Probabilistic Learning Activities Network

TEACHING WITH PLANETQHE

This resource is primarily focussed on the Group 5 subjects of the International Baccalaureate diploma: Mathematical studies, Mathematical Methods and Higher Mathematics in particular. The philosophy behind P-L-A-Net is intended to help meet the IB Group 5 Aims and Objectives, not least the Aim to "Develop logical, critical and creative thinking in mathematics".

The activities are divided into four sections. The first, Random Behaviour, is designed to help students gather experience with the eccentricities of random behaviour. The second, Experimental vs Theoretical, introduces one method humans have adopted in making sense of the probabilistic world. The subsequent sections continue this theme into compound events. I have included where possible in the spreadsheets the experimental data to emphasise the point that probability 'calculations' that basically boil down to fraction forming must compare closely to the experimental data - in the long run! The final section reflects the whole topic of probability back onto the 'real world' through web-based research. This section is under construction and is open for contributions.

The prerequisites expected for students to use this site effectively really depends on your own modes of teaching, the kind of classroom you run and the motivation of your students. I'd say that knowledge of how to calculate simple probabilities, knowledge of the use of tree diagrams and tables to find compound probabilities would be most useful. Also, for the Chevalier de Mere and Birthday Problems, knowledge of complementary events is useful. The multiplication rule for independent events is another fundamental result that is of great utility. It could be argued that the site could be used with very little pre-requisites, as a motivation to learn the methods of probability theory. I'd love to hear your experiences with this site, whatever your teaching philosophy. NONE OF THE ACTIVITIES COME WITH ANSWERS. I have deliberately left in the activities one or two bits of 'blockage' and one or two 'dead ends.'

The activities in this website are of two types. The QHE Questions follow the framework offered by Hatano and Inagaki (1987), who advocate a "Hypothesis-Experiment-Instruction" model for learning mathematics, described by Brown and Campione (1990:115) as: Students are given a problem with three or four alternative answers. the alternatives include some choices based on common misconceptions or 'bugs'. Students make independent choices, and consider alternatives in whole class discussions. They may change their choices. Finally, students' choices are tested by observing an experiment: After presenting the problem, the teacher's role is to act as a neutral chairperson; the students take control of the discussion and experimentation." Hatano and Inagaki claim that "Peer interaction, or dialogical interaction in general...invites students to 'commit' themselves to some ideas by asking them to state their ideas to others, thereby placing the issue in question in their domains of interest. In addition, the social setting makes the enterprise of comprehension more meaningful." (p115) Get the idea? I have therefore deliberately omitted answers.

The other type are the Essential Questions. They are more general questions aimed at unifying probabilistic themes across a range of activities and curriculum areas.

How you use these activities in your classroom is of course up to you - they could, I hope, be used to motivate learning the 'formal' methods of Probability theory as found in the Group 5 subjects.

Update on advice for implementing this site in your classroom- February 2003. The following is an extract from my entry proposal for the European Teachmath Excellence award; Read the entire proposal here

Description of the used materials, availability, and classification according to the process followed (v. gr.: initial assessment, classroom presentation, homework, group work, final assessment, etc.)

Each lesson consists of a question and a simulation. Put the question to the class and ask the students to individually make their initial hypotheses.

Then ask each student for their response and collate their answers on the board.

This gives the impetus for an initial discussion in which opposing views can be explored and compared. Then ask students to run the simulation and collect data, perhaps finding the mean, median and standard deviation or an experimental probability so that their results can be compared with their colleagues.

A further discussion can take place and the students can be encouraged to try to find the theoretical probability that answers the question.

It is often appropriate to ask students to complete this part for homework and/or prepare a presentation for to give in the next lesson.

Advice for implementation of the TEACHMATH Action.

As with any new teaching initiative, thorough preparation is essential - especially if you have planned in advance enough to book a computer lab.

Make sure you have checked the lab's machines for compatibility with Java and Microsoft Excel. Further, ensure that the random number generating

add-in is installed, as described in the tech help section. You should also make sure that you have tried out the simulations for yourself.

Ensure that the students know how to calculate mean, median, mode, standard deviation, box and whisker etc. I have also found it best to teach the basics of probability theory before and during using planetqhe; complementary events, venn diagrams, compound events - the intersection and union of events - tree diagrams and tables. All these skills can be useful in the write-up of each question. Your students could use a template to present their work, perhaps under the headings Question, Hypothesis, Experimental Data, then Theoretical Result, in which they write a 'proof' of the theoretical result.

I have found that a concrete experience is the best way to start the discourse.

For example use dice to run the Chevalier de Mere experiment a few times at the start of that lesson.

Or ask the class their birthdays at the start of either of the birthday problem lessons. I created a cardboard set of three doors to introduce the monty hall problem.

You may want to prepare students for these sessions by telling them that it is ok to discuss and share methods across groups; in fact they are encouraged and

expected to do so. These are not problems where they can seek the answer in the back of their textbook - in fact planetqhe deliberately avoids giving answers.

Make sure that you are clear on how each student will record and share their data collection - by floppy disc? Email? Network?