groups?” fraction division problems
  • Understand the connection between invert and multiply procedure and solving “how many groups/” fraction division problems using a picture
  • Identify and write “How many in one group?” fraction division story problems, (and recognize them as ratio problems also)
  • Understand the connection between the invert and multiply procedure and solving “how many in one group?” fraction division problems using a picture
  • Distinguish fraction division story problems from fraction multiplication story problems (distinguish dividing in half from dividing by one half.)
  • Use double number lines to solve “how many in 1 group?” fraction division problems and explain how this is related to the “invert and multiply” algorithm.
  • Write story problems involving decimals using both perspectives of division;
  • Justify why we can divide two decimals by shifting the decimal point of both the dividend and divisor the same number of decimal points by:
    • Using equivalent fractions/ratios
    • Using a picture
  • Solve ratio and proportion problems in multiple ways
    • ratio table
    • double number line
    • cross multiplying with a proportion
    • equivalent fractions
  • Use strip diagrams to solve ratio problems and explain their use
  • Identify situations that can’t be solved with a proportion
  • Solve problems involving inversely proportional quantities using logical reasoning

Unit 3

  • Be familiar with how the various number systems (counting numbers, the whole numbers, the integers, the rational numbers, and the real numbers) are related
  • Give examples of problems that can be formulated within a number system but require a larger number system to solve.
  • Convert fractions to decimals using long division and understand what the resulting reminders tell you about the fraction’s decimal representation and about whether it will terminate or repeat
  • Justify why the prime factorization of the denominator of a simplified fraction allows us to determine if a fraction will terminate or repeat.
  • Given a terminating or repeating decimal, use algebraic equations to write it as a fraction (Know the value of 0.999… )
  • Use a decimal square to “see” a fraction as a decimal
  • Solve story problems
    • using strip diagrams,
    • using algebraic equations, and
    • identifying connections between the two methods.
  • Justify the formula for the area of a rectangle
  • Use moving and additivity principles to solve area problems
  • Understand a variety of ways to develop the formula for the area of a triangle
  • Given any base of a triangle, draw its corresponding altitude and use it to compute area
  • Use appropriate units for area
  • Understand a method of developing the formula for the area of a parallelogram
  • Understand several ways to develop the formula for the area of a trapezoid
  • Investigate the relationship between the circumference and diameter of a circle
  • Develop a rationale for the formula of the area of a circle
  • Develop a proof for the Pythagorean Theorem and use the theorem to solve problems
  • Determine if a square of given area can be made on a geoboard and explain why or why not.
  • Know that while perimeter and area are not directly related, knowing the perimeter gives insight into the maximum area of a shape.
  • Draw a net for a solid and use it to find the solid’s surface area
  • Understand the rationale for and use the formula for the volume of prisms and cylinders
  • Understand how volume can be calculated using repeated fillings of water.
  • Relate formulas for the volume of a prism to the volume of a pyramid that has the same base and the same height, or of a cone that has the same base and height as a cylinder
  • Find volume of various solids, including pyramids and cones
  • Use the scale factor and internal factor methods of solving problems about lengths in similar shapes, and understand conceptual differences between the two methods
  • Solve problems involving area and volume for similar shapes.

Unit 4

  • Understand how sampling design affects the data that is collected
  • Display both categorical and numerical data in a variety of appropriate ways.
  • Recognize mean, median and mode as measures of center and identify which is most appropriate to use for a given data set
  • View the mean of numerical data in two ways:
    • as the “leveling out” of data values and see how this corresponds to the way we calculate it
    • as the “balance point” for data when looking at dot plots and histograms
  • Solve problems involving the mean and median
  • Solve problems involving GPA
  • Compute a weighted course percent grade
  • Understand percentile language
  • Create and interpret box plots
  • Understand how volume can be calculated using repeated fillings of water.
  • Relate formulas for the volume of a prism to the volume of a pyramid that has the same base and the same height, or of a cone that has the same base and height as a cylinder
  • Find volume of various solids, including, prisms, pyramids, cylinders and cones
  • Display both categorical and numerical data in a variety of appropriate ways.
  • If two outcomes of an experiment/situation are equally likely, then their probabilities are equal
  • Given a set of equally likely outcomes to an experiment, the probability of a given event, E, occurring is the number of outcomes satisfying E divided by the total number of outcomes. For equally likely outcomes ONLY: P (2H) = # of ways 2H can occur/# of outcomes possible

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Students With Disabilities

Iowa State University is committed to assuring that all educational activities are free from discrimination and harassment based on disability status. Students requesting accommodations for a documented disability are required to work directly with staff in Student Accessibility Services (SAS) to establish eligibility and learn about related processes before accommodations will be identified. After eligibility is established, SAS staff will create and issue a Notification Letter for each course listing approved reasonable accommodations. This document will be made available to the student and instructor either electronically or in hard-copy every semester. Students and instructors are encouraged to review contents of the Notification Letters as early in the semester as possible to identify a specific, timely plan to deliver/receive the indicated accommodations. Reasonable accommodations are not retroactive in nature and are not intended to be an unfair advantage. Additional information or assistance is available online at, by contacting SAS staff by email at, or by calling 515-294-7220. Student Accessibility Services is a unit in the Dean of Students Office located at 1076 Student Services Building.

More information about disability resources in the Mathematics Department can be found at