Teaching Mathematics to Prospective Elementary

School Teachers

Presentation at the PMET Summer WorkshopHumboldt State University

Arcata, CAJune 13, 2004

Dr. Randy PhilippSan Diego State University

Rphilipp@mail.sdsu.edu(619) 594-2361

Session OutlineSession Outline

•A Secondary Example from Trigonometry•An Elementary Example: Place Value•The Role of the Unit in Fractions•Preparing for our Lesson on FractionsTo ground our work, we’ll look to:1) Examples from Preassessment and

Prior Research2) Video Clips of Children’s Reasoning

What do Mathematics Educators Think About?

A Secondary Example

•8th grade geometry student, Rick

•Introduction to trigonometry

•Rick can compute sin, cos, tan, and inverse functions. For example, he can determine the sin (38˚), or the angle whose cosine is .58.

•Rick is supposed to understand the fundamentals of right angle trigonometry.

My goal

Trig functions depend upon the ratios of the lengths of the sides.

Question: Rick, you have a right triangle, and one angle is 38˚. Can you find the ratio of the opposite side of the 38˚ angle to the hypotenuse of the triangle?

Rick draws a right triangle, using a protractor to make one acute angle approximately 38. He measures the two sides, and divides the opposite side by the hypotenuse.

I ask him if he realizes that the opposite side divided by the hypotenuse is the same as the sin 38˚.

He says he does.

I ask, “Rick, can you do the same for another angle, say, 21˚?”

Pause. No, not without knowing the lengths of the sides.

He draws another right triangle, and again uses a protractor to make one acute angle approximately 21˚. He measures the two sides. He divides. He sees it is the same as Sin 21˚.

This goes on. And on.

What’s going on here?•Rick knows that Sin (x) = Opp/Hyp

•Rick knows that the value he got by dividing Opp/Hyp is the same thing he got when he punched “sin 21˚” into his calculator.

•Why then, when I ask him to determine the ratio of the opposite side to the hypotenuse of a particular angle, must he draw the triangle?Sketchpad

Why then, when I ask him to

determine the ratio of the opposite side to the hypotenuse of a particular angle, must he draw the triangle?

What is a ratio?

Rick was conceptualizing a ratio as the result of dividing two quantities. He was not conceptualizing it as a single quantity.

If he can not conceptualize a ratio as a single quantity, can he possibly understand triangle trigonometry?

A related example

There are students who can determine the volume of the prism on the left, but not the prism on the right. Do you see why?

8 units 8 units 3 units3 units

6 units6 units

3 units3 units

17 square units17 square units

What do Mathematics Educators do?

Among other things, we try to go beyond determining whether a student is right or wrong, and try to understand how students are looking at something.

Then we try to share this information with teachers, because we have found that, at least at the elementary school level, when teachers understand their students’ thinking, they are able to better support their students’ learning.

The Situation with Prospective Elementary School Teachers

We know that developing deep understanding of the mathematics of elementary school is far more difficult than was once thought. -(Ball, 1990; Ma, 1999; Sowder, Philipp, Armstrong, & Schappelle, 1998)

Even when PSTs attend a thoughtfully planned course designed to engage them in rich mathematical thinking, too many of them go through the course in a perfunctory manner.

Example: Place Value

5 27-135

Can PSTs apply the standard subtraction algorithm?

Yes.

100% of the15 PSTs in Eva Thanheiser’s study could correctly apply the whole-number multidigit subtraction algorithm.

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Eva then asked:Does Regrouping Change the

Value of the Numbers?

100% of the 15 PSTs stated that the value was the same.

But 10 of the 15 could not explain why.

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Related Example: Place Value

Below is the work of Terry, a student who solved

one addition and one subtraction problem.

A) 1 B) 2 5 9 3412 9

+ 3 8 - 3 4 2 9 7 3 9 5

a) Does each 1 in these problems mean the same thing?

b) Terry knows that when adding, as in Problem A, he adds 1 to the 5, but when subtracting, as in Problem B, he adds 10 to the 2, but he doesn’t know why. Can you explain the reasons to Terry?

What type of response from your college students would

please you?

Does each 1 mean the same thing? Humboldt Students: NO:11 Yes: 4

Comments for “NO”

(5) In A we add 1 to the tens column.

In B we add 10 to the tens column.

(3) The 1 in the addition problem is meant to be added (1 + 5 + 3), and the 1 in the subtraction problem is changing the 2 to a 12, it is “borrowed” from the 4 adding 10 to the 2.

A) 1 B) 2 5 9 3412 9 + 3 8 - 3 4 2 9 7 3 9 5

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1.How would you solve it?

2.How would you explain it?

3.How would you like your child’s teacher to think about it?

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-135

5 27

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2

5 27

-135

2

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How would you like children to be able to think about the red numbers?

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Questions for PSTs:

Look at the red numbers.

Can you see 12 of something? If so, what?

Can you see 120 of something? If so, what?

Eva’s Findings

2 of 15 students could only see ones.45127

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392

Carmen: “I don’t get how the 1 can become a 10. One and 10 are two different numbers. How can you subtract 1 from here and then add 10 over here? Where did the other 9 come from?”

Eva’s Findings

2 of 15 students could only see ones.45127

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392

Carmen: “I don’t get how the 1 can become a 10. One and 10 are two different numbers. How can you subtract 1 from here and then add 10 over here? Where did the other 9 come from?”

Eva’s Findings

2 of 15 students could only see ones.45127

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392

Carmen: “I don’t get how the 1 can become a 10. One and 10 are two different numbers. How can you subtract 1 from here and then add 10 over here? Where did the other 9 come from?”

Unit is only “1”•All groupings are only in ones.

•These two students knew that there were ten ones in one ten, but they did not connect that knowledge to the algorithm.

•How can a person “seeing” only ones correctly add multi-digit numbers?

36

+ 57

The same way that most of us think when using algorithms.

36

+ 57

The same way that most of us think when using algorithms.

36 + 57

“Six plus seven is Thirteen. Write down the 3 and carry the 1.”

1 6 + 7

3

36

+ 57

13 6

+ 5 + 7

9 3

Eva’s Findings8 of the 15 PSTs were aware that a digit in the hundred’s place represented 100s, but viewed the digits in the ten’s place as representing ones rather than tens.

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Delia: “When you are borrowing it [1 in the hundred’s place] from over here … it’s a hundred. But once you put it [the borrowed 100] into the number, it becomes a ten.”

Eva’s FindingsThe last 5 provided valid explanations for why the value of the number stayed the same when it was regrouped. Two of those 5 reasoned in terms of groups of 100 ones, not ten tens.

Seeing “Groups of Ones”45127

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Veronica: “So in reality when you are taking a 1 from the 5 you are not really just taking a 1 and putting it next to the 2, you are taking 100 from the 5 – so that is making it 400 and you are borrowing, you are adding the 100 to the 20, so in reality it is 120.”

Seeing “Groups of Ones,” But Not Seeing Tens

When asked about the 12, Veronica did not show any evidence of thinking in terms of 12 tens.

She referred to the 12 as “the 12 part of the 120.”

Veronica: “Just out of reflex it’s 12, but if you like, look at it and you think about it, it’s actually 120 and 30. If you separate it into the separate components, but like out of just like reflex and human nature it’s a 12 and you are just subtracting 3 from it.”

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Seeing “Groups of Ones,” But Not Seeing Tens

When asked whether she could think of it as 12, she covered up the one’s column and stated “we can just like pretend that you’re taking 10 from the 5 and then just adding it to the 2 and then it would be 12.”

When asked whether she could think of 12 without covering up the last digit she stated “no.”

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Explaining the Students’ Conceptions

Different Conceptual Structures for one Hundred

5 2 7How might one think about the 5?

45 12 7

45 12 7

45 12 7

Rich Student Whole-Number Reasoning

VC#6, Javier, (Gr 5), (0-1:09)

VC #5, Zaneida (Gr 3),(0 - 1:23)

(Both of these videoclips are on the Select CD of Students’ Reasoning that I am giving to you.)

Select Video of Children’s Reasoning

• 25 Video Clips

• Video Guide

• Instructor Interview Materials

• Applets Prentice Hall, ISBN #0-13-119854-8.

Rational Numbers

First Example: EstimationFirst Example: Estimation

NAEP ITEM: 13-year oldsEstimate the answer to 12/13 + 7/8. You

will not have time to solve the problem using paper and pencil.

a) 1 b) 2 c) 19 d) 21 e) I don’t know c) 28% d) 27% b) 24% e) 14%a) 7%

#7 From AssessmentEstimate the sum of 12/13 + 7/8.How did you get this? (n = 15)

Not Close to 2

No response (5)

19/21 (2)

9/10 (1)

15/21 (1)

TOTAL: 9 PSTs

Close to 2Close to 2

187/104-LCD (4)187/104-LCD (4)

Estimated to 2 (2)Estimated to 2 (2)

TOTAL: 6 PSTs TOTAL: 6 PSTs

Example of PST & Estimation

Video #522

#14 From AssessmentWithout actually doing the calculations, put the decimal point in the answers. (n = 15)

a) 77.5 x 2.84 = 2 2 0 1No response (2)

2.201 (6)

“In multiplication you count how many numbers are behind the decimal point and that’s where you put the decimal.”

22.01 (1)

222.1 (6)

Only one of the students provided an explanation indicated he or she estimated.

Fraction SubtractionAssessment #9, 10

#9: Write relevant story problem for 1/2 - 1/3

#10, Calculate 1/2 - 1/3

Predict Performance on Each

#10: 13 of 15 correctly solved

#9: Virtually no one could do this

Examples of 1/2-1/3 Problems

If you had 1/2 of a sandwich and you ate 1/3 of that, how much would be left?

Example of “correct” story problem:

Judy and Sally are playing with 6 marbles. Judy had 3 of the 6, but Sally just won two of them in her last round. How many does Judy have left now?

Difficulty with the Role of the Unit in Rational Numbers

Assessment #12

No: 2

Yes: 12

No response: 1

Results of large scale study are similar:Score Pretest Posttest

0 94% 73%3 6% 27%

AllyAn average 5th-grade student at a high-

performing local school

(1:37 - 2:38)

Three Conceptions Driving Ally’s Reasoning

1) Fractions are smaller than 1.

2) Fraction numerators and denominators may be compared additively

3) With fractions, the number that looks smaller is (usually?) larger.

Video Example 3

Felisha, VC #15, End of Grade 2 (1:39)

Felisha learned fraction concepts using equal-sharing tasks in a small group over 14 sessions (7 days). She had not been taught any procedures for operating on fractions.

Concepts/Procedures:Consequences for Which is

Taught FirstRachel, 5th grader, in a class where teacher is conceptually-oriented.We asked teacher to teach procedures for converting between improper fractions and mixed numbers. Then, several weeks later, the teacher taught a conceptually-oriented lesson. We’ll see a slice of interviews after each lesson.VC#13, 0:50 - 1:54

2:22 - 6:21

The Issue Goes Even Deeper:

How Do We Conceptualize Fractions?Assessment #3a, b#3a: No: 4Yes: 2Yes and No: 8

#3b: No: 0Yes: 14No Response: 1

Jacky VC #12 (0:00 - 2:00)

Are Our Courses Doing Enough?

nPretestmean

Posttestmean

Meanchangescore

Varianceof change

scoresAll 159 36.9 50.1 13.2 65.3

Maximum score possible = 82

The Situation with Prospective Elementary School Teachers

Even when PSTs attend a thoughtfully planned course designed to engage them in rich mathematical thinking, too many of them go through the course in a perfunctory manner.

4 Noteworthy Documents With Important Implications for our

Work

“Two general themes that guide this report are: (i) the intellectual substance in school mathematics; and (ii) the special nature of the mathematical knowledge needed for teaching. There has been a widespread assumption that because the topics covered in school mathematics are so basic, they must also be easy to learn and to teach. We owe to mathematics education research of the past decade or so the realization that substantial mathematical understanding is needed even to teach even whole number arithmetic well.” (preface)

Conference Board of the Mathematical Sciences. (2001). The mathematical education of teachers. Washington DC: American Mathematical Society.http://www.cbmsweb.org/MET_Document/index.htm

1. “Learning with understanding is more likely to promote transfer than simply memorizing information from a text or a lecture” (p. 224).

2. “Knowledge that is taught in a variety of contexts is more likely to support flexible transfer than knowledge that is taught in a single context” (p. 224).

3.“Teachers are learners, and the principles of learning and transfer for student learners apply to teachers” (p. 231).

National Research Council. (1999). How people learn: Brain, mind, experience, and school. Washington, DC: National Academy Press.

http://stills.nap.edu/books/0309065577/html/

1. ”Teacher education must no longer be viewed as a set of disconnected phases for which different communities assume the primary responsibility “ (p. x).

2. “Responsibility for teacher education in science, mathematics, and technology can no longer be delegated only to schools of education and school districts. Scientists, mathematicians, and engineers must become more informed about and involved with this effort” (p. xi).National Research Council. (2001). Educating teachers of science, mathematics, and technology. Washington, DC: National Academy Press.

http://www.nap.edu/books/0309070333/html/

The Five Components of Mathematical Proficiency

•Conceptual Understanding•Procedural Fluency•Strategic Competence•Adaptive Reasoning

•Productive Disposition These 5 strands might be thought of as intertwined to form a rope, and to become mathematically proficient, one must possess all 5 components.

National Research Council. (in press). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press.

http://www.nap.edu/books/0309069955/html/index.html

A Student reflecting on a Problem-Solving Oriented Mathematics Course in response to the question:

“What should I tell a group of mathematics instructors?”

(Paraphrased) Kasey: You can just fool the professors, just BS them. They think that you

understand when you don’t. When I was asked to show with a drawing how I got an answer, I would do the algorithm to get the answer then make the drawing to fit. Then I’d erase the algorithm. I did the same thing for estimates. I’d do the calculations.

Randy: What if you were required to give a verbal explanation?

Kasey: I’d memorize it.

Note: As Kasey spoke, almost every other student in this small class of 12 was nodding in agreement.

What should I tell a group of Professors? (Paraphrased)

Catherine: The math classes need to start with basics. Start with 210 (the first class) and pretend like we never learned algorithms at all. I was hearing the music when we worked in small groups the last week before break, but I fell right back into my old ways of just using algorithms after the break.

Many PSTs report having had bad experiences learning mathematics (Ball, 1990).

Should we try to get them to care more about mathematics for mathematics sake?

No! (Noddings, 1984)

Why not start with what PSTs care about?

Children! -Darling-Hammond & Sclan, 1996

Start with that about which prospective teachers care.

Children

When PSTs engage children in mathematical problem solving, the PSTs’ circles of caring expand to include children’s mathematical thinking.

Children

Children's Mathematical Thinking

When PSTs come to recognize that children solve problems in varied and sometimes mathematically powerful ways, their circles of caring extend to mathematics, because they realize that to be prepared to understand the depth and variety in children’s mathematical thinking, they must themselves grapple with the mathematics.

Children

Children's Mathematical Thinking

Mathematics

This idea is at least 100 years old.

John Dewey, 1902

Every subject has two aspects, “one for the scientist as a scientist; the other for the teacher as a teacher” (p.

351).

“[The teacher] is concerned, not with the subject-matter as such, but with the subject-matter as a related factor

in a total and growing experience [of the child]” (p. 352).

Dewey, J. (1990). The child and the curriculum.

Chicago: The University of Chicago Press.

Placing Children Between PSTs and the Mathematics

Looking at the mathematics through the lens of children’s mathematical thinking helps PSTs come to care about mathematics, not as mathematicians, but as teachers.

MathematicsChildren

Where Might we Infuse Children’s Mathematical Thinking?

• Mathematics Methods Courses

• Separate Course (CMTE) (SDSU & Local Community Colleges)

• Into Content Courses (using CD)

The Problem? (Paraphrased)

Megan: In working with the child today, I found that I need a better understanding to explain more. I can do the problems, but I don't have enough foundation to be able to get the child to see it. I can do 1/2 - 1/3, but just with numbers. I'm realizing I need a better grasp, a way to help [the child] other than just with numbers. In 313 we have to look at problems as if we are were children, but we did not do that in the first three math classes [for prospective elementary school teachers], and we should have been doing that all along. I couldn't get to the 1/6 in explaining 1/2 minus 1/3 to Liz [fifth grader] today.

Comment From a Mathematics Professor

I have used the tape to show my prospective elementary teachers the kind of creative and "different" thinking students use to reason and make calculations. The video clips became motivational clips and saved me having to make the argument for PUFM. I showed one or two clips per class over a 2- or 3-week period. We had fun discussing them . . . and discussing the teachers.  For me the tapes are extremely helpful.  

-George Poole

Professor of Mathematics, East Tennessee State University

Planning for Lesson