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Facilitating Knowledge Development and Refinement in Elementary School Teachers

Denise A. SpanglerUniversity of Georgia

USA

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Teacher Education ContextElementary in the US means different things in

different places. K-5 (ages 5-11) K-6 (ages 5-12) K-8 (ages 5-14) Rarely K-9 (kindergarten through first year of high school) Increasingly includes prekindergarten (PreK = age 4)

University of Georgia: PreK-5

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Teacher Education ProgramAll courses are 3 semester hours or 45 contact

hours2 years liberal arts curriculum–60 semester hours

2 mathematics classes (modeling, precalculus, statistics, calculus)

1 mathematics course for elementary teachers (number and operations)

Special education, foundations of education, educational psychology, diversity/equity

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TE Program, continued2 years of the teacher education program

2 mathematics courses for elementary teachers Geometry, measurement Algebra, statistics

2 mathematics methods courses Children’s mathematical thinking with respect to

numbers and operations (whole & rational) Curriculum, assessment, teaching of other

content areas (geometry, measurement, algebra…)

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First

sem

este

r • ECE• Community experience

• Math content (geometry/ measurement)

• Math methods

Seco

nd se

mes

ter • ECE

• ECE field experience

• Math methods• Reading methods

• Language arts methods

• Social studies methods Th

ird se

mes

ter • ECE

• ECE field experiences

• Math content (algebra)

• Reading methods

• Language arts methods

• Science methods

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GoalExamine where the mathematical knowledge

needed for teaching in elementary schools comes into play.

Look at an effort to assess it in preservice upper elementary teachers.

Look at an effort to develop it in preservice elementary teachers/

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Mathematical Knowledge for Teaching

Knowing mathematics to pass a test ≠ knowing mathematics in the ways needed to teach it.

Teaching mathematics involves knowing Representations Analogies Illustrations Examples Explanations Demonstrations

Shulman, 1986

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And alsoKnowing

What makes a topic easy or hard Students’ preconceptions and misconceptions Strategies to address misconceptions

Shulman, 1986

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MKT as defined by Ball, Thames & Phelps 2008

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A recent study of MKTConducted by Jisun Kim, University of Georgia, 2011Preservice teachers of mathematics, grades 4-8

Calculus I Numbers & Operations

Content course and methods course Geometry & Measurement

Content course and methods course During study

Goal: investigate multiple aspects of MKT

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5 geometry/measurement tasks, given one at a time over a semester

Preservice teachers had to Solve the task Examine 4-5 student solutions (from research projects) to

determine if they were correct Identify causes of errors Propose instructional strategies to address the causes of the

errors

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Area of triangles

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Types of Triangles

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Similar Figures

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Volume

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Area/Perimeter

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Your turnSolve the taskHypothesize errors and causes of errorsPropose instructional strategies to address the

causes of the errors

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Student response 1

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Student response 2

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Student response 3

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Student response 4

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Findings of the studyPreservice teachers generally got the correct answer themselves, but in some

cases they exhibited the same misconceptions as the students even though the topic had been addressed in a course.

focused on the answer, not the solution path. (Students 3 & 4)

attributed errors mostly to student’s faulty procedural knowledge.

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Findings, Cont’d.Diagnoses did not match prescriptions.

Example Diagnosis: only thinking about squares Prescription: review area formula

Small, weak repertoire of instructional strategies; often wanted to “tell” students the correct answer/formula.

Used examples from class for instructional strategies.

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Implication for teacher educationFocus on planning but shift

Away from lesson planning Toward task planning

Task dialogues (Crespo, Oslund, & Parks, 2011) Create plausible teacher/student conversation Equal sign a balance point vs. “do something”

5 + 3 = _____ + 7 Common student answer is 8

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Task Dialogues Task–we solve it in class and discuss multiple solution

strategies I give them possible student solutions

1 correct 2-3 incorrect or incomplete

What mathematical thinking could be behind that response? What question could I ask next to test whether or not that is

what the child was thinking? How would the child respond if it was or was not what she was thinking?

What is my next move?

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Seeing the child through the mathematics

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Seeing the mathematics through the child

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Field ExperienceThey do the task dialogue task with childrenThey also do other tasks that day, and in their

plans THEY have to posit student responsesTwo formats

My class: Do the task one week with one child Allyson Hallman’s class: Do the task 3 weeks in a

row with 3 different children to build MKT (study in progress)

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Example Task In a soccer championship there are 6 teams. If

all teams are going to play each other, how many games will there be in the championship?

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Response 1There will be 3 games because

Team A will play Team B Team C will play Team D Team E will play Team F .

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PST A dialogue excerptWhy don’t you set up your diagram like this:

ABCDEF

And now make sure team A plays all the teams. Team B, team C, and team D also play all the teams.

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PST B dialogue excerptYou have the right idea by going in order, but maybe you should try looking at just one team at a time so that it’s easier to see the number of games they play.

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Response 2There will be 30 games because each team

plays 5 other teams. There are 6 teams so 5 + 5 + 5 + 5 + 5 + 5 = 30.

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PST A dialogue excerptT: Let’s draw out your diagram.S: Okay.T: Now, if each team can only play each team one time does your diagram still work?S: Yes.T: Do you agree that Team 1 vs. Team 2 is the same as Team 2 vs. Team 1?

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PST B dialogue excerptT: Do you think you could draw a picture to show that?

S: Yeah, I could. It would look like this [draws picture].

T: You did a great job on your math and adding correctly. But let’s look back at what a tournament means.

S: It means every team plays all the other teams.

T: Good, but there was one other part of it, too. Do you remember?

S: Oh, yeah. They only play each other once.

T: So let’s take a look at your picture. Did we stick to the rules of that kind of tournament?

S: (after looking at the picture) No. There are repeats.

T: So, you definitely have the right idea with the way you solved the problem. What do you think we could do to solve the issue with the repeats?

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Response 3There will be 15 games:AB BC CD DE EFAC BD CE DFAD BE CFAE BFAF

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PST A dialogue excerptCan you show me another way of getting there?

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PST B dialogue excerptWhat if there were 10 teams?What about 20 teams? Do you see a pattern in the solutions for 6 teams and 10 teams that would help you solve 20 teams without writing them all out?What if I told you there were 45 games. How many teams would that be?

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ObservationsPST with lower content knowledge tend to

Have difficulty seeing children’s mathematical thinking, especially when it’s different from their own

Assume they know what children are thinking and do not ask

Push children to do it their way (the PST’s way) Ask bite-sized questions, leading/directive

questions Start over rather than building from existing ideas Don’t push on correct answers Don’t make an effort to connect solution strategies

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Observations, cont’d.PST with higher content knowledge tend to

Ask more open questions Try to get students to figure things out for

themselves Push students to analyze their solutions and go on

from there rather than starting over Pay attention to process as much as final answer Link solution strategies Extend correct solutions to push for

generalizations

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ConclusionFocusing on preservice or inservice teachers’

content knowledge is necessary but not sufficient.

Need to develop OUR repertoire of tasks/activities to tap into the application of that content knowledge (and study them!)