Anna Verren



Statistics and Probability – Probability 1Language analysisProbability is a way of mathematically communicating chance. Just like any communication, the language needs to be understood to understand the concepts being explained. This language is important for both teaching and understanding the mathematics involved CITATION Ran02 \l 1033 (Rangecroft, 2002).Probability can be quite difficult for students to grasp; language plays a strong role in this. One of the reasons for this is that probability introduces many new concepts which use new words or words found outside of maths CITATION Boo02 \l 1033 (Booker, 2002). The new words and terminologies can present difficulties for students; however, most of the problems occur when the students come across words which are often found outside of maths CITATION Ott09 \l 1033 (Otten & Herbel-Eisenmann, 2009). The definitions for these words in Mathematical English (ME) often differ from the definitions in Ordinary English (OE) CITATION Ran02 \l 1033 (Rangecroft, 2002), these differences can be small or they can be rather large. The reasons that students struggle more with words that they already know in OE is quite simple. When a word is found in OE then the students have probably herd it used prior to meeting it in mathematics class. However, since these words have a different meaning, the students have their own constructed understanding of the word CITATION Kaz07 \l 1033 (Kazima, 2007). When these same words are presented to students in a mathematical context with a new meaning and use, then the students will often struggle with the conflict in what they know to be true and what they are being told CITATION Kaz07 \l 1033 (Kazima, 2007). When working with these words students will often try to relate the words to their prior understandings, causing problems in their understanding of mathematics.It is important to remember that the nature of words in OE and ME differs itself. Words found in OE are more of a social construct, something that is malleable. Much of our understanding of OE comes from hearing words and the context they are used in and not looking at the definition CITATION Rub02 \l 1033 (Rubenstein & Thompson, 2002), where ME is based on words that have set meanings and concepts attached to them. The Definitions of words from ME are often hundreds of years old and quite ridged CITATION Jam00 \l 1033 (Jamison, 2000). This adds to students’ difficulties in understanding the language in mathematics. It may seem obvious but all the same, it is very that mathematics teachers understand the meanings of the words that they are teaching CITATION MaL99 \l 1033 (Ma, 1999). If the teachers do not have a deep understanding of these words then how can they expect their students to understand what they are teaching?Once a teacher has a sound understanding of the language of a topic then there are two approaches that they can take to teaching it; they can try to just us OE when teaching mathematics and leave out the ME or they can use the proper ME vocabulary and try to make the meanings of these words as clear as possible CITATION Ran02 \l 1033 (Rangecroft, 2002). Ma (1999) suggests that mathematics teachers should try to use the proper vocabulary where possible. She found that the use of the proper vocabulary when teaching mathematics was one of the strengths of Chinese mathematics teachers and a possible contributor to why Chinese students out preformed American mathematics students CITATION MaL99 \l 1033 (Ma, 1999).As students study and learn the ME, it is important that they have lots of opportunities to discus and use these words CITATION Ran02 \l 1033 (Rangecroft, 2002). By practicing and using mathematics vocabulary, students will slowly become used to the use and more comfortable with these words. The discussions that students have about the ME words can help build their understanding of the mathematical uses of words CITATION Boo02 \l 1033 (Booker, 2002), these discussions should be amongst the students as well as with the teacher and they should look at the origins of words as well as the different uses and meanings. However, it is important not to neglect the OE use of words. This in fact can be a good place to start from, a student’s prior understanding of words can be built upon to help a student evolve their understanding of the word to incorporate the ME meanings CITATION Kaz07 \l 1033 (Kazima, 2007). After all, for the most part words that are shared in both OE and ME have the same roots.When teaching this subject, like in all teaching, it is important to remember that all students are different CITATION Sno09 \l 1033 (Snowman, et al., 2009). These differences will impact on how well a student learns the subject, especially if consideration is not given to these diversities.When it comes to the difficulties in mathematics due to the ME language, the students who struggle the most are those who have English as a second language CITATION Kaz07 \l 1033 (Kazima, 2007). The way that words are shared across OE and ME with different meanings can be particularly had for these students so work with as they are often still struggling with the OE and have to mentally translate back and forth between the two CITATION Kaz07 \l 1033 (Kazima, 2007). This will need to be taken into consideration when working with such students. More time may be needed with these students and explanations need to be clear and concise.Student diversity can be quite large and varied which can make it hard to plan ahead without being aware of the particular students involved, this contributes to why it is important to know your students CITATION lat06 \l 1033 (Latham, et al., 2006). One way of planning ahead, for student’s diversity, is to use a variety of approaches to teaching a topic CITATION Sno09 \l 1033 (Snowman, et al., 2009). Try to use pictures and diagrams to demonstrate what is happing, or use a model or demonstrate the event. In probability, this can be done using tree diagrams and table along with class experiments of probability to demonstrate what is meant by each word.It is important that students to properly the vocabulary of probability so that they can move beyond doing, into actually understanding this mathematics CITATION Boo02 \l 1033 (Booker, 2002).Unit PlanTopic: Statistics and Probability Probability 1Duration: 5 × 45 minute periodsStage: 4Year: 7Syllabus: NSW Draft K–10 Syllabuses for Mathematics (version 2) CITATION Boa12 \l 1033 (Board of Studies NSW, 2012)Unit Outcomes: A student:? communicates and connects mathematical ideas using appropriate terminology,diagrams and symbols MA4-1WM? applies appropriate mathematical techniques to solve problems MA4-2WM? recognises and explains mathematical relationships using reasoning MA4-3WM? represents probabilities of simple and compound events MA4-21SPRelated Life Skills outcomes: MALS-1WM, MALS-2WM, MALS-3WM, MALS-35SP, MALS-36SPLesson 1Outcomes: Construct sample spaces for single-step experiments with equally likely outcomes (ACMSP167)use the term ‘chance experiment’ when referring to occurrences such as tossing a coin, rolling a die, randomly selecting an object from a bag [L]use the term ‘outcome’ to describe a possible result of a chance experiment and list all possible outcomes included in a single-step experiment [N]use the term ‘sample space’ to describe a list of all possible outcomes of a chance experiment, eg if a fair six-sided die is rolled once, the sample space is {1,2,3,4,5,6} [L, N] Language Issues and Teaching Strategies: In this lesson students will be introduced to three terms which have meanings that differ in ME and OE, these are; ‘chance experiment’, ‘outcome’ and ‘sample space’. To help minimise language problems an example experiment will be run to demonstrate what each of these terms looks like. Students will also have to write down a definition of each term. This demonstration will give students a concrete link to what each of these terms mean.Lesson 2Outcomes: Construct sample spaces for single-step experiments with equally likely outcomes (ACMSP167)distinguish between equally likely outcomes and outcomes that are not equally likely in singlestep chance experiments [L, N]describe single-step chance experiments in which the outcomes are equally likely, eg the outcomes of a single toss of a fair coin (Reasoning) [N]describe single-step chance experiments in which the outcomes are not equally likely, eg ‘The outcomes of a single roll of a six-sided die labelled {1, 2, 3, 4, 4, 4} are not equally likely since the outcome ‘4’ is three times more likely to occur than any other outcome.’ (Communicating, Reasoning) [N, CCT]design a spinner given the relationships between the likelihood of each outcome, eg design a spinner with three colours red, white and blue, so that red is twice as likely to occur than blue, and blue is three times more likely to occur than white (Problem Solving) [N, CCT]Language Issues and Teaching Strategies: In this lesson students will be introduced to the new terms; ‘equally likely’, ‘not equally likely’, ‘fair’ and ‘single-step’, they will also be using terms from the last lesson. During the lesson, a class discussion about what these terms may mean will lead to the class developing definitions for these terms. This will help the students to evolve their understanding of these words CITATION Kaz07 \l 1033 (Kazima, 2007). Also during the lesson, when last lessons terms are mentioned, students will be reminded of their meanings and they will be encouraged to use these terms. This will help familiarise the students with the words in this context CITATION Ran02 \l 1033 (Rangecroft, 2002).Lesson 3Outcomes: Assign probabilities to the outcomes of events and determine probabilities for events (ACMSP168)use the term ‘event’ to describe either one outcome or a collection of the outcomes in the sample space of a chance experiment, eg in the experiment of rolling a single fair-sided die once, getting the number ‘1’ is an ‘event’; similarly, getting a number divisible by three is also an event [L]explain the difference between an experiment, outcomes, the sample space and events in chance situations (Communicating) [L]assign a probability of zero to events that are impossible and a probability of one to events that are certain [L, N] 1explain the meaning of a probability of 0, and 1/2 in a given situation (Communicating) [N, 2 CCT]assign probabilities to simple events by reasoning about equally likely outcomes, eg the probability of randomly drawing a diamond card from a standard pack of 52 playing cards is 13/52 = 1/4 [N]Language Issues and Teaching Strategies: During this lesson students will be introduce to the concept of ‘probability’ and to the term ‘event’. Students will be asked to discuss what they think probability to be and stories will be made up to demonstrate different events and probability situations. The discussion will lead to a development of a definition which the students can relate to CITATION Kaz07 \l 1033 (Kazima, 2007) and the stories will give students practice working with these concepts CITATION Ran02 \l 1033 (Rangecroft, 2002).Once again, students will be reminded of the language from the previous lessons and encouraged to use it, as a way of familiarising them to this language CITATION Ran02 \l 1033 (Rangecroft, 2002).Lesson 4Outcomes: Assign probabilities to the outcomes of events and determine probabilities for events (ACMSP168)express the probability of an event A given a finite number of equally likely outcomes asP(A) = number of favourable outcomes/nwhere n is the total number of outcomes in the sample space [L, N]interpret and use probabilities expressed as percentages or decimals (Communicating) [N, CCT]solve probability problems involving single-step experiments such as card, dice and other games [N]establish that the sum of the probabilities of all possible outcomes of a single-step experiment is 1 [N]Language Issues and Teaching Strategies: In this lesson students will be introduced to the terms “favourable outcomes’ and probability as a precent or decimal. This lesson relies heavily on the terms taught in the last three lessons, for this reason students will be encouraged to use these ME terms and be reminded of their ME meaning. This is to promote familiarity with the ME uses of these term CITATION Ran02 \l 1033 (Rangecroft, 2002).The teacher will discuss the origins of the word ‘precent’, to help students understand what this word means in a ME context. This should help promote the students understanding of this term CITATION Rub00 \t \l 1033 (Rubenstein & Schwartz, 2000).Lesson 5Outcomes: Identify complementary events and use the sum of probabilities to solve problems (ACMSP204)identify and describe the complement of an event, eg the complement of the event ‘rolling a six’ when a die is thrown is ‘not rolling a six’ [L]establish that the sum of the probability of an event and its complement is 1,ie P(event A)+ P(complement of A) = 1 [N]calculate the probability of a complementary event using the fact that the sum of the probabilities of complementary events is 1, eg calculate the probability of ‘rolling a six’ when a die is thrown is 1/6 the probability of the complement ‘not rolling a six’ is 1 – 1/6 = 5/6 [N]Language Issues and Teaching Strategies: During this lesson, the terms ‘complement’ and ‘complementary’ will be introduced to the students. To help the students understand these terms, a class discussion will be led; which will result in an introduction of the terms and their definitions. This is to help students build an understanding of these terms CITATION Bar04 \l 1033 (Barnett-Clarke & Ramirez, 2004), these concepts will be demonstrated with real world examples. Once again, students are reminded of terms from previous lessons and encouraged to use them to help build the students familiarity with this terms CITATION Ran02 \l 1033 (Rangecroft, 2002)References: BIBLIOGRAPHY Barnett-Clarke, C., & Ramirez, A. (2004). Language pitfalls and pathways to mathematics. Perspectives on the teaching of mathematics, 66, 56-66. Retrieved October 27, 2012, from of Studies NSW. (2012, February 13). Mathematics K–10 Draft Syllabus Version 2. Retrieved October 26, 2012, from Educational Resources: , G. (2002). Valuing language in mathematics: Say what you mean and mean what you say. Valuing mathematics in society, 11- 29. Retrieved October 26, 2012, from , R. (2000). Learning the Language of Mathematics. Language and Learning Across the Disciplines, 4(1), 45- 54. Retrieved October 27, 2012, from , M. (2007). Malawian students' meanings for probability vocabulary. Educational Studies in Mathematics, 64(2), 169- 189. Retrieved October 26, 2012, from , G., Blaise, M., Dole, S., Faulkner, J., lang, J., & Malone, K. (2006). Learning to teach: New times, new practices. Hong Kong: Oxford Unerversity Press.Ma, L. (1999). Knowing and teaching elementary mathematics. Mahwah, NJ: Lawrence Erlbaum Associates.Otten, S., & Herbel-Eisenmann, B. A. (2009). Multiple meanings in mathematics: Beneth the surface of area. In S. L. Swars, D. W. Stinson, & S. Lemons-Smith, Proceedings of the 31st annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 296-303). Atlanta: GA: Georgia State University.Rangecroft, M. (2002). The language of statistics. Teaching statistics, 24(2), 34-37. Retrieved October 26, 2012, from , R. N., & Schwartz, R. K. (2000). Word Histories: Melding Mathematics and Meanings. The Mathematics Teacher, 93(8), 664-669. Retrieved October 27, 2012, from , R. N., & Thompson, D. R. (2002). Understanding and Supporting Children's Mathematical Vocabulary Development. Teaching Children Mathematics, 9(2), 107-112. Retrieved October 27, 2012, from , J., Biehler, R., Dobozy, E., Scevak, J., Bryer, F., & Bartlett, B. (2009). Pyschology applied to teaching. Brisbane: John Wiley & Sons Australia Ltd. ................
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