Teaching Mathematics Students with Real-World Applications Stefan Baratto Clackamas CC AMATYC Webinar (AAS Committee) 12 May 2015
Download ReportTranscript Teaching Mathematics Students with Real-World Applications Stefan Baratto Clackamas CC AMATYC Webinar (AAS Committee) 12 May 2015
Teaching Mathematics Students with Real-World Applications Stefan Baratto Clackamas CC AMATYC Webinar (AAS Committee) 12 May 2015 Skills (of course) Problem Solving (reduce fear) Critical Thinking (advance and improve) Technical Communication (the point of it all) Other? (barriers and anxieties) Do we have any additional goals for our developmental math students? Learn Math Skills through Applications Learn their skills Students become more comfortable reading and solving problems Lead students to think critically about the content Difficulties Posed by Reading Level Developmental math & reading Relevant applications are easier to understand Difficulties Posed by Abstract Math We are not good at this Wason Selection Tasks Activity 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 Activity Teaches them my expectations Careful reading; critical thinking Easy, short Provides a wealth of teachable moments on the first day Especially when they ask, “what does this have to do with math?” Surprising how many get this activity wrong! Math is relevant, even to the developmental student Millennial generation needs relevance Finding applications that students care about and are within their abilities is a challenge Set of transferable skills that B&I feels that students should have when they graduate college B&I felt (1990s) that too many graduates didn’t possess these skills Some of these skills are best learned in math classes Communicating technical information is high on the SCANS skills list This 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! Reading Percent Problems 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. B $200 We began with this. A $40 $40 is a part of the original base. R 20% 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? R 60% B 80 60% is the percent. 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 30GB 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 R 107.25% 1.0725 The base is the list price, B $299. We use the percent relationship to solve the problem. A RB 1.0725 299 R B 320.6775 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 R 7.25%, but then, after computing the amount, we would need to add it to the original price to get the actual selling price. Relevance They all own iPods or the like; they can relate They all pay sales tax Critical Thinking Skills Rate: 107.725% If using 7.725%, then add back at the end Developmental Algebra Skill Distributing a negative sign always gives them trouble Easier to teach in context Distributing a Negative 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 x 285 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 x 6.27 x 285 14.95 x 6.27 x 285 8.68 x 285 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 application Demonstrate how the application requires the math they are about to learn Move to the math content, in the abstract 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 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? 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 class I think they keep expecting me to switch to more abstract math; we don’t If I give students an input and ask them for profit, most students will compute the revenue and cost separately and then find the difference This works well for a single input This does not work well when constructing a model to use for many inputs Our job is to help them learn to generalize so that they can take advantage of the power of algebra 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 revenue They understand this My students gain a better understanding of the need to wrap the entire cost in parentheses More 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 We moved on to abstract examples More students were able to demonstrate mastery of this skill A month later, on their final exams, more students demonstrated that they had learned this well than in any other class I could remember This continues term after term I am stunned. And thrilled!!! I shouldn’t be This is classic Wason Test material It is still awesome! What works? What does not work? It is not enough to simply bring applications into the classroom Students need to be interested in the application But, inherent interest is merely a beginning Sometimes, I think that my students come to class just to tell me how much they hate math in general and word problems in particular Why is it that humanities and social science majors (students who “read for a living”) hate word problems? Teach students to read a word problem Help them discern the important “math” in a paragraph Separate the math into the parts of a problem and help students to understand how these parts fit together This is the basic challenge Of course, in order to motivate students, they need to find an application relevant How do you find your applications? Using what they know can help with topics that they find especially difficult We introduce students to functions in developmental algebra Difficult Idea No element in the domain can be paired with more than one element in the range More than one element in the domain can be associated with the same range element They are very familiar with grades Spreadsheets work with functions This example uses a real-life function whose range is not a set of numbers Functions and Grades Activity: Functions in Developmental Math A student’s course grade is determined by adding up the total number of points the student earned that term and dividing by the total number of points available. This number is then multiplied by 100 to put the student’s raw score on a 100-point scale (assume there is no extra credit available). A computer program assigns a letter grade to the student based on their raw score. It assigns A for raw scores between 90 and 100 B for scores greater than or equal to 80, but less than 90 C for scores greater than or equal to 70, but less than 80 D for scores greater than or equal to 60, but less than 70 F for scores less than 60 a. What letter grade would be assigned to a student whose raw score is 92? b. What letter grade would be assigned to a student whose raw score is 96? Activity: Functions in Developmental Math c. Consider a set of students with raw scores 74, 86, 92, 86, and 96. What grades would be assigned to these five students? d. Write the raw scores and letter grades from exercise (c) as ordered pairs. e. In this context, are letter grades a function of raw scores? f. Are raw scores a function of letter grades? g. Use the definition of a function to justify your answers to exercises (e) and (f). h. What is the domain and range of the function that assigns a letter grade to each raw score? Modern Items and brands Relevant Students “want” them Careers Surprises Up-to-date Prices Students know about gas prices and traveling Use what they know Rates 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 0.1455 57, 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 22.7 57, 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 6.87 8,336 The plane uses nearly 7 gal of fuel for each mile flown. Government National Weather Service (NWS) Bureau of Labor Statistics (BLS) National Center for Education Statistics (NCES) Commercial Amazon and Google Most popular lists Zillow Popular (upscale and trendy brands) Use the News Elections Events Math & Statistics 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 4 5 5 11 Source: National Weather Service 2001-05 14 2006-10 11 Construct a bar graph to display this information. Severe Atlantic Hurricanes 16 14 12 10 8 6 4 2 0 1981-85 1986-90 1991-95 1996-00 2001-05 2006-10 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. Use what they like Math & Statistics Activity: Developmental 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 applications Use this same application when teaching pie charts Students understand and you do not need to completely introduce a new application Math & Statistics Activity: Developmental Math and Statistics The pie chart shows the top cocoa producing nations in 2011. Cocoa Production 2011 Other 23% Ivory Coast 30% Brazil 5% Cameroon 6% Ghana 16% Indonesia 20% 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. 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 Speak with faculty in other disciplines on campus They know the employers that will hire their students You might be surprised at how willing they are to help Campus Data My students couldn’t care less about acid concentrations or alcohol solutions They do not envision themselves as chemists Nor do they care about the number of student or general admission tickets sold Count the tickets They do see themselves owning a business or in the health sciences fields Real-World Applications 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? 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? 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? Courses are sometimes jam-packed with content It 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 time Applications and problem solving are what math is all about Apps are all about developing critical-thinking skills Good applications increase student interest Increased student interest increases student attentiveness Increased attentiveness increases student learning and retention 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? I want students to answer apps with sentences I’ve learned to actually write sentences on the board when I complete an app in class Students should write the answer to an application as a sentence This “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! 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? Do you model what you want students to do? Do your students get to practice in class? 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 Students who watch us do math problems learn to watch us do math problems Always follow an instructor-led example with student work Many faculty members need to be taught this Use a computer, document camera, or hand- outs Give the students time to work on a problem, then ask questions about how to proceed Strongly encourage them to work with their neighbors Complete the problem for those who had trouble; write the answer as a complete sentence You can design these questions to elicit certain outcomes You 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 Critical Thinking 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. R x 14.95 x C x 6.27 x 285 P x 8.68 x 285 a. How many do they need to sell if they need at least $750 in revenue? R x 14.95 x 14.95 x 750 750 50.17 14.95 51 Round up! 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? C x 6.27 x 285 6.27 x 285 1, 200 6.27 x 915 915 145.9 6.27 145 Round down! 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? Applications make the math feel important to the students They can relate to real world applications Build on an application They will feed off of your enthusiasm Working in groups gives the class a community feel and spirit Thank you for joining and participating Please Email me with thoughts, ideas, comments, or to request these files Stefan Baratto [email protected] Clackamas Community College Oregon City, OR Thank you for your hospitality Enjoy the rest of the weekend