Teaching Mathematics Studentswith Real-World Applications

Content in Context

Stefan Baratto Clackamas CC

AMATYC Conference (S178)

16 November 2014

Skills(of course)

Problem Solving(reduce fear)

Critical Thinking(advance and improve)

Developmental Math: Goals

Technical Communication(the point of it all)

Other?(barriers and anxieties)

Developmental Math: Goals

Learn Math Skills through ApplicationsLearn their skillsStudents become more comfortable

reading and solving problemsLead students to think critically about

the content

Meeting the Goals

Difficulties Posed by Reading LevelDevelopmental math & readingRelevant applications are easier to

understandDifficulties Posed by Abstract Math

We are not good at this

Applications

Wason Selection Tasks Activity

Relevant or Abstract

Activity: Wason Selection Tasks Four cards are laying on a table. Each card has a letter of the alphabet on one side and a number on the other side. You are given the rule: If there is a vowel on one side of the card, then there must be an even number on the other side. You are to determine which (if any) of the cards must be turned over in order to determine whether the rule is being followed. You want to flip the minimum number of cards necessary to accomplish this. The cards on the table are as follows:

T 6 E 9 You want to turn over the fewest number of cards possible to determine whether the rule is being followed. For each card, determine if you need to turn it over, and write a short sentence justifying your choice.

Activity: Wason Selection Tasks As a bartender in an all-ages club that serves alcohol, your job is to ensure compliance with the following rule: Patrons under 21 may not drink alcohol. Toward that end, you may ask an individual their age, or check what they are drinking, but you may not be more intrusive than absolutely necessary. Four people sitting at the bar are shown below. #1: Drinking Water #2: Over 21 #3: Drinking Beer #4: Under 21 In which cases (if any) should you ask a patron’s age, or determine what beverage they are consuming? For each patron, write a short sentence justifying your choice.

Wason Selection Tasks ActivityTeaches them my expectations

Careful reading; critical thinking Easy, shortProvides a wealth of teachable moments

on the first dayEspecially when they ask, “what does this

have to do with math?”Surprising how many get this activity wrong!

Relevant or Abstract

Math is relevant, even to the developmental studentMillennial generation needs relevance

Finding applications that students care about and are within their abilities is a challenge

Relevance

Set of transferable skills that B&I feels that students should have when they graduate collegeB&I felt (1990s) that too many

graduates didn’t possess these skillsSome of these skills are best learned

in math classes

SCANS Skills

Communicating technical information is high on the SCANS skills listThis involves the ability to read, take in,

and think critically about technical writing

It also involves the ability to communicate technical information

This is important!It can be dull!

Technical Communication

Reading Percent Problems

Reading and Understanding

Activity: Reading Percent Problems Definition: Base, Amount, and Rate The base B is the whole in a problem. It is the standard used for comparison. Often, the base is the initial quantity and may be paired with the word “of.” The amount A is the part of the whole being compared to the base. The amount is often paired with the word “is.” The rate R is the ratio of the amount to the base. The rate is usually given as a percent. Most any percent problem comes down to finding one of these three elements. Given two of them, we can always find the third. The first step is to identify the elements in a problem. Example 1: Identifying the Elements of a Percent Problem Identify B, A, and R in the statement: $40 is 20% of $200.

$200B We began with this. $40A $40 is a part of the original base. 20%R 20% is the percent. Often, it will be written as 0.20.

Activity: Reading Percent Problems Example 2: Identifying the Elements of a Percent Problem with

an Unknown Identify A, R, and B in the following percent problem. 60% of the 80 students who took MTH 050 last term are now

working. How many are working?

60%R 60% is the percent. 80B The base is 80 students; this is the total number of

students we began with. A is unknown

We do not know how many of the original number (80) are working.

Activity: Reading Percent Problems A state adds a 7.25% sales tax to the price of most goods. If a 30-GB iPod is listed for $299, how much will it cost after the sales tax has been added?

If we use the price, including tax, as the unknown amount, then the rate is

107.25% 1.0725R The base is the list price, $299.B We use the percent relationship to solve the problem.

1.0725 299

320.6775BR

A R B

Because our answer refers to a selling price, we round to two decimal places. The iPod sells for $320.68, after the sales tax has been included. Note: We could use 7.25%,R but then, after computing the amount, we would need to add it to the original price to get the actual selling price.

RelevanceThey all own iPods or the like; they can

relateThey all pay sales tax

Critical Thinking SkillsRate: 107.725%If using 7.725%, then add back at the

end

Reading and Understanding

Developmental Algebra SkillDistributing a negative sign always

gives them troubleEasier to teach in context

Distributing a Negative

Skills and Critical Thinking

Activity: Distributing a Negative Funghi Books pays $6.27 for each copy of The Forager’s Mushroom Cookbook (wholesale cost). They estimate that the weekly cost of selling the book is $285. The bookstore sells each copy for $14.95. a. Write an expression that describes the revenue Funghi Books

earns from selling this book. Let x be the number of books sold. 14.95x b. How much does the bookstore bring in if they sell 75 copies one

week? 14.95 75 1,121.25

They bring in $1,121.25 if they sell 75 copies. c. Write an expression for the weekly cost of selling this book. 6.27 285x

Activity: Distributing a Negative d. How much does it cost the bookstore to sell 75 copies in one

week? 6.27 75 285 755.25

It costs the store $755.25 to sell 75 copies in a week. e. Construct a simplified profit model for the sale of this book.

14.95 6.27 285 14.95 6.27 285

8.68 285

x x x x

x

f. How much profit does the store earn if they sell 75 copies in one

week? 8.68 75 285 366

They earn $366 if they sell 75 copies in a week.

Introduce a topic with an applicationDemonstrate how the application

requires the math they are about to learn

Move to the math content, in the abstract

Past Practices

Do examples; give students examples to do

Return to the application and complete the math

If time permits, students would have the opportunity to practice with an application, as well

Past Practices

Do your students maintain their motivation when you transition to more abstract math?

What have you done to help students be more successful with this skill?

Your Experiences

I introduce students to the basic business models of revenue, cost, and profit early in the course

Students maintain their interest when we discuss these models in classI think they keep expecting me to switch

to more abstract math; we don’t

Present Practice

If I give students an input and ask them for profit, most students will compute the revenue and cost separately and then find the differenceThis works well for a single inputThis does not work well when constructing

a model to use for many inputsOur job is to help them learn to generalize

so that they can take advantage of the power of algebra

Understanding Students

I now allow them to find a profit by finding the difference between revenue and cost one time

We talk about how they are subtracting the entire cost from the revenueThey understand this

Present Practice

My students gain a better understanding of the need to wrap the entire cost in parenthesesMore of them seem to understand that

they need to distribute the negative sign to encompass the entire cost

They gained a better understanding of the role of parentheses, in general

Present Practice

We moved on to abstract examplesMore students were able to demonstrate

mastery of this skillA month later, on their final exams,

more students demonstrated that they had learned this well than in any other class I could rememberThis continues term after term

The Upshot

I am stunned. And thrilled!!!I shouldn’t beThis is classic Wason Test materialIt is still awesome!

The Upshot

What works? What does not work?It is not enough to simply bring

applications into the classroomStudents need to be interested in the

applicationBut, inherent interest is merely a

beginning

Using Applications

Sometimes, I think that my students come to class just to tell me how much they hate math in general and word problems in particularWhy is it that humanities and social

science majors (students who “read for a living”) hate word problems?

Using Applications

Teach students to read a word problemHelp them discern the important “math”

in a paragraphSeparate the math into the parts of a

problem and help students to understand how these parts fit togetherThis is the basic challenge

Sound Fundamentals

Of course, in order to motivate students, they need to find an application relevant

How do you find your applications?

Your Experiences

ModernItems and brands

RelevantStudents “want” themCareersSurprises

Up-to-datePrices

Applications

Students know about gas prices and travelingUse what they know

Rates

Surprises: Thinking Critically

Activity: Rates A Boeing 747 can travel 8,336 mi on one 57,285-gal tank of airplane fuel. a. Report the gas mileage of this plane.

8,336

0.145557,285

The plane gets 0.1455 miles to the gallon. b. Report the passenger-miles flown if a Boeing 747 carries 156

passengers for one full tank. 156 8,336 1,300,416 The plane flies 1,300,416 passenger-miles. c. Report the passenger-miles per gallon.

Compare this to your answer in part a.

1,300,416

22.757,285

The plane gets about 22.7 passenger-miles per gallon. This is a much more reasonable means of determining the plane’s

fuel efficiency.

Activity: Rates d. A Ford Explorer gets about 15 mpg. If the driver is the only

person in the Explorer, how does its fuel efficiency compare to your answer in part c?

The plane is more fuel efficient. e. Compute the rate of gallons of fuel used by the plane per mile

flown.

57,285

6.878,336

The plane uses nearly 7 gal of fuel for each mile flown.

GovernmentNational Weather Service (NWS)Bureau of Labor Statistics (BLS)National Center for Education Statistics

(NCES)Commercial

Amazon and GoogleMost popular lists

ZillowPopular (upscale and trendy brands)

Finding Applications

Use the NewsElectionsEvents

Math & Statistics

Relevance

Activity: Developmental Math and Statistics The total number of severe Atlantic hurricanes (Categories 4 and 5) are

given for each five-year period. Severe Atlantic Hurricanes 1981-85 1986-90 1991-95 1996-2000 2001-05 2006-10 4 5 5 11 14 11 Source: National Weather Service Construct a bar graph to display this information.

How many severe Atlantic hurricanes occurred between 2006 and 2010? There were 11 severe Atlantic hurricanes between 2006 and 2010. What was the percent increase in the number of severe Atlantic hurricanes

between the periods 1996-2000 and 2001-2005 (to the nearest whole percent)?

There was a 27% increase in the number of severe Atlantic hurricanes. Which period saw the largest increase over the period that came before it?

The largest increase occurred between the 1991-1995 and 1996-2000 periods.

0

2

4

6

8

10

12

14

16

1981-85 1986-90 1991-95 1996-00 2001-05 2006-10

Severe Atlantic Hurricanes

Use what they like

Math & Statistics

Students

Activity: Developemental Math and Statistics The table gives the cocoa bean production in a recent growing season, in millions of pounds, along with value of that season’s crop, in millions of dollars. Cocoa Bean Production and Value Production Share of Value Nation (millions of pounds) World’s Total (millions of dollars) Ivory Coast 2,706 34.75% $3,287 Ghana 1,608 20.62% 1,951 Indonesia 1,078 13.84% 1,309 Cameroon 462 5.93% 561 Nigeria 462 5.93% 561 Brazil 363 4.66% 441 Ecuador 286 3.67% 347 Malaysia 70 0.90% 86 Other 755 9.69% 917 Source: International Cocoa Organization; IndexMundi What was the world’s total production of cocoa beans that growing season? 7,788,000,000 lb of cocoa beans were grown. What was the total value of the world’s cocoa production that season? That year’s crop was worth $9,460,000,000

Activity: Developmental Math and Statistics In a subsequent year, the Ivory Coast’s production fell to 2,688 million

pounds. Find the percent decrease this represents (round to the nearest hundredth of a percent).

The percent decrease was 0.67%. In that same year, Indonesia’s production increased to 1,760 million pounds.

What percent increase does this represent (round to the nearest whole percent)?

This was a 63% increase.

Build on applicationsUse this same application when

teaching pie chartsStudents understand and you do not

need to completely introduce a new application

Math & Statistics

Students

Activity: Developmental Math and Statistics The pie chart shows the top cocoa producing nations in 2011.

Source: U.N. Food & Agricultural Organization

Which country was the largest cocoa bean producer in 2011? What

percent of the world’s cocoa beans were grown by this country?

The Ivory Coast was the largest producer with 30% of the world’s total production.

A global total of 8,980 million pounds of cocoa beans were grown

in 2011. How many pounds did Cameroon produce?

Cameroon produced about 539,000,000 lb of cocoa beans in 2011.

Ivory Coast30%

Indonesia20%Ghana

16%

Cameroon6%

Brazil5%

Other23%

Cocoa Production 2011

Many students who are less than strong in math see themselves as business majors.This explains a lot about our nation’s

economic situation over the last decade plus

Business and Finance can provide you with a rich source of applications which build one off of the other

Other Sources

Speak with faculty in other disciplines on campusThey know the employers that will hire

their studentsYou might be surprised at how willing

they are to helpCampus Data

Campus Resources

My students couldn’t care less about acid concentrations or alcohol solutionsThey do not envision themselves as chemists

Nor do they care about the number of student or general admission tickets soldCount the tickets

They do see themselves owning a business or in the health sciences fields

Likes and Dislikes

Real-World Applications

Likes and Dislikes

Activity: Real-World Applications Mixture Problems A coffee reseller wishes to mix two types of coffee beans for the House Blend. The Kona bean that she wants to use wholesales for $4.50 per pound; the Sumatran bean wholesales for $3.25 per pound. If she wishes to mix 200 pounds of beans for a wholesale price of $4 per pound, how many pounds of each type of coffee bean should she include in the mix? Check Yourself Minh splits his $20,000 investment between two funds. At the end of a year, one fund grows by 3.25% and the other grows 4.5%. If the total earnings on his investment came to $793.75, how much did he invest in each fund? Check Yourself Currently, 8% of a 42-gal mixture of patching compound is water. Local conditions require the mixture to be 13% water. How much water needs to be added to the mix in order for it to be 13% water (round to three decimal places)? What will the total volume of the mixture be after the water is added?

Activity: Real-World Applications Products A toy store is selling the Fisher-Price Rollin’ Rumblin’ Dump Truck at a 10% discount for $16.19. How much does the toy normally sell for? In order to make room for the new fall line of merchandise, a proprietor offers to discount all existing stock by 15%. How much would you pay for a Fendi handbag that the store usually sells for $229? A store sells a certain Kicker amplifier model for a car stereo system for $249.95. If the store pays $199.95 for the amplifier, what is their markup percentage for the item (to the nearest whole percent)?

Have you worked with faculty in other departments to improve course content?Have you worked with them to find

applications for your classroom?What careers are the developmental

math students at your college likely to enter?

Your Experiences

Courses are sometimes jam-packed with contentIt is difficult getting to it all in a term

without having to try and find the time to also do applications with students

Do not think of apps as cutting into class timeApplications and problem solving are what

math is all aboutApps are all about developing critical-

thinking skills

Motivating an Instructor

Good applications increase student interestIncreased student interest increases

student attentivenessIncreased attentiveness increases

student learning and retention

Apps in the Classroom

How much of what you teach do your students actually learn and retain?Even your best students?

Is it better to increase the proportion of content they learn by doing a better job with fewer topics?

Your Experiences

I want students to answer apps with sentencesI’ve learned to actually write sentences on the

board when I complete an app in classStudents should write the answer to an

application as a sentenceThis “forces” them to go back and re-read the

problem“Did they ask for the discount or the original

price?”Go back and see what the original question

asked!

Modeling for Students

Why is it that we assume they need help with the math (they do), but we also assume they are able to describe math in sentences?Why do we model the former but not the

latter?

Modeling for Students

Do you model what you want students to do?

Do your students get to practice in class?

Your Experiences

Just like working with abstract math, students need classroom practice working with apps

Strongly encourage students to work with their neighbors, especially when working on an application

Using Apps in Class

Students who watch us do math problems learn to watch us do math problems

Always follow an instructor-led example with student workMany faculty members need to be

taught thisUse a computer, document camera, or

hand-outs

Doing is Learning

Give the students time to work on a problem, then ask questions about how to proceedStrongly encourage them to work with

their neighborsComplete the problem for those who

had trouble; write the answer as a complete sentence

Doing is Learning

You can design these questions to elicit certain outcomesYou can write a revenue problem so that

they need to round up, even though the fraction part is less than a half

You can write a cost problem so they need to round down, even though the fraction part is greater than a half

Never pass up a teachable moment

Teachable Moments

Teachable MomentsCritical Thinking

Teachable Moments

Activity: Teachable Moments & Critical Thinking Recall Funghi Books pays $6.27 for each copy of The Forager’s Mushroom Cookbook (wholesale cost). They estimate that the weekly cost of selling the book is $285. The bookstore sells each copy for $14.95.

14.95

6.27 285

8.68 285

R x x

C x x

P x x

a. How many do they need to sell if they need at least $750 in

revenue?

14.95

14.95 750

75050.1

Round u

714.

!9

1 p5

5

R x x

x

x

They need to sell 51 books in order to earn at least $750 in revenue.

Activity: Teachable Moments & Critical Thinking

b. How many can they sell if their costs cannot exceed $1,200?

6.27 285

6.27 285 1,200

6.27 915

915145.9

6.27Round down!145

C x x

x

x

x

They can sell up to 145 books without costs exceeding $1,200.

Do your students even reach really teachable moments when working with math in the abstract?

Your Experiences

Applications make the math feel important to the students

They can relate to real world applications

Build on an applicationThey will feed off of your enthusiasmWorking in groups gives the class a

community feel and spirit

Summary

Thank you for joining and participating

Please Email me with thoughts, ideas, comments, or to request these files

Stefan Barattosbaratto@clackamas.eduClackamas Community College Oregon City, OR

Thank you for your hospitalityEnjoy the rest of the weekend

Thank You