Candace Walkington- Problems vs PBI

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



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



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



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)



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?



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.




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.




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?




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?



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?




Instruction Students applied well-know processes

to solving story problems that didn¶t fit

the story and what was being asked.




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.




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 .




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.




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.




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?



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.



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.

Candace Walkington- Problems vs PBI

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