**Probability & Problem Solving**

Probability is useful in daily life, plays an instrumental role in other disciplines, is needed in many professions, and probability reasoning has an important role in decision making (Contreras, Batanero, Diaz, & Fernandes, 2011).

Probability is challenging. See the following article: “Why Do People Find Probability Unintuituve and Difficult?” at http://nrich.maths.org/7326

Problem Solving

A. If you have a probability bag with 10 marbles in total (4 red, 3 blue, 2 green, and 1 yellow) what is the probability of drawing a green marble?

VS.

B. Ten marbles of varying colours are chosen at random and placed in a bag. What are the colours of the 10 marbles in the bag?

Which question provides a better opportunity for problem solving?

References

Batanero, C., Biehler, R., Maxara, C., Engel, J., Vogel, M. (2004 or later). Using simulation to bridge teachers´ content and pedagogical knowledge in probability, in 15th ICMI Study Conference: The Professional Education and Development of Teachers of Mathematics. http://stwww.weizmann.ac.il/G-math/ICMI/strand1.html.

Batanero, C. and Diaz, C. (2012). Training school teachers to teach probability: Reflections and challenges. Chilean Journal of Statistics. 3(1), 3-13.

Batanero, C., Godino, J. D., and Roa, R. (2004). Training teachers to teach probability. Journal of Statistics Education. 12(1). www.amstat.org/publications/jse/v12n1/batanero.html

Baturo, A. R., Cooper, T. J., Doyle, K. M., & Grant, E. J. (2007). Using three levels in design of effective teacher-education tasks: the case of promoting conflicts with intuitive understandings in probability. Journal of Mathematics Teacher Education, 10(4-6), 251-259.

Contreras, J. M., Batanero, C., Diaz, C., & Fernandes, J. A. (2011). Prospective teachers’ common and specialized knowledge in a probability task. Proceedings of Working Group 5 on Stochastic Thinking. CERME 7. Retrieved from: http://www.cerme7.univ.rzeszow.pl/?id=call-for-papers

Jones, G. A. (2005). An overview of research into the teaching and learning of probability. Exploring Probability in School: Challenges for Teaching and Learning. http://books.google.ca/books?id=d0ybYaoKmegC&pg=PA64&lpg=PA64&dq=Jones+Thornton++%22An+overview+of+research+into+the+teaching+and+learning+of+probability%22+Exploring+probability+in+schools:+Challenges+for+teaching+andlearning&source=bl&ots=_tXipcChFH&sig=KzHgyR9CX7mWxn1umcXAZu8KIPk&hl=en&sa=X&ei=wYAJUpSyI4O4yQGqjoDQDQ&ved=0CCoQ6AEwAA#v=onepage&q&f=false

Ontario Ministry of Education and Training. (2005). The Ontario curriculum: Grades 1-8: Mathematics. Queens Park, Toronto: Author.

Reston, E. (2012). Exploring inservice elementary mathematics teachers’ conceptions of probability through inductive teaching and learning methods. 12th International Congress on Mathematical Education. 8 July – 15 July, 2012, COEX, Seoul, Korea.

Rigelman, N. R. (2007). Fostering mathematical thinking and problem solving: The teacher’s role. Teaching Children Mathematics. The National Council of Teachers of Mathematics, Inc.

Why Probability?

Probability can be challenging for students

•“Counterintuitive results in probability are found even at very elementary levels (for example, the fact that having obtained a run of four consecutive heads when tossing a coin does not affect the probability that the following coin will result in heads is counterintuitive)” (Batanero, & Diaz, 2012, p.5).

•“In arithmetic or geometry, an elementary operation (like addition) can be reversed and this reversibility can be represented with concrete materials". For example: 2 oranges + 3 oranges = 5 oranges, 3 oranges + 2 oranges = 5 oranges, 5 oranges - 2 oranges = 3 oranges, etc. (A child always obtains the same result, no matter how many times this operation is repeated.)

•Whereas, “in the case of random experiment, we obtain different results each time the experiment is carried out and the experiment cannot be reversed (we cannot get the first result again when repeating the experiment). This makes the learning of probability comparatively harder for children.”

Probability can be challenging for teachers

•“Teaching probability and statistics is not easy for mathematics teachers” (Batanero, Godino, and Roa, 2004).

•Studies of preservice teachers suggest that they have poor common and specialized knowledge of elementary probability and are thus likely to fail in “figuring out what students know; choosing and managing representations of mathematical ideas; selecting and modifying textbooks; deciding among alternative courses of action” (Ball, Lubienski, & Mewborn, 2001, p. 453 as cited in (Contreras, Batanero, Diaz, & Fernandes, 2011).

So, there is a need for further professional development about probability, but what format should this take?

“Since students build their knowledge in an active way, by solving problems and interacting with their classmates, we should use this same approach in training the teachers” especially if we want them to use a “constructivist and social approach in their teaching”; see Jaworski (2001) (Batanero, & Diaz, 2012, p.9).

“We should give teachers more responsibility in their own training and help them to develop creative and critical thinking,” by creating “suitable conditions for teachers to reflect on their previous beliefs about teaching and discuss these ideas with other colleagues” (Batanero, & Diaz, 2012, p.9).

“One fundamental learning experience that teachers should have to develop their probability thinking is working with experiments and investigations. To teach inquiry, teachers need to learn skills “such as the ability to cope with ambiguity and uncertainty; re-balance between teacher guidance and student independence and deep understanding of disciplinary content” (Batanero, & Diaz, 2012, p.9).

As cited in (Reston, 2012) Page 3:

•Keeler & Stein (2001, p.6) suggested using an inquiry-based approach that includes strategies that provide opportunities for students to confront misconceptions on the topic.

•Batanero and Diaz (2011) recommended implementing an experimental approach to probability through experiments and simulations in order for students to grasp the connections between the notions of relative frequency and probability.

•Chaput, Girard & Henry (2011) suggested the use of modelling approaches and simulation of models using computer tools in teaching statistics and probability.

•Inquiry learning problem-based learning and investigations, where students are presented with questions to be answered, problems to be solved or a set of observations to be explained (Bateman, 1990).

As cited in (Reston, 2012) Page 4:

•Effective implementation of inquiry learning will enable students to “formulate good questions, identify and collect appropriate evidence, present results systematically, analyze and interpret results, formulate conclusions, and evaluate the worth and importance of these conclusions” (Lee, 2004; cited in Prince and Felder, 2006, p. 9). These expected outcomes are in line with the goals for teaching probability and statistics courses (Gal & Garfield, 1997). (Reston, 2012, p.4)

•Problem-based learning (PBL) is recognized as another student-centered inductive teaching method in which students learn about a subject in the context of complex, multifaceted, and realistic problems. The teacher presents an open-ended, ill-structured, authentic (real world) problem to the class and students work in teams to identify their prior knowledge and learning needs including how and where to access new information that may lead to a viable solution or resolution of the problem with instructors acting as facilitators (Prince and Felder, 2006 as cited in Reston, 2012, p.4). In mathematics teaching, the problem may be a mathematical problem, task, real-world situation, case, phenomenon or event, which requires students to answer or find a solution.

Randomness

1.Which of the following are random events? Why?

a. Customers arriving at a mall on weekends

b. Patients walking in a doctor’s clinic

c. Bank posting of interest in a savings account

d. Trading at the New York Stock Exchange

e. Winning in the lottery or LOTTO

2. Given the following events, which of the following are random properties associated with the event? Explain your answer.

a. Weather in London, ON: precipitation, seasonal change, temperature on specific days

b. Car accidents on Highway 401: driving practices, specific cars or conditions met on road

c. Arrival of customers at the Eaton Centre: hours open, time of day, specific pattern of customer arrival

d. Lottery (LOTTO): prizes, numbers drawn, winning patterns on tickets

A version of these questions based on specific real world events were highlighted in Reston (2012) and adapted from the Real-Life Math: Probability Series (2007)

Reston (2012) noted “the responses generated from the teacher-participants, ideas on chance, uncertainty variability, and unpredictability of events evolved into more formal definitions of randomness and probability”.

They also discussed “the concept of probability of occurrence of events. For example, in the discussion of weather, precipitation was identified as a random property and the follow-up question “What is the probability that it will rain in Cebu City today?” elicited various responses showing intuitive and subjective conceptions of probability as participants evaluated the likelihood of rain by looking up the sky or by reasoning based on the occurrence of rain within the last few days” (Reston, 2012).

Batanero, Godino, and Roa (2004) investigated student’s perceptions of randomness using an activity taken from Green (1991), in order to “reflect on the complexity of the meaning of stochastic notions, particularly that of randomness, show their practical utility and predict some learning difficulties” (p. 3).

Item 1: Some children were each told to toss a coin 40 times. Some did it properly. Others just made it up. They put H for Heads and T for tails.

These are Daniel and Diana's results:

Daniel: HTHTTHHTHTHHTTHTTHHTTHTHHTTHTHTHTHTHTTHT

Diana: HTTTHTTHTHTTTHTTTTHHTTTHTTHTTHTTTTHTTTHT

Did Daniel or Diana make it up? How can you tell?

Page. 4:

Some reasons given by the students to justify that Daniel or Diana were cheating were the following:

a) The pattern of the sequence is too regular to be random, the results almost alternate;

b) The frequencies of heads and tails are too different;

c) There are too long runs; heads and tails should alternate more frequently.

Which of these arguments are right ?

How would you explain the wrong answers?

Which of these arguments do you think was mainly employed in each item?

What other correct and wrong arguments would you expect in this item?

Are these arguments similar or different to those that a professional statistician uses in testing randomness?

For Question A, there is only one correct answer.

Question B allows for multiple strategies and solutions for solving the problem. The teacher who asks this question views problem solving as a process of exploring, developing methods, discussing methods, and generalizing results.

Suggestions for Effective Problem Solving:

Give students a chance to explore the problem individually.

Have them share their ideas and thinking.

Give them an opportunity to individually reflect on what has been shared, explore whether they can create general statements or formulas, and construct their own meaning of the information.

(Fostering Mathematical Thinking and Problem Solving, Rigelman, p.309)

Minds On

Place 10 marbles (or counters, multilink cubes...) in a bag/envelope/hat and ask students if they can predict the colours of the marbles.

To help them make accurate predictions allow them to pick one marble, record its colour and return it to the bag before repeating the process.

After this has been done 10 times, ask the following questions:

"Do you now know the colours of the 10 marbles?"

"If not, why not?"

"Is there anything we can be sure of?"

"What questions do we have?"

Record the results of another 10 viewings.

"Are the results the same/different?" "Why?"

"Can you now predict the colours of the 10 marbles?"

"How has our prediction changes from the first 10 viewings?

"What could we do to help us improve our predictions?

Repeat a few more times.

"What can you do with the different sets of results?"

"How are the different sets of results helpful?"

When the class become fairly confident that they can predict the contents of the bag, show them what the bag contains.

"How accurate were our predictions?"

"What could have made our predictions more accurate?"

"What do you notice?" "Who else had something different?"

**?**

In the Bag Interactive:

Ten marbles of varying colours are chosen at random and placed in a bag. Can you guess the colours of the 10 marbles in the bag?

To help you make accurate predictions you can choose to see the results of 10 viewings - each viewing removes a marble out of the bag, records its colour and returns it to the bag before repeating the process.

You can choose to have as many viewings as you like before deciding to "Make a Guess".

Each Round concludes when you have made a guess.

Can you reach a score of 1000 points?

You start with 500 points. Each run of 10 viewings "costs" you 10 points but you can "earn" points by predicting accurately the contents of the bag:

150 points for a perfect match

100 points for 9 matching marbles

0 points for 8 matching marbles

-100 points for 7 or fewer matching marbles.

Can you develop an effective strategy for reaching 1000 points?

Can you develop an effective strategy for reaching 1000 points in the least number of rounds? What is the least number of rounds you can reach 1000 points using your strategy?

Describe your strategies.

What methods of collection, analysis and representation were most appropriate and effective in communicating your findings?

What are the merits and pitfalls of different approaches?

Is it better to have just a few viewings before making a guess or is it better to have more viewings and improve your chances of guessing correctly?

How does having more viewings influence the data?

How do you decide on the most effective strategy?

Highlight good choices and appropriate use of the data.

Bring students back together (possibly in a follow-up lesson).

Discuss, refine and list possible strategies. Who else had a different strategy? Ask students (possibly working in small groups) to select different strategies to test.

How could you test the effectiveness of your strategy?

What data will you need to collect?

What amount of data is needed to ensure meaningful results?

Ask groups of students to write down a plan for what they will do to test their strategy before they carry out the investigation. Give them time to collect, analyse and interpret the data before presenting their findings to other groups.

Discuss the value of averaging the results.

Key Questions for Students

Possible Solutions

1. Guess based on results of first 10 viewings only.

2. Always guess all four colours.

3. Press the 10 viewings buttons 3 times, each time recording the differences. Add up each difference and then divide the total by the number of times you pressed the 10 viewings button (in this case 3).

For Example:

Viewing 1: 5r + 2b + 0g + 3y

Viewing 2: 1r + 4b + 1g + 4y

Viewing 3: 3r + 4b + 1g + 2y

Average: (9r +10b + 2g + 9y)/3 = 3r + 3b + 1g + 3y [note that we round each term to the nearest integer, and make sure our number of marbles adds up to 10]

Reflection Questions for Teachers about the Lesson:

• Were students’ thinking and reasoning guiding the discussion or evident in the written work? Why or why not?

• Were all students actively engaged in the discussion (individually solving, sharing and comparing their solution strategies, listening attentively, building on one another’s ideas, synthesizing the results)?

• Were students sharing both how and why their methods work? Were students able to convince others of the correctness of their solution?

http://nrich.maths.org/7250

Try This Experiment Out to Examine What Random Looks Like:

Curriculum Connections:

Some possible Ontario Math Curriculum (2005) connections, taken from the Data Management & Probability strand by grade.

By the end of Grade 4, students will:

-predict the frequency of an outcome in a simple probability experiment, explaining their reasoning; conduct the experiment; and compare the result with the prediction;

-determine, through investigation, how the number of repetitions of a probability experiment can affect the conclusions drawn

By the end of Grade 5, students will:

- pose and solve simple probability problems, and solve them by conducting probability experiments and selecting appropriate methods of recording the results (e.g., tally chart, line plot, bar graph)

By the end of Grade 6, students will:

– express theoretical probability as a ratio of the number of favourable outcomes to the total number of possible outcomes, where all outcomes are equally likely;

By the end of Grade 8, students will:

– determine, through investigation, the tendency of experimental probability to approach theoretical probability as the number of trials in an experiment increases, using class-generated data and technology-based simulation models

http://nrich.maths.org/6016

Other Resources

Adjustable Spinner: http://www.shodor.org/interactivate/activities/AdjustableSpinner/

Experimental Probability dice and spinners - http://www.shodor.org/interactivate/activities/ExpProbability/

For slightly older students, I really like the Prize Giving question because it discusses it is an authentic problem that discusses the idea of fairness and allows students to develop multiple strategies/solutions: http://nrich.maths.org/9843

To learn about Conditional Probability - My Dog Ate my Homework and Which Team Will Win are interesting problems: http://nrich.maths.org/probabilityApps/#/prob9525

A possible worksheet for students to use while testing Solution 3. Otherwise students can track the viewings and rounds using tally marks.

Theoretical vs. Experimental Probability

This is a great resource to learn about theoretical versus experimental probability: http://www.shodor.org/interactivate/discussions/TheoreticalVsExperimental/

The following link allows students to experiment with how as the number of trials increases, the experimental probability gets closer to the theoretical probability: http://www.shodor.org/interactivate/activities/CrazyChoicesGame/

This site will allow you to run up to 10,000 trials at one time.

Prize Giving

Ms Dickens' class has submitted business plans in a national enterprise competition, and to their great delight have been awarded two of the prizes.

However, 8 students - 2 boys and 6 girls - were involved in putting together their winning entries, so who should get the prizes?

Alesha and Jack felt they'd done the most work, so they ought to be chosen.

After considerable discussion the group came up with three ways of choosing who will get the prizes.

Which would you go for if you were Alesha or Jack?

Which do you think would be most fair if actually they hadn't done any more than anyone else?

Which do you think is the most representative of the whole group?

Excellent Interactives

Common Misconceptions

to Discuss

If a coin has given a run of heads, many people believe that a tail is 'overdue'; this is known as the gambler's fallacy.

A coin has no memory, and the chance of getting a head does not change. Any single sequence of heads and tails is as likely as any other.

However, if you flip a coin three times, these are the possible outcomes:

HHH, HHT, HTH, THH, HTT, THT, TTH, TTT

Out of these possible outcomes, there are three which have two heads and one tail, and three which have two tails and one head, whereas there is only one outcome of three heads and one of three tails. This gives the appearance that some sequences are more likely than others, particularly where the three coins are flipped simultaneously rather than sequentially.

Another example of this misconception is that some sequences of numbers are more likely to be winning combinations in the lottery. Again, any one sequence is as likely - or unlikely - as any other. It makes no difference if you choose a sequence like 1, 2, 3, 4, 5, 6 or 17, 23, 45, 9, 11, 27 - both are equally likely (although you will have to share any prize you win with more people if you choose a popular sequence).

Students' Misconceptions

Jones and Thorton in Jones (2005) highlighted a few misconceptions that students often have with respect to probability, based on work by Kahneman and Tversky (1972) about Heuristics:

Representatives – negative or positive recency – The idea that past flips effect future flips. i.e. There have been 2 heads, so the next flip with also be heads. Or heads and tails should alternate rather having a long run of the same side.

Availability – based on ease of which students remember earlier occurrences of the event. i.e. When playing board games my sister always wins because she always rolls sixes.

What Participants and Students Learn

These activities help teachers learn about selecting effective problems, think about common misconceptions in probability, and reflect on what is required to prevent and avoid misconceptions in probability. It also enables teachers to confront the difficulties in a probabilistic rather than deterministic mathematical task. Effective discussion and questioning for problem solving is also modeled.

Both teachers and students learn about probability, randomness, making predictions, and theoretical and experimental probabilities.

How do our beliefs about probability and randomness influence our interpretation of the results?