ࡱ > \ ^ [ a < jbjbA]A] 4^ +? +? 5 % t t t
~ 6 6 6 8 n $ \ e ( " " D D D D D D , " R S$ R
D D D D D
D D D D D ( T | 6 T 6 D : x , r v 6
5 0 e R % % 6 6 Mathematical literacy: issues for engagement from the South African experience of curriculum implementation
A core paper prepared for Discussion Group 3: Math education: for what and why?
by: Mellony Graven & Hamsa Venkat, Marang Centre, Wits University
Introduction
Due to the restricted length of this paper we will focus our attention on giving a very brief story of what our research is indicating in relation to the implementation of Mathematical Literacy as a new subject in the SA curriculum, and the issues arising within this implementation. For further reading we suggest reference to Graven & Venkat (2007) and Venkat & Graven (2007) attached to this DG3 website.
This will be followed by introducing some key insights and issues which our research has highlighted and we believe require further exploration, discussion and investigation in the DG3. Thus each insight/issue raised will be accompanied by a leading question and subset of questions which we will engage with in DG3 at the ICME conference.
Mathematical Literacy the South African case
What the curriculum says:
Mathematical literacy (ML) was introduced in schools in the Further Education and Training (FET) phase (grades 10-12, learners mainly aged 15-18) in South Africa in January 2006. The subject is structured as an alternative option to mathematics, and all learners entering the FET phase since January 2006 are required to take one or other of these two options. ML is defined in the curriculum statement in the following terms:
Mathematical Literacy provides learners with an awareness and understanding of the role that mathematics plays in the modern world. Mathematical Literacy is a subject driven by life-related applications of mathematics. It enables learners to develop the ability and confidence to think numerically and spatially in order to interpret and critically analyse everyday situations and to solve problems. ADDIN EN.CITE DoE2003175, p91756DoE,National Curriculum Statement Grades 10-12 (General): Mathematical Literacy2003Department of Education(DoE, 2003, p9)
While the phrase applications of mathematics allows for an interpretation where mathematical language, conventions, algorithms, theorems and practices can be learnt first and then applied to life related problems and everyday situations, the post-amble headed Context following the Learning Outcomes, emphasizes that contexts should be engaged with in a way which enables and drives mathematical learning:
The approach that needs to be adopted in developing Mathematical Literacy is to engage with contexts rather than applying Mathematics already learned to the context. (DoE, 2003a, p.42)
This is re-emphasised in the Teachers Guide although here there seems to be an attempt to highlight the importance of striking a balance between contextual and mathematical learning and an emphasis on the dialectical relationship between the two:
the challenge for you as the teacher is to use situations or contexts to reveal the underlying mathematics while simultaneously using the mathematics to make sense of the situations or contexts ADDIN EN.CITE DoE2006195, p41956DoENational Curriculum Statement Grades 10-12, Teacher Guide, Mathematical Literacy2006Department of Education(DoE, 2006, p.4)
Our research
Our work within the Mathematical Literacy (ML) thrust in the Marang Centre at Wits University involves a range of strands research, lecturing and teacher development, and raising public awareness. Our research work centrally involves a longitudinal case study, now in its third year, tracing the experiences of educators and the first cohort of learners taking ML in one inner city Johannesburg school. This work has involved weekly visits to the three ML classes in this cohort across grade 10, grade 11 and now grade 12, as well as questionnaire and interview data from learners and teachers. Additionally, we have conducted a series of classroom observations, learner and teacher interviews in three other Johannesburg schools. We also draw upon feedback from the teachers we interact with as part of our lecturing, supervision and thrust work.
Our findings
We have identified a spectrum of agendas that teachers work with as they navigate their teaching of Mathematical Literacy across the FET band (see Graven & Venkat, 2007). While the spectrum is likely to be refined and reviewed over time it has proven useful as a tool for analysing mathematical literacy practices:
Table 1: A spectrum of agendas
Context driven (by learner needs)
Content in service of contextContent & context driven
Content & context in dialectical relationship Mainly content driven
Context in service of contentContent driven
No clear need for contextDriving agenda:
To explore contexts that learners need in their lives (current everyday, future work & everyday, and for critical citizenship) and to use maths to achieve this.Driving agenda:
To explore a context so as to deepen math understanding and to learn maths (new/GET) and to deepen understanding of that context.Driving agenda:
To learn maths and then to apply it to various contexts.Driving agenda:
To give learners a 2nd chance to learn the basics of maths in GET band (grades 0-9)
Our documentary analysis of the curriculum suggests that in intention at least agenda 2 is foregrounded.
Our classroom observations and interviews to date seem to suggest that learners have tended to be largely positive about ML. We are well aware of the fact that this may well not be the case more broadly and that in the classrooms that we have observed teachers have tended to work with agendas towards the left hand side. In these classrooms both learners and teachers have noted substantive differences (compared to mathematics experiences) in: the nature of tasks in ML (engagement with a scenario rather than application of maths in word problems) and the nature of interaction in ML (much slower pace, more discussion and group work). See Venkat & Graven (2007) for elaboration and evidence of such changes.
Some teachers on our ACE and postgraduate courses have reported low levels of motivation and lack of interest amongst learners in their ML classes. Anecdotal evidence from educators suggests that this kind of ongoing negativity is associated with a lack of substantive change in pedagogic practice. In some cases this is due to an interpretation by the teacher of ML as involving basic maths (towards the right of the spectrum), and consequentially, teaching that incorporates the kinds of tasks and pedagogic practice that have predominated within learners earlier experiences with mathematics.
Our table highlights various issues that are experienced by teachers when working with a particular agenda. In particular we point to the issues of authenticity of context, development of mathematical progression and discrepancies of continuous and summative assessments. These issues are experienced by teachers in different ways depending on their primary driving agenda. Below we summarise these issues and raise some questions for discussion in the DG.
Issues for discussion
* Authenticity & mathematical progression
Agendas on the left hand side demand a certain degree of authenticity in the tasks and scenarios that learners engage with. This can sometimes raise tensions in relation to mathematical progression. Thus many teachers might wish to trim off certain aspects of a context as they deal with mathematics which they feel learners are not yet ready to deal with. Trimming such contexts affects the authenticity of the task. Other teachers feel that mathematical progression is less of an issue in mathematical literacy and that one draws on the maths you need when and where you need it (and then only as far as you need it and not further).
Curriculum documents struggle to build mathematical progression into assessment standards from one grade to another so for example assessment standards from one grade to the next simply say in more complex contexts or provide what they believe are more complex contexts from one grade to the next.
DG Topic question
In what ways might teachers and materials developers work with such tensions between authenticity and mathematical progression?
How should mathematical literacy year to year planning deal with progression (contextual and/or mathematical)?
* Discrepancies in performance between continuous and summative assessments
In ML there is an increased amount of project work or extended activity work which is often done over an extended period of time, is mediated by the teacher and draws on other learners ideas (often in group work). Success in such work sometimes contrasts with weaker performance in time constrained, unmediated and individualized assessments.
DG Topic question
What do such discrepancies tell us about learner ML competences? Are such discrepancies a problem?
Is there a place for individualized, time constrained, summative assessments in ML? If so, what is their place and how should performance on these be weighted against performance on other types of activities?
What about issues of assessment validity? [E.g. in SA the externally set and marked Grade 12 examination carries the most weight - seen to have greater validity and reliability but such assessments sometimes reduce the scope of ML]
* ML in its own right versus ML in contrast Mathematics
Much of the data that we have received to date on ML is in contrast to learner and teacher experiences of Mathematics. Thus many of the successes for learners are often in contrast to their largely negative experiences of learning mathematics. While this provides interesting comparative data it skews the data towards aspects which contrast to mathematics. ML is now in its third year of implementation. We are beginning to get comments from learners which critique the issues arising in ML classrooms in their own right without comparison to Maths comments about ML involving the need to make sense of situations for example. As researchers too, we are aware of the mathematical lenses that we bring to our observations. Depending on how one views ML such lenses can be obstacles in the way of being able to see the range of non mathematical outcomes achieved in classes.
DG topic question
In what ways do our mathematical experiences/ lenses through which we view ML an influence what we see and how we interpret what we see? How do we reflect on the influence of these experiences?
How do varying definitions of ML affect the appropriateness of the influence of mathematical lenses on its research? [E.g. If ML was instead incorporated into or renamed Life Orientation would we research it through a different lens?]
How (if we see this as desirable) do find ways to research ML in its own right without ongoing comparison to mathematics?
* Language issues
In our research to date there is a conspicuous absence of comments by learners and teachers noting any language difficulties (note in SA the majority of learners are taught in English which is not their first language). Much has been written on the difficulties of learning mathematics in this context (see Setati, 2005), and many SA educators have noted that integration with contexts could be problematic due to the increased English language demands.
In contrast learner interviews have highlighted accessibility in relation to language demands in ML with comments such as Its in English unlike Mathematics which was compared to a language they did not speak or understand (e.g. Latin). Our hypothesis is that changes in both the nature of activities and mediation of those activities (increased discussion in class with learners around scenarios and contexts used) are supporting learners in the language demands of ML and the use of a more everyday, rather than more technical, register alleviates some of the language demands that are noted in mathematical classes.
DG Topic question
What is the nature of the language issues that emerge in the teaching and learning of mathematical literacy? How are these similar or different from those that emerge in the teaching of mathematics?
How should language issues be researched in ML? How (if at all) should such research differ from research on language issues in mathematics classes?
References:
DoE. (2003). National Curriculum Statement Grades 10-12 (General): Mathematical Literacy: Department of Education.
DoE. (2006). National Curriculum Statement Grades 10-12, Teacher Guide, Mathematical Literacy: Department of Education.
Graven, M. & Venkat, H. (2007) Emerging pedagogic agendas in the teaching of Mathematical Literacy. AJRMSTE, Vol 11 (2), pp. 67-86
Venkatakrishnan, H. & Graven, M. (2007) Insights into the implementation of Mathematical Literacy. Proceedings of the Thirteenth Annual National Congress of the Association for Mathematics Education of South Africa (AMESA), Uplands College, Mpumalanga, Vol 1, pp 72-83.
Setati, M. (2005) Teaching Mathematics in a Primary Multilingual Classroom. JRME. NCTM, USA. 36(5), 447-466.
Many assessment standards (ASs) are the same for all three grades with different contextual examples given in each grade. No detail is given on what makes one context more complex than another, or what progression within a context might entail.
PAGE
PAGE 5
l s
*
+
D
E
X Y h i [ i y z { | W X e f h i j k w ºªªª£££££££ h-i h |f 5mH sH j hc
z h |f U hc
z h |f h h |f j h |f Uh#O[ h |f 5h& h |f 5h |f h h |f 5mH sH h |f mH sH h& h |f 5mH sH h |f 5mH sH h& h |f 5CJ aJ mH sH 0 l
*
+
E
j j ]^ gd |f ]^ gd |f d ]^ gd |f $ ]^ a$gd |f ; < < j k x 4 W u
$If ^ gd |f $ d $7$8$If ^ a$ gd |f ]^ gd |f $ ]^ a$gd |f $]a$gd |f w x 4 W u N R - 2 5 " " h$ ~$ Q( R( 8 8 8 8 ϩϩϩzne hm h |f CJ h-i h |f 5mH sH h |f 5mH sH j h |f 0J Uh-i h |f 5hL0 h |f CJ H*aJ h |f CJ aJ hL0 h |f CJ aJ hL0 h |f CJ PJ aJ hL0 h |f 5CJ PJ aJ hL0 h |f 5CJ aJ h-i h |f 5mH sH h |f mH sH h |f h#O[ h |f 5mH sH ' , _ o ^ Q Q Q Q Q Q Q
$If ^ gd |f kd $$If l \ 4 ( @ @ p p
t 0 `'6 4
l 4 a
y z E U E E E E $ ]^ a$gd |f kdt $$If l \ 4 ( @ @ p p
t 0 `'6 4
l 4 a $If gd |f E F " " g$ h$ ~$ $ $ %' T( f( ( U) $q x $If ]q ^ a$gd |f $ x ]^ a$gd |f $ ]^ a$gd |f
U) V) W) ) *
+ p+ A, +- o o o o $q x $If ]q ^ a$gd |f $ x ]^ a$gd |f g kd $$If l hD% #
t 0 6 4
l 4 a +- ,- -- e- 0 0 1 2 3 o o o o $q x $If ]q ^ a$gd |f $ x ]^ a$gd |f g kd6 $$If l hD% #
t 0 6 4
l 4 a 3 3 3 "3 4 V7 h7 /8 8 o o o $q x $If ]q ^ a$gd |f $ x ]^ a$gd |f g kd $$If l h% $
t 0 6 4
l 4 a 8 8 8 8 8 F9 9 ^ J J J d ]^ ` gd |f ' $
, 0p@ P !$ ]^ a$gd |f $ x ]^ a$gd |f g kd $$If l hD% #
t 0 6 4
l 4 a 8 ,9 9 9 9 ): A: B: C: P; Q; R; ; ; ; ;< < < < < < < < < < < < < < < < < < < < < < < < ôå㛗}}}}smsmsmsbsm h |f 0J mH nH u
h |f 0J j h |f 0J Uj h |f Uh h |f mH sH hn h |f h |f j h |f 0J U hk h |f CJ aJ mH sH hm h |f CJ aJ mH sH h |f CJ aJ mH sH hm h |f 6CJ aJ h |f CJ aJ hm h |f CJ aJ hm h |f CJ h h |f CJ &9 B: C: Q; R; ; < < < < < < < < < < < < < < < < < h]hgd |f &`#$gd |f $ x ]^ a$gd |f $ ]^ a$gd |f < < < hm h |f CJ aJ h |f < < $ x ]^ a$gd |f 2 1h:p |f / =!n"n#n$n% r $$If !v h5 @ 5@ 5p5p#vp#v @ :V l
t `'6 5p5 @ a r $$If !v h5 @ 5@ 5p5p#vp#v @ :V l
t `'6 5p5 @ a L $$If !v h5 ##v #:V l
t 6 5 #aL $$If !v h5 ##v #:V l
t 6 5 #aL $$If !v h5 $#v $:V l
t 6 5 $aL $$If !v h5 ##v #:V l
t 6 5 #a @ @ @ N o r m a l CJ _HaJ mH sH tH D A@ D D e f a u l t P a r a g r a p h F o n t R i@ R T a b l e N o r m a l 4
l 4 a ( k ( N o L i s t H O H @ Q u o t e $ d x 7$ 8$ ^ a$ aJ N O N @ l o n g q u o t a t i o n $^a$ mH sH D O D @ q u o t e $]^a$ mH sH > @ "> &
F o o t n o t e T e x t CJ aJ @ &@ 1@ & F o o t n o t e R e f e r e n c e H* F Q@ BF " B o d y T e x t 3 x CJ aJ mH sH j @ Sj "
T a b l e G r i d 7 :V 0 N O N " F i g T i t l e $d@ x x 7$ 8$ a$ CJ aJ f O rf HA
R e f e r e n c e s ( $ d x 7$ 8$ ^ `a$ CJ aJ mH sH B '@ B \F C o m m e n t R e f e r e n c e CJ aJ < < \F C o m m e n t T e x t CJ aJ @ j @ \F C o m m e n t S u b j e c t 5\H H \F B a l l o o n T e x t CJ OJ QJ ^J aJ 4 @ 4 5 F o o t e r
! . )@ . 5 P a g e N u m b e r Q" 6 6 ^ z z z z z x ,' 2 6 R l
* + E j j k x 4 W u , _ o
y z E F g h ~ %! T" f" " U# V# W# # $
% p% A& +' ,' -' e' * * + , - - - "- . V1 h1 /2 2 2 2 2 2 F3 3 B4 C4 Q5 R5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 $ @ 8 r@ $ @ 8 $ @ 8 $ @ 8 $ @ 8 $ @ 8 $ @ 8 $ @ 8 $ @ 8 $ @ 8 $ @ 8 $ @ 8 $ @ 8 $ @ 8 $ @ 8 $ @ 8 $ @ 8 $ @ 8 $ @ 8 $ @ 8 $ @ 8 $ @ 8 $ @ 8 $ @ 8 $ @ 8 $ @ 8 $ @ 8 $ @ 8 $ @ 8 $ @ 8 > > W > > W > > W > > W > > W > > W > > W > > W 9 )( ` l $ @ 8 ^ , ^ , ^ , ^ , ^ , ^ , ^ , ^ , xS $ @ 8 $ @ 8 $ @ 8 $ @ 8 $ @ 8 $ @ 8 $ @ 8 0 P 0 P 0 P 0 P 0 P 0 P 0 P 0 P 0 P 0 P 0 P 0 P 0 P 0 P 0 P 0 P 0 P 0 P 0 P 0 P 0 P 0 P 0 P 0 P 0 P 0 P 0 P 0 P 0 P 0 P 0 P 0 P 0 P 0 P 0 P 0 P 0 P 0 P 0 P 0 P 0 P 0 P 0 P 0 P 0 P 0 P @ 8 0 @ 0 @ 0 @ 0 @ 0 0 $@ 8 $ $ $ ' w 8 < < # + - j E U) +- 3 8 9 < < " $ % & ' ( ) * , . < ! X h {
W h 6 QQ ' ! ! ' ) ^ ` M T Z ] j x z & ) ) +4 .4 45 >5 @5 C5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 2 2 F3 K3 "4 +4 24 B4 C4 4 4 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 : : {6Tp C ? @ A B C D E F G H I J L M N O P Q R T U V W X Y Z ] R o o t E n t r y F _ D a t a
0 1 T a b l e 8 % W o r d D o c u m e n t 4^ S u m m a r y I n f o r m a t i o n ( K D o c u m e n t S u m m a r y I n f o r m a t i o n 8 S C o m p O b j X O b j e c t P o o l F Microsoft Word Document NB6W Word.Document.8