Abstract:

Focusing on the use of language is a crucial strategy in good
mathematics teaching and a teacher's guidance can assist students
to master the language of mathematics. This article discusses the
statements with reference to recent Tear 7 and 9 NAPLAN numeracy tests.
It draws the readers' attention to the complexities of language in
the field of mathematics. Although this article refers to NAPLAN
numeracy tests it also offers advice about good teaching practice.

Article Type:

Report

Subject:

Literacy
(Australia)

Literacy (Research)

Educational evaluation (Research)

Mathematics (Study and teaching)

Mathematics (Methods)

Literacy (Research)

Educational evaluation (Research)

Mathematics (Study and teaching)

Mathematics (Methods)

Authors:

Quinnell, Lorna

Carter, Lyn

Carter, Lyn

Pub Date:

02/01/2011

Publication:

Name: Literacy Learning: The Middle Years Publisher: Australian Literacy Educators' Association Audience: Academic Format: Magazine/Journal Subject: Education Copyright: COPYRIGHT 2011 Australian Literacy Educators' Association ISSN: 1320-5692

Issue:

Date: Feb, 2011 Source Volume: 19 Source Issue: 1

Topic:

Event Code: 310 Science & research

Product:

Product Code: 9105111 Educational Quality Assessment NAICS Code: 92311 Administration of Education Programs

Geographic:

Geographic Scope: Australia Geographic Code: 8AUST Australia

Accession Number:

249223413

Full Text:

Background of NAPLAN tests

NAPLAN is an acronym for National Assessment Program: Literacy and Numeracy which came into being nationally in Australia in 2008. Year 3, 5, 7 and 9 students have taken NAPLAN tests every year since 2008. Test results give information about how students are performing in Literacy and Numeracy and are used to inform parents/carers of student's progress and to provide information to schools to help rectify literacy and numeracy problems in different stages of schooling.

Australian Curriculum Assessment and Reporting Authority (ACARA) manage the national testing and Queensland Studies Authority (QSA) administers, marks and reports on the testing in Queensland. Previous test papers, sample questions, a summary of previous results and other relevant information can be found at the following website: http://www. naplan.edu.au/tests/tests_landing_page.html

In this article the language used in Year 7 and 9 NAPLAN numeracy tests will be investigated. In both Years 7 and 9 students complete two numeracy papers of 32 questions with a time limit of 40 minutes. In one paper they may use a calculator and in the other calculator use is not permitted.

Referring to previous NAPLAN numeracy tests it is evident that every question demands an understanding of everyday language and mathematical language which includes specific mathematics terminology and the concise use of vocabulary as well as symbols, graphs and other representations of mathematical operations and concepts.

Focusing on the use of language is a crucial strategy in good
mathematics teaching. Issues of language in the Year 7 and 9 NAPLAN
numeracy tests will be discussed with reference to previous research in
the teaching and learning of mathematical language.

Language use in NAPLAN Numeracy Tests

'Language is a medium through which students learn mathematics' (Leach & Bowling, 2000, p. 24) and a large amount of the literature has focused on language leading to problems with the learning of mathematics.

Many authors have discussed the difficulties of words that have multiple meanings (Pierce & Fontaine, 2009; Saxe, 1988). For instance, words used in English often have a different meaning in mathematics. Teachers need to 'recognise and make explicit the difference between 'mathematical' English and 'natural' English' (Dawe & Mulligan, 1997 cited in Frigo, 1999, p. 15). Examples of such words from the Year 7 and 9 tests are mean, grid, rate, volume, balance, scale, key, face, head, tail, capacity, mode, range, die, positive, product, expression, prime, regular, right and rule. Even within mathematics words such as scale, cube and square have more than one meaning. In some cases, the same word functions as a different part of speech, for instance square can be a noun, verb or adjective.

Often many different words can be used to describe the same concept in mathematics (Spanos, Rhodes, Dale & Crandall, 1988). For instance subtraction, minus, take-away and difference are used for the same concept but the word difference is most often used in the NAPLAN testing. It should also be noted that the word chance in lower years becomes increasingly replaced by the word probability and the word average becomes mean in later years.

Knowledge of the meanings of prefixes like bi- and di- both meaning two, tri- meaning three, quad-meaning four are also important. This can be confusing because often two prefixes indicate the same number for instance both sex- and hex- represent six (sex- originating from Latin and hex- from Greek). Examples of words using hex- and sex- are hexagon, hexagonal and sextant, sextet, sextuplet. Bi- and di- are found in binary, bisect, binomial, bias, biannual, bilateral, bilingual, bicycle, bipolar and diagonal, diameter, divide, dialogue. Other important prefixes are circum- meaning around tetra- meaning four, pent(a)- meaning five, kilomeaning thousand, mega- meaning million and micro- meaning a millionth. More prefixes are shown in the table below, many of which are utilised in the NAPLAN numeracy tests.

The word altogether is used in a number of questions. Altogether is often taken to imply addition. However, this is not always the case in these questions. In the same way, the word total does not necessarily imply addition. This indicates the pitfalls of teaching that a particular word always implies a specific mathematics operation.

Mathematical text is lexically dense which means that it contains a minimum of redundant words, that is, contextual clues (NSW Department of School Education, 1997). Students' attention needs to be drawn to dense phrases which contain multiple concepts which can pose difficulties to students for example reflex angle, closest to, possible outcomes, exactly halfway, number sentence, per person, satisfies equations, best estimate, number line, equal length, regular hexagon, percentage decrease, square based pyramid, positive whole number, satisfy the inequality, average daily saving, stem and leaf plot, standard six sided die, product of prime factors, right angled isosceles triangle, four consecutive whole numbers, three quarter turn clockwise, sum of dots on opposite faces.

Symbols in mathematics can also pose problems for students. In the NAPLAN numeracy tests for Year 7 and 9 many symbols or words for symbols or combinations of words and symbols are used. Examples of the included symbols in one or both Year 7 and 9 are: the four operations (+, -, +, x), am and pm, symbols for units such as mL, kg, mg, m, km, cm, mm, [m.sup.2], [cm.sup.2], [degrees]C, symbols such as $, <, [check], [10.sup.3] and %,--and [degrees] for percentage, negative and degrees, 3D for three dimensional, N, S, E, W for the four directions, x, h for variables, fractions written as numbers and grid references such as A3. A combination of words and symbols is seen in 6-sided, 6 metres, 3 times, litres per 100 km and words are used when writing cubic metres, minutes, metres per minute, cents. Square centimetres, litres, grams, kilograms, dollars and grams and numbers such as seventy-five are written as either words or symbols. Large numbers are often written in prose form.

In addition to words and symbols causing misunderstandings, mathematics also introduces graphs, diagrams and other representations which add to the complexity (Lowrie & Diezmann, 2009). Lowrie and Diezmann (2009) pointed out that students may have problems interpreting graphics in word questions and they stated, 'Students' performance may thus be a measure of their ability to comprehend the graphical (or linguistic) components of a task rather than their knowledge of the mathematics within the task' (Lowrie & Diezmann, 2009, p. 146). Students need practice interpreting information in the many different visuals in NAPLAN tests and they need to examine these representations carefully for help with meaning. However, in many cases the words and visual image duplicate each other and it is possible to answer the question without understanding every aspect of the question,

Common themes in the NAPLAN tests are shopping involving prices (costs), sale prices, percentage increases, travelling questions involving maps, distances, speeds, petrol consumptions and directions, areas and perimeters of land (paddocks), recipes involving masses, volumes, capacities and ratios, questions based on sport/game scores giving rise to scores, means, medians and so on and probability questions based on marbles or lollies such as jelly beans, questions and movies, timetables, ticket sales, temperatures, surface areas, population figures. Teachers need to expose students to word questions, including reading the questions, especially on the above topics.

The NAPLAN tests allow just over a minute per question which does not seem long for students who are struggling to understand and complete the questions in the allocated time. However, special provisions such as readers, scribes, use of dictionaries and extra time are available for some students for instance ESL and special needs students if applications are forwarded to the relevant bodies. Current information on this can be found in the Test preparation handbook on the NAPLAN website.

Students should be introduced to resources such as dice, coins (emphasising the head and the tail), spinners, money, maps, recipes, games, calendars and clocks as they are referred to in the test questions.

As expected, more context-based questions appear in the Year 9 than the Year 7 tests. Examples in the Year 7 and 9 NAPLAN questions refer to recipes, floodlights, smudges, conveyor belts, satellite dishes, air-conditioners, paddocks, planks, rockets, aeroplane seating, exchange rates, films, sticks, watches, alarm clocks, DVD players, leaflets and the summit of a mountain all of which are probably unfamiliar to some students especially rural and LBOTE students. Some of the vocabulary and several contexts may be unfamiliar to certain groups of students, raising equity questions. This issue has been highlighted by many researchers such as Zevenbergen, Dole and Wright (2004). Although the NAPLAN tests have been attractively presented, they have not been designed to be inclusive of all Australians. Questions which contain familiar content and are attractive to indigenous peoples should be included in each paper.

Numeracy is more than straightforward mathematical computation and includes the interpretation of information presented in printed and other forms. It is reasonable that NAPLAN numeracy tests present students with age-appropriate challenges in interpreting information in print form. However, Abedi argued that

There is a difference between language that is an essential part of the content of the question and language that makes the question incomprehensible to many students ... While it is important to understand and value the richness of language in an assessment system; it is also important to make sure that ... students ... not be penalised for their lack of English proficiency in areas where the target of assessment is not language. Though we understand the views of some language modification critics in not 'dumbing down' assessment questions by simplifying the language, we also recognise the distinction between necessary and unnecessary linguistic complexity. (Abedi, 2009, p. 173)

Conclusion

NAPLAN tests from 2008, 2009 and 2010 have similar layouts, questions and use of language. The NAPLAN test papers and some additional trial questions are freely available on the web making it possible for students to practice on previous tests.

Teachers need to ensure that students become familiar with the necessary mathematical language. This article should not be seen as implying that teachers should aim to teach primarily for test purposes, but rather that vocabulary is an important part of mathematics teaching and learning. In terms of the NAPLAN testing, it is important that students have a sound knowledge of mathematics language in order to feel well prepared for the testing. This must occur in mathematics lessons--English teachers cannot be expected to teach the nuances of mathematical language. This means that mathematics teachers must also become teachers of language and literacy.

Teachers' guidance can assist students to master the language of mathematics. And as Murray, 2009 stated, 'The language, and progressively the more specialist language, is necessary for learning and even more, understanding mathematics' (Murray, 2009, p. 6).

References

Abedi, J. (2009). Validity of assessments for English language learning students in a national/ international context. Estudiossobre Education, 16, 167-183.

Frigo, T. (1999). Resources and teaching strategies to support Aboriginal children's numeracy learning. A review of the literature. Retrieved October 18, 2010, from http://research.acer.edu.au/ indigenous_education/11/

Leach, S. & Bowling, J. (2000). A classroom research project : ESL students and the language of mathematics. Australian Primary Mathematics Classroom, 5(1), 24-27.

Lowrie, T. & Diezmann, C.M. (2009). National numeracy tests: A graphic tells a thousand words. Australian Journal of Education, 53 (2), 141-158.

Ministerial Council for Education Early Childhood Development and Youth Affairs (MCEECDYA). (2010). National assessment program literacy and numeracy. Viewed 10 March, 2010, at http:// www.naplan.edu.au/home_page.html

Murray, J. (2009). Two heads are better than one. Retrieved October 18, 2010, from http://nrich. maths.org/public/viewer.php?obj_id=6383

NSW Department of School Education. (1997). Teaching literacy in mathematics in Tear 7. Sydney: Author.

Pierce, M.E. & Fontaine, L.M. (2009). Designing vocabulary instruction in mathematics. Reading Teacher, 63(3), 239-243.

Saxe, G.B. (1988). Linking language with mathematics achievement. In R.R. Cocking & J.P. Mestre (Eds.), Linguistic and cultural influences on learning mathematics. Hillsdale, NJ: Lawrence Erlbaum Associates.

Spanos, G., Rhodes, N.C., Dale, T.C. & Crandall, J. (1988). Linguistic features of mathematical problem solving: Insights and applications. In R.R. Cocking & J.P. Mestre (Eds.), Linguistic and cultural influences on learning mathematics (pp. 221-240). Hillsdale NJ: Lawrence Erlbaum Associates.

Zevenbergen, R., Dole, S. & Wright, R. (2004). Teaching mathematics in primary schools. Crows Nest, NSW: Allen and Unwin.

NAPLAN is an acronym for National Assessment Program: Literacy and Numeracy which came into being nationally in Australia in 2008. Year 3, 5, 7 and 9 students have taken NAPLAN tests every year since 2008. Test results give information about how students are performing in Literacy and Numeracy and are used to inform parents/carers of student's progress and to provide information to schools to help rectify literacy and numeracy problems in different stages of schooling.

Australian Curriculum Assessment and Reporting Authority (ACARA) manage the national testing and Queensland Studies Authority (QSA) administers, marks and reports on the testing in Queensland. Previous test papers, sample questions, a summary of previous results and other relevant information can be found at the following website: http://www. naplan.edu.au/tests/tests_landing_page.html

In this article the language used in Year 7 and 9 NAPLAN numeracy tests will be investigated. In both Years 7 and 9 students complete two numeracy papers of 32 questions with a time limit of 40 minutes. In one paper they may use a calculator and in the other calculator use is not permitted.

Referring to previous NAPLAN numeracy tests it is evident that every question demands an understanding of everyday language and mathematical language which includes specific mathematics terminology and the concise use of vocabulary as well as symbols, graphs and other representations of mathematical operations and concepts.

Language use in NAPLAN Numeracy Tests

'Language is a medium through which students learn mathematics' (Leach & Bowling, 2000, p. 24) and a large amount of the literature has focused on language leading to problems with the learning of mathematics.

Many authors have discussed the difficulties of words that have multiple meanings (Pierce & Fontaine, 2009; Saxe, 1988). For instance, words used in English often have a different meaning in mathematics. Teachers need to 'recognise and make explicit the difference between 'mathematical' English and 'natural' English' (Dawe & Mulligan, 1997 cited in Frigo, 1999, p. 15). Examples of such words from the Year 7 and 9 tests are mean, grid, rate, volume, balance, scale, key, face, head, tail, capacity, mode, range, die, positive, product, expression, prime, regular, right and rule. Even within mathematics words such as scale, cube and square have more than one meaning. In some cases, the same word functions as a different part of speech, for instance square can be a noun, verb or adjective.

Often many different words can be used to describe the same concept in mathematics (Spanos, Rhodes, Dale & Crandall, 1988). For instance subtraction, minus, take-away and difference are used for the same concept but the word difference is most often used in the NAPLAN testing. It should also be noted that the word chance in lower years becomes increasingly replaced by the word probability and the word average becomes mean in later years.

Knowledge of the meanings of prefixes like bi- and di- both meaning two, tri- meaning three, quad-meaning four are also important. This can be confusing because often two prefixes indicate the same number for instance both sex- and hex- represent six (sex- originating from Latin and hex- from Greek). Examples of words using hex- and sex- are hexagon, hexagonal and sextant, sextet, sextuplet. Bi- and di- are found in binary, bisect, binomial, bias, biannual, bilateral, bilingual, bicycle, bipolar and diagonal, diameter, divide, dialogue. Other important prefixes are circum- meaning around tetra- meaning four, pent(a)- meaning five, kilomeaning thousand, mega- meaning million and micro- meaning a millionth. More prefixes are shown in the table below, many of which are utilised in the NAPLAN numeracy tests.

The word altogether is used in a number of questions. Altogether is often taken to imply addition. However, this is not always the case in these questions. In the same way, the word total does not necessarily imply addition. This indicates the pitfalls of teaching that a particular word always implies a specific mathematics operation.

Mathematical text is lexically dense which means that it contains a minimum of redundant words, that is, contextual clues (NSW Department of School Education, 1997). Students' attention needs to be drawn to dense phrases which contain multiple concepts which can pose difficulties to students for example reflex angle, closest to, possible outcomes, exactly halfway, number sentence, per person, satisfies equations, best estimate, number line, equal length, regular hexagon, percentage decrease, square based pyramid, positive whole number, satisfy the inequality, average daily saving, stem and leaf plot, standard six sided die, product of prime factors, right angled isosceles triangle, four consecutive whole numbers, three quarter turn clockwise, sum of dots on opposite faces.

Symbols in mathematics can also pose problems for students. In the NAPLAN numeracy tests for Year 7 and 9 many symbols or words for symbols or combinations of words and symbols are used. Examples of the included symbols in one or both Year 7 and 9 are: the four operations (+, -, +, x), am and pm, symbols for units such as mL, kg, mg, m, km, cm, mm, [m.sup.2], [cm.sup.2], [degrees]C, symbols such as $, <, [check], [10.sup.3] and %,--and [degrees] for percentage, negative and degrees, 3D for three dimensional, N, S, E, W for the four directions, x, h for variables, fractions written as numbers and grid references such as A3. A combination of words and symbols is seen in 6-sided, 6 metres, 3 times, litres per 100 km and words are used when writing cubic metres, minutes, metres per minute, cents. Square centimetres, litres, grams, kilograms, dollars and grams and numbers such as seventy-five are written as either words or symbols. Large numbers are often written in prose form.

In addition to words and symbols causing misunderstandings, mathematics also introduces graphs, diagrams and other representations which add to the complexity (Lowrie & Diezmann, 2009). Lowrie and Diezmann (2009) pointed out that students may have problems interpreting graphics in word questions and they stated, 'Students' performance may thus be a measure of their ability to comprehend the graphical (or linguistic) components of a task rather than their knowledge of the mathematics within the task' (Lowrie & Diezmann, 2009, p. 146). Students need practice interpreting information in the many different visuals in NAPLAN tests and they need to examine these representations carefully for help with meaning. However, in many cases the words and visual image duplicate each other and it is possible to answer the question without understanding every aspect of the question,

Common themes in the NAPLAN tests are shopping involving prices (costs), sale prices, percentage increases, travelling questions involving maps, distances, speeds, petrol consumptions and directions, areas and perimeters of land (paddocks), recipes involving masses, volumes, capacities and ratios, questions based on sport/game scores giving rise to scores, means, medians and so on and probability questions based on marbles or lollies such as jelly beans, questions and movies, timetables, ticket sales, temperatures, surface areas, population figures. Teachers need to expose students to word questions, including reading the questions, especially on the above topics.

The NAPLAN tests allow just over a minute per question which does not seem long for students who are struggling to understand and complete the questions in the allocated time. However, special provisions such as readers, scribes, use of dictionaries and extra time are available for some students for instance ESL and special needs students if applications are forwarded to the relevant bodies. Current information on this can be found in the Test preparation handbook on the NAPLAN website.

Students should be introduced to resources such as dice, coins (emphasising the head and the tail), spinners, money, maps, recipes, games, calendars and clocks as they are referred to in the test questions.

As expected, more context-based questions appear in the Year 9 than the Year 7 tests. Examples in the Year 7 and 9 NAPLAN questions refer to recipes, floodlights, smudges, conveyor belts, satellite dishes, air-conditioners, paddocks, planks, rockets, aeroplane seating, exchange rates, films, sticks, watches, alarm clocks, DVD players, leaflets and the summit of a mountain all of which are probably unfamiliar to some students especially rural and LBOTE students. Some of the vocabulary and several contexts may be unfamiliar to certain groups of students, raising equity questions. This issue has been highlighted by many researchers such as Zevenbergen, Dole and Wright (2004). Although the NAPLAN tests have been attractively presented, they have not been designed to be inclusive of all Australians. Questions which contain familiar content and are attractive to indigenous peoples should be included in each paper.

Numeracy is more than straightforward mathematical computation and includes the interpretation of information presented in printed and other forms. It is reasonable that NAPLAN numeracy tests present students with age-appropriate challenges in interpreting information in print form. However, Abedi argued that

There is a difference between language that is an essential part of the content of the question and language that makes the question incomprehensible to many students ... While it is important to understand and value the richness of language in an assessment system; it is also important to make sure that ... students ... not be penalised for their lack of English proficiency in areas where the target of assessment is not language. Though we understand the views of some language modification critics in not 'dumbing down' assessment questions by simplifying the language, we also recognise the distinction between necessary and unnecessary linguistic complexity. (Abedi, 2009, p. 173)

Conclusion

NAPLAN tests from 2008, 2009 and 2010 have similar layouts, questions and use of language. The NAPLAN test papers and some additional trial questions are freely available on the web making it possible for students to practice on previous tests.

Teachers need to ensure that students become familiar with the necessary mathematical language. This article should not be seen as implying that teachers should aim to teach primarily for test purposes, but rather that vocabulary is an important part of mathematics teaching and learning. In terms of the NAPLAN testing, it is important that students have a sound knowledge of mathematics language in order to feel well prepared for the testing. This must occur in mathematics lessons--English teachers cannot be expected to teach the nuances of mathematical language. This means that mathematics teachers must also become teachers of language and literacy.

Teachers' guidance can assist students to master the language of mathematics. And as Murray, 2009 stated, 'The language, and progressively the more specialist language, is necessary for learning and even more, understanding mathematics' (Murray, 2009, p. 6).

References

Abedi, J. (2009). Validity of assessments for English language learning students in a national/ international context. Estudiossobre Education, 16, 167-183.

Frigo, T. (1999). Resources and teaching strategies to support Aboriginal children's numeracy learning. A review of the literature. Retrieved October 18, 2010, from http://research.acer.edu.au/ indigenous_education/11/

Leach, S. & Bowling, J. (2000). A classroom research project : ESL students and the language of mathematics. Australian Primary Mathematics Classroom, 5(1), 24-27.

Lowrie, T. & Diezmann, C.M. (2009). National numeracy tests: A graphic tells a thousand words. Australian Journal of Education, 53 (2), 141-158.

Ministerial Council for Education Early Childhood Development and Youth Affairs (MCEECDYA). (2010). National assessment program literacy and numeracy. Viewed 10 March, 2010, at http:// www.naplan.edu.au/home_page.html

Murray, J. (2009). Two heads are better than one. Retrieved October 18, 2010, from http://nrich. maths.org/public/viewer.php?obj_id=6383

NSW Department of School Education. (1997). Teaching literacy in mathematics in Tear 7. Sydney: Author.

Pierce, M.E. & Fontaine, L.M. (2009). Designing vocabulary instruction in mathematics. Reading Teacher, 63(3), 239-243.

Saxe, G.B. (1988). Linking language with mathematics achievement. In R.R. Cocking & J.P. Mestre (Eds.), Linguistic and cultural influences on learning mathematics. Hillsdale, NJ: Lawrence Erlbaum Associates.

Spanos, G., Rhodes, N.C., Dale, T.C. & Crandall, J. (1988). Linguistic features of mathematical problem solving: Insights and applications. In R.R. Cocking & J.P. Mestre (Eds.), Linguistic and cultural influences on learning mathematics (pp. 221-240). Hillsdale NJ: Lawrence Erlbaum Associates.

Zevenbergen, R., Dole, S. & Wright, R. (2004). Teaching mathematics in primary schools. Crows Nest, NSW: Allen and Unwin.

Table 1. More examples of prefixes Prefix and meaning Examples of words which include the prefix mon(o)- and uni- meaning one or monolith , monorail , monotony , single monocle, monocycle, monologue, monotreme, unit, uniform, unification, unicorn, unique tri- meaning three triangle, trillion, triomino, trinomial, third, tripod, triplets, tricycle, tricolour, triple, triad quadr- or quart- meaning four quadrilateral, quadrillion, quarter; quartile, quadrant, quadratic, quartet, quart, quadbike, quadruplets, quadrangle oct- meaning eight octagon, octopus, octane, octopod October dec(a) or dek(a)- meaning ten decade, decagon, decillion deci- meaning one tenth decimal system, decimetre, decilitre cent- or centi- meaning hundred cents, centimetre, century or hundredth centenary, centennial, centipede midcentury, percentage, percent, milli- meaning thousand or millimetre, millennium thousandth de- meaning taking something decrease, deduce, descend, decline away the opposite decentralisation , deforestation poly- meaning many polygon(2D), polyhedron (3D) , polymer polyomino, polynomial, polygamous, Teachers should encourage students to take note of common roots in words, for example, rectangle and rectangular, cylinder and cylindrical, parallel and parallelogram, clock and clockwise or anticlockwise, reflect and reflection, half and halfway and division and divisible or divisor.

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