D2.1 use mathematical language, including the terms “impossible”, “unlikely”, “equally likely”, “likely”, and “certain”, to describe the likelihood of events happening, and use that likelihood to make predictions and informed decisions.
Skill: Using Mathematical Vocabulary to Express the Probability of an Event Occurring
In the primary grades, students develop probabilistic thinking on a daily basis through the learning of certain words or phrases that describe the frequency or likelihood of certain events (for example, We always have Saturday and Sunday off from school. It's possible I'll play ball at recess. It's unlikely to rain today.) Students also develop an understanding of the concepts of variability and chance by conducting simple probability experiments. Although their understanding of theoretical probability is still in its infancy, students can recognize, for example, that by spinning the arrow on a spinner that is divided into thirds, there is a one-in-three chance that the pointer will come to rest on the third that is red.
Source: translated from Guide d’enseignement efficace des mathématiques, de la 4e à la 6e année, Traitement des données et probabilité, p. 128.
In the primary grades, one of the first manifestations of the meaning of variability occurs when students are able to indicate whether it is certain, possible or impossible that a given event will occur (for example, It is impossible that cows begin to fly). Some young students find it very difficult to do so. For many, what is possible becomes certain. For example, if the teacher tells them that it is possible that they will bring their dog to class the next day, the students will be very disappointed if this does not happen, since in their minds, it is certain that the teacher will bring it. Students need to understand that if an event is possible, it can just as well happen as not happen. Other students have difficulty distinguishing between what has never happened and what may never happen. For these students, if an event has never happened, it is impossible. For example, students may think it's impossible for them to go a whole night without sleep because they've never done it. However, such an event is not impossible.
Source: translated from Guide d’enseignement efficace des mathématiques, de la 4e à la 6e année, Traitement des données et probabilité, p. 132-133.
The notion of probability can be represented on a continuum from impossible to certain, each term qualifying the degree of probability: impossible, unlikely, equally likely, very likely and certain.
Skill: Making Predictions and Informed Decisions
As with the Big Idea Data Processing, the recommended problem-solving approach uses the process of inquiry. This approach fosters critical probabilistic thinking by encouraging students to formulate conclusions from data collected in probability experiments and to question their intuition about the probability of one of the outcomes. Such an approach helps to avoid the development of some of the misconceptions about probability that are too often found in both students and adults.
Knowledge of concepts related to probability helps students better understand all kinds of everyday situations, such as understanding weather forecasts, the possible outcomes of an experiment, the probability of winning at the time of a drawing or game. The big idea of Probability emphasizes the importance of probabilistic thinking in informing decision making in situations where the outcome is uncertain due to the fact that it is related to chance.
Source: translated from Guide d’enseignement efficace des mathématiques, de la 4e à la 6e année, Traitement des données et probabilité, p. 127.
Concept of Probability
Probability describes the degree of certainty with which a particular outcome or event can be predicted to occur in a situation of variability. Developing an understanding of the concept of probability is a long process that begins in the early grades. However, it is very important to build on an intuitive understanding of the concept first. In the primary grades, students are able to intuitively recognize that in a given situation, certain events are possible and that the degree of certainty that they will occur is on a continuum from impossible to certain. The certainty line model is an effective visual means of describing this continuum.
Students approximate a point on this continuum to describe how certain they are that a particular event will occur. The further to the right of the line, the more certain they are. If students place it in the middle, it means that they believe that the event has as much chance of occurring as not.
By the end of the primary grades, students begin to replace the word possible with probable. They can then label an event as unlikely or very likely (for example, It is unlikely to rain today.) and represent it as a point on a probability line. In the junior grades, students continue to develop their understanding of the concept of probability. In Grade 3, students compare the probability of two different events using the terms more likely than, less likely than, or equally likely. They can also do this by placing the events on a probability line. At this stage of the development of the concept of probability, there is still no question of quantifying probability precisely, so the events are placed approximately on the line.
Students also begin to pay more attention to the possible outcomes of a given situation, so teachers need to present them with simple situations that use concrete materials and encourage them to list these outcomes and compare their probability.
Example
The three spinners below are presented to students and they are asked to compare the probability of some of the outcomes.
image Three wheels with a needle are placed side by side. Roller A is separated into three equal parts of different colors: one purple, one red and one yellow. The needle points in the yellow part. Wheel B is separated into three parts: the yellow part takes half of the space, while the red and purple parts each take a quarter. The needle points in the yellow part. Wheel C is separated into 4 equal parts: red, purple, yellow and white. The needle points in the white part.
Here are some possible answers.
- With spinner B, it is more likely to obtain the result yellow than to obtain the result blue or the red result.
- With spinner A, the blue, red and yellow results are equally likely.
- With spinner C, the four outcomes are also equally likely.
- It is more likely to get the red result with roulette A than with roulette C.
Teachers can use this situation to encourage students to think about the concept of equally likely outcomes, which is fundamental to the development of an understanding of theoretical probability. They can point out that spinners A and B each have three possible outcomes (red, blue and yellow) and ask them to explain why the outcomes of spinner A are equally likely and those of spinner B are not. This type of questioning allows teachers to help students understand the importance of always challenging their intuition and thinking things through.
The teacher can then have the students, as a team, conduct a small probability experiment in which each spinner is spun a number of times (for example, 100 times) and the results are recorded. The teacher then asks the students to check whether the results support the answers given earlier. The teacher uses the variability in the results of the different teams to re-emphasize the fact that in any probability situation, the results are random and cannot be determined with certainty.
Probability experiments are used to support or challenge intuitive reasoning and are central to the development of probabilistic thinking.
Source: translated from Guide d’enseignement efficace des mathématiques, de la 4e à la 6e année, Traitement des données et probabilité, p. 134-136.
As students gain experience in solving situations of variability, some tend to develop misconceptions. One of the most common misconceptions relates to the imposition of implicit limits on variability. For example, in a game of heads or tails, students get tails five times in a row. If they are then asked to predict the result of the next toss, many will tend to indicate that it will certainly be heads side because their intuition leads them to believe that there is a limit to chance favouring the tails. side. Teachers should help them understand that chance has no memory, that is, each result is independent of previous results and cannot be predicted with certainty. It is therefore just as likely to get tails as to get heads on the next roll. This kind of misunderstanding persists among many adults who believe, for example, that certain numbers have a better chance of being part of the winning lottery numbers simply because they have come up more often in the past. A good understanding of the concept of variability makes it possible to recognize that this is not the case. To help students fully grasp this concept, teachers should introduce them to many simple probability experiments and encourage them to question themselves and remain objective with regard to their intuition.
Source: translated from Guide d’enseignement efficace des mathématiques, de la 4e à la 6e année, Traitement des données et probabilité, p. 133-134.