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Mathematics Live Satellite PresentationJanuary 11th, 2012 @ 4:00 p.m.

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Mathematics Live Satellite PresentationJanuary 11th, 2011@ 4:00 p.m.

Today’s Topic

Mathematics: Context & Connections

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MathematicsContext & Connections

Putting learning in context & making connections to the experiences of learners are powerful processes in developing mathematical understanding. When mathematical ideas are connected to each other or to real-world phenomena, students begin to view mathematics as useful, relevant & integrated.

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The Mathematical Processes

CommunicationContext & ConnectionsMental Math & EstimationProblem SolvingReasoningTechnologyVisualization

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The Mathematical Processes

Communication

Students need opportunities to read about, represent, view, write about, listen to, & discuss mathematical ideas using both personal & mathematical language & symbols. Concrete, pictorial, physical, verbal, written, & mental representations of mathematical ideas should be encouraged & used to help students make connections & strengthen their understandings.

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The Mathematical Processes

Mental Math & Estimation

Mental mathematics is a combination of cognitive strategies that enhance flexible thinking & number sense. It is calculating mentally & reasoning about the relative size of quantities without the use of external memory aids. Estimation is used to make mathematical judgements & develop useful, efficient strategies for dealing with situations in daily life.

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The Mathematical Processes

Problem Solving

In order for an activity to be problem based, it must ask students to determine a way to get from what is known to what is sought. If students have already been given ways to solve the problem, it is not problem solving but practice. Problem solving is a powerful teaching tool that fosters multiple & creative solutions.

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The Mathematical Processes

ReasoningMathematical experiences in & out of the classroom should provide opportunities for students to engage in inductive & deductive reasoning. Inductive reasoning occurs when students explore & record results, analyze observations, make generalizations from patterns, & test these generalizations. Deductive reasoning occurs when students reach new conclusions based upon what is already known or assumed to be true.

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The Mathematical Processes

TechnologyTechnology tools contribute to achievement of a wider range of outcomes, & enable students to explore & create patterns, examine relationships, test conjectures, & solve problems. Calculators, computers, & other forms of technology can be used to explore & demonstrate mathematical patterns, organize & display data, extrapolate & interpolate, decrease the time spent on computations when other mathematical learning is the focus.

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The Mathematical Processes

Visualization

Visual images & visual reasoning are important components of number sense, spatial sense, & logical thinking. Number visualization occurs when students create mental representations of numbers & visual ways to compare those numbers.

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The Mathematical Processes

CommunicationContext & ConnectionsMental Math & EstimationProblem SolvingReasoningTechnologyVisualization

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Context & Connections

Putting learning in context & making connections to the experiences of learners are powerful processes in developing mathematical understanding. When mathematical ideas are connected to each other or to real-world phenomena, students begin to view mathematics as useful, relevant & integrated.

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Mathematics in ContextReal World Events

http://www.youtube.com/watch?v=jRMVjHjYB6w

http://blog.mrmeyer.com/

The video that Dan Meyer presents demonstrates a context for mathematics.

While using media can be effective, mathematics can be put into relevant context for students in a variety of ways.

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Mathematics Connected to the Experiences of the Learner

Many of our rural students have a connection to the agricultural economy of our province. There is an abundance of mathematics in daily life on the farm, especially related to measurement outcomes:

linear measurementsurface areavolumeratio & ratesbudgets & cost analysis

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Activity 1A Farmer’s Dilemma

Mathematics in the Workplace

Joe is a 56 year-old farmer. In 2004 his wheat harvest was larger than expected and, as a result, he ran out of bin space. In a panic to complete the harvest he augered several truck loads of grain directly onto the ground forming a cone shaped pile.

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Activity 1A Farmer’s Dilemma

Mathematics in the Workplace

His dilemma …. Leave the grain out over winter or purchase more bins.

He decided to purchase more grain bins to get his wheat under cover before winter. Unsure of how many truck loads he had dumped on the ground, he needed to calculate the number of bushels of wheat in the pile in order to purchase the correct number & size of grain bins.

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A Farmer’s Dilemma – Task 1

Discuss what information Joe will need, & how he might get that information.

Mathematical thinking break (3 minutes)

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A Farmer’s Dilemma – Task 1

Useful Information Ways to Get This Information

various formulae textbook, teacher, parent, web, other students

circumference of the pile of wheat pace it, string, tape measure, trundle wheel

diameter & radius estimate, referent, tape measure, formula

angle of incidence with the ground estimate, home made device, from auger

height of pile estimate, pole probe, referent, trigonometry

slant height (slant distance) tape measure, trigonometry, Pythagorean Th.

linear conversions Metric ↔ Imperial textbook, teacher, parent, web, other students

volume conversions .. ex .. cubic feet to bushels textbook, teacher, parent, web, other students

grain bin sizes community, teacher, parent, web, other students

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A Farmer’s Dilemma – Task 2

If the circumference of the pile was 60 meters, & the cone was shaped with an angle of 35 degrees, how many bushels would be in the pile? How many bins might he purchase?

Or ……

If the circumference of the pile was 195 feet & the slant height was 38 feet, how many bushels would be in the pile? How many bins might he purchase?

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Mathematics in ContextReal World Events

Pat Kiernan, a journalist that used to work in Edmonton has “Pat’s Papers” & he wrote about a graph that Epcor produced around water consumption.

proportional reasoning related to ratesdata collectiondata analysisdata displaystatistical analysis

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Who used more water during breaks in the game?

Which city had the biggest range of water consumption?

What was the biggest range in the shortest amount of time?

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A question posed in context can have connections to multiple curricular outcomes.

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http://pods.dasnr.okstate.edu/docushare/dsweb/Get/Document-2216/BAE-1501web.pdf

http://ks.water.usgs.gov/waterwatch/flood/conv.html#factors

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Mathematics in ContextReal World Events

Elections provide an excellent source of data & information for mathematical exploration, activity & discussion:

data collectiondata analysisdata displaystatistical analysis

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• The chart clearly shows that substantial variations are found using different survey techniques by the different companies, even when polling companies conduct interviews on the same days. For example, there was a 10 point (46-35) spread among polls reporting Conservative support in the closing days of the campaign, April 28-29, & an 8 point (33-26) spread for the NDP. Seven polls were conducted in the period covering April 14-7, & yet the results provided an 8 point (17-24) spread for the NDP that included a healthy overlap with Liberal support. Also, polls conducted April 5-6 produced a 10 point spread for the Conservatives & an 8 point span for the Liberals. Such wide divergences seriously undermine the public`s ability to gain clear insights into party support. The most likely culprit in this variations is how different survey companies deal with the "soft" voter, but there are other explanations as well.

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• These variations are the result of quite different methodologies adopted by the polling companies. One difference lies in what specific questions are asked. For example, the Nanos polls simply asks an open-ended question about which party a person is likely to vote for. Ekos asks that question, but then provides a list of parties, including the Green Party. As a result, Ekos usually reports much higher Green support than Nanos. A third way of putting the questions is to attach the party leaders' names to the list of parties: i.e. "Stephen Harper's Conservative Party.”

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• A further difference among the companies is how aggressively they try to uncover the preferences of voters who initially say that they haven't made up their minds. If a person answers "don't know" to the first question. Most companies will follow up with a question asking about the party a person may be leaning towards. Different variations of this question may be posed repeatedly. A few companies also use a person's answers to other questions to statistically predict how they would vote, based on a model of how other voters with those share traits are known to be voting. The result is that some companies will report much lower levels of "undecided" voters than others. But the parties with the lowest levels of support are likely to include a substantial amount of soft support, voters who are only marginally attached to this party & who might easily switch to another.

• While the actual level of support could have been anywhere in that range, it is more likely to lie somewhere in the middle. Alternatively, the highest extreme poll results could have been the legendary "20th poll;" the highest is mentioned here simply because those are the ones that stand out from the rest of the poll results. Statistically, polls are reliable within their margin of error 19 times out of 20; thus it is quite possible for an occasional poll to report out of range results that do not reflect reality.

• http://www.sfu.ca/~aheard/elections/polls-scatter-plot-2011.html

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• What are other examples in print or media that contain margin of error, confidence intervals, or confidence level?

• For homework, find an example for the class to discuss.

• How its Made: Baseballs

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http://www.madehow.com/Volume-1/Baseball.html

• A statistically representative sample of each shipment of baseballs is tested to measure Co-Efficient Of Restitution (COR), using Major League Baseball's officially sanctioned testing procedures. Essentially, the COR is an indication of the resiliency of a baseball.

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• The COR test involves shooting a baseball from an air cannon at a velocity of 85-feet-a-second (25.90-meters-a-second) at a wooden wall from a distance of eight feet (2.43 meters), & measuring the speed with which the ball rebounds off the wall. Major League COR specifications stipulate that a baseball must rebound at 54.6 percent of the initial velocity, plus or minus 3.2 percent.

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• A baseball must also retain its round shape after being hit 200 times by a 65-pound (29.51 kilograms) force. As proof of its strength, a baseball must distort less than 0.08 of an inch (.20 centimeter) after being compressed between two anvils.

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How a baseball is made video

• http://www.youtube.com/watch?v=mfPuRoStEdw

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Mathematics in ContextStudents making Connections

There are few topics that carry as many existing ideas as probability. By the time students reach a 30 level course they will all have made personal connections to a variety of contexts involving probability.

probabilityreasoningpsychology

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The Monty Hall Problem• You are shown 3 identical doors. Behind one of them

is a car. The other 2 conceal goats. You are asked to choose, but not open, one of the doors. After doing so, Monty, who knows where the car is, opens one of the two remaining doors. He always opens a door he knows to be incorrect, & randomly chooses which door to open when he has more than one option. After opening an incorrect door, Monty gives you the option of either switching to the other unopened door or sticking with your original choice. You then receive whatever is behind the door you choose. What should you do?

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Student Task

• Draw & explain your solution for whether you should switch or stay.

• Be prepared to convince others of your answer.

• Mathematical thinking break (3 minutes)• http://connect.edonline.sk.ca/acmed-prolearn

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• Students can conduct an experiment using cards. One player is Monty & knows what is behind all three doors.

• Have students collect data always choosing to switch or alternately always choosing to stay. In groups of two, give one student from each group three cards.

• Record how many times you won the car!

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The Sample Space

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• “the monty hall problem” written by Jason Rosenhouse presents a look at the remarkable story of this problem.

• Included in this book are quotes from mathematicians who responded to Marilyn vos Savant, a Q & A columnist for Parade magazine, who answered 2/3 & 1/3 to this problem

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Since you seem to enjoy coming straight to the point, I’ll do the same. In the following question & answer, you blew it! Let me explain. If one door is shown to be a loser, that information changes the probability of either remaining choice, neither of which has any reason to be more likely, to ½. As a professional mathematician, I’m very concerned with the general public’s lack of mathematical skills. Please help by confessing your error & in the future being more careful.

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• You blew it, & you blew it big! Since you seem to have difficulty grasping the basic principle at work here, I’ll explain. After the host reveals a goat, you now have a one-in-two chance of being correct. Whether you change your selection or not, the chances are the same. There is enough mathematical illiteracy in this country, & we don’t need the world’s highest I.Q. propagating more. Shame!

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Everyone can make a mistake.

• It is ok to question the position that others take on a problem, & even the best in the field will make a mistake or struggle with a math problem. If there was no struggle, it wouldn’t be called a problem.

• Defending an answer, even if it is incorrect has value & can lead to the correct answer.

• If everyone had the correct answer, then what would we spend our time in class learning about?

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Paul Lockhart

• “Mathematics is the art of explanation. If you deny students the opportunity to engage in this activity- to pose their own problems, to make their own conjectures & discoveries, to be wrong, to be creatively frustrated, to have an inspiration, & to cobble together their own explanations & proofs – you deny them mathematics itself.” (p. 29, 2009)

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Post-broadcast questions can be emailed to Lisa Eberharter at

lisa.eberharter@gov.sk.ca http://connect.edonline.sk.ca/acmed-prolearn