Teaching Probability Essay
Teaching ProbabilityIntroductionProbability appeals to the common sense–reason why people often ‘know’ an answer to a question of chance even if they do not understand its mathematical explanation.
Judgment may be entirely heuristic, and results are correct often enough to make decisions or conclusions based on inherent or personal judgment. This can be helpful or a deterrent to mathematizing probability, the key is to utilize the strength of everydayness of probability in the classroom.The challenge for educators is to help students formulate, systematize, and make judgments from conditions of reality through dialectical reasoning (Skovmose, 1994) and consequently replace ‘commonsense’ rules to inferences made by mathematical logic.Teaching probability or statistics to young students must explore conjectures and develop the students’ ideas in a socially interactive teaching environment (Lopes, 2008), because probability discusses common situations students encounter in their daily lives, and will prove important in everyday decision-making.Reducing UncertaintyThe most effective way of reducing the students’ uncertainty to a mathematical problem is to help them solve them mathematically. Inform students that mathematics is not only numbers, but a tool to interpret life.
Reassure them that in probability there is no right or wrong answer–there are only answers that are more likely than others.A good method of getting sixth-grade students interested in probability is to provide them something that would stir their imagination or invoke personal experiences. Teachers could pose questions such as, “If I toss a coin and conceal it in my hand, is it correct to say that the coin is now a half-head and a half-tail?” or, “How many chances are there that Mom won’t ground me for getting home after 9?” Give students time to ponder, and ask what their ideas are. They could give out different answers, perhaps “Mom will let me stay out late because it’s a holiday.” or even, “A head or a tail will depend on what hemisphere you are at.
” The important task for teachers here is to get students to think analytically of everyday situations, and then reveal that there is a mathematics of ‘likeliness’ of an event happening.Gathering their thoughts on how they arrived at such a solution is the next important step. The instructor could follow up with, “How many holidays are there in a year?” to figure out the likeliness of being grounded; or tell students “That proves that there are two possible outcomes, that this coin has 1/2 possibility of being a head or a tail, whichever hemisphere you are at.”The question “Is getting a five, a six, or a seven more likely in a two-dice toss?” would prove a bit more challenging for students. Although sixth-graders have enough knowledge a priori to figure out the answer must be 7, teachers must supply the reason why the answer is correct.
Mere reassurance that 7 is the correct answer increases the students’ confidence that probability is not so difficult after all, and reduces the uncertainty. However, a good follow-up explanation would seal that confidence.Mathematizing conjecturesAfter gathering their ideas on why they think the answer must be 7, the next step is to help students create a plan to find out why their guess was right. Diagrams are a good tool for teaching probability. The abstraction of numbers is translated into actual, observable phenomena, which are more appealing to students beginning to learn probability. The teacher could draw a simple diagram of the first die thrown, after asking students “What numbers should we draw on first die if we need to get a five? How about a six? A seven?” This reduces other possibilities, and points out that there are more possible numbers that could get a 7. Students could supply the numbers themselves.
At this point, the teacher could introduce the mathematical equation that represent the problem, after asking students how to figure out probability without the painstaking task of illustrating possibilities, “What if the problem involves 1,000,000 possibilities?” the teacher could ask. Beside the first diagram, the teacher could write its fractional counterpart.The teacher could explain how the denominator 6 represents all possibilities (being the number of faces in the die); therefore there is 1 possibility (because the die can only give one side) in 6 that you can roll a 1, or a 2, a 3, and so on in every dice roll. The same goes for the second die, also one in 6 chances that you may get the number you need to get the sum of five. To get 1 and 4 together means multiplying all the chances of getting the numbers, that is 36. Adding the possibilities of all number sets in the chart will prove that mathematically, 7 has more chances of coming up with 6 chances in every 36 rolls.
References:Lopes, C. E. (2008). Prof. Dr. Retrieved July 15, 2009, from Department of Statistics Faculty of Science University of Auckland: http://www.
stat.auckland.ac.nz/~iase/publications/1/6e1_lope.pdfSkovmose, O. (1994).
Towards a philosophy of critical mathematical education. Dordrecht: Kluwer Academic Publishers.