candace walkington- problems vs pbi
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8/9/2019 Candace Walkington- Problems vs PBI
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Candace Walkington
PhD Candidate in Math Education, UT Austin
Associate Professor of Mathematics, Collin College
8/9/2019 Candace Walkington- Problems vs PBI
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Why put mathematics in
context? i.e., Why give students ³real world´
mathematics problems instead of just
symbolic/numeric problems?
y Solve the ³transfer problem´
y Provide accessibility
y Enhance motivation
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How is mathematics put in
context in schools today?
A machine called the
Crawler, which moves spaceshuttles, travels at the rate of
3 feet per second. The
Crawler is currently 10 feet
from the hanger, movingtoward the launching pad.
vs.y = 3x+10
Story problems
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Although story
problems sometimes
get a ³bad rap,´
research shows that
students¶ competence
to solve verbal story
problems develops
BEFORE their
symbolic competence
± i.e. for algebra
students, story
problems are EASIER.(Koedinger & Nathan,
2004)
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The 9th grade TAKS test is around half story problems.
If our goal is to teach students how mathematics is used to solve real
problems, and to allow them to use their real life experiences to gain
access to problems, how do scenarios like this really measure up?
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Research on Traditional Instruction
I spent one year in a 9th grade algebra
classroom in an ³Academically
Unacceptable´ school in this area.
As part of the study, some (24) students
solved story problems on linear
functions.
We also questioned some (29) studentson their beliefs about mathematics in
context.
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Research on Traditional Instruction
After a year of traditional algebra
instruction, many students had not
learned the basics:
Many students (55%) were unable to
solve simple symbolic problems like:
y=2x, if x=3 what is y?
Oooh... if x equals 3, what is y? I
don't like these problems! I don't
know how to do these problems.
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Research on Traditional Instruction
Most students (80%) were not
comfortable with equation solving.
y For scenarios like 275=4x+175, used
informal/invented arithmetic strategies,
especially for story problems.
y = -1.5x ± 35 If y =0, what is x?
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Research on Traditional Instruction
Most students (80%) were not
comfortable with equation solving.
y For scenarios like 275=4x+175, used
informal/invented arithmetic strategies,
especially for story problems.
You have 175 song s d ow nloaded ont o y our iP od f rom
Li mewi r e and iT unes. You d ow nload 4 mor e ever y week. If
y ou have 275 song s, how many weeks have passed?
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Research on Traditional
Instruction Students used ³nonsense´ strategies (12-
28% of time) to solve story problems, like
plugging in the numbers they saw randomly.
An object moves at 1500
mph. It has already moved
500 miles. How far will it
have moved total 30minutes from now?
I: Can you tell me what you think this
story is about?
S: I have no idea. So, if it moves at
one thousand five hundred miles per
hour, that means«so I think I would
just divide 500 and 30. I¶m not sure.
Alright, so 16. So maybe for that one,
16 miles?
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Research on Traditional
Instruction Students applied well-know processes
to solving story problems that didn¶t fit
the story and what was being asked.
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Research on Traditional
Instruction Students become so accustomed to solving
multiple choice TAKS story problems, sometimes
sense-making is completely suspended.
«And (b) is wrong because it¶s even more minutes than 43 minutes. I¶m gonna go
with (d) because it says if her bill was 38.13 dollars, and it¶s trying to find how many
minutes she¶s talking so you just divide it by .23, and I came up with 165.7 minutes.
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Research on Traditional
Instruction Students had problems reading and
interpreting story problems (16% of time).
The 1.25 is the«per mi nute, how muc h it costs per mi nute.
But I thi nk that ³i nitial´ is plus tax, or the whol e thi ng t og ether .
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Research on Traditional
Instruction Students rarely used their real world
knowledge when solving story problems
(4% of time). Real-world knowledge could
be disruptive when used.
The number of st udents
g etti ng A or B i n alg ebr a
cl ass is g iven by the equati on
y = .25x wher e x is the t otal number of st udents taki ng
alg ebr a. If 40 st udents
earned an A or a B i n Alg ebr a
l ast year , how many t otal
st udents wer e enroll ed?
S: 80 students were enrolled.
I: How did you get 80 for that one?
S: Just times the 40 students times 2,
because there¶s always a half thatdoesn't get the full stuff done, pretty
much there¶s so many students and
then, it divides how many students get
an A or a B, and the other students
don¶t get an A or B.
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Research on Traditional Instruction
Students had little understanding of the
idea that algebra could be used as a tool
to solve complex, real problems
I: H ow d o y ou thi nk alg ebr a wi ll
be usef ul t o y ou i n the f ut ur e?
S: Bec ause when y ou w or k at
st or es and l ike as a c ashier y ou have t o add all the st uff
and d o the pr i c es I guess, I
d on' t k now.
I: H ow d o y ou thi nk alg ebr a wi ll be usef ul
t o y ou i n the f ut ur e?
S: Hel p me i n g eomet r y, that ' s f or sur e. I
thi nk that ' s the onl y thi ng it 'll probabl y hel p
me with.
I: What about outside of sc hool , what about i n r eal l ife, d o y ou thi nk it wi ll hel p
y ou?
S: N o pe.
I: S o why d o y ou thi nk y ou have t o l earn
alg ebr a i n the 9th gr ade?
S: Bec ause it ' s a r equi r ement.
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Research on Traditional
Instruction These results undermine ALL of the
reasons we place mathematics in
context in the first place.
y Why bother if students are no longer making
sense of the stories we give them?
Clearly, traditional instruction using story
problems is not enough. So what¶s the alternative to teach
mathematics in context?
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Project-Based Instruction
Students can use their everyday experiences toaccess mathematics, since authentic, realistic, andinteresting scenarios are used.
PBI is open-ended, so students can explore and
solve problems in different ways, and come todifferent answers that are valid. They can alsodiscuss different ideas and incompleteunderstandings in groups.
PBI allows students to see how mathematics is
actually used in the real world to solve problems ±answers the ³so what.´
Students are conditioned to be independent andcreative thinkers, rather than to stop thinking andstop sense-making when they see a problem.
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How can we make a PBI that
³works´ in mathematics? Start with standards, and keep your project
tightly centered around the content.
Think hard about the ways those standards
are ac t uall y used by people who usemathematics to solve complex problems.
Leave the project open-ended enough thatstudents learn to think and figure things out for
themselves, as it is in real problem solving. Focus on developing student understanding of
the ³big ideas´ of your topic.