aa
LitMaths 1 Teaching literacy in mathematics in Year 7 bpryor Neals footer T 2 Acknowledgements Lesley Swan Fairfield High School Carolyn McGinty Fairfield High School Susan Busatto Curriculum Directorate Peter Gould Curriculum Directorate Penny Hutton Curriculum Directorate © 1997 NSW Department of School Education Curriculum Directorate RESTRICTED WAIVER OF COPYRIGHT The printed material in this publication is subject to a restricted waiver of copyright to allow the purchaser to make photocopies of the material contained in the publication for use within a school, subject to the conditions below. 1. All copies of the printed material shall be made without alteration or abridgment and must retain acknowledgement of the copyright. 2. The school or college shall not sell, hire or otherwise derive revenue from copies of the material, nor distribute copies of the material for any other purpose. 3. The restricted waiver of copyright is not transferable and may be withdrawn in the case of breach of any of these conditions. SCIS Order Number: 908063 ISBN 073 1308204 3 Contents • Chapter 1: The literacy demands of mathematics 5 • Chapter 2: The continuum of literacy development 10 • Chapter 3: Assessing, planning and programming for explicit teaching 18 • Chapter 4: Units of work Unit one: Numbers 22 Resources 32 Unit two: Fractions 46 Resources 53 Unit three: Geometry 62 Resources 66 • Chapter 5: Planning a whole-school approach to literacy 82 T 4 5 : Chapter 1: The literacy demands of mathematics I was made to learn by heart: “The square of the sum of two numbers is equal to the sum of their squares increased by twice their product”. I had not the vaguest idea what this meant, and when I could not remember the words, my tutor threw the book at my head, which did not stimulate my intellect in any way. (Bertrand Russell, Autobiography, 1986, p. 34) In a Year 7 class learning about directed numbers, the following discussion unfolds: Teacher: You will remember yesterday when we started to look at numbers that showed direction. Who can tell me how we would show a loss of $40 as a directed number? Student 1: Take away $40. Teacher: Yes… We could say minus $40 to remind us of the operation but to record this as a number we say negative 40. How could we write this as a number? Student 2: With a minus sign in front of the 40. Teacher: (Teacher writes -40 on the board.) We say that directed numbers have both size (points to the 40) and direction (points to the - sign). Teacher: How would we show a change of temperature if the temperature fell from 37° to 26°? (Writes question on the board and underlines “from” and “to”.) In this brief transcript we can see that the teacher is providing scaffolding for the students’ learning in a number of ways: • The teacher makes links with and activates prior learning. • The teacher provides explicit teaching of the subject-specific vocabulary and moves the students from their commonsense understandings of the topic to the technical understandings required. • The teacher provides a visual model of the structure of the language of the question. T 6 Students need explicit instruction to enable them to read, write and interpret basic mathematical symbols and prose with confidence. Words in mathematics that have different meanings in everyday language often confuse students. Where words have mathematical and non- mathematical meanings, students should know both and be able to interpret the meaning correctly in the appropriate context. For example: In everyday use the word “table” refers to a piece of furniture. In mathematics the meaning is quite different. Similarly, the word “leaves” most often refers to parts of a plant as a noun or frequently as a verb means “departs”. In the mathematics statement “eight minus six leaves two” the meaning is different again. • This book will highlight the explicit and systematic teaching of the literacy demands of mathematics so that the teaching of content is not impeded by students’ lack of ability to read and write appropriately or to use mathematical langauge. Definition of literacy Literacy is the ability to read and use written information and to write appropriately, in a range of contexts. It is used to develop knowledge and understanding, to achieve personal growth and to function effectively in our society. Literacy also includes the recognition of numbers and basic mathematical signs and symbols within text. Literacy involves the integration of speaking, listening and critical thinking with reading and writing. Effective literacy is intrinsically purposeful, flexible and dynamic and continues to develop through an individual’s lifetime. All Australians need to have effective literacy in English, not only for their personal benefit and welfare, but also for Australia to reach its social and economic goals. Australia’s Language and Literacy Policy, Companion Volume to Policy Paper, 1991, p.9. Successful Year 7 students in mathematics need to demonstrate a variety of literacy skills in order to develop and convey their knowledge, skills and understandings of mathematics. Talking In studying mathematics students are expected to: • discuss • explain • describe and • argue a particular point of view (for example, justifying a strategy for solving a problem). The learning of mathematics relies heavily on oral and written explanations. Working in small, collaborative groups, students can be given the opportunity to communicate orally, join in discussions constructively, and express ideas and opinions without dominating. This may help them to make the link between language and meaning. 7 : By encouraging students to talk you can assess the link between the students’ prior understandings of mathematics and the new concepts being introduced. Discussions between the teacher and students can also be beneficial as preparation for reading or writing activities, since they can assist in increasing students’ understandings before undertaking the task. Requiring students to present a verbal report to the class provides an opportunity for students to choose an appropriate language form for the audience. All lesson types in mathematics can support the development of literacy skills. Cooperative learning activities can be designed to focus on the acquisition of mathematical language and concepts. The teaching of mathematical literacy is part of teaching mathematics. Leaving out the words or avoiding the language has short-term benefits but ultimately doesn’t work. We need to develop teaching strategies that address the specific mathematical language needs of our students. Listening When studying mathematics, students are expected to listen in order to gain information and follow instructions. This means students will have opportunities to ask questions (of the teacher and peers) to clarify meanings, respond positively to alternative viewpoints, and make brief notes based on a spoken text. While students are listening, the teacher could write on the board words that may cause difficulties for some students. Words that may be misinterpreted because of the similarity of their sounds include ankle for angle, and size for sides. Reading In studying mathematics students are expected to read to locate specific information, and understand concepts and procedures, as well as to interpret problems. When reading familiar texts we often leave out words, change their order or even substitute words. Language is normally full of redundant information. This allows us to understand by skim reading or to gain meaning from the use of key words and contextual clues. Mathematics texts, however, are often lexically dense. This means that few words are used, all essential to the meaning. Consequently, as part of the literacy demands of mathematics, word order is very important. Consider the following two questions which contain exactly the same words: • Sixty is half of what number? • Half of sixty is what number? Apparently otherwise insignificant small words such as to, of or by become vitally important for making sense in mathematics. Compare “increase by one-third” to “increase to one-third”. Similarly, the description of change is often dependent on the use of prepositions: • The temperature increased to 5 degrees. • The temperature increased by 5 degrees. • The temperature increased from 5 degrees. T 8 The demands in processing such language are often far more complex than the underlying number facts suggest. The following question demonstrates this difficulty. Mary is 35 years younger than Tom. Fred is half the age of Mary. Judy is 17 years older than Fred. If Judy is 35, how old is Tom? McSeveny, A, Conway, R and Wilkes, S (1987) Signpost Mathematics Year 7, p.43. Each sentence is short and compares the ages of two people. The comparisons are younger than, half the age of and older than. Beyond the use of three different comparisons, the order of reference of the people presented in pairs is intentionally designed to increase the difficulty of the question. Students can use several strategies when reading difficult texts. These include talking to others about information in the text, re-reading parts of the question, making notes about key features, using diagrams which accompany the text or using diagrams to make sense of the text. The order in which information is presented in language is often at odds with the order in which it is processed in mathematics. This mismatch occurs even with very simple questions such as “Take 6 from 12”. Weaker readers process information in the order in which it is encountered. Even students fluent in everyday spoken English may still have problems with “The number 5 is 2 less than what number?”. The 5, 2 and less in that order suggest the answer is 3. The way the words are put together (the syntax) produces a different result. The mental restructuring that is necessary to recover the meaning may overload a student’s processing and memory capabilities. Students often give up and simply guess what to do with the numbers. The structure of everyday language can affect the translation of a situation from natural language into an algebraic statement. This occurs in the well-known “students-and-professors problem”: Write an equation to show that there are six times as many students as there are professors. The common variable-reversal error, 6S=P, appears to stem from using a left-to-right translation of the problem statement. Literal translation aligned with the syntax results in an incorrect mathematical statement. Confusion over the order for processing information in text may lead to inappropriate simplifying strategies. This is common with students attempting division questions. Not only is there no consistent left-to-right processing of meaning in English: • What is 3 divided by 6? • Divide 3 into 6. • Divide 3 into 6 equal parts. but this lack of ‘order’ is perpetuated by two different