M670 Special Topics: Algebra and Logical Reasoning for the Elementary and Middle School Classroom
Course Description:
Graduate Math content class aimed primarily for elementary and middle school teachers providing a broad foundation in the concepts and applications of algebra and associated logical reasoning.
This course explores the elementary and middle school teacher’s use of algebra and logic and the algebra and logic taught across grade levels from K to grade 8.
Students in this course will develop an understanding of the mathematical concepts and processes of algebra used in grades K – 8, and will develop and express those mathematical activities using formal logical statements.
The course promotes the use of algebraic and logical thinking through a combination of exploration, inquiry, and a “total emersion” in an algebra and logic speaking class environment, and through personal practice in the near yearlong class activity project.
In our use the word “algebra” is understood to be a “strand” of mathematics running through all grade levels, and is not restricted to the symbolic manipulation that is the content traditionally known as “Algebra”. For our purposes we follow the idea of algebra expressed by Van de Walle et al.: “Algebraic thinking or algebraic reasoning involves forming generalizations from experiences with number and computation, formalizing these ideas with the use of a meaningful symbol system, and exploring the concepts of pattern and functions.”[1]
Class Meetings:
The meeting times are June 27, 28, 29, & 30 3:30 pm to 8:30 pm.
July 11 8:30 am to 3:30 pm.
There will also be follow-up meetings during the school year.
September 24 8.30-3.30
October 22 8.30-3.30
Additional information and materials available through:
Course Objectives:
- Develop and or reinforce math content knowledge in algebraic and logical reasoning to the point of comfortable use in the classroom
- Examine the latest research on BEST practices in the area of algebraic and logical reasoning.
- Examine the National Council of Teachers of Mathematics [NCTM] Principles and Standards for SchoolMathematics and Curriculum Focal Points and their impact on mathematics education specific to algebra and logical reasoning.
- Examine the Common Core State Standards for Mathematicsand its impact on mathematics education specific to algebra and logical reasoning.
- Develop, teach, reflect upon, and adjust lesson plans appropriate for K-8 mathematics instruction.
- Develop strategies for integrating problem-solving into mathematics lessons.
- Develop strategies for assessing students’ prior learning and ongoing learning in order to inform instructionand enhance students’ learning.
- Develop an understanding of the role of various methods, materials, manipulatives, and technologies inelementary and middle school mathematics curriculum and instruction specific to algebra and logical reasoning.
- Develop fluency in using logical statements with particular emphasis on statements of equality.
- Learn to identify places where the written logical and mathematical statements are left implicit and develop skills at teaching students to make these statements explicit.
- Reflect on mathematics teaching and learning to enhance teaching practices and students’ learning.
- Develop enthusiasmfor teaching and learning mathematics.
Anticipated outcomes for students of our participants:
- higher overall achievement in math scores
- higher achievement in word problem solutions
- diminished achievement gaps in math
- markedly higher achievement in targeted math concepts and principles, all reflected in increased CMT scores
Student LearningOutcomes / Objectives
Teachers will demonstrate that they can:
- Describe the “strand” concept of algebra, noting similarities and differences with traditional “Algebra”.
- Spot patterns and describe the patterns they see.
- Abstract, decontextualize, and build mathematical models (representing situations symbolically), and perform symbolic manipulations.
- Identify activities where the “equals” is read as “and the numerical answer is”.
- Identify activities where the “equals” is read as a statement of balance.
- Grade children’s work in a way that reinforces the balance concept of the equals sign.
- Drop, or almost totally ignore, the “and the numerical answer is” use of equation sign.
- Completely avoid accidently leaving a written equals sign in a false statement.
- Differentiate between, and correctly use, variables and constants (a.k.a. parameters).
- Derive meaningful consequences from an equation with three variables.
- Correctly combine measurement units and statements of equality.
- Perform dimensional analysis and unit analysis to check the plausibility of simple equations.
- Correctly and habitually:state assumptions, present logical arguments, and express connections between algebraic statements of equality using such terms and symbols as: let, = , if, then, set, when, for, therefore, because, so, and, or.
- Correctly manipulate equations, perceiving and thereby expressing the mathematical actions they take, producing written explanations of what they did.
- Correctly change units in an equation, leaving written explanation of what was done.
- Create a glossary of mathematical terms and symbols relating to algebra and logical reasoning.
- Correctly use pronouns, I, you, you, they, we, one to indicate the levels of conformity with accepted mathematical practice.
- Correctly use hedges in teaching mathematics.
- Recall research conclusions on the deleterious effects of knowledge of operational patterns on equation use.
- Work quickly with equations (thereby demonstrating that they are not relying on a conscious application of a mnemonic for the order of operations) identifying the structure of equations in terms of “terms”. For example, in the equation (2+3)/5+1 = 2 x (5-4) – (6 – 4 – 2) seeing:
(2+3) as a single object
(2+3)/5 as a single object
(2+3)/5+1 as two objects
The form of the equation as: T1 + T2 = T3 – T4 , whereTxis a term (x = 1,2,3, etc)
- See, without conscious reference to order of operations, that parentheses, division signs, and multiplication signs (including multiplication implicit in the use of exponents and proximity) all fail to separate an expression into separate terms.
- Add parentheses at will, for example, rewriting the above equation as:
((2+3)/5) + (1) = (2 x (5-4)) – (6-4-2) to make the separate terms more obvious.
- That the so-called “order of operations” does not give the time sequence order in which operationsare performed when working with algebraic expressions.
- Teach without rewarding a child’s knowledge of the mnemonic for order of operations.
- Replace classroom use of PEMDAS (as a mnemonic to associate with the order of operations) with mathematically based ideas.
- Align their classroom instruction with the Common Core State Standards for Mathematics (CCSSM).
Do the following for all algebraic tasks identified in the CCSSMfor all grade levels k-8:
- Perform all algebraic tasks.
- Describe the mathematical abstractions that children must perform.
- Locate and describe the location of algebra concepts using the CCSSM language of Domains & Standards.
- Identify and use the properties of operations in Table 3 of the CCSSM Glossary p90.
- Make explicit the properties of operations (CCSSM p90) that are implicit in our writing of algebraic expressions and in our decimal place value notation.
Do the following for all algebraic tasks identified in the CCSSMfor the grade(s) they teach:
- Describe and provide examples for all algebra strand tasks.
- Identify algebraic thought when expressed in the naïve or emergent language of children.
- Correct common student misconceptions.
- Generate grade level exercises that incorporate the use of statements of equality.
- Generate grade level exercises that incorporate logical reasoning with equations.
- Generate grade level exercises that incorporate measurement units.
- Engage a student in working on a math question with explicit written statements of the logic the student applied.
- Assist a student by deliberately placing one task both in explicit written form and separately in the students’ inner mental actions.
Core Text Books:
1)VANDE WALLE, John A., KARP, Karen S., & BAY-WILLIAMS, Jennifer M. Elementary and Middle School Mathematics: Teaching Developmentally (7th Edition)Ch. 8,9,10,14, & 23
2)BAY-WILLIAMS, Jennifer M., & VAN DE WALLE, John A.Field Experience Guide for Elementary and Middle School Mathematics: Teaching Developmentally
3)FOSNOT Catherine and JACOB Bill. Young Mathematicians at Work: Constructing Algebra
4)Common Core State Standards Initiative (National Governors Associationand Council of Chief State School Officers):Common Core State Standards in Mathematics (CCSSM)
Additional Readings (for Sept 24th essay)
5)WHITEHEAD Alfred North. An Introduction to Mathematics. Chapter 5 (pages 58-70) “The Symbolism of Mathematics”
6)KLINE Morris. Mathematics and the Physical World. Chapter 5 (pages 56-72) “Numbers, Known and Unknown”.
7)ROWLAND Tim (Cambridge). Between the Lines: The Language of Mathematics. Chapter 4 of Children’s Mathematical Thinking in the Primary Years. J Anghileri (Ed.)
8)OWEN Annie (Cambridge). “In Search of the Unknown: A Review of Primary Algebra” Chapter 8 of Children’s Mathematical Thinking in the Primary Years. J Anghileri (Ed.)
9)McNEILNicole and ALIBALI Martha (2005) “Why Won’t You Change Your Mind? Knowledge of Operational Patterns Hinders Learning and Performance on Equations” Child Development, July/August 2005, Volume 76, Number 4, (pages 883-889)
10)BAROODY Arthur and GINSBURG Herbert (1983) “The Effects of Instruction on Children’s Understanding of the “Equals” Sign”. The Elementary School Journal, Volume 84, Number 2
11)From Proc. of a Nat. Symp. on the Nature and Role of Algebra in the K-14 Curriculum:
DOSSEY John.Making Algebra Dynamic and Motivating: A National Challenge.
KAPUT James.Transforming Algebra from an Engine of Inequity to an Engine of Mathematical Power by “Algebrafying” the K-12 Curriculum.
Resource/Reference Books:
12)Curriculum Focal Points: for Prekindergarten though Grade 8 Mathematics, A Quest for Coherence. National Council of Teachers of Mathematics. 2006.
13)Principles and Standards for School Mathematics. NCTM Mathematics. 2000.
14)Your school’s current textbook or classroom mathematics resource material
15)Hass H. (Columbia Uni.) Algebra with Integrity and Reality. Proceedings of a National Symposium on The Nature and Role of Algebra in the K-14 Curriculum. Keynote address
Assignments and contribution to Final Course Grade:
Due July 11th15%
Prepare a glossary of algebraic reasoning terms used in the summer course, the reading materials, and especially the Common Core State Standards for Mathematics (CCSSM). For each term give a brief description and also identify the standard where the mathematical concept is first introduced in the Common Core State Standards for Mathematics (CCSSM).
Due Sept. 24th10%
Write a 5 page essay based on reading assignments listed as “additional readings” 5) to 11). Addresses how teachers can effectively communicate algebraic concepts in the classroom. Emphasize 1) the balance or equivalence concept (rather than the “and the numerical answer is” concept for the equals sign, 2) the use of pronouns, and hedges 3) the use of logic in connecting one statement of equality to another. Incorporate the main theme or finding from each article or chapter and include two direct quotes from each article or chapter. Those with a more abstract bent might also include the Keynote address by Bass, but this is optional.)
Due Oct 22nd 10%
Read the book “Young Mathematicians at Work: Constructing Algebra” by Catherine Fosnot and Bill Jacob. In a five page essay describe what elements of Fosnot and Jocab’s philosophy of teaching math you will incorporate, or are you already using, in your math teaching. Also describe how you can pass on this philosophy to other teachers through your coaching.
Due Feb 10th 201230%
Answer all “Writing to Learn” questions from Van de Walle chapters 8, 9, 10, 14, & 23.
Due May 9th30%
Select 10 “Activities at a Glance” from your grade appropriate “Teaching Student-Centered Mathematics” book by Van de Walle and Lovin. Perform ten activities in class with students keeping a portfolio of student work samples. Submit the portfolio with a 1 page analysis for each of the 10 activities. Include in your analysis what worked (or didn’t) and what mathematical insights the children developed. Given the volume that is most grade appropriate for you, work with the following chapters: Volume one (chapters 2,3,4,&10) or Volume Two (2,3,4,& 10) or Volume Three (chapters 2,5,9,&10).
In class May 9th 5%
MSP Algebraic Reasoning Post Assessment
Course Topics
- Understanding algebra as a K-12 strand (the “new algebra” concept)
- Developmental analysis of Algebra through the grades
- Looking longitudinally through the Common Core State Standards
- Abstraction: always thinking at two or more levels
- Algebraic expression of math teaching
- Higher algebraic analysis of your classroom text
- Teacher to teacher “peer” communication through algebraic symbolism
- Visualizing algebraic concepts and number theory
- Algebraic languages
- Physical
- Diagramed
- Symbolic
- Algebraic terminology
- Problem based algebra teaching
- The use of units with algebraic expressions
- Errors and misconceptions arising from the use of units
- Assessing and selecting algebra problems
- Misreading algebra, common misconceptions
- Understanding and correcting student errors
- Role of talking in learning algebra (student self-expression, student peer discussion, student teacher discussion)
- Status of ideas and the manipulation of ideas (students naively manipulate what is in mind, they do not easily transition to manipulate symbols on paper)
- Resources for the algebra teacher
- Use ofmanipulatives
- Review of word problem books
APPENDIX
Elements of the Common Core State Standards for mathematics to be addressed
Mathematics | Standards for Mathematical Practice:
This course will strongly address practice standards: 1, 2, 3, 4, 5, 7, & 8
Mathematics | Standards for Mathematical Content: K-8
This course will strongly address both the student more concrete tasks and the abstract teacher understanding of content within the following K-8 domains clusters:
Counting and Cardinality:
Compare numbers
Operations and Algebraic Thinking
Represent and solve problems involving addition and subtraction
Understand and apply properties of operations and the relationship between addition and subtraction
Work with addition and subtraction equations
Understand properties of multiplication and the relationship between multiplication and division
Solve problems involving the four operations, and identify and explain patterns in arithmetic
Use the four operations with whole numbers to solve problems
Gain familiarity with factors and multiples
Generate and analyze patterns
Write and interpret numerical expressions
Analyze patterns and relationships
Number and Operations in Base Ten
Work with numbers 11 – 19 to gain foundations for place value
Understand place value
Understand the place value system
Use place value understanding and properties of operations to add and subtract
Use place value understanding and properties of operations to perform multi-digit arithmetic
Generalize place value understanding for multi-digit whole numbers
Perform operations with multi-digit whole numbers and with decimals to hundredths
Number and Operations - Fractions
Develop understanding of fractions as numbers
Extend understanding of fraction equivalence and ordering
Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers
Understand decimal notation for fractions, and compare decimal fractions
Use equivalent fractions as a strategy to add and subtract fractions
Apply and extend previous understandings of multiplication and division to multiply and divide fractions
Ratios and Proportional Relationships
Understand ratio concepts and use ratio reasoning to solve problems
Analyze proportional relationships and use them to solve real-world and mathematical problems
The Number System
Apply and extend previous understandings of multiplication and division to divide fractions by fractions
Apply and extend previous understandings of numbers to the system of rational numbers
Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers
Know that there are numbers that are not rational, and approximate them by rational numbers
Expressions and Equations
Apply and extend previous understandings of arithmetic to algebraic expressions
Reason about and solve one-variable equations and inequalities
Represent and analyze quantitative relationships between dependent and independent variables
Use properties of operations to generate equivalent expressions
Solve real-life and mathematical problems using numerical and algebraic expressions and equations
Work with radicals and integer exponents
Understand the connections between proportional relationships, lines and linear equations
Analyze and solve linear equations and pairs of simultaneous linear equations
Functions
Define, evaluate, and compare functions
Use functions to model relationships between quantities
Measurement and Data
Describe and compare measurable attributes
Measure lengths indirectly and by iterating length units
Measure and estimate lengths in standard units
Relate addition and subtraction to length
Tell and write time
Work with time and money
Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects
Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit
Convert like measurement units within a given measurement system
Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures
Geometric measurement: understand concepts of volume and relate volume to multiplication and addition
Geometry
Graph points on the coordinate plane to solve real-world and mathematical problems
Solve real-world and mathematical problems involving area, surface area and volume
Solve real-world and mathematical problems involving angle measure, area, surface area, and volume
Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres
Understand and apply the Pythagorean Theorem
The CCSS content standard domain of Statistics and Probability and clusters relating to data representation and interpretation will be covered in a cursory way if at all.
Mathematics | Standards for Mathematical Content for High School
This course will address elements of the following domains:
The Real Number System
Quantities
The Complex Number System
Seeing Structure in Expressions
Arithmetic with Polynomials and Rational Expressions
Creating Equations
Reasoning with Equations and Inequalities
Interpreting Functions
Building Functions
Linear and Exponential Models
[1]Van de Walle, J., Karp, K., and Bay-Williams, J. Elementary and Middle School Mathematics; Teaching Developmentally. 7th