In this paper, I argue that the establishment of after-school mathematics clubs in early grades holds rich potential for supporting the development of increasingly participatory and sense-making maths learning dispositions. Within the South African Numeracy Chair project, lead by the author, multiple after-school mathematics clubs have been set up for learners in Grades 3–6 across Eastern Cape schools. These clubs are a complementary initiative to teacher development, aimed at improving low levels of numeracy learning across the majority of schools in the province. Two sources of data, learner interviews and teacher questionnaires, from one case study club, are shared in this article to illuminate the potential such clubs hold in developing increasingly participatory mathematics learning dispositions.
South African mathematics education is widely acknowledged to be in crisis. Both performance on international assessments such as TIMSS (see Reddy
Whilst a range of complex systemic factors challenge mathematical learning (e.g. social disadvantage, language issues, teacher knowledge, and absence of teacher support), attention is increasingly being drawn to the limited opportunity to learn mathematics. Classroom-based research points to teaching that foregrounds ritual participation (including chanting), passive listening and little sense-making, resulting in many failing to progress beyond inefficient one-to-one counting methods, even by Grade 7 (Heyd-Metzuyanim & Graven
Adler argued in her presentation to the South African Mathematics Education Chairs’ initiative in 2010 that we have to ‘interrupt’ the learning and teaching culture in schools, where learners are passive, learning is teacher-dependent and the focus of teaching is on compliance (compliance to national assessments and producing documentation for district officials). Because our recent apartheid history systematically disempowered people politically, economically, socially and educationally, I have argued (Graven
Many learners have negative mathematical experiences, beginning in the early primary years, that impact on their mathematical learning trajectories throughout schooling. Earlier research I conducted with mathematical literacy learners revealed that many learners expressed helplessness and hopelessness in accessing mathematics (Graven & Buytenhuys
A sociocultural and socioconstructivist perspective of learning informed the design of clubs and related research. Mathematics learning is considered a social activity in which learners actively construct meaning. Additionally, learning involves developing dispositions and ways of being (Gresalfi & Cobb
Because mathematics is a strongly hierarchical subject, learners build new knowledge on previously learnt knowledge as they progress towards increasingly abstract levels and stages of numerical strategies, conceptual place value and multiplicative reasoning. Wright
Additionally, the five-stranded definition of mathematical proficiency put forth by Kilpatrick, Swafford & Findell (
refers to the tendency to see sense in mathematics, to perceive it as both useful and worthwhile, to believe that steady effort in learning mathematics pays off, and to see oneself as an effective learner and doer of mathematics (p. 131).
Kilpatrick
This coheres with the South African Curriculum and Assessment Policy (Department of Basic Education
Being mathematically proficient (numerate) means:
Understanding what you are doing (conceptual understanding);
Being able to apply what you have learnt (strategic competence);
Being able to reason about what you have done (adaptive reasoning);
Recognising that you need to engage (productive disposition) with a problem in order to solve it; and of course
Being able to calculate/compute (procedural fluency) with confidence. (Emphasis in original.)
However, a willingness to engage
Resilience: ‘the inclination to take on (at least some) learning challenges where the outcome is uncertain, to persist with learning despite temporary confusion or frustration and to recover from setbacks or failures and rededicate oneself to the learning task’. (p. 14)
Playfulness: ‘means being ready, willing and able to perceive or construct variations on learning situations and thus to be more creative in interpreting and reacting to problems. In our current conceptualisation we identify three different types of playfulness, which we refer to as mindfulness, imagination and experimentation’. (p. 14)
Reciprocity: ‘The most valuable learning resources, especially for the young, are, of course, other people. Those who lack the awareness to articulate their own learning processes and problems, the ability to communicate these to others or the inclination or the courage to do so are inevitably handicapped as learners’. (p. 15)
Whilst not included within the above definitions of productive learning dispositions, I consider developing enjoyment and passion for mathematical engagement and learning an essential part of our work with learners in clubs. This is an under-represented aspect in majority of mathematics education literature on both teaching and learning (Graven & Schafer
The South African Mathematics Education Chair initiative has established six chairs since 2010. I hold the SANC at Rhodes University in the Eastern Cape (one of the poorest and lowest performing provinces in South Africa). The Chairs differ and have the dual mandate of both research and development. All are involved in ongoing teacher development programmes with teachers from at least 10 schools (with learners from predominantly ‘disadvantaged’ backgrounds). It is beyond the scope of this paper to outline the work of the broader development project suffice it to say that the project has two interrelated professional learning communities. The first is our research community of master’s and doctoral students (many of whom are also teachers) and the second community consists of more than 40 Grades R to 6 teachers from 12 schools participating in the Numeracy Inquiry Community of Leader Educators (NICLE) program. Both communities meet regularly and work together to find ways forward to the challenges of mathematics teaching and learning.
Whilst NICLE is the key intervention programme of the SANC project, I chose a complementary learner-focused strategy, in the form of the implementation of multiple after-school primary mathematics clubs in order to support both teachers and learners in our project schools. The rationale for implementing clubs (focused at Grade 3 and 4) as an intervention strategy emerged from the context discussed above. Because most of the learners were way below the expected level of performance for their grade, the district-provided schemes of work and workbooks would constrain opportunities for remediating learning in classrooms. Furthermore, large classes, with little space for teacher movement between desks, often work against opportunities for individualised learner attention. Thus, whilst the SANC project works with teachers in NICLE to enable the development of key foundational numeracy skills, I believed that much could be gained by working with learners in after-school spaces in which club facilitators (project team members and NICLE teachers) would be free to work with learners in smaller groups focusing on specific learner needs informed by his/her competence level and learning disposition. Additionally, it was hoped that these club learners might become catalysts for demonstrating new ways of engaging in their mathematics lessons. This was seen as important as some teachers noted that when they shift their practices towards more learner-centred pedagogies through, for example, asking learners to find and explain alternative solution methods, learners often do not know how to respond. Feedback from teachers about club learners indicated that club learners often modelled for others ways of responding to new forms of teacher questioning.
Furthermore, club sessions and the take-home activities, games and worksheets provided would extend the time learners engaged with mathematics and would promote continued mathematical learning outside of clubs and classrooms. The cross-border Botswana–South Africa study by Carnoy
Another advantage for the SANC project is that clubs provide important learning spaces for project researchers and NICLE teachers through providing access to engagement with learners in smaller groups and without concern for disruptions to school time. Clubs can serve as explorative spaces (or ‘labs’) to try out new activities and ways of teaching and recovering early numeracy concepts, testing theories, frameworks and assessments before sharing them with a whole class of learners or teachers in NICLE workshop sessions. Thus, the clubs provide a safe space for trying new pedagogical approaches for teachers and additionally provide researchers with an empirical field for their studies. For many postgraduate researchers in SANC project (which includes several teachers and two NICLE teachers), the clubs provide empirical fields for their research (Hewana
Despite an absence of research (other than that generated by SANC project researchers) on
The methodology of the broader research combines qualitative and quantitative research methods across learners in clubs and schools. Qualitative methods include case study classroom observation, club observations (with transcriptions), individual club learner mathematics interviews and dispositional interviews (also video recorded and transcribed). Two sources of data are shared in this paper. Firstly, teacher questionnaire responses are shared that relate to possible influences of club participation on learner participation in mathematics class. Secondly, the responses of one learner to a learning dispositional questionnaire/interview instrument administered to the same learners three times over a 12-month period are shared. The dispositional instrument is given below. All data shared in this paper relate to one case study club of six learners in which I was the primary facilitator from May 2012 to November 2013 (
The dispositional instrument.
The club ran at an after-school development centre that supported learners who were most in need of after-school care because of circumstances at their homes. All Grade 3 learners at the centre were invited to participate in the club and parental permission was sought for both their participation and the related research. Six learners participated regularly in the club. All six club learners were in Grade 3 in 2012 and Grade 4 in 2013. There were three girls, Jaya, Bernie and Agnes, and three boys, Leonard, Thato and Sonny (these are all pseudonyms). In 2013, Agnes and Leonard moved to another town and were no longer in the club.
At the end of 2012 (after I had run the club on a weekly basis for 5 months), a questionnaire was provided to the teachers of each of the learners asking them whether they noted any influence of participation in after-school clubs on the learners in their class. The questions asked whether teachers thought the club had any influence on learners’ ways of participating in class, learners’ mathematical understanding and learners’ performance on assessments. All teachers, except Agnes’s, stated that the clubs improved learners’ performance on written assessments (and I have other data indicating improved performance shifts across other mathematics assessments). However, because some improvement in mathematics performance on assessments over time would be expected irrespective of learner club participation, here I include only responses in relation to learners’ ways of participating and understanding.
Teacher responses to questionnaire on club learners in their class.
Teacher | Ways of participating/understanding |
---|---|
Leonard and Bernie’s teacher | - Leonard’s maths improved a lot. He is enjoying every lesson. |
Jaya’s teacher | - Jaya and the other learners definitely benefit from club. Although she struggles, I can see she is doing her best to participate. |
Thato and Sonny’s teacher | - Yes, they were very quiet. Ever since they joined the club they are lively. |
Agnes’s teacher | - Yes, Agnes has become confident in answering questions verbally. The fear of giving a wrong answer (hesitating) has decreased. She starts to take chances. |
The teachers’ comments point to learners shifting ways of participating and understanding. They refer to increased enjoyment (three of the four teachers refer to this), increased willingness to verbalise/discuss and engage mathematically (three of the four teachers) and increased willingness to try without fear of being wrong (two of the four teachers), which was also accompanied by use of the term increased ‘confidence’ (three of the four teachers). Each of these relates, respectively, to the following aspects of productive disposition discussed above and derived from the literature: enjoyment/excitement, reciprocity (Carr & Claxton
Below, I present data from one club learner (Jaya) to illustrate shifts in ways of talking about mathematical disposition that were captured through administering a dispositional interview/questionnaire instrument. Similar shifts were noted for other learners in the club, although only Jaya admitted to not liking mathematics in the May 2012 questionnaire. Thus, her increasing love of mathematics was more visible over time.
The learner disposition interview was designed as both a written response (though orally administered) questionnaire and an interview schedule. The instrument involved several ‘complete the sentence’ items. The items included in this paper involved completing the sentence ‘Maths is …’ locating oneself on a spectrum of nine learners (the first indicating the weakest learner and the ninth the strongest in a class); completing the sentence ‘Sam is …’ after being told Sam is a strong learner who does well at maths. Another item asked learners how they felt about mathematics and a final question asked learners about what they do if they do not know an answer in maths class. (Sam was deliberately chosen as a gender-neutral name, although Jaya assigns male gender to this name in her responses.)
The dispositional instrument was administered to learners three times over a 1-year period (although two learners moved to a different town in 2013). All learner responses were transcribed. (Whilst there is much more data, including transcriptions of learners’ shifting mathematical participation and proficiency in club sessions and mathematical task–based interviews, it is beyond the scope of this paper to share these data.) This is a clear limitation of these data, and as Lahire (
Jaya’s responses to mathematics dispositional questions.
Question | May 2012 | Nov 2012 | May 2013 |
---|---|---|---|
Maths is … | Phonics | When you are learning and the teacher tells you more things to learn. Tells you times tables, minus and divide sums. | Learning a lot of sums and you have to listen because the next day she (teacher) comes. (You must) concentrate on what the teacher tells you. Learning lots of fun maths games, cards to play, times table, division sums and multiply. |
Self-evaluation | 2 | 9 | 9 |
Sam is … | Good at maths | Is very good in the class. His teacher is very proud of him. He always does the things the teacher tells him to do. | Listens a lot and can remember what the teacher tells him. He is working everyday in his book. If he has done everything he shows the teacher and then fixes mistakes. |
Feelings about maths | I don’t like maths | I’m loving maths. It’s so nice to be in maths class. It’s my favourite thing. I have a (good work) sticker in my book. When I am big I will study it, maybe become a teacher, it will be fun for me to do maths with children. With other children we play school in class and I give them maths problems. Its lovely to do maths. | I love maths because its nice to do it and we learn more (in club) than what we do at school. |
What do you do if you don’t know answer? | I ask the teacher | I write down the sums on the paper then I halve this … I draw some children when we do problems. Share the remainder in pieces. | Then I’m getting scared because the teacher says you didn’t listen but last year the teacher said we must tell her and she would give a green counting card. In maths club I ask you and then I will try figure out the answer and use a paper. In class my friend explains and I ask for a piece of paper. |
Jaya’s initial responses in May 2012 indicate a dislike of mathematics and a lack of confidence in her ability. She identifies herself as second from the bottom of nine performance-ranked learners in a class. Her description of Sam reflects the dominant finding in the data of 1208 Grade 3 and 4 learners in project schools where the same instrument was administered in May 2012 (including to Jaya as a learner in one of these schools) orally to a whole class, but learners provided individually written responses on the questionnaire. In this larger data set, most learners described Sam in terms of either innate cleverness or as someone who listens and/or is well behaved in class. Only 1% of learners described Sam as hard working and only 2% described Sam as someone who thinks or makes sense of mathematics (Graven & Heyd-Metzuyanim
Whilst there are clear limitations to the instrument, especially when orally administered with written learner responses (as in the May 2012 responses), it does reveal some shifts in terms of Jaya’s stated relationship with mathematics. Her first written response to what maths is indicates a difficulty with providing a written description of it and was assumed to have been copied from the word phonics that was written on the board at the time. Whilst her description of what maths is continued in 2013 to emphasise the importance of the teacher and listening and complying with what she ‘tells you’, she is able to describe a range of activities of the mathematics class including some fun maths games. Whilst Jaya’s self-evaluation is overstated in relation to her performance positioning in the class, it coheres with her own improvement in mathematical proficiency (which in our task-based interview assessments showed improvement from 65% to 83% over 2012). Additionally, it shows Jaya’s increasing confidence and sense of self-efficacy (a key aspect of a productive disposition in the definition provided by Kilpatrick
Similarly, Jaya’s description of what behaviours constitute being a good math learner (i.e. Sam is …) shifted from simply being ‘good at maths’ or being compliant and pleasing the teacher (always doing what ‘the thing the teacher tells him’) to listening and ‘working every day in his book’ (steady effort) as well as a focus on correcting his errors (pointing to both steady effort and resilience). In terms of Jaya’s relationship to mathematics, she shifts from saying she does not like maths to loving it and elaborates on how ‘loving it’ plays out with her friends. She indicates that she ‘plays school’ with other children by giving them maths problems and has additionally developed an imagined trajectory of studying it when ‘she is big’. In relation to the discussion made above, one could argue that, at least in terms of what Jaya says, her mathematical participation and relationship with mathematics in terms of key dispositional traits has shifted.
Of interest, in 2014, she mentions that in clubs she does harder activities but adds that this is fun, indicating some resilience in relation to mathematical challenge. Similarly, her response in November 2012 indicates a wider range of options that draw on her own resources rather than only asking the teacher (May 2012) (e.g. writing and using drawings), when not knowing an answer. However, her May 2013 response shows that her disposition in class is strongly influenced by the teacher. Thus, with her Grade 4 teacher, she says she is ‘scared’ if she does not know an answer as she will be in trouble for ‘not listening’, whilst she says that her Grade 3 teacher would provide her with support resources such as counting cards if she did not know an answer. In maths club, she adds that whilst she will ask me, she will also try to ‘figure it out’. That one’s learning disposition is affected by one’s teachers is unsurprising and indeed an assumption of this development and research work is that dispositions can be proactively shaped.
In this paper, I shared a sample of data to illustrate that after-school mathematics clubs can provide rich opportunities for developing increasingly productive learning dispositions. The data, exemplified by one learner, point to several indicators of shifting disposition. These indicators, as drawn from the Kilpatrick
Whilst I have argued that the after-school club I ran enabled shifting dispositions, this is not to imply that there is anything inherent in an after-school club that will ensure increasingly productive dispositions, nor that mathematics classrooms cannot and should not provide these opportunities. Indeed, supporting teachers to strengthen their student learning dispositions is critical, and within our broader work with teachers (in the NICLE project), we work towards promoting teacher practices that enable increasingly productive mathematics learning dispositions. However, I have argued that after-school clubs, because of their extra-curricular freedom, hold rich potential for explicit focus on strengthening learning dispositions. Club learners might then become catalysts in their mathematics classrooms for modelling increasingly active, participatory, sense-making, hardworking and resilient ways of engaging mathematically (as was indicated to some extent in the teacher data on shifting classroom participation of the six learners in this case study club).
The authors declare that they have no financial or personal relationships that may have inappropriately influenced them in writing this article.