What Important Mathematics do

1

• What Important Mathematics do
• Elementary School Teachers
• Need to Know?
• (and)

2

• How Should
• Elementary School Teachers
• Come to Know Mathematics?
• Jim Lewis
• University of Nebraska-Lincoln

3
Math Matters

• A Mathematics Mathematics Education Partnership
  at the
• University of Nebraska-Lincoln
• Ruth Heaton and Jim Lewis

4
Whats the big deal? Cant any good elementary
school teacher teach children elementary school
mathematics?
5
What is so difficult about the preparation of
mathematics teachers?

• Our universities do not adequately prepare
  mathematics teachers for their mathematical needs
  in the school classroom. Most teachers cannot
  bridge the gap between what we teach them in the
  undergraduate curriculum and what they teach in
  schools.
• We have not done nearly enough to help teachers
  understand the essential characteristics of
  mathematics its precision, the ubiquity of
  logical reasoning, and its coherence as a
  discipline.
• The goal is not to help future teachers learn
  mathematics but to make them better teachers.
• H. Wu

6
What is so difficult .?

• For most future elementary school teachers the
  level of need is so basic, that what a
  mathematician might envision as an appropriate
  course is likely to be hopelessly over the heads
  of most of the students.
• Forget what seems like a good thing to do in some
  idealized world, and get real adapt the courses
  to the actual level and needs of the students.
• The mathematics taught should be connected as
  directly as possible to the classroom. This is
  more important, the more abstract and powerful
  the principles are. Teachers cannot be expected
  to make the links on their own.

7

• Get teacher candidates to believe, that
  mathematics is something you think about - that
  validity comes from inner conviction that things
  make sense, that mathematical situations can be
  reasoned about on the basis of a few basic
  principles.
• The goal is to have them develop some flexibility
  in their thinking, to be able to reason about
  elementary mathematics.
• Roger Howe

8
Adding it Up argues that mathematical proficiency
has five strands

• Conceptual understanding
• Comprehension of mathematical concepts,
  operations, and relations
• Procedural fluency
• Skill in carrying out procedures flexibly,
  accurately, efficiently, and appropriately

9
mathematical proficiency

• Strategic competence
• Ability to formulate, represent, and solve
  mathematical problems
• Adaptive reasoning
• Capacity for logical thought, reflection,
  explanation, and justification
• Productive disposition
• Habitual inclination to see mathematics as
  sensible, useful, worthwhile, coupled with a
  belief in diligence and ones on efficacy.

10

• Educating Teachers of Science, Mathematics, and
  Technology
• New Practices for the New Millennium
• A report of the National Research Councils
• Committee on Science and Mathematics
• Teacher Preparation

11
Educating Teachers

• argues
• 1) Many teachers are not adequately prepared to
  teach science and mathematics, in ways that
  bolster student learning and achievement.
• 2) The preparation of teachers does not meet the
  needs of the modern classroom.
• 3) Professional development for teachers may do
  little to enhance teachers content knowledge or
  the techniques and skills they need to teach
  science and mathematics effectively.
• and recommends
• a new partnership between K-12 schools and the
  higher education community designed to ensure
  high-quality teacher education and professional
  development for teachers.

12
The Mathematical Education of Teachers

• Is guided by two general themes
• the intellectual substance in school mathematics
  and
• the special nature of the mathematical knowledge
  needed for teaching.

13
The MET recommends

• Prospective teachers need mathematics courses
  that develop a deep understanding of the
  mathematics they will teach.
• Prospective elementary grade teachers (K-4)
  should take at least 9 semester-hours on
  fundamental ideas of elementary school
  mathematics.

14

• Mathematics courses should
• focus on a thorough development of basic
  mathematical ideas.
• develop careful reasoning and mathematical
  common sense in analyzing conceptual
  relationships and in solving problems.
• develop the habits of mind of a mathematical
  thinker and demonstrate flexible, interactive
  styles of teaching.

15

• The mathematical education of teachers should be
  based on
• partnerships between mathematics and mathematics
  education faculty.
• collaboration between mathematics faculty and
  school mathematics teachers.

16
The Math Matters Vision

• Create a mathematician mathematics educator
  partnership with the goal of improving the
  mathematics education of future elementary school
  teachers
• Link field experiences, pedagogy and mathematics
  instruction
• Create math classes that are both accessible and
  useful for future elementary school teachers

17
MATH MATTERS Timeline

• Project began 01/01/2000 with NSF support.
• AY 2000/2001, first Math Matters cohort (16
  students) participated in a year-long, 18-hour
  block of courses. Each semester had a math class,
  a pedagogy class, and a field experience.
• Significant progress in learning mathematics and
  in developing a positive attitude towards
  teaching math
• Significant growth in potential to be an
  outstanding teacher
• Strong bonding between students and faculty

18
MATH MATTERS Timeline

• Math Matters was repeated with a second cohort
  (16 students) during AY2001/2002
• Findings similar to first cohort
• AY 2002/2003 Two experiments with a
  one-semester, 12 hour block of courses (28
  students, 19 students)
• Fall 2003 The Mathematics Semester begins
• Current program is a one-semester, 10-hour, math,
  pedagogy, field experience program.

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Barriers to a Successful Partnership

• Math expectations seem to overwhelm students in
  Elementary Education
• Student evaluations critical of math faculty
• Type of Course Faculty GPA Students
• Honors class 3.20 1,367
• All faculty courses 3.04 16,693
• Large Lectures 2.88 6,060
• Education Majors 2.48 726

20
Comments from a math class for elementary school
teachers (the course GPA was 2.93)

• This wasn't a course where we learn to teach
  math. Why do we have to explain our answers.
• I did not like getting a 0 on problems that I
  attempted. I could have just of left the problem
  blank then.
• tests are invalid. They ask questions we have
  never seen before. It would help if we knew more
  about the questions on the exams - If examples in
  class were used on the exams.

21
Comments from a math class for elementary school
teachers (the course GPA was 2.93)

• Her way of assessing her class aren't fair.
• there was never partial credit. When 20 people
  drop a class ... there is an obvious problem.
  Note Only 3 of 33 students dropped the class.
• Test materials were not consistent or reliable
  with the material covered in class. Grading was
  very biased.

22
Comments from a Contemporary Math class

• (She) does a good job making the subject matter
  interesting. She always seems very enthusiastic
  about the class and and actual work. More
  teachers should be like her.
• (She) is a great teacher with a love for her
  subject that becomes addictive. It has really
  been my lucky pick to have gotten her as an
  instructor.
• (She) made the class exciting. It is obvious she
  enjoys math and teaching. She was always clear
  in her expectations and directions.

23
Comments from a Contemporary Math class

• This was a very useful class. I also think that
  (she) is a great teacher.
• (She) was one of the best teachers I have had
  here at UNL. She was always available for
  questions!
• This was a very good class. I failed the class
  last semester with a different teacher but (she)
  did a much better job and I am doing great in the
  class!

24
Barriers to a Successful Partnership

• Students took math courses before admission to
  Elementary Education Program
• Math for Elementary Education was often taught by
  graduate students or part-time lecturers
• Cultural differences in how instruction delivered
  and students assessed
• Fall 2000 Undergraduate GPA by Dept.
• Math 2.53 (UNLs lowest)
• Curr Inst 3.64 (among highest)

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Beliefs of Math and El Ed Faculty1
2 3
4Strongly Disagree
Disagree Agree Strongly Agree

• El Ed Math Question
• 1.71 2.12 Stated from Traditional
  Viewpoint
• -------------------------------------------------
  -----------------------------
• 2.00 2.92 Algorithms are best learned through
• repeated drill and practice.
• 1.57 2.55 An advantage of teaching math is that
  there is one correct answer.
• 2.00 3.22 Frequent drills on the basic facts
  are
• essential in order for children to learn
  them.
• 1.83 2.70 Time should be spent practicing
• computational procedures before children
  are expected to understand the
  procedures.
• 2.83 2.14 The use of key words is an effective
  way for children to solve word problems.

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Beliefs of Math and El Ed Faculty1
2 3
4Strongly Disagree
Disagree Agree Strongly Agree

• 3.27 3.07 Stated from Reform Viewpoint
• -----------------------------------------------
  -------------------------
• 3.29 2.25 Teachers should let children work
  from their own assumptions when solving
  problems.
• 3.86 2.78 Mathematics assessment should occur
  every day.
• 2.71 3.40 Leading a class discussion is one of
  the most important skills for a math
  teacher.

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What is the typical ability of a future
elementary school teacher at UNL?

• How would our students do on two famous problems?
• 1) How do you solve the problem 1 ¾ / ½ ?
• Imagine you are teaching division with
  fractions. Teachers need to write story-problems
  to show the application of some particular piece
  of content.
• What would be a good problem for 1¾/½ ?

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• In Knowing and Teaching Elementary Mathematics
  this question was asked of 23 US teachers. Liping
  Ma reports
• 9 of 23 US teachers could perform the calculation
  using a correct algorithm and provide a complete
  answer.
• Only 1 of 23 US teachers could create a valid
  story problem for the computation. It was
  pedagogically problematic in that the answer
  involved 3 ½ children.

29
Surely our teachers and future teachers can do
better?

• In the summer of 2002, Ruth interviewed a group
  of Lincoln Public School teachers who were
  participating in a summer workshop Jim had
  organized.
• 14 of 17 (elem and middle level) LPS teachers
  tested could perform the division of fractions
  algorithm correctly.
• Only 4 out of 17 could create a valid word
  problem.
• 3 of these 4 were middle school teachers
• We also asked our students to work this problem.
• Algorithm Word Problem
• Fall 2003 Students 23 of 25 8 of 25
• Year 2 (early in sem) 10 of 16 6 of 16
• Year 1 (mid year) 12 of 16 10 of 16

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Sample solutions from our class last Fall

• Among the better answers we received were these
• Bobby only ran ½ the distance that he hoped to
  run in practice. If Bobby ran 1 ¾ miles and that
  was only ½ of the amount he wanted to run, what
  was his original goal?
• Bill is running a race that is 1 ¾ miles. The
  director of the race wants to put markers every ½
  mile. How many markers does he need?

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Sample solutions from our class this Fall

• Other solutions included the following
• Suzie has one and ¾ pies. Ed is hungry and
  decides he is going to eat ½ of what Suzie has.
  When Ed is finished, how many halves of the pie
  does Suzie have left?
• Jane has 7 quarters. She wants to put the
  quarters into groups of 2 quarters each. How many
  groups can she make?
• You have 1 ¾ of a pie! That is not enough. You
  want ½ more of what they started with. How much
  more will you need?
• Mom baked 2 pies and divided the pies in
  quarters. After giving away one slice shes left
  with 1 ¾ pies. She then divides each left over
  piece into halves. How many pieces does she have
  now?

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Perimeter and Area

• 2) A student comes to class excited. She tells
  you she has figured out a theory you never told
  the class. She says she has discovered that as
  the perimeter of a closed figure increases, the
  area also increases. She shows you a picture to
  prove what she is doing.

• The square is 4 by 4
• Perimeter 16
• Area 16
• The rectangle is 4 by 8
• Perimeter 24
• Area 32

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• The first day of our Fall 2002 class, we asked
  our (geometry) students to respond to the area
  perimeter question.
• 24 of 32 future elementary teachers believed the
  childs theory was correct and indicated that
  they would congratulate the child.
• Eight of 32 future elementary teachers questioned
  the child's theory about area and perimeter.
• Only six of the eight explained that the theory
  was definitely not true.

34
Mental Math Quiz (Work all problems in your
head.)

• 1) 48 39 2) 113 98
• 3) 14 x 5 x 7 4) is closest to what integer?
• 5) 4 x 249 6) 6(37 63) 18
• 7) .25 x 9 8) 12.03 .4 2.36
• 9) 1/2 1/3 10) 90 of 160
• 11) The sum of the first ten odd positive
  integers
• (1351719) is equal to what integer?
• 12) If you buy items (tax included) at 1.99,
  2.99 and 3.98, the change from a 10 bill would
  be?
• 13) To the nearest dollar, the sale price of a
  dress listed at 49.35 and sold at 25 off is
  _____?

35
Mental Math Quiz (Work all problems in your
head.)

• 14) The area of a square of perimeter 20 is ___?
• The ratio of the area of a circle of radius one
  to that of a circumscribed square region is
  closest to?
• a) .5, b) .6, c) .7, d) .8, e) .9
• 16) The average (arithmetic mean) of 89, 94, 85,
  90, and 97 is _____ ?
• 17) If 4/6 16/x, then x _____ ?
• 18) If 2x 3 25, then x _____ ?
• 19) The square root of 75 is closest to what
  integer?
• 20) To the nearest dollar, a 15 tip on a
  restaurant bill of 79.87 is _____ ?

36
How do UNLs future elementary school teacher do
on the Mental Math Quiz?

• Group Number Median Average
• Correct Correct
• MM Fall 2000 16 15.5 14.8
• GW Fall 2003 21 13 13.1
• LOK Fall 2003 19 12 12.7
• SW Fall 2003 23 12 12.4
• MM Fall 2001 16 12 12.3
• BH Fall 2000 32 12 11.9
• JL Fall 2003 24 12 11.8
• MM Fall 2002 28 12 11.6
• MG Fall 2001 35 12 11.5
• WH Fall 2000 24 11 11.0
• WH Fall 2001 32 10 10.9
• TM Fall 2003 20 10.5 10.4
• Total 290 12 11.9
• LPS Sum Wks. 22 13.5 13.7
• Have already taken the Arithmetic course.

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The Mathematics Semester(For all Elementary
Education majors starting Fall 2003)

• MATH
• Math 300 Number and Number Sense (3 cr)
• PEDAGOGY
• CURR 308 Math Methods (3 cr)
• CURR 351 The Learner Centered Classroom (2 cr)
• FIELD EXPERIENCE
• CURR 297b Professional Practicum Exper. (2 cr)
• (at Roper Elementary School)
• Students are in Roper Elementary School on
  Mondays and Wednesdays (four hours/day)
• Math 300 CURR 308 are taught as a 3-hour block
  on Tuesday and Thursday
• CURR 351 meets at Roper on Wednesdays

38
A look inside Math Matters(and The Mathematics
Semester)

• Early math assignments establish our expectations
• Professional/Reflective Writings
• Curriculum Materials
• Number and Number Sense Items
• Working together as a team
• The Curriculum Project
• Some Geometry Assignments
• Activities at Roper
• Teaching a Math Lesson
• Child Study
• Learning and Teaching Project

39
A look inside Math Matters(and The Mathematics
Semester)

• Early math assignments establish our expectations
• Professional/Reflective Writings
• Curriculum Materials
• Number and Number Sense Items
• Working together as a team
• The Curriculum Project
• Some Geometry Assignments
• Activities at Roper
• Teaching a Math Lesson
• Child Study
• Learning and Teaching Project

40
Fall 2001 The Rice Problem

• (This assignment builds on a NNS problem.)
  Recall our discussion about the game of chess and
  how a humble servant for a generous king invented
  it. The king became fascinated by the game and
  offered the servant gold or jewels in payment,
  but the servant replied that he only wanted
  riceone grain for the first square of the chess
  board, two on the second, four on the third, and
  so on with each square receiving twice as much as
  the previous square. In class we discussed how
  the total amount of rice was 264 grains of rice.
  (To be completely precise, it is this number
  minus one grains of rice.) Suppose it was your
  job to pick up the rice. What might you use to
  collect the rice, a grocery sack, a wheelbarrow,
  or perhaps a Mac truck? Where might you store the
  rice?

41
Fall 2001 A letter from 1st 2nd graders

• Dear Math Professors,
• We are 1st and 2nd graders in Wheeler Central
  Public School in Erickson, Nebraska. We love to
  work with big numbers and have been doing it all
  year! Every time we read something with a big
  number in it we try to write it. Then our teacher
  explains how to write it. We are getting pretty
  good at writing millions and billions!
• We have a problem that we need your help with.
  We were reading amazing Super Mom facts in a
  Kid City magazine. It told how many eggs some
  animals could lay. We came across a number that
  we dont know. It had a 2 and then a 1 followed
  by 105 zeros!! We wrote the number out and it
  stretches clear across our classroom! We know
  about a googol. We looked it up in the
  dictionary. A googol has 100 zeros. Then what do
  you call a number if it has more than 100 zeros?
  Is there a name for it?

42

• Another problem is that we learned about using
  commas in large numbers. In the magazine article
  they used no commas when writing this large
  number. That confused us. Also, if you write a
  googol with 100 zeros, how do you put the
  commas in? It doesnt divide evenly into groups
  of 3 zeros. There will be one left over.
• We appreciate any help you can give us solving
  this big problem. Thank you for your time.
• Sincerely,
• Mrs. Thompsons 1st 2nd graders
• Megan Kansier, Mark Rogers
• Marcus Witt, Ashley Johnson

43
clipping from Kid City magazine

• Apple Of My Eye
• The tiny female apple aphid is a champ
• as an egg-layer. This insect can lay as
• many as 21000000000000
• 00000000000000000000000
• 00000000000000000000000
• 00000000000000000000000
• 00000000000000000000000
• 0000 eggs in 10 months.

44
Fall 2003 Making change

• What is the fewest number of coins that it will
  take to make 43 cents if you have available
  pennies, nickels, dimes, and quarters? After you
  have solved this problem, provide an explanation
  that proves that your answer is correct?
• How does the answer (and the justification)
  change if you only have pennies, dimes, and
  quarters available?

45
A look inside Math Matters(and The Mathematics
Semester)

• Early math assignments establish our expectations
• Professional/Reflective Writings
• Curriculum Materials
• Number and Number Sense Items
• Working together as a team
• The Curriculum Project
• Some Geometry Assignments
• Activities at Roper
• Teaching a Math Lesson
• Child Study
• Learning and Teaching Project

46
Professional/Reflective Writings

• Purpose To make connections within and across
  mathematical, pedagogical, and field experiences
  through writing.

47
Professional/Reflective Writings

• Read What do Math Teachers Need to Be? The
  author is Herb Clemens, a mathematics professor
  at The University of Utah, and the article was
  published in 1991 in Teaching academic subjects
  to diverse learners (pp. 84-96). In this article,
  Herb Clemens lists what he thinks teachers of
  mathematics need to be. After reading his
  article and his meaning and use of these words,
  where does your own practice of teaching
  mathematics stand in relationship to what Clemens
  says mathematics teachers need to be  unafraid,
  reverent, humble, opportunistic, versatile, and
  in control of their math. On p. 92, Clemens
  lists four fundamental questions about
  mathematics teaching that matter to him. If he
  came to your practicum classroom and watched you
  teach a math lesson tomorrow, how would he answer
  his own last question about your practice Can
  this teacher teach it math with conviction, and
  with some feeling for its essence? Explain.

48
Professional/Reflective Writings

• Some educators argue that there is real value in
  teaching children mathematics in diverse,
  heterogeneous classrooms. Some teachers may
  counter this position, contending that it is best
  for children if students are homogeneously
  grouped for mathematics instruction. Pick a
  position in this argument and articulate it in
  writing. State your position and explain why you
  believe what you do. Your reasons for believing
  what you do may come from past teaching and
  learning experiences (your own and others), and
  things that youve read, or learned in other
  courses.

49
Professional/Reflective Writings

• Read "Teaching While Leading a Whole-Class
  Discussion," Chapter 7 from Lamperts book. In
  this chapter Lampert examines problems of
  practice that arise while addressing a whole
  group of students or choosing students to answer
  questions. As you read the chapter find places
  in the chapter where you can relate Lampert's
  writing to your own experiences in the practicum
  setting while teaching math and maybe even other
  subjects. Use quotes from the text that connect
  to your experiences. Explain how and why they
  relate.

50
A look inside Math Matters(and The Mathematics
Semester)

• Early math assignments establish our expectations
• Professional/Reflective Writings
• Curriculum Materials
• Number and Number Sense Items
• Working together as a team
• The Curriculum Project
• Some Geometry Assignments
• Activities at Roper
• Teaching a Math Lesson
• Child Study
• Learning and Teaching Project

51
Curriculum Materials

• Adapting NSF funded curriculum materials
• Schifter, D., Bastable, V., Russel, S.J.
  (1999).
• Number and operations, part I Building a
  system of tens. Parsippany, NJ Dale Seymour.
• Schifter, D., Bastable, V., Russel, S.J.
  (2001).
• Geometry Examining features of shape.
  Parsippany, NJ Dale Seymour.
• Sowder, J. et al. (2000).
• Number and number sense. San Diego State
  University.
• Shapes and measurement. San Diego State
  University.

52
Curriculum Materials

• Other materials used
• Ladsen-Billings, G. (1997). The dreamkeepers
  successful teachers of African American children.
  Sanfrancisco Jossey-Bass.
• Lampert, M. (2001). Teaching problems and the
  problems of teaching. New Haven, CT Yale
  University.
• Charney, R. (1992). Teaching children to care.
  Greenfield, MA Northeast Foundation for
  Children.
• Reys, R., Lindquist, M., Lambdin, D.V., Smith,
  N.V., Suydam, M.N. (2003). Helping children
  learn mathematics. NY John Wiley Sons, Inc.

53
A look inside Math Matters(and The Mathematics
Semester)

• Early math assignments establish our expectations
• Professional/Reflective Writings
• Curriculum Materials
• Number and Number Sense Items
• Working together as a team
• The Curriculum Project
• Some Geometry Assignments
• Activities at Roper
• Teaching a Math Lesson
• Child Study
• Learning and Teaching Project

54
A Typical Weekly Homework All Shook Up

• Five couples met one evening at a local
  restaurant for dinner. Alicia and her husband
  Samuel arrived first. As the others came in some
  shook hands and some did not. No one shook hands
  with his or her own spouse. At the end Alicia
  noted that each of the other 9 people had shaken
  the hands of a different number of people. That
  is, one shook no one's hand, one shook one, one
  shook two, etc., all the way to one who shook
  hands with 8 of the people. How many people did
  Samuel shake hands with?

55
Sample Test Items

• 1) Give an example of one number that you are
  sure is an irrational number. Explain why you
  know that it is irrational.
• 2) Let B 11232. Factor B into a product of
  powers of prime numbers. Then factor B2 into a
  product of powers of prime numbers.
• 3) What is the smallest positive integer with
  exactly 10 factors?
• 4) Is 250 a factor of 10030? Explain.

56
Is 250 a factor of 10030? Explain.

• 8.881784197 E44

57
Is 250 a factor of 10030? Explain.

• 8.881784197 E44

58
A look inside Math Matters(and The Mathematics
Semester)

• Early math assignments establish our expectations
• Professional/Reflective Writings
• Curriculum Materials
• Number and Number Sense Items
• Working together as a team
• The Curriculum Project
• Some Geometry Assignments
• Activities at Roper
• Teaching a Math Lesson
• Child Study
• Learning and Teaching Project

59
"Why is this stuff so hard?"

• I believe this test, this class, this subject,
  are all difficult because they involve thinking
  in different ways than what we are used to. We
  have all been conditioned, in our own education
  to believe that things are the way they are, and
  that's all there is to it. We haven't challenged
  ideas and proofs nearly as much as we should
  have. Asking "Why" to an idea or trying to
  understand the reasoning behind something is just
  not something most of us are used to doing.
  That's why this stuff is hard. Miss A

60
"Why is this stuff so hard?"

• I don't have a difficult time with abstract
  ideas. I love it when we work with new concepts.
  I just want you to know that I have almost
  always been able to figure math problems out and
  I get VERY frustrated when I get stumped. I am
  very stubborn like that. Please don't take my
  temper personally. Miss J

61
"Why is this stuff so hard?"

• The major problem that I had was my reasoning
  for the factoring problem. I started off thinking
  that I should try dividing 250 into 10030, but
  the large numbers were daunting, so I panicked
  and tried using my calculator. The answer it gave
  me did not look pretty, which I think is what
  triggered my fall down a road of insanity (see my
  test for more details). Bad, bad calculators
  ....once you started to explain the problem on
  the board, I wanted to smack myself in the head
  for being so silly. Miss P

62
A look inside Math Matters(and The Mathematics
Semester)

• Early math assignments establish our expectations
• Professional/Reflective Writings
• Curriculum Materials
• Number and Number Sense Items
• Working together as a team
• The Curriculum Project
• Some Geometry Assignments
• Activities at Roper
• Teaching a Math Lesson
• Child Study
• Learning and Teaching Project

63
Curriculum Project

• The goal is to investigate a new mathematical
  area of the elementary curriculum and consider
  what teachers need to know as well as what
  children need to learn the topic in the deep and
  meaningful ways suggested by the NCTM Standards
  (2000).
• 1) Pick a topic data analysis and probability,
  geometry, reasoning and proof, or algebra.
• 2) Read, analyze, and synthesize the mathematical
  topic in the NCTM Standards.
• 3) Analyze and synthesize the topic in a set of
  reform curriculum materials (Everyday Math, Trail
  Blazers, Investigations, local curriculum).
• 4) What do teachers need to know to teach this?
• 5) Create 5 math problems that would help
  teachers learn. Create 5 math problems that would
  help children learn.

64
A look inside Math Matters(and The Mathematics
Semester)

• Early math assignments establish our expectations
• Professional/Reflective Writings
• Curriculum Materials
• Number and Number Sense Items
• Working together as a team
• The Curriculum Project
• Some Geometry Assignments
• Activities at Roper
• Teaching a Math Lesson
• Child Study
• Learning and Teaching Project

65
Geometry Problems

• A) How many different squares are there on a
  checkerboard? If the number of rows and columns
  of the checkerboard is doubled, how does the
  number of squares change?
• B) Let T be the lower left-hand corner of a
  checkerboard and let B be the intersection of
  the 6th vertical line and the 5th horizontal line
  counting from the lower left-hand corner. Suppose
  you want to move from T to B and all movement
  must be horizontal or vertical along lines formed
  by the various squares on the checkerboard. The
  shortest path has length 11 in terms of a unit
  equal to a side of a small square. How many
  different paths have length 11? Can you prove it?
  
• C) Consider a large circle and pick n points on
  the circle. Here n might be 2, 3, 4, and so
  forth. Connect each pair of points with a chord.
  Notice that if n 2 the circle is cut into 2
  regions. If n 3 the circle is cut into 4
  regions and if n 4 the circle is cut into 8
  regions. How many regions do you get if n 6?
  What about n 8?

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A look inside Math Matters(and The Mathematics
Semester)

• Early math assignments establish our expectations
• Professional/Reflective Writings
• Curriculum Materials
• Number and Number Sense Items
• Working together as a team
• The Curriculum Project
• Some Geometry Assignments
• Activities at Roper
• Teaching a Math Lesson
• Child Study
• Learning and Teaching Project

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Math Lesson 1

• You need to teach a math lesson. It should
  connect to the curriculum in the classroom in
  which you are working. You should make use of the
  textbook and other resources from your
  cooperating teacher and what you are learning and
  reading in your courses with Jim and Ruth this
  semester.

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Child Study

• This assignment will give you some experience
  watching, listening to, probing, and assessing
  one childs understanding of several math
  problems focused on a particular area of
  mathematics. You will write a report of your
  interview and suggest instruction for the child
  based on the information you gathered in the
  interview session.

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A look inside Math Matters(and The Mathematics
Semester)

• Early math assignments establish our expectations
• Professional/Reflective Writings
• Curriculum Materials
• Number and Number Sense Items
• Working together as a team
• The Curriculum Project
• Some Geometry Assignments
• Activities at Roper
• Teaching a Math Lesson
• Child Study
• Learning and Teaching Project

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Learning and Teaching Project Area and Perimeter

• The task began with a homework problem. It is
  taken from Reconceptualizing Mathematics
  Courseware for Elementary and Middle School
  Teachers, Center for Research in Mathematics and
  Science Education, 1998.
• Is there a relationship between the area and the
  perimeter of a polygonal shape made with
  congruent square regions? (For fixed area, find
  the minimum and maximum perimeter. For fixed
  perimeter, find the minimum and maximum area.)
  Squares must be joined complete-side to
  complete-side. The outside boundary should be a
  polygon. In particular, this would not permit a
  shape with a hole in the middle.

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A couple of weeks later we told our students

• We want to revisit the Area and Perimeter
  problem. This is to be the basis for a
  mathematics lesson that you will videotape
  yourself teaching to one elementary school
  student.
• How can you present this task to the student you
  will teach? How can you set the stage for the
  student to understand the problem? How far can
  the student go in exploring this problem?
  Remember that you want your student to discover
  as much as possible for himself (or herself). But
  there may be some critical points where you need
  to guide the student over an intellectual bump
  so that he (she) can move on to the next part of
  the problem.
• Finally, produce a report analyzing the
  mathematics and your teaching experience.

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Shapes from Four Triangles

• The task began with a problem on a midterm exam.
  It is taken from Reconceptualizing Mathematics
  Courseware for Elementary and Middle School
  Teachers, Center for Research in Mathematics and
  Science Education, 1998.
• Given four congruent isosceles right triangles,
  how many different polygonal regions can you
  make, using all four triangles each time?

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• Several weeks later we told our students
• We want to revisit the Shapes from Four
  Triangles problem. This is to be both a
  mathematical task for you and the basis for a
  mathematics lesson that you will teach to one
  elementary school student.
• How many different polygonal regions can be made
  using all four isosceles right triangles each
  time? How do you give a mathematical argument
  that you have found a complete set of shapes and
  that you have no duplications?

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• How can you present this task to the student you
  will teach? How can you set the stage for the
  student to understand the problem? How far can
  the student go in exploring this problem?
• Remember that you want your student to discover
  as much as possible for himself (or herself). But
  there may be some critical points where you need
  to guide the student over an intellectual bump
  so that he (she) can move on to the next part of
  the problem.

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Is Math Matters a Success?

• The response from students has been positive.
• Students are engaged in research projects
• Each year several students have sought to
  participate in a 1- 2 year research project
  funded by UNLs UCARE program or NSF-REU funds
• Yr 1 4 students
• Yr 2 4 students
• Yr 3 4 students
• Yr 4 2 students

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• The project evaluator reports
• significant cohesion among the students a
  community of professionals with high standards
• Cooperating teachers working with the elementary
  teacher education program assess current or
  former MM students as better prepared to be a
  teacher than a control group at a comparable
  point in their teacher education program.
• MM graduates are finding jobs. Several indicate
  their preparation to be an outstanding math
  teacher was key to their successful job search.

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What are we learning?

• Integrate content and pedagogy course offerings.
• Keep expectations of students high.
• Emphasize learning how to learn and offer
  continued opportunities.

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• Build on existing relationships.
• Commitment to the partnership need to be long
  term.
• Partnership relationships need to extend beyond
  the relationship of two individuals.

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What do we want to learn from our teacher
education efforts?

• What happens to our students when they leave our
  project and start teaching in their first jobs?
• How do our former students use (or do they use)
  the mathematical knowledge they learned in
  teacher education for teaching? What kinds of
  challenges do they face?

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• What are these new teachers abilities to
• Choose and use mathematical representations?
• Give and evaluate mathematical explanations?
• Choose and use precise mathematical definitions?
• (Ball, Lubienski, and Mewborn, 2001, Handbook
  of Research on Teaching)

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A visit to Carlas 1st Grade Class

• Carla was in our first cohort of Math Matters
  students. She volunteered for the program because
  she considered herself weak in mathematics and
  she was highly motivated to be a good teacher.
• This Spring, Dr. Heaton and I visited Carlas 1st
  Grade Class. The school was located in a
  low-income area of Lincoln.

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Carla was teaching a geometry lesson

• The lesson was masterfully crafted. Time was used
  efficiently. For over an hour the students were
  eagerly engaged in learning mathematics. Carlas
  explanations were clear and her use of
  terminology was careful. Most importantly, the
  classroom was filled with questions
• Why?
• How do you know?
• What does that mean?
• etc.

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• There was lots of encouragement and plenty of
  praise for a job well done. One comment stood
  out. Repeatedly, these 1st grade students were
  rewarded with the encouragement
• You are so smart!
• Thats why you are mathematicians!