The Role of Metacognition in Mathematical Problem Solving: A Study

DOC WENT RESUME ED 314 255

SE 051 112

....-

AUTHOR TITLE

INSTITUTION

SPONS AGENCY PUB DATE GRANT NOTE PUB TYPE EDRS PRICE DESCRIPTORS

IDENTIFIERS

Lester, Frank K., Jr.; And Others The Role of Metacognition in Mathematical Problem Solving: A Study of Two Grade Sever. Classes. Final Report. Indiana Univ., Bloomington. Mathematics Education Development Center. National Science Foundation, Washington, Jun 89 NSF-MDR-85-50346 284p.; Drawings may not reproduce well. Reports - Research/Technical (143) MFO1 /PC12 Plus Postage.

*Eelle±s; *Cognitive Processes; Grade 7; Junior High Schools; Mathematical Applications; Mathematics Achievement; Mathematics Education; *Mathematics Instruction; Mathematics Skills; *Metacognition; Problem Sets; *Problem Solving; *Oualitativ Research Mathematics Education Research

ABSTRACT This project was designed zo: (1) assess 7th-graders' metacognitive beliefs and processes and investigate how they affect problem-solving behaviors; and (2) explore the extent to which these students can be taught to be more strategic and aware of their own problem-solving behaviors. The primary assessment was conducted by analyzing video tapes of 1. ividual students and pairs of students working 3n multi-step problems and non-standard problems, and subsequently being interviewed about their performance. The interview probed the students' mathematical knowledge, strategies, decisacns, beliefs, and affects. A variety of ''ritten protocol data, including pre- and post-tests, homework assignments, and In-class assignments were collected. The instruction was presented by onc of the investigators to a regular-level and an advanced-level 7th-grade class about 3 days per week for a period of 12 weeks. Among the four ca-.egories of the cognitive-metacognitive framework, which included orientation, organization, execution, and verification, the orientation category had the most important effect on students' problem-solving. Finally, it appeared that instruction was most likely to be effective when it occurs over a prolonged period of time and within the context of regular day-to-day mathematics instruction. (Author /YP)

xxXXxx************txxxxx, she drew 3 circles and counted, connected lines "only need 3". Further DISCUSSION showed confusion between connections and lines..clarifying discussion followed, but she still thought 240. Didn't see doubles. Needed some more analysis.

D-8 r) r, 4, 4 A::

Kennedy collected 225 tape cassettes and 4 old shce boxes to put them in. If he puts the same number of cassettes in each box, how many extra cassettes will there be?

Li

\

messa did not reread the problem, she just looked back at It to s : the "right numbers". She knew it was division because "you ca.i't times it cause it would be more". She divided and gave 56 Ri. During discussion she commented that she sometimes talks to herself when she doesn't understand because "it straightens me out so I know what I'm doing". No analysis, little or no assessment, no chsck of anything - she was confident.

4 _,,

The sixth grade math teacher did an experiment with her students. One at a time, students were to give her change for a 50 cent piece without using pennies. No student could use the same set of coins as someone else. How many students will be able to give her change?

A

Ss

0

e/-T to

4

ks

S

S.

;5

S

50

10

IC C

Sot'

Tessa did not reread the problem, she looked at it briefly "it didn't give me a specific way...it didn't come to me...didn't have another number". She started calculating sets, some adding, some multiplying. She seemed to have some understanding, "see what numbers made 50". She got 4 sets - "all I can think of". She did not look back, was not very organized, and had little concern for completeness. I asked if there could be any more "yeah...probably 10 or something". SECOND TRY. She started getting new sets, multiplying and adding together with running totals, using pen movements to "make sure it's right". She lost sight of coin condition (using 4, 7, 9) - just remembered she had to get 50 (the "big scene"). I has her reread it, she only caught that she used pennies, not that she used non-coin 'mounts. I REWORDED - THIRD TRY. She continued along, with no systematic plan, "I would use coins", just trying different combinations, but again lost sight of coin condition (using 15). No rereading, no analysis, no concern for completion, little monitoring, not systematic. D-10 FA., A:,

Atlas Steel makes 4 different types of steel. From a shipment of 300 tons of raw steel, the factory produced 60 tons of type I, which sold for $60 a ton; 75 tons of type II, which sold for $65 a ton; 120 tons of type III, which sold for $72 a ton; and 45 tons of type IV, which sold for $85 a ton. Raw steel costs $40 a ton. It costs the factory $2,500 to convert every 300 tons of raw steel into the 4 types. How much profit did Atlas make on this shipment?

\

(3,(5

;-5 CC'

F

Tessa did not reread the problem, so she did no analysis of conditions, and had no understanding. She just started dividing 4 into 2500 because "the 2500 and the 4 caught my eye, I'd try it 1st, I was not sure I did it right" - but did no assessment or looking back. She divided, using side multiplications to help out, made a mistake but thought it was correct. She acted as fast as she did because "usually I can just look at a problem and tell, if it doesn't work, I'll try another" - an assess by carryout strategy, but she didn't assess this one. Sh,- had little concern for reasonableness, and ignored much important data. No final check on anything. D-11

22

There are 10 people at a party. If everyone shakes hands with everyone else, how many handshakes will there be?

ea

C7 C,,C)

h s

(ONE

1"/

t

!D

51.--405

---,. (.7):"\

\

q(5crs-

2 ' ilqpec (i Cc .

4,

.3QC4C aiLkt,e()1Q

77V(T,r,

tfx.

CP

Tessa did not reread the problem, but remarked "its only got one number in it". No analysis oL conditions, no thinking, she just multiplied 10x2=20. No checking of anything. SECOND TRY - later when discussing, she drew 4 circles, connected them, got "12". She then got 6 for three people, but got 3 when modelled. When pointed out to her she said "but you can't do that, shake both hands" - she was lost even with a lot of discussion and modelling.

D-12

Name

Felipe's typewriter sticks when the 7,8, or 9 key is typed. If he types each number from 100 to 200, how many times will his.typewriter keys stick?

`1-

ti

3

14

`7

;i

38

Gig

39 -19 57

f7

15 8 )59

4,0

IO

Co

4,5t;v0f3 arc

SO

ls:119110

\;), 116

1?ri

\IL9

a

\

v;_s

\34 135

\-73 \

\

\AS

\L.\2

\Q\-k

\s

.1s(-\ \

".0S

1'19

i?0

19

/5'

\ ,0.1

1'14

"-.)

isza 173 D

11

V-0$

\(0c\

1`10

I))

18=1

19 2 193

I

\53

\5

5q

\S

\(0(.0

13k°

\-\?,)

\u'

\L

\c".1

V

\I%

i')1

1`10 \'11-

1?-1

{9

-200

Tessa did not reread the problem, just "scanned it". She began writing out numbers in rows, counting with her pen. No analysis - her plan was to list and count. She had no consideration of completeness, some organization, some tapping while counting. No assessment, no final check. During discussion, she thought she made a mistake (100-200), assessed it, OK. When omissions were pointed out (70,...doubles...), she gave it a SECOND TM. "Better start over" not to "get messed up". No rereading or analysis of conditions. Her new plan was tci list out all numbers (assess: too many problems with shortcut?). She listed them in order, counted with pen "to make it easier". No check, no final evaluation. Later mentioned she doesn't feel good when she gets many wrong.

D-13

2 2r:'

There were 347 people at a $150-a-plate luncheon to raise money for charity. Expenses were $5000. How much -ually went to the charity?

.23 3L-0-1

1So

SS 3 (4. 1a

S

o0

Tessa did not read the problem, she just looked back for the numbers ("took out parts"). She understood what to do - she multiplied "just a guess...how much it was" then subtracted expenses - "couldn't divide...couldn't add". She had the correct idea, but no analysis or assessment or check.

D-14

fA.,

Name

Mr Shuttlemeier's English class is printing a newspaper for the school and giving the proceeds to student government. The local newspaper is charging them $0.15 a paper for the first 100 papers printed, $0.10 each for next 200, and $0.06 for each paper thereafter. The class has ordersthe for 625 papers. What will they have to charge for each paper in order to give student government at least $100.00?

" Y. ICS

I

ti

ICS

0

c).D -t-

t

5

t) 0

0 0

-?.1.511.)--'"")

k

C

t

(CO

.13%.

C.,N

ja../

e

Tessa did not reread the problem, she looked back to list numerical data on the side "so I don't have to keep looking at it": (Frank told them to consider "what I know"...). No analysis of conditions/meaning of data. She had some understanding/plan - multiplied amounts x costs to get "how much it cost", this was done semi-systematically but not according to exact conditions (100, 200). She kept track of "copies made so far" so she would "know where I was". She was a little unsure of ner strategy - she "just guessed...if you try something id it doesn't work you can try something else". At least some reason in method. Added in 100, but lost sight of question asked - never tried to find cost of each paper, never looked back to check. Assess by carry out, with no assessment. No analysis, no check.

D-15

22c

Name

There are 16 football teams in the National ferent city. T6 conduct t heir annual draft, Football League, each in a difeach team has a direct telephone line to each of the other teams. How many direct telephone lines must be installed by the telephone company to accomplish this? Suppose the league expands to 24 to ams?

S

L,_,K

14

3/)

-1 A-

I

14

C

/7 4S