Literacy in the mathematics classroom Aaron Wilson

1.Why focus on literacy in Mathematics? 2.Activating prior knowledge 3.Text features and purposes 4.Vocabulary 5.Strategies for “translating” word problems

1.Why focus on literacy in Mathematics?

Inquiry focus  “Since any teaching strategy works differently in different contexts for different students, effective pedagogy requires that teachers inquire into the impact of their teaching on their students.” (NZC, p.35)

NZ Curriculum  Each learning area has its own language. As students discover how to use them, they find they are able to think in different ways, access new areas of knowledge, and see their world from new perspectives (NZC, p.16)

NZ Curriculum Students need specific help from subject area teachers as they learn:  Specialist vocabulary  How to read and understand its texts  How to communicate knowledge and ideas in appropriate ways  How to listen and read critically, assessing the value of what they read and hear (NZC, p.16)

Shanahan & Shanahan (2008)

Cross-curricular literacy  “Many literacy messages fail to resonate with mathematics educators because they neglect, deemphasize, or misrepresent the nature and content of the discipline of mathematics”. - Siebert and Draper (2008, p. 231).

Why focus on literacy in Mathematics?  Assessments written in English will always be, to some extent, assessments of English (Abedi, 2004; Martiniello, 2007  Lower language proficiency tends to be associated with poorer mathematics performance (Cocking & Mestre, 1988; Wiest, 2003).

Why focus on literacy in Mathematics?  Research indicates that students peform 10% to 30% worse on arithmetic word problems than on comparable problems presented in a numeric format (Abedi & Lord, 2001; Carpenter, Corbitt, Kepner Jr, Lindquist, & Reys, 1980,Neville- Barton & Barton, 2005).

 Children were shown 5 birds and 3 worms and asked, “How many more birds are there than worms?” 17% of the nursery school children and 64% of the first graders correctly answered the question correctly. When the problem was rephrased as, “Suppose the birds all race over and each one tries to get a worm! How many birds won’t get a worm?” 83% of the nursery school children and 100% of the first graders answered correctly” (Hudson,1983).

Talking point:  “Students generally find it harder to solve a word problem than they would if the same problem was presented in a mathematical format”.

The language of mathematics  Mathematics has a unique linguistic register with special features that must be mastered by students of mathematics.  A mathematics register is the variety of language oriented to mathematics activities comprising the meanings and uses of the various linguistic forms that appear in the context of these activities (Halliday,1975).

The language of mathematics  Quality teaching bridges students’ intuitive understandings and the mathematical understandings sanctioned by the world at large. Language plays a central role in building these bridges: it constructs meaning for students as they move towards modes of thinking and reasoning characterised by precision, brevity, and logical coherence (Marton & Tsui, 2004).

The language of mathematics  In order to enculturate their students into the mathematics community, effective teachers share with their students the conventions and meanings associated with mathematical discourse, representation, and forms of argument (Cobb & Yackel, 1996; Wood, 2002).

Activity  Read the three NCEA texts and identify aspects of language your students might find challenging

2. Activating prior knowledge and building necessary background knowledge

Why activate prior knowledge?  It is (relatively) simple to do and can pay big dividends  Reading comprehension always involves making links between new and existing knowledge  Students’ comprehension will be greater when they know in a broad way what the text is going to cover or be about  Students’ motivation to read is greater when they have a purpose for reading  Helps teachers identify gaps or misunderstandings that might cause students’ problems.

Student attitudes Types of texts Reading Challenges Teaching challenges Teacher attitudes Reading in Mathematics

Identifying problems  Analyse the NCEA texts and explain examples where students’ experience or lack of experience of the context might affect their understanding.  Discuss what you could do as a teacher to prepare students for situation where they encounter unfamiliar contexts.

What are some features of mathematics word problems?  Word problems are “stylized representations of hypothetical experiences- not slices of everyday existence” (Lave, 1992, p. 77).  “One of the most significant problems provided by many of the contexts used in mathematics classrooms occurs when students are required to engage partly as though a context in a task were real whilst simultaneously ignoring facts pertinent to the real life context” (Boaler, 1994, p. 554).

 Meaney and Irwin (2005) found that Year 8 NZ students were far more successful at recognising the need to ‘peel away’ the story shell of word problems.  Students’ real world concerns sometimes get in the way of their mathematical problem solving, For example, when asked to describe, “How much of the pizza is left? A year 4 student responded, “All the herbs.” !  Lower socio-economic students were more likely to focus on the contextual issues of a problem at the expense of the mathematical focus, (Lubienski, 2000) Other issues

The mathematics or the context: What knowledge to activate?  Solving word problems involves individuals analysing them in a specialised way to build a task- or content-specific mental representation that is most conducive to successful problem solution (Kintsch & Greeno, 1985).  Effective reading of a word problem will involve identifying what type of problem it is so appropriate schema can be activated. Readers need to abstract the problem type (Paris, 2010).

Prior knowledge of word problems In terms of the 3 NCEA texts,  What prior knowledge is it important for students to activate? i.e. knowledge of the context or mathematical knowledge or both?  What likely gaps in experience might affect students’ reading?  Can students ‘real life’ experiences and understandings hinder rather than help their problem- solving?  How can teachers prepare students to cope with a lack of knowledge about context?  Are literacy mantras about activating prior knowledge useful (or even harmless) in a mathematics context?

 Hypothesis: Students might find that their familiarity with a particular context is actually a barrier to solving a problem because they might apply everyday rather than ‘mathematical’ solutions to the problem e.g. “I’d calculate the distance between those two points using a tape measure” (rather than by applying a theorem).

Word problem problems  Hypothesis: When some students encounter an unfamiliar context in a word problem they might react by not attempting the problem, or giving up too easily e.g. “I couldn’t solve it because I’ve never played golf before.”

Experience and knowledge of context  Teach ‘predictable’ contexts & their associated vocabulary  Develop students’ strategies for coping with unpredictable contexts.

TRIGONOMETRY A wallerer is at the top of a vertical clanker. The top of the wallerer is 60m above the ground at the base of the clanker. Sione walks away from the base of the clanker along horizontal ground until he comes to a jumba. He measures the angle of elevation from the ground to the top of the wallerer as 69 degrees. He then walks in the same direction until the angle of elevation is 40 degrees and stops. How far from the jumba did Sione walk?

‘Predictable’ contexts (90151)

3. Text features and purposes Expert readers read different text types in different ways because we know they have:  Different features  Different purposes

Activity  You have 45 seconds to get as much important information from the news story as you can. Your time starts now….  What are the main ideas?  What types of knowledge did you draw on to locate this information so quickly?

Hypothesis  Students will be better at reading word problems when they are explicitly taught about the purpose and text features of this genre.

Organisational features Organisational features that may be useful for students to focus on in this context include:  the name and description of the achievement standard they are attempting. For example, a word problem presented in a standard headed ‘right angle triangles’ is likely to have variables that refer to the length of lines, degrees of angles and location of points (and relationship between them).  words that are italicized, underlined or in bold as the author is probably signalling that these are important  Headings, subheadings, labels (e.g. on rows, columns, axis)

Understanding the ‘word problem’ genre or text type. To what extent do students understand that word problems:  Are “stylized representations of hypothetical experiences- not slices of everyday existence”?  Have unique features and purposes?  Demand a special way of reading that may be quite different than other texts?

‘Think aloud’  The think-aloud is a technique in which students and teachers verbalise their thoughts as they read and thus bring into the open the strategies they are using to understand a text.  This metacognitive awareness (being able to think about one's own thinking) is a crucial component of learning, because it enables learners to assess their level of comprehension and adjust their strategies for greater success.

Activity  In pairs practice a ‘think aloud’ to model how you read one of the NCEA mathematics texts

4. Vocabulary

Vocabulary key concepts  Comprehension appears to depend on knowing between 90 – 95% of words in a text  Students need frequent and repeated opportunities to experience and use new vocabulary  Vocabulary is best learned in context  Amplify rather than simplify vocabulary

Receptive & productive vocabulary  Receptive = what you receive (through reading and listening)  Productive = what you produce (through writing and speaking)  Both are important and mutually beneficial  Talking point: “Students are better at understanding mathematics vocabulary than they are at using it e.g. when explaining their problem-solving?”

Key types  Subject specific  General academic  General vocabulary (especially low- frequency vocabulary) Activity: Identify 3 examples of each type that you think might be problematic for students

A vocabulary learning sequence  Inquiry to identify existing knowledge and needs  Explicit instruction  Repeated opportunities to practice – both receptive and productive  Metacognition e.g. –reflecting on strategies –‘think alouds’  Inquiry into effectiveness of teaching sequence, and planning next steps.

Challenging aspects of vocabulary in mathematics  Lots of complex new technical mathematics vocabulary e.g. ‘inverse’, ‘binomial’, ‘coefficient’ and ‘denominator’.  A wide number of synonymous words and phrases e.g. ‘add, ‘plus’, combine’, ‘sum’, ‘more than’ and ‘increase by’ are all synonymous terms related to addition.  Terms that are challenging in isolation are commonly part of more complex strings of words or phrases e.g. ‘least common denominator’.

Challenging aspects of vocabulary in mathematics contd.  Terms that are familiar from everyday contexts but which have a very different meaning in a mathematics context. E.g. ‘square’, ‘rational’, ‘volume’ and ‘equality’.  Use of symbols and mathematical notation as ‘vocabulary’ e.g. =,, ( )  Similar terms but with different functions e.g. ‘less’ vs ‘less than’, the ‘square’ vs ‘square root’, ‘multiply’ vs ‘multiply by’

Specialist mathematics vocabulary Inquiry  How do you identify important topic- related terminology and make this explicit to students?  How do you inquire into students’ strengths and needs in relation to this?

Specialist mathematics vocabulary Direct and explicit teaching  What are some effective ways of initial/direct/explicit teaching of new vocabulary?

Specialist mathematics vocabulary Sufficient opportunities  to practice reading, writing, hearing and speaking the new terminology e.g.  Vocabulary jumble  Matching  Bingo  Clustering  Cline

Specialist mathematics vocabulary Encouraging reflective thought and action Times when it may be useful for students to reflect on their own or others thinking strategies include:  Strategies for approaching text e.g. what did you look at first? Why?  Strategies for remembering e.g. vocab  Strategies for making sense of an unfamiliar word e.g. context clues, morphemes  Strategies for recognising problem types

Activity  Develop an essential word list of key mathematics vocabulary for a unit.  Incorporate vocabulary activities into the unit following the framework of: –Inquiry –Direct teaching –Practice –Metacognition

Vocabulary Jumble Purpose This activity will help you to:  Check that you know the sound, spelling and meaning of some key words for this topic  Use the key word list to predict what the text or topic will be about

Vocabulary Jumble Instructions 1.Fold your arms and study the words on the OHT for TWO minutes. Try to remember all the words on List One so that you will be able to write them down later with correct spelling. Do the same for the Level Two words if you have time.

Level One Words right angleparallelperpendicular verticalpolygonbearings degreestrianglequadrilateral hexagonequilateralpentagon heptagonnonagondecagon isosceles octagon scalene lengthangle clockwise Level Two Words revolutionacutereflex obtuse vertexvertices

Activity 1.Tick all words you know well 2.Circle words that are new to you 3.Predict what you will learn about today

Polygons A polygon is a closed figure with three or more sides. Generally, a n-agon has n sides. E.g. a ‘3-agon’ is called a triangle; an ‘8-agon’ is called an octagon. If a polygon has all sides the same length, and all angles the same size, it is called regular. A square is a regular quadrilateral.

Polygons A ________is a closed figure with three or more sides. Generally, a n-agon has __ sides. E.g. a ‘3-agon’ is called a_____; an ‘8-agon’ is called an _____. If a polygon has all sides the same length, and all angles the same size, it is called regular. A square is a regular quadrilateral.

Polygons A ________is a closed figure with three or more _____. Generally, a n-agon has __ sides. E.g. a ‘3-agon’ is called a_____; an ‘8-agon’ is called an _____. If a polygon has all sides the_____ ______, and all angles the _______ ______, it is called regular. A square is a regular quadrilateral.

Polygons A ________is a closed figure with three or more _____. Generally, a n-agon has __ sides. E.g. a ‘3-___’ is called a_____; an ‘8-____’ is called an _____. If a polygon ___ all sides the_____ ______, and all angles the _______ ______, it is called regular. A square is a ________ _____________.

Prepositions Prepositions locate nouns, noun groups, and phrases in time, space or circumstance e.g.  The temperature fell to 10 degrees  The temperature fell by 10 degrees  The temperature fell from 10 degrees  The temperature fell 10 degrees

Prepositions contd.  Four into nine equals...  Four divided by nine…  Two multiplied by three…  Four exceeds three by …  Ten over twenty equals…  His pay rate increased from…. to…

Nominalisation  Nominalisation is the process by which verbs or adjectives are transformed into nouns  E.g. Sione works at a petrol station after school. His pay rate has increased from $11.20 per hour to $12.10 per hour. Calculate the percentage increase in his pay rate.

Nominalisation  Nominalised phrases abstract away from immediate lived experiences to build abstractions or generalisations  Knowledge becomes distilled & much of the semantic information gets hidden e.g. ‘gross hourly pay rate’ = ‘the amount a worker gets paid by their employer for each hour that they work, before tax is deducted’.  Readers who understand how nominalisation works are likely to be in a better position to work out the meaning of abstract nouns.

Activity: annotate a mathematics text with examples of:  Prepositions  Nominalisations  Other features that might ‘get in the way’

Avoid nominalisation (and other types of linguistic complexity)?  Some studies show that English Language Learners and students in average to low-level mathematics classes perform better on linguistically simplified mathematics assessments (Abedi and Lord 2001; Martiniello, 2007)  Teacher simplification of texts has risks but teaching students strategies for simplifying the language themselves may well be very useful.

Rewrite the problems by changing the nominalised parts 1.Today the cost of petrol has gone up by 6 cents per litre. What percentage increase is this? 2.Investigate the usefulness of the walkie- talkies in this situation. 3.Calculate the horizontal distance between the edges of the two steps.

Reversal errors 1.a is seven less than b  Correct equation: a = b – 7  Incorrect equation: a= 7 – b or a -7 = b 2. There are five times as many students as professors in the mathematics department  Correct equation: 5p = s  Incorrect equation: 5s = p

5. Strategies for “translating” word problems

‘Source’ and ‘target’  One way to conceptualise figurative language is as ‘source’ and ‘target’.  ‘Source’ is the original denotation of the word/phrase  ‘Target’ is what the writer is describing e.g. “Her eyes (target) are like limpid pools (source).” Boers, 2000

Ski Jump Nose SourceTarget

“Eyes empty like knot holes in a fence…” Source Target

Table activity ExampleTypeSourceTargetEffect

Abstracting the mathematics Sione has two savings accounts. One is for his university fees and the other is for his holiday. He divides the money between the university fees account and the holiday account in the ratio 5:2. Last week Sione banked $95 in his university fees account. Calculate the amount he banked in his holiday account.

Abstracting the mathematics Sione has two savings accounts. One is for his university fees and the other is for his holiday. He divides the money between the university fees account and the holiday account in the ratio 5:2. Last week Sione banked $95 in his university fees account. Calculate the amount he banked in his holiday account.

Abstracting the mathematics Sione has two bags. One bag is for bus fare (Bag A) and the other is for junk food (Bag B). He divides the money between Bag A and Bag B in the ratio 5:2. Last week Sione put $95 in Bag A. Calculate the amount he put in Bag B.

University feesHoliday 52 95?

TotalUniversity feesHoliday ??

Activity  Locate a complex word problem and practice a ‘think aloud’ for simplifying it or changing its context

Activity  Find examples of problems where it might be helpful for students to translate into a:  Mathematical sentence  A diagram  A table  Other?