Literacy in the mathematics classroom Aaron Wilson SLP February 2011

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)

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)

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).

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

2. Vocabulary in mathematics

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’

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.

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…

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.

3. Activating prior knowledge and building necessary background knowledge

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

 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.

‘Predictable’ contexts (90151)

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?

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

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  words that are italicized, underlined or in bold  headings & subheadings,  labels (e.g. row, column, 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?

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

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.

University feesHoliday 52 95?

TotalUniversity feesHoliday ??

‘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