Department of Mathematics

Recommendations for Early Childhood, Elementary Education, and Special Education Majors

 

Department of Mathematics

9 October, 2009

 

Since January 2009, the General Curriculum MTEL examination has a significantly expanded Mathematics subtest (40 questions) which is being scored separately and must be passed for students to pass the examination.  The Mathematical Foundations courses that have been in place here at Westfield State College are entirely compatible with the new “Guidelines for the Preparation of Elementary Teachers” that accompanied this change in MTEL. (Available online at http://www.doe.mass.edu/mtel/MathGuidance.pdf.)  Indeed, we are one of the very few Massachusetts colleges that are not scrambling to completely revamp their mathematics content offerings for future elementary teachers in response to these new changes.

 

We are grateful to all those who continue to provide invaluable advising, wonderful support and appropriate course registration of our pre-service elementary teachers.  We are happy to share this updated version of our recommendations.  They are unchanged from last semester.

 

For Early Childhood, Elementary Education, and Special Education majors, our department recommends the following coursework in mathematics:

 

1)    Math 150 (Foundations:  Mathematical Reasoning) as a first mathematics course taken during the student’s first year.  This is a core course.  It is required for Early Childhood, Elementary Education, and Special Education majors. 

2)    A second mathematics core course which is chosen from among the following:

a)     Math 251  (Foundations:  Geometry).

b)    Math 252  (Foundations:  Probability and Statistics).

c)     Math 253  (Foundations:  Number Systems). 

These courses can be taken in any order following Math 150.  While other mathematics core courses may be appropriate on rare occasions, these courses are recommended. Please note that Math 110 and Math 111 are not recommended.

3)    As many of the Foundations courses that a student can fit into their program. 

a)     From the outset of their development, Math 150, Math 251, Math 252, and Math 253 have been compatible with both the Massachusetts Curriculum Frameworks and the National Council of Teachers of Mathematics Standards for School Mathematics.  They are designed precisely to provide the necessary mathematical preparation for pre-service teachers in a context that nurtures their interest in teaching mathematics.

b)    The courses Math 251, Math 252, and Math 253 are the only mathematics courses that satisfy the Mathematics/Science elective listed under “Additional Arts and Sciences Coursework” listed on p. 99 of the Bulletin.

c)     We realize that not all education students can take four mathematics courses.  However, to have comprehensive coverage of the content strands of the Massachusetts Curriculum Frameworks students must take Math 150, Math 251, Math 252, and Math 253. 

d)    Students must take Math 150, Math 251, Math 252, and Math 253 to have coverage of all of the areas that make up the mathematics portion of the Massachusetts Tests for Educator Licensure (MTEL). 

 

More information about these courses can be found at the URL

http://www.wsc.ma.edu/math/courses.asp

 

Questions, comments, concerns, and/or suggestions are appreciated.  Please direct all such matters to Volker Ecke at vecke@wsc.ma.edu or x5348.

 

 

Math 150                                                        (Revised: 2/23/06)

Foundations: Mathematical Reasoning

 

                                                                                               

Course Description

An introductory course.  Topics include: finding, analyzing, and describing patterns; sets and classification; functions and relations; inductive and deductive reasoning; problem solving; and logic.  Students will develop a conceptual understanding of the course material in a learning environment that models the pedagogical foundations of the Massachusetts Curriculum Frameworks for Mathematics and the National Council of Teachers of Mathematics (NCTM) Standards.

 

Pedagogy

Part of the course objective is to introduce students to some of the teaching pedagogy outlined by the NCTM.

Following the NCTM Standards, the course is designed to educate students to become active participants (rather than passive observers) in mathematical thinking, and to encourage them to educate their future students in the same spirit.  This may be done using some or all of the following approaches:

 

o      Small group work

o      Emphasis on student verbal explanation of problem-solving processes rather than just providing answers.   Verbally explaining or defending solutions can help develop students’ mathematical thinking, as well as articulative skill.          

o      Increasing students’ self-reliance on checking solutions leads to deeper mathematical thinking.  They have to think about whether or not their answers make sense; they have to think about the problem further to devise a way to check the solution; and their understanding and thinking about the problem will be deepened.  If the professor simply says, “That’s right” or “That’s wrong,” thinking about the problem will immediately cease.

o      The Constructivist approach to learning is emphasized:  students discover and build mathematical concepts themselves, rather than just memorizing them without really understanding.    According to schema learning theory, knowledge is acquired in greater depth and is more efficiently retained if it can be connected with the learner’s pre-existing knowledge.  It cannot be assumed that learners will make the connections without help.  The connections of new material to other knowledge structures must be made; and in keeping with the self-reliance issue raised earlier, the connections should be made by the students themselves whenever possible.

o      Using manipulatives as a tool for understanding mathematical concepts and for solving problems.

o      Developing different problem-solving strategies   (estimating, drawing a diagram, discovering patterns, constructing a table, etc.) that are applicable to a wide variety of situations.

 

Other objectives vary from instructor to instructor.  There is a real concern that many students preparing to be elementary school teachers lack basic arithmetic skills that they need to teach their students.  This issue is frequently addressed in the “plus” sections of MA 150.  Another concern is that the attitude they as teachers have about mathematics is likely to carry over into their own classroom; keeping a journal is one way of helping students deal with this issue.

 

Course Objectives

Required topics in MA 150 include:

 

o      Problem solving techniques

o      Patterns

o      Sets and classification

o      Functions and relations

o      Inductive and deductive reasoning

o      Logic

        

Instructional Objectives

Upon completion of the course, students will be able to:

 

o      Appropriately select and apply different problem-solving techniques

o      Clearly articulate the problem-solving method(s) used

o      Utilize mathematical and logical reasoning

o      Construct simple proofs

o      Use basic set theory to describe different  types of sets and their relationships

o      Use functions to describe and solve applied problems

o      Describe mathematical patterns and incorporate them in problem-solving

o      Describe the difference between deductive and inductive reasoning, including the strengths and limits of each

 

Resources

o      Bassarear, T.  Mathematics for Elementary School Teachers, 3rd Ed., Houghton-Mifflin Co. 

o      Bennett and Nelson.  Mathematics for Elementary Teachers, 6th Ed., McGraw Hill Publishing; ISBN: 0-072-53298-X + supplements & manipulative kit.

 

It is strongly recommended that anyone teaching the course for the first time read one or  more of the following three books:

o      Ohanian, S.  Garbage Pizza, Patchwork Quilts, and Math Magic, W. H. Freeman and Co., 1992.

o      Schifter, Deborah (ed.).  Reconstructing Mathematics Education, Teachers College Press, 1993.

o      Ma, Liping.  Knowing and Teaching Elementary Mathematics, Lawrence Erlbaum Associates, Inc., 1999.

 

The following videotape resources are available from the math department:

o      Teaching Math: a Video Library, K-4, Annenberg/CPB Mathematical Science Collection, WGBH, 1995. ( A collection of 26 tapes and 11 DVDs)

o      Challenge in the Classroom, Mathematical Association of America (no date given).

o      Mathematics: Making the Connection, NCTM, 1991.

o      Discovery Workshop, NCTM, 1991.

o      Using Numbers: Real Data in the Classroom, Dale Seymour Productions, 1990.

o      Learning Games, Frog Publications, 1997.

 

 

Math 251                                                                 (Revised: 3/12/09)

Foundations: Geometry

 

 

Course description:

An introductory course on geometry and measurement.  Topics will include: Euclidean geometry, characteristics and properties of 2- and 3-dimensional shapes, topology, symmetry and transformational geometry, the development of measure, and the derivation of measurement formulae.  Students will develop a conceptual understanding of the course material in a learning environment that models the pedagogical foundations of the Massachusetts Curriculum Frameworks for Mathematics and the NCTM Standards.

Prerequisite:  Math 150 or equivalent.

 

Course Objectives:

 

Upon completion of this course students will have learned to:

1. draw and recognize two- and three-dimensional shapes
2. identify the defining properties of two- and three-dimensional shapes
3. employ spatial perception in a variety of contexts
4. understand the notion of dimension
5. be able to create, identify, and characterize tessellations
6. understand and be able to employ the fundamental ideas of transformational geometry (e.g. translations, rotations)
7. understand the notion of measure
8. understand, derive, and utilize measurement formulae
9. be able to identify characteristics, properties, and symmetries of two- and three-dimensional shapes
10. understand and utilize some of the basic elements of topology including Euler characteristic, graphs and networks, knots and surfaces.
11. explore geometry using content specific pedagogical practices and methodologies, including:  dynamic geometry; tessellation software such as Tesselmania and Logo; physical models; classic geometry tools; and the van Hiele levels of geometric thought.

 

Pedagogy

One of the course objectives is to introduce students to some of the teaching pedagogy outlined by the NCTM. Following the NCTM Standards, the course is designed to educate students to become active participants (rather than passive observers) in the mathematical experience, and to encourage them to educate their future students in the same spirit.  This may be done using some or all of the following approaches:

 

o      Using geometry-based manipulatives and technology as tools for understanding mathematical concepts, solving problems, and preparing them to employ these manipulatives effectively in their classrooms in the future.  These manipulatives should include several of the following:  unit cubes, geoboards, pattern blocks, regular polygon templates, tangrams, geometric algebra blocks, models of the conic sections, Legos, Tangles, flexagons, Zome geometry, origami and paper-folding, geometric solids, conic section models, and geostrips.  Appropriate technology may include:  scientific calculators, Geometer’s Sketchpad, Cabri Geometry, Tesselmania, Logo, CAD programs, and choices from the wealth of interactive Java scripts that enable explorations of geometry in real time on the Internet;

o      Small group work;

o      Emphasis on student verbal explanation of problem-solving processes rather than just providing answers; verbally explaining or defending solutions can help develop students’ mathematical thinking, as well as articulation skills; 

o      Increasing students’ self-reliance on checking solutions leads to deeper mathematical thinking.  They have to think about whether or not their answers make sense.  They also have to think about the problem further to devise a way to check the solution.  As a result their understanding and thinking about the problem will be deepened.  If the professor simply says, “That’s right” or “That’s wrong,” thinking about the problem will immediately cease;

o      The Constructivist approach to learning is emphasized:  students discover and build mathematical concepts themselves, rather than just memorizing them without really understanding. According to schema learning theory, knowledge is acquired in greater depth and is more efficiently retained if it can be connected with the learner’s pre-existing knowledge.  It cannot be assumed that learners will make the connections without help.  The connections of new material to other knowledge structures must be made, and in keeping with the self-reliance issue raised earlier, the connections should be made by the students themselves whenever possible;

o      Developing different problem-solving strategies (estimating, drawing a diagram, discovering patterns, constructing a table, etc.) that are applicable to a wide variety of situations.

 

Appropriate Texts

 

Michael Serra, Discovering Geometry:  An Investigative Approach, Key Curriculum Press, 2003.

David Gay, Geometry by Discovery, John Wiley & Sons, 1998.

Phares G. O’Daffer and Stanley R. Clemens, Geometry:  An Investigative Approach, Addison-Wesley Publishing Company, 1992.

L. Christine Kinsey and Teresa E. Moore, Symmetry, Shape and Space:  An Introduction to Mathematics Through Geometry, Key Curriculum Press, 2001.

 

Math 252                                                                 (Revised: 3/12/09)

Foundations: Probability and Statistics

 

 

Course description:

The study of the foundations of Probability and Statistics. Topics will include understanding, constructing, and computing data graphs and numerical summary measures; probability models; and statistical inference. Students will develop a conceptual understanding of the course material in a learning environment that models the pedagogical foundations of the Massachusetts Curriculum Frameworks for Mathematics and the NCTM Standards.  Prerequisite: Math 150 or equivalent

 

Course Objectives:

Upon completion of this course students will have learned about:

    1.   bar graphs/histograms.

    2.   circle graphs (pie charts).

    3.   mean, median, mode.

    4.   stem-and-leaf displays.

    5.   standard deviation.

    6.   box plots.

    7.   line plots.

    8.   randomness.

    9.   probability models.

  10.   probabilities of equally likely outcomes.

  11.   probabilities of events.

  12.   product tables.

  13.   tree diagrams.

  14.   statistical inference: confidence intervals and/or hypothesis testing.

  15.   (optional) simple linear regression.

 

Pedagogy

One of the course objectives is to introduce students to some of the teaching pedagogy outlined by the NCTM. Following the NCTM Standards, the course is designed to educate students to become active participants (rather than passive observers) in the mathematical experience, and to encourage them to educate their future students in the same spirit.  This may be done using some or all of the following activities:

 

M&M Bar Graphs                              Coin Flip (3 flips)

“Fox in Sox” line plots                        Real Words (tree diagram activity)

Horse Race (exploratory game)           “Pokemon” (geometric probability distribution)

Sum of Dice Roll                                Missing Monsters (attribute activity)

Minimum of Dice Roll

 

 

Math 253                                                                 (Revised: 3/12/09)

Foundations: Number Systems

 

 

 

Course description:

An introductory course on number systems.  Topics include the development and properties of various number systems (such as integers, rational, real and complex numbers), as well as operations and different representations in these number systems (e.g. in bases other than 10). Students will develop a conceptual understanding of the course material in a learning environment that models the pedagogical foundations of the Massachusetts Curriculum Frameworks for Mathematics and the NCTM Standards.

Prerequisite:  Math 150 or equivalent.

 

Course Objectives:

 

Upon completion of this course students will have learned:

12. the conceptual, mathematical, and historical development of several important numbers systems including several of the following:  natural numbers, integers, rational numbers, real numbers, complex numbers, transfinite ordinal numbers, transfinite cardinal numbers, and modular number fields
13. different representations of numbers in these number systems, including:  base 10 numeration, representations in bases other than 10, fractions, decimals, and scientific notation
14. operations in these number systems, including:  addition, subtraction, multiplication, division, and exponentiation
15. about key properties in these number systems, including:  divisibility and factorization, associativity, commutativity, distributivity, closure, and density

 

Pedagogy

One of the course objectives is to introduce students to some of the teaching pedagogy outlined by the NCTM. Following the NCTM Standards, the course is designed to educate students to become active participants (rather than passive observers) in the mathematical experience, and to encourage them to educate their future students in the same spirit.  This may be done using some or all of the following approaches:

 

• Using number systems related manipulatives and technology as tools for understanding mathematical concepts, solving problems, and preparing them to employ these manipulatives effectively in their classrooms in the future.  These manipulatives should include several of the following:  unit cubes, pattern blocks, Cuisenaire rods, fraction tiles, number line strips, fraction strips, geoboards, geostrips, and base ten blocks.  Appropriate technology may include:  scientific calculators, scales, and a variety of measurement tools;

o      Small group work;

o      Emphasis on student verbal explanation of problem-solving processes rather than just providing answers; verbally explaining or defending solutions can help develop students’ mathematical thinking, as well as articulation skills;       

o      Increasing students’ self-reliance on checking solutions leads to deeper mathematical thinking.  They have to think about whether or not their answers make sense.  They also have to think about the problem further to devise a way to check the solution.  As a result their understanding and thinking about the problem will be deepened.  If the professor simply says, “That’s right” or “That’s wrong,” thinking about the problem will immediately cease;

o      The Constructivist approach to learning is emphasized:  students discover and build mathematical concepts themselves, rather than just memorizing them without really understanding. According to schema learning theory, knowledge is acquired in greater depth and is more efficiently retained if it can be connected with the learner’s pre-existing knowledge.  It cannot be assumed that learners will make the connections without help.  The connections of new material to other knowledge structures must be made, and in keeping with the self-reliance issue raised earlier, the connections should be made by the students themselves whenever possible;

o      Developing different problem-solving strategies (estimating, drawing a diagram, discovering patterns, constructing a table, etc.) that are applicable to a wide variety of situations.

 

 

 

Math 352                                                                          (Revised: 3/12/09)

Foundations of Teaching Mathematics: PreK-6

 

 

Course description:

Designed to introduce the prospective early childhood, elementary, and special education school teacher to the teaching of mathematics. An activity-based format will be used to create a learning environment that fosters an exploration of the processes of mathematics. Emphasis will be placed on the role and use of manipulatives in a laboratory setting that encourages the development of fundamental concepts in mathematics. Topics may include: the inductive and deductive processes, measurement, graphing, cognitive development theory, the learning cycle, discussion of innovative projects, state and national frameworks, techniques for assessment, number and arithmetic operations, patterns, variables, modeling and geometry. Three contact hours per week, including substantial laboratory/activity time. Prerequisites: Math 150 and one of the Math 25x  mathematics subject matter courses, or permission of instructor  

 

Course Objectives:

The student will use course content and experiences to develop the following understandings and abilities:

1)    An understanding of current trends in mathematics education policy and goals.

2)    An understanding of constructivist learning theory and practices that promote mathematics literacy.

3)    An awareness of the variety of curricular approaches available to elementary mathematics educators, including inquiry, discovery, and interdisciplinary curricula.

4)    An ability to design mathematics lessons and units that are developmentally appropriate and sensitive to the needs, values, and interests of a diverse group of students.

5)    An ability to construct assessment plans that are compatible with teaching goals and methods and that allow for multiple ways of representing knowledge.

6)    An ability to use multimedia technologies to support meaningful learning.

7)    An understanding of the role of reflection in professional development and lifelong learning.

8)    An awareness of organizations and resources that serve the professional development of elementary mathematics teachers.

 

Potential Textbook:

Elementary and Middle School Mathematics—Teaching Developmentally, John A. Van De Walle, 7^th edition, Pearson Allyn and Bacon.

 

Potential Topics:

·     Understand different learning styles.

·     Comparing different teaching styles.

·     Inquiry-based Learning in Mathematics.

·     Planning in The Problem-Based Classroom. (Chapter 5)

·     Building Assessment into Mathematics Instruction. (Chapter 6)

·     Using Technology in Mathematics Instruction. (Chapter 8)

·     Using Manipulatives in Mathematics Instruction.

·     Memorizing versus Understanding, e.g. for the multiplication facts using memorization only versus understanding the use of the distributive property.

·     Differentiated Instruction in the Mathematics Classroom.

·     Teaching Mathematics in the Era of the NCTM Standards and the Massachusetts Curriculum Frameworks, e.g. Content Areas investigated by Grade Level and Teaching Methods. (Parts of Chapters 1, 9 - 23)

 

o      Number Sense (9-14)

o      Algebra and Fractions (15-18)

o      Geometry and Measurement (19-21)

o      Probability and Statistics (22, 23)