What do Elementary Teachers Need to Know to Teach Math?Know to Teach Math?

Terry Goodman (University of Central Mi i)Missouri)

Susie Katt (Lincoln Public Schools)Sharon Katt (Nebraska Department of

0011 0010 1010 1101 0001 0100 1011Sharon Katt (Nebraska Department of

Education)Michael Matthews (University of Nebraska -

O h )Omaha)Ira Papick (University of Nebraska-Lincoln)

Janice Rech (University of Nebraska-Omaha)Janice Rech (University of Nebraska Omaha)

0011 0010 1010 1101 0001 0100 1011

Call for Content Intervention0011 0010 1010 1101 0001 0100 1011

►Around the turn of the century, national organizations involved in teacher education calledorganizations involved in teacher education called for a shift, they called for special content courses (Conference Board of Mathematical Sciences(Conference Board of Mathematical Sciences, 2001, and National Research Council, 2001.)

►Specifically they called for teacher education►Specifically, they called for teacher education institutions to bring in three content courses covering the topics:g p Number Sense Geometry and Measurement Algebra, Statistics, and Probability

Nebraska NCLB RequirementBy 2005-06, all teacher education programs in

Nebraska institutions of higher education were to have revised their elementary education preparation

program to include the following: "At least 30 semester hours distributed across the four academic

0011 0010 1010 1101 0001 0100 1011core curriculum areas reflected in the Nebraska K-12 standards. The core academic areas are language arts (reading/writing), math, sciences, and social

studies. Each core area will require a minimum of six credit hours."

0011 0010 1010 1101 0001 0100 10110011 0010 1010 1101 0001 0100 1011

0011 0010 1010 1101 0001 0100 10110011 0010 1010 1101 0001 0100 1011

0011 0010 1010 1101 0001 0100 10110011 0010 1010 1101 0001 0100 1011

0011 0010 1010 1101 0001 0100 10110011 0010 1010 1101 0001 0100 1011

0011 0010 1010 1101 0001 0100 1011

Future Trends0011 0010 1010 1101 0001 0100 1011

• Find ways to include more algebra/probability/ and statistics– Third class?Third class?– Revamp 2nd Math course

M i li d ff i h• More specialized course offerings such as courses for K-4 teachers only and K-6 mathematics specialists courses

Sh K tt

0011 0010 1010 1101 0001 0100 1011

• Sharon Katt– sharon.katt@nde.ne.gov

S i K tt0011 0010 1010 1101 0001 0100 1011• Susie Katt

T G d• Terry Goodman– goodman@ucmo.edu

• Ira Papick– ipapick2@math.unl.edu

• Janice Rech– jrech@unomaha.edu

• Michael Matthews– michaelmatthews@unomaha.edu

Mathematics Courses and Materials for Middle Grade Teachers

Ira PapickUniversity of Nebraska LincolnUniversity of Nebraska-Lincoln

December 14, 2009

N b k S it M th ti Ed tiNebraska Summit on Mathematics Education

Before It’s Too Late: Glenn Commission Report, 2000

Primary message:Primary message: “America’s students must improve their

f i

Second message: “The most direct route to improving performance in

mathematics andscience if they are

p gmathematics and science achievementto succeed in

today’s world and if the United States is

achievementfor all students is better mathematics and scienceto stay competitive

in an integrated global economy.”

and scienceteaching.”

global economy.

Improving K-12 Mathematics Teaching

How can we best help teachers experienceHow can we best help teachers experience and learn mathematics in a way that would help them understand it more deeply andhelp them understand it more deeply and teach it more effectively?

How can we best help teachers become lif l l f th ti d i tilllifelong learners of mathematics, and instill a passion for learning mathematics with their students?

Teacher Knowledge : Some Basic Findings from the Mathematics Education LiteratureMathematics Education Literature

A substantial number of classroom teachers lack the A substantial number of classroom teachers lack the mathematical knowledge to teach mathematics well. (Borko & Putnam, 1995; RAND, 2003)

Cl t h h k l d f l d d lClassroom teachers have knowledge of rules and procedural skills, but are deficient in conceptual understanding and reasoning abilitiesreasoning abilities.(Wilson, Floden & Ferrini-Munday, 2001)

Most teachers see very little connection between the ymathematics they study as undergraduates and the mathematics they teach. (Cuoco, 2001)

Many teachers perceive their college mathematics preparation as an end rather than as the beginning of lifelong p p g g gmathematical learning. (Cuoco, 2001)

Teaching Without Understanding Leads to Rote Learning and Mindless Manipulation

Mathematical Knowledge for Teaching

Teachers of mathematics need a knowledge of mathematics that differs from other users of mathematics. This knowledge must prepare them to:Deeply understand the mathematical ideas (concepts, procedures, reasoning skills) that are central to the grade levels they will be teaching (as well as the mathematics their students have learned and will learn)

Represent and connect mathematical ideas so that students may understand them and appreciate the power, utility, and diversity of these

(as well as the mathematics their students have learned and will learn)

ideasUnderstand student thinking (solution strategies, misconceptions, etc), and address them in a manner that supports student learning

Make mathematical curricular decisions (choosing and implementing curriculum), understand the mathematical content of state standards and

Assess student learning through a variety of methods

curriculum), understand the mathematical content of state standards and grade level expectations, communicate mathematics learning goals to parents, principals, etc)

This kind of mathematical knowledge is beyond what most teachers experience in standard pre-service mathematics courses in the United States. (Principles and Standards for School Mathematics, NCTM, 2000)

A Sampling of Some Basic Mathematics Questions that Middle Grade Teachers Regularly Encounter in their

Teaching Practice 1. My teacher from last year told me that I whatever I do to one side of an equation, I must do the same thing to the other side to keep the equalityequation, I must do the same thing to the other side to keep the equality true. I can’t figure out what I’m doing wrong by adding 1 to the numerator of both fractions in the equality 1/2 =2/4 and getting 2/2 = 3/4?2 Y l k t l i thi ki I k th t t f ti2. You always ask us to explain our thinking. I know that two fractions can be equal, but their numerators and denominators don’t have to be equal. What about if a/b =c/d, and they are both reduced to simplest form. Does a=c and b=d, and how should we explain this?3. I don’t understand why (–3)x(–5)=15. Can you please explain it to me?

4. The homework assignment asked us to find the next term in the list of numbers 3,5,7,… ? John said the answer is 9 (he was thinking of odd numbers), I said the answer is 11 (I was thinking odd prime numbers), and ) ( g p )Mary said the answer is 3 (she was thinking periodic pattern). Who is right?5. My father (who is very smart) was helping me with my homework last

i ht d h id th b k i H id th t 4 2 d 4 2night and he said the book is wrong. He said that 4=2 and 4=–2, because22=4 and (–2)2=4, but the book says that 4≠–2. He wants to know why we

Mathematics Courses forMathematics Courses for TeachersCBMS, 2001, Chair of Steering Committee-Jim Lewis

Courses for teachers should involve:Courses for teachers should involve:+ foundational mathematics+ careful reasoning careful reasoning + connections of principles / practice+ mathematical ‘common sense’ + habits of mind / inquiry+ utility, power and elegance+ multiple ways to engage students+ multiple ways to engage students+ technology-computation /exploration

Mathematics Courses and Materials for Middle Grade

Teachers2001 2005 Teachers2001-2005

•Algebra Connections

G C i•Geometry Connections

D t A l i d P b bilit•Data Analysis and ProbabilityConnections

•Calculus ConnectionsCalculus Connections

The textbooks utilize middle school curricular materials in

multiple waysmultiple ways

∑ As a springboard to college level mathematics∑ As a springboard to college level mathematics

∑ To expose future (or present) teachers to current middle grade curricular materialsmaterials

∑ To provide strong motivation to learn more and better mathematics

∑ To support curriculum dissection—critically analyzing middle school curriculum content—developing improved middle grade lessons through lesson study approachpp

∑ To use college content to gain new perspectives on middle grade content and vice versa

∑ To apply middle grade instructional strategies and multiple forms of assessment to the college classroom

Algebra ConnectionsChapter 1: Patterns

This discussion is directly connected to the unit Say It With Symbols, which is part of the y y , pmiddle school curriculum Connected MathematicsMathematics.

Connected Mathematics, Say It With Symbols, p. 20

QuickTime™ and aPhoto - JPEG decompressor

are needed to see this picture.

Connected Mathematics, Say It With Symbols, p. 21

QuickTime™ and aPhoto - JPEG decompressor

are needed to see this picture.

Student ThinkingMeaghan: I drew out the first three poolsand noticed that for each time you add af yfoot to the side of the pool the number oftiles goes up by 4tiles goes up by 4.

Recursive ruleRecursive ruleT1 = 8

T T 4 ( 1)Tn =Tn-1+4 (n>1)

Student ThinkingReese: I noticed that for a pool of any size you willalways have a tile for each foot of theperimeter, or 4n, and then you need 4 moretiles for the corners, so I added 4.tiles for the corners, so I added 4.

Explicit ruleExplicit ruleT(n)=4n+4

Student Thinking

• T(n) = 4n+4• T(n) = 4(n+1)• T(n) = 4(n+2) 4• T(n) = 4(n+2)–4• T(n) = 2(n+2)+2n• T(n) = (n+2)2–n2

The Middle Grade Connection Series

For desk copies contactFor desk copies contact:Marnie Greenhut

Project Manager, Statistics & Service MathematicsPrentice Hall

marnie.greenhut@pearson.com