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2 ago. dos professores de matemático do ensino fundamental, tomou como base . propostas pelos PCN do 1º e 2º ciclos do ensino fundamental aos. Mathematics PCN consist of content blocks; and suggest that probability and Ensinando Probabilidade no ensino fundamental (Teaching Probability in the. ensino fundamental e médio vem se intensificando desde a década de 70, Na prática, toda família realiza a educação sexual de suas crianças e jovens.

The results indicate that the instructional unit and the use of Scratch enable the learning of basic computing concepts specifically programming in an effective and entertaining way and attract the interest and motivation of students to this knowledge area. Goal question metric paradigm.

Encyclopedia of Software Engineering, 1 1 , Branch, R. How to create and use rubrics for formative assessment and grading. Acesso em: mar. Franklin, D. Animal tlatoque: attracting middle school students to computing through culturally-relevant themes.

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Joystick nation: How videogames ate our quarters, won our hearts, and rewired our minds. Jimenez, Y. The Johns Hopkins University Press. Porto Alegre: Artmed. Scratch: A web tool to automatically evaluate Scratch projects. Before beginning the first session, a survey was conducted Annex A , consisting of three expository questions. Each question corresponds to each one of the topics of the three meetings, the aim being to assess the level of knowledge and degree of significance of the topics for the teachers.

In outlining the third question which is linked to the 3rd session, Table 1 below shows statistics from the State Highway Dept. Victims of accidents by state of the driver — data from the State Highway Dept. Would you say that the condition of the driver — drunk or sober — affects the occurrence of fatalities? We expected that teachers would calculate the conditional probability in an informal way and would conclude that intoxicated drivers were more likely to be involved in fatal accidents.

But, in fact, only eleven teachers approx. Out of the teachers who responded incorrectly, some based their wrong answer on the following calculation: This kind of confusion between conditional probability - P A B - and the probability of intersection - The results of the pre-test served as a diagnosis, as well as a guide for the theoretical discussions during the session.

Pedagogical Workshop — 3rd Session: Influence of Previous Knowledge in a Bayesian Approach Initially, there was a discussion on whether prior information or previously acquired knowledge may be of help at the time someone makes a determined statement.

That is, it was asked whether the experience and information conveyed to someone may enable that person to reach at quantitative judgments about the possibility of an event occurring, even in a rather subjective way.

A number of questions were raised to the teachers including: For a coffee crop planted in southern Minas Gerais, Brazil, what is the chance of a successful yield? And how about success of the same crop planted in Bangladesh in center-southern Asia?

Without hesitation, the teachers expressed a strong likelihood that the crop would be successful in southern Minas, but that they could not provide an immediate reply regarding Bangladesh because they had no information on coffee production in that country.

We should mention that coffee crops are an important local matter in Lavras and the local media would report almost daily on crop yield estimation and coffee wholesale prices. In addition, at school, pupils learn about the particular climate and soil types of the region and that Southern Minas is very suitable for coffee cultivation.

The teachers thus realized that their previous knowledge of coffee production in southern Minas had influenced their responses, although they did not know how to link this information to a quantitative value probability estimation. Teachers expressed that it could well that the production of coffee might be more prosperous in Bangladesh than in south Minas Gerais as the general conditions might be better for plantation, but they stated that this was something they knew nothing about.

Then, it was discussed that some information should be named as prior information only when it influences the estimate of some probability in which we may be interested. For example, if teachers knew that a coffee pest infested plantations in south Minas Gerais, this would probably lead them to a change of mind in their original opinion.

On the other hand, that very same information could not be of any use in the context of Bangladesh, since teachers did not even have any certainty that coffee is produced there.

After that stage, five didactic activities were set up to apply the concepts of the 3rd session. We should stress that our main concern was not to present questions of a great theoretical complexity in any of the didactic sequences, and not even to observe only numerical results; we focused primarily on the solving strategies that were employed by the teachers.

The methodological procedures adopted to carry out the didactic activities follow guidelines laid down in the work by Cordani , Lopes , , Mendes and Brumati and Coutinho et al Table 2. Elementary II teachers by gender and school system. Gender School Female Male Public 11 3 Private 3 4 Based on the data from Table 2, we asked teachers to calculate the probabilities of a randomly selected teacher to be a male; b from the public network of schools; c female and from a public school; d male or from a private school; e from the public network of schools provided the person is female.

In other words, the pupils can reach various conclusions just by relying on intuitive notions of probability so that the theory can be formalized later. In their study, Silva, Kataoka and Cazorla demonstrated that the two-way table is a representational tool that allows one to carry out an easier conversion to a symbolic record of the probability of simple events, conditional probability and the probability of intersection. Further points discussed with the teachers were: How can information on an event that has already occurred be incorporated into the calculation of the probability of another event?

Should events such as school and gender be regarded as dependent or independent? Second Didactic Sequence In the second didactic sequence, by manipulating dice, we resumed the discussion on conditional probability; we examined the concepts of dependent and independent events. Teachers worked in pairs to answer the following two questions. A die was actually handed out to them allow also for experimental or visual work. In this situation, should events A and B be regarded as dependent or independent?

Due to a discussion about conditional probability in the first didactic sequence, the answers to the first part of the question were satisfactory. Furthermore, in this inquiry, there was an argument on how the prior information, for instance a number less than 3, has a direct influence on the probability of an even number appearing, by the reduction of the sample space of the second event Table 3.

Table 3. Different sampling spaces dependent on the result of the toss.

Condition or information on the toss Sample space S Probability of B: After the discussion of those types of independence of events and the way the calculations were set out, the teachers said to have understood the questions.

Imagine that women supposedly being pregnant underwent a test for diagnosing pregnancy with the following results Table 4. Table 4. Results of the pregnancy test for women.

Carzola The teachers were given the following question: In our example, supposing that a randomly selected person received the test result and it is positive, what is the probability that the result is wrong? The teachers were told that, in this situation, they should take into account that the group investigated consisted of women who thought they might be pregnant and that a sample of had been chosen at random from this group.

Before attempting to find any kind of answer, the following concepts were discussed with the teachers: The first two probabilities regarding the false negative and the false positive were obtained quite quickly. However, the results regarding the specificity and sensitivity led to a lengthy discussion, even after they had supposedly understood the concepts in the first didactic sequence. With regard to the calculation of the probability of interest, that is, given the fact that a randomly selected person received the test result and it is positive, what is the probability that the result being wrong P A' B , our intervention was also needed.

We demonstrated the result both by using the two-way table and the tree diagram Figure 2. The diagram was also used to discuss the Total Probability Theorem in the calculation of P B , even though the teachers had found the result of 0.

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Tree diagram for the data shown in Table 3. For the teachers, both the use of the tree diagram and the two-way table were enhancing for the resolution of the example. Although the teachers observed that the use of a formula was not required for this example, they stated that it was important to show it first with two events, and only then to progress to the generalization.

After this preliminary discussion, the third didactic sequence was presented with a similar problem, this time about breast cancer Pena, , and with the difference that the data were not displayed in the form of a two-way table, but in a written form as follows: Based on this information, what is the probability that your aunt indeed suffers from breast cancer?

Some teachers commented that in this situation, unlike the example shown in Table 3, many women undergo preventive breast cancer examinations regardless of symptoms, and in the case of women over 40, this becomes, at least in theory, a mandatory annual examination, thus the ideal studied population would be in that age range.

In the course of that didactic sequence, after about 10 minutes we verified that most teachers had not managed to give a satisfactory answer; they reported difficulties to organize the data in a two-way table as well as to understand the factors that would condition the event as — differing from sequences 1 and 2 and unlike the example given in Table 3 — the question was not asked in a direct way. That is, in order to calculate all probabilities involved in this situation, the teachers had to perform a conversion from natural language in which the problem was presented, to either a reading table graphic representation or to symbols; such difficulties have also been observed by Silva et al as key problems in addressing this kind of questions.

Therefore, our intervention was required for the initial development of the sequence, but it allowed the teachers to have an opportunity to look at the concepts again and discuss the results together. Fourth Didactic Sequence In order to start the discussion on the concepts of geometrical and frequentist probability, the teachers were asked the following questions: What is the probability that a car will be stolen?

We asked the teachers to identify similarities or discrepancies between the two questions, and, after some discussion, they realized that the first question could be solved with the aid of geometrical concepts, while to solve the second question, it was necessary to observe how often the incident occurred. The teachers also noticed that it was necessary to define other — complementary — information: For instance, in the first question, the distance of the person from the target and the height of the target — that is, to determine how randomness of the throw can be ensured, since just having the person blindfolded does not in itself guarantee the experiment to be random.

For the second question, it was necessary to define the site of the investigation as well as how much time it would take to perform it. After this stage, the concept of geometrical probability was discussed, a very old concept, used by Buffon in and again by Betrand in , and that, according to Tunala , may be characterized by the fact that some probability problems correspond to a random selection of points in a sample space represented by geometrical figures.

In the models in question, the probability of a specific event can be reduced to the relation between homogeneous geometrical measurements, such as length, area or volume. The main feature of this approach is that the mathematical value of the probability emerges from the experimental process, thus characterizing the so- called frequentist probability.

We ended the theoretical discussion by stressing that some probability problems may be solved by adopting a geometrical approach, where the result can be interpreted and checked in the original approach — that of probability. The idea was to introduce geometrical probability, as well as a confrontation of results with the concept of frequentist probability.

In that way, the fourth and fifth didactic sequences were carried out in practice. The probability of the needle In the fourth sequence, we performed Buffon's needle experiment, in which a needle of length l is thrown crossing a line is asked for see also Tunala, ; we employed geometrical and probabilistic concepts to estimate this probability.

To start with the sequence, we briefly told the story of the French mathematician and naturalist George Louis Leclerc, the Count of Buffon, and the origin of the problem. We also showed the results of simulations carried out by other researchers such as Wolf , Smith , Lazzerini , or Reina Next, pairs of teachers carried out the activity step by step, making use of the room floor to determine the parallel straight lines Figure 3 left.

Each pair performed 50 throws, with results ranging from 2. The teachers were enthusiastic about the activity, and it was interesting to see their expectations over Figure 3. In addition, we carried out , using the animation package and the buffon. A simulation with 50 throws and another an online simulation available at http: It was interesting to start a discussion on the phenomenon of convergence of estimations, based on the law of large numbers.

Due to time constraints, it was not possible to outline the way the value for the geometrical probability was obtained and we had to refer the teachers to the material used by Wagner They understood the time constraint, but were curious about the geometrical explanation.

As the teachers were unable to explain what had happened, it was necessary for us to intervene. We explained that the occurrence was due to factors inherent to the experiment, since it is normal for people to break a piece of spaghetti into three parts that are all about the same size they break the spaghetti first into two pieces and tend to take the longer stick for the second break , and one would rarely obtain three pieces that do not form a triangle Figure 5. To conclude, in carrying out those two last didactic sequences, we emphasized the importance of randomness by drawing a parallel between the results of geometrical and frequentist probability.

A further advantage of the activity was seen in its potential to explore mutually related concepts such as estimates, the variability of small samples, and the use of games in simulation processes. Examples of a success, and b failure in the spaghetti experiment.

Evaluation of the Workshop Final Assessment of the 3rd Session After the five didactic sequences, we asked the teachers to work on the pre-test shown in Table 1 again and found that, in most cases, their approach for tackling problems and reasoning had changed a great deal. Only two teachers were still uncertain about which had been the conditional event, that is, they still did not know whether they should investigate the probability of accidents to be fatal due to the fact that the driver was drunk, or to calculate the probability that the driver was drunk in cases when there were fatal victims.

We then closed the meeting with a group discussion about how to solve this question. It was also evident from the analytical reports carried out by the teachers themselves that all the discussions were valuable and that all the activities could feasibly be incorporated into classroom practice.

This was partly because some of the teachers spoke about their initial difficulties in understanding some of the concepts, such as conditional probability and the independence of events and how to connect them to other subjects. Final Assessment of the Pedagogical Workshop As had already been found from the analysis of the entrance questionnaire, most of the teachers stated that, although they had a sound knowledge of mathematics acquired during their basic education and professional training, they were unfamiliar with the basic concepts of probability and statistics.

This was due to the slight contact with those areas during their teacher training, as well as the fact that their teaching experience had given them little opportunity to get familiarized with didactic features connected to teaching probability and statistics. The teachers completed an evaluation form on the workshop Table 5. Table 5. Evaluations from pedagogical workshops delivered to 30 mathematics teachers, Elementary II and Secondary School.

One teacher made the following statement: And the most important thing was qualifying us to use the results in teaching. Final Considerations We deem the results attained during the workshops as rather positive, but we found the need to prepare further workshops focused both on probability and on statistics, as well as planning other didactic sequences which would allow teachers to upgrade their probabilistic reasoning. That is, the exchange of experiences with teachers should not finish at the end of the workshop itself in order to allow further cycles of improvement of the activities and to get the teachers continuously involved into issues of their professional improvement.

Lastly, this work may be considered as a pilot-plan, with real possibilities for further investigation. We also expect that the work will serve, at least, as an indication for actions by other researchers in different places. Therefore, we emphasize that researches and projects really matter for the continued formation of mathematics teachers in probability. References Ainley, J. A Focus on Graphing. Challenges for teaching and teacher education.

Batanero, C. Universidad de Granada. Online http: Proyecto Edumat maestros Stochastics and its didactics for teachers: Edumat-Teachers project. Granada, Universidad de Granada.

Training teachers to teach probability. Journal of Statistics Education, Concept of probability. IV, Itabuna, Bahia, Brasil: Via Litterarum. Cazorla, I. Teaching statistics in Brazil. In Rossman, A.

Seventh Intern. Teaching Statistics. Education, Salvador Brazil. CD ROM. Coutinho, C. PhD thesis. Fourier, Grenoble, France. Tese de Doutorado. Cordani, L. Salvador, Brasil.

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The national curriculum for maths. Projeto Gestar: Teaching of probability. Diaz, C. Flores e J. An exploratory study with psychology students. Bayer, A.

Computing probabilities from two way tables. An exploratory study with future teachers.Explain the reasons for your reply.

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Teachers mentioned they liked geometric shapes because of the options that can be explored with the kids, not only in Mathematics; a teacher mentioned the two- dimensional representation of solid objects 3D as a possible activity.

Aprendizagem colaborativa: o professor e o aluno ressignificados.

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