Literacy in math

1. Literacy in the mathematics
classroom
Aaron Wilson
SLP
February 2011

2. 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)

3. 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)

4. Shanahan & Shanahan (2008)

5. 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).

6. 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).

7. 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).

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

9. 2. Vocabulary in mathematics

10. 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’.

11. 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’

12. 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.

13. 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.

14. 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.

15. 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.

16. 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 ________
_____________.

17. 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

18. 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…

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

20. 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.

21. 3. Activating prior
knowledge and building
necessary background
knowledge

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

23. 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.

24. 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).

25.  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

26.  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).

27. 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.”

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

29. ‘Predictable’ contexts (90151)
Context Examples 2006 2007 2008 2009
Discounts Specials
Sale price
√ √ √
Price increases √ √ √
Prices
Price per unit √
Income Pay rate
Hourly rate
√
gst √ √
Tax on income PAYE
Taxation
Tax (on interest) √
Reduce in value √
Change in
value Increase in value
investment √ √
interest √ √
Compound interest √ √
Investments
Retirement √
Annual turnover √
Profit
Business
Loss

30. 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?

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

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

33. 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)

34. 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?

35. 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

36. 5. Strategies for “translating”
word problems

37. 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.

38. University fees Holiday
5 2
95 ?

39. Total University fees Holiday
5 2
133 ? ?

40. ‘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.

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