Systemic functional linguistics in the Australian Curriculum: English

The Australian Curriculum: English (AC:E) combines traditional Latin-based grammar with Systemic Functional Linguistics (SFL) theory. The syllabus-supporting material refers to traditional grammar as ‘standard grammatical terminology’, and to SFL as its ‘contextual framework’. Functional grammar is introduced across all three English strands Language, Literature and Literacy starting in the Foundation year. However, the curriculum language and terminology does not always make this explicit (Exley, 2016). This is because a conscious attempt was made to write content descriptors that ‘describe the knowledge, understanding, skills and processes that teachers are expected to teach and students are expected to learn’ in a metalanguage readily accessible to all teachers.

This post assesses the relevance of the functional model of language (SFL) across all 237 AC:E content descriptors for primary schools (Foundation to Year 6). The analysis is based on AC:E (v8.1) content descriptors and elaborations that are thematically grouped by year level, English strand and sub-strand by the Australian Curriculum, Assessment and Reporting Authority (ACARA). The ‘English: Sequence of content‘ document is annotated using three levels of SFL relevance:

  1. not applicable (red)
  2. somewhat applicable (orange)
  3. very applicable (green).

Here are some examples for how SFL terminology has been translated in the Australian Curriculum (Derewianka, 2012; Exley, 2016):

The Register of language is described in the following words:

  • Field – ‘topics at hand
  • Tenor – ‘relationships between the language users
  • Mode – ‘modalities or channels of communication

Metafunctions of language are specifically addressed in the following Language sub-strands:

  • Expressing and developing ideas unpacks the functions of language, i.e. ideational, interpersonal and textual
  • Text structure and organisation unpacks the thematic structures of text, i.e. how to create coherent and cohesive texts
  • Language for interaction unpacks the ‘Mood system’ and ‘System of Appraisal’ of language (Martin & White, 2005), language functions that establish and maintain relationships, including expressing graduations in feelings, emotions, opinions and judgements (Tenor).

The examples for AC:E language relating to the ‘System of Appraisal’ analysing Attitude, Graduation and Engagement (Martin & White, 2005) are compiled by Beryl Exley (2016):

  • appreciating … the qualities of people’ (ACELA1462) – i.e. expressing ‘judgement’
  • evaluations of characters’ (ACELA1477) – i.e. expressing ‘judgement’
  • judgement about … events’ (ACELA1484) – i.e. expressing ‘appreciation’
  • exploring examples of language which demonstrate a range of … positions’ (ACELA1484)- i.e. expressing ‘appreciation’
  • feelings’ (ACELA1484, ACELA1518) – i.e. expressing ‘affect’
  • engages us emotionally’ (ACELT1606) – i.e. expressing ‘affect’ and ‘engagement’
  • empathy’ (ACELT1610, ACELY1698, ACELA1518) – i.e. expressing ‘affect’ and ‘engagement’
  • identifying the narrative voice’ (ACELT1610, ACELY1698) – i.e. expressing ‘focus’ and ‘engagement’
  • point/s of view’ and ‘viewpoints of others’ (ACELT1603, ACELT1609, ACELY1698, ACELA1518) – i.e. expressing ‘appreciation’
  • concern for their welfare’ (ACELA1518) – i.e. expressing ‘affect’
  • make connections between students’ own experiences and those of characters and events represented in texts’ (ACELT1613) – i.e. expressing ‘engagement’
  • attitudes we may develop towards characters’ (ACELT1613) – i.e. expressing ‘judgement’ and ‘engagement’
  • build emotional connection’ (ACELT1617) – i.e. expressing ‘affect’ and ‘engagement’

Statistical analysis of the 237 annotated AC:E primary school content descriptors (CD) highlights some interesting facts. Teaching and learning opportunities related to the functional model of language increase from Foundation (12 or 33% of all CD) to Year 6 (21 or 67% of all CD). Only half of all AC:E CD in the Foundation year have no links to SFL. This number is gradually reduced to just 13% in Year 6! A more detailed analysis of CD by English strand and sub-strands highlights that SFL teaching and learning is very applicable across all three strands: Language (47%), Literature (62%), and Literacy (44%). However, due to the large number of Language CD (49%), nearly half of all very applicable CD (47%) fall into the Language Strand.

Systematic functional linguistics relevance to Australian Curriculum (v8.1): English content descriptors

Systematic functional linguistics relevance to Australian Curriculum (v8.1): English content descriptors by English strand

Systematic functional linguistics relevance to Australian Curriculum (v8.1): English content descriptors by Sub-strands

The results suggest that SFL, in particular transitivity, system of appraisal, and genre writing approaches, plays a significant role in the teaching and learning of English at Australian primary schools. The functional model of language is particular important in the AC:E Language strand, most prominently in the sub-strands “Expressing and developing ideas“, “Text structure and organisation“, and “Language for interaction“. Beverly Derewianka (2012) explains that the new Language strand, designed to teach and learn specific knowledge about the English language, requires a robust, future-oriented, unifying model of language that can meaningfully link grammatical form with function from the level of discourse (genre) to individual phonemes. This is achieved through the introduction of SFL, as this functional model adequately describes how language is used in social contexts.

References:

  • Derewianka, B. (2012). Knowledge about language in the Australian curriculum: English. Australian Journal of Language and Literacy, 35(2), 127-146.
  • Exley, B. (2016). Secret squirrel stuff in the Australian curriculum English: The genesis of the ‘new’ grammar. Australian Journal of Language and Literacy, 39(1), 74.
  • Martin, J.R. & White, P.R.R. (2005). The language of evaluation: Appraisal in English. London: Palgrave.

Very relevant AC:E content descriptors by year level:

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What mindfulness practices work best in an Australian Year 5 classroom?

According to the Melbourne Declaration on Educational Goals for Young Australians, one key goal of school education is to enable young learners to manage their emotional, mental, spiritual and physical wellbeing (Barr et al., 2008). The Australian Curriculum, Assessment and Reporting Authority (ACARA) addresses this goal in the focus area ‘Mental health and wellbeing’ of the new Health and Physical Education (AC:HPE) syllabus (ACARA, 2017a), as well as in the general capabilities ‘personal and social capability’, which supports students in “. . . recognising and regulating emotions, [and] developing empathy for others” (ACARA, 2017b), and ‘ethical understanding’ through fostering “. . . attributes such as honesty, resilience, empathy . . .” (ACARA, 2017c).

An increasing body of international research highlights the role mindfulness practices can play in supporting mental health across all age groups in school settings. Mindfulness is also discussed to increase students’ socio-emotional wellbeing, behaviour and academic performance. However, when it comes to the implementation of mindfulness practices in the classroom, not all approaches are equally suitable and effective. Teachers are faced with the task to select and implement the components of mindfulness intervention programs that are most responsive to their specific school community, year level and classroom (Gould, Dariotis, Greenberg, & Mendelson, 2016). This post provides new insights into which mindfulness approaches work best in an Australian Year 5 classroom. It is based on student responses throughout a four-week mindfulness program that investigated students’ experiences of and perceptions towards sensory awareness practices, movement awareness practices, and attention and emotion regulation practices. The mindfulness practices conducted for this research directly benefited all participating students and the outcomes can inform implementations of mindfulness programs in upper primary schools across Australia.

Literature review

Over the past decade, mindfulness has developed into a buzzword closely associated with mental and emotional health interventions, as demonstrated in the tenfold increase in related internet search queries, with disproportionately high numbers from Australia (Google Inc., 2017). An identical trend is recorded by the research platform The Web of Science® (Thomson Reuters, 2017), with 20% (>1,500) of academic papers investigating the benefits of mindfulness published in 2016 alone.

Back in 2004, a panel of clinical psychologists agreed on a widely-cited operational definition of mindfulness:

mindfulness guides and sustains awareness of the present moment, the thoughts, feelings, and sensations that arise, through self-regulation of attention on the immediate experience through a mental orientation of curiosity, openness, and acceptance (Bishop et al., 2004).

Two schools of mindfulness interventions dominate the research landscape (Ivtzan & Hart, 2016). The first is led by Jon Kabat-Zinn from the University of Massachusetts, who developed mindfulness programs drawing on ancient Buddhist meditative practices (e.g. Kabat-Zinn, 2004). The second is led by Ellen Jane Langer from Harvard University, who developed a “psychology of possibility” that questions preconceived mindsets and encourages acting on new observations (Langer & Moldoveanu, 2000). For this research report, twelve formal mindfulness practices mostly aligned with the first approach were curated, developed and delivered in separate lessons (Appendix A).

Despite the methodological shortcomings of individual studies, mindfulness can now be considered a scientifically-proven method for improving psychological aspects of wellbeing and executive functioning (Gu, Strauss, Bond, & Cavanagh, 2015). For example, research into neuroplasticity based on electroencephalographic measurements of the brain activity of experienced meditators, established that sustained mindful awareness can improve the depth of information processing and the speed of attention allocation (Van Leeuwen, Singer, & Melloni, 2012). In another seminal work, based on correlational, quasi-experimental and laboratory studies, mindfulness is shown to improve psychological wellbeing by reducing negative emotions such as anxiety, hostility and depression, while enhancing positive affectivity, including self-awareness, self-regulation, and emotional intelligence (Brown & Ryan, 2003).

In the school context, the potential of mindfulness-based interventions has recently been established in a systematic literature review and statistical meta-analysis to improve students’ cognitive performance, emotional wellbeing, and behavioural self-regulation (Zenner, Herrnleben-Kurz, & Walach, 2014). Applied to primary schools settings, mindfulness is still an emergent field of research, with a growing number of case studies suggesting similar benefits (e.g. Arthurson, 2015; Flook et al., 2010). The most comprehensive independent study so far is based on an eight-week mindfulness program delivered in five primary schools in New Zealand (Rix & Bernay, 2015). The results corroborate the positive findings for enhancing psychological wellbeing and executive functioning previously established for adolescents and adults. In Australia, a pilot mindfulness-based teaching program conducted in Year 7 reported similar results and further highlighted the high approval rates by participating students (Arthurson, 2015).

These promising research outcomes inspire an increasing number of mindfulness-based educational programs advocating for a ‘mindfulness curriculum’ in Australia. Smiling Mind (2016) is a prominent example of a not-for-profit organisation promoted by KidsMatter (2013), an Australian mental health and wellbeing initiative set in primary schools. The program is codeveloped by Australian psychologists with knowledge of developmental stages, and its modules are aligned with the AC:HPE bands. A critical review of existing mindfulness programs for schools however highlights the wide range of different mindfulness approaches offered in each age-based module (e.g. Meditation Capsules, n.d.; Smiling Mind, 2016). So far, little evaluation has been conducted into the different types of mindfulness activities that are most responsive to particular age groups (Burke, 2010). There is still a gap in our understanding of how effective and engaging particular mindfulness approaches and activities are in different year levels (Arthurson, 2015).

This study aims at narrowing this gap through participatory research into age-appropriate and responsive mindfulness practices in a Year 5 class. The researcher, a pre-service teacher and Yoga Australia- accredited yoga and meditation teacher, introduced a wide range of mindfulness activities following a strength-based progression recommended by child and adolescent psychologist Karen Hooker and school psychologist Iris Fodor (Hooker & Fodor, 2008). The sequenced introduction of mindfulness activities facilitated a systematic inquiry into which types of mindfulness practices Year 5 students consider most beneficial and engaging.

Research questions

Australia’s diverse and inclusive classrooms mandate teaching and learning approaches that are responsive to individual learners’ needs and their social and cultural backgrounds (AITSL, 2014, APST 1). The design and implementation of an Australian ‘mindfulness curriculum’ therefore requires that informed choices are made on the range and type of mindfulness practices taught. At present, there is a lack of published research from Australia or elsewhere on what mindfulness practices work best with primary school students. No information is available on how an Australian Year 5 student cohort experiences mindfulness practices that focus on sensory awareness, movement awareness, and attention and emotion regulation. Besides, little information is available on how effective these methodologically distinct practices are in promoting socio-emotional wellbeing and academic learning in a primary school setting. In order to answer which mindful practices work best in an Australian Year 5 classroom, more information is required on which mindfulness experiences are considered most effective and enjoyable by participating students.

Consequently, three research questions were asked:

  1. How effective are particular mindfulness practices in terms of improving the socio-emotional wellbeing of a Year 5 student cohort?
  2. How effective are particular mindfulness practices in terms of improving the executive functioning of a Year 5 student cohort?
  3. Which mindfulness practices are the most and least popular with a Year 5 student cohort, and why?

Methodology

The research was carried out over the course of four weeks (15th May – 2nd June, 2017) in a large (Australian Government, 2016) Brisbane metropolitan Primary State School with a ranking on the Index of Community Socio-Educational Advantage (ICSEA) slightly above national average (ACARA, 2017d). The mindfulness program was designed to address the five key propositions informing the AC:HPE (ACARA, 2017a), namely:

  • educative purpose
  • health literacy
  • strengths-based approach
  • valuing movement
  • and critical inquiry

The program was divided into four thematic modules that sequentially introduce a wide range of mindfulness practices (Hooker & Fodor, 2008):

  • Module 1: health literacy education on mindfulness and related neurobiological concepts
  • Module 2: sensory awareness practices
  • Module 3: movement awareness practices
  • Module 4: attention and emotion regulation practices

Individual mindfulness activities were selected for educational value, variety and in-depth exploration to cater for student diversity and promote practical skill development (Kaiser Greenland, 2016; Thompson & Gauntlett-Gilbert, 2008) (Appendix A). Adhering to the school policy and the Queensland school legislation (Queensland Government, 2006, chapter 5), the teaching researcher selected only mindfulness practices that were strictly secular and used instructional language that refrained from any terminology or sounds (e.g. ‘Om’) that could be considered religious or culturally sensitive in the Australian school context.

The exploratory case study was carried out as action research with the objective to generate data applicable to the specific classroom. In education, action research typically involves small-scale systematic inquiries conducted by teachers with the aim to improve aspects of their teaching and learning practice (Milton-Brkich, Shumbera, & Beran, 2010). The unit of analysis was a mixed Year 5 class of twenty-eight students from various backgrounds. Mindfulness practices were taught after break time for approximately fifteen minutes three times a week.

Action research is cyclical and participatory in nature, where action is planned, observed, discussed, reflected and revised for subsequent rounds (Milton-Brkich et al., 2010). The students were invited to participate in all aspects of data generation, evaluation and reflection. Towards the end of the school day, participating students completed a questionnaire. They ticked and submitted anonymous exit slips with three times three emoticons to indicate what best expressed their current emotional and mental states, and their level of enjoyment of the mindfulness practice (Appendix B). The questionnaire is an adaption of the Smiling Mind “How do you feel?”-question (Smiling Mind, 2017). The use of emoticons to express feelings and emotions has previously been evaluated as a safe and fun way by most participants in a workplace environment (Huang, Yen, & Zhang, 2008), and was proposed as a suitable self-report measurement for young children (Allen, et al., 2017). Questionnaire participation was optional, anonymous, and exit slips were folded and collected in a closed box for additional privacy.

The daily activity-specific questionnaires were complemented by a single qualitative twenty-minute group discussion at the end of the program. The discussion took place in the format of a dialogue circle to promote cogenerative whole-class dialogue in a safe and inclusive environment (Queensland Studies Authority, 2010). More specifically, the ‘Curriculum into the Classroom’ (C2C) Education Queensland yarning circle protocol (Department of Education and Training, 2017) was followed to address the qualitative research questions in a semi-structured discussion with all case study participants in an environment of openness, respect and equitable participation (Roth, Robin, & Zimmermann, 2002). The teaching researcher sequentially introduced stimulus questions and took written (de-identified) notes of students’ answers on why they considered particular mindfulness practices most or least helpful and engaging (see Appendix C). This critical inquiry approach is in alignment with the final AC:HPE syllabus key propositions (ACARA, 2017a). The qualitative data helped to explain trends in the quantitative data, thereby limiting potential bias and subjectivity in the interpretations of the participating researcher (Snoeren, Niessen, & Abma, 2012).

Data analysis

Throughout the twelve rounds of mindfulness activities, all the students took the opportunity to provide feedback on their emotional and mental wellbeing and engagement with the activity using the exit slips (Appendix C). Valid exit slips were counted and analysed each day, tallying up student responses in each of the nine fields. The total number of responses varied between twenty-four and twenty-eight. The data resulted in daily absolute frequency distributions of socio-emotional states (‘happy’; ‘okay’; ‘sad or angry’), mental states (‘clear and alert’; ‘okay’; ‘stressed and confused’), and student engagement with the activity (‘liked activity’, ‘okay’, ‘disliked activity’). In order to compare responses between different mindfulness activities, the data had to be normalised by scaling individual data sets to 100% (Lambert, 2012).

The exit slip responses to individual activities (Table 1) indicate that the student cohort socio-emotionally felt best after practicing postural yoga (‘asana postures’). The lowest levels of emotional wellbeing were recorded after exploring the visual sense in a ‘human camera’ activity that involved being guided with closed eyes by a friend to three “sights” in the classroom. Postural yoga practice also resulted in the highest levels of mental clarity, a 40% increase when compared to an activity that took inventory of student mental presence at the beginning of the unit using Mindfulness jars (‘awareness meters’). The student cohort also liked postural yoga practice more than any other activity. In contrast, the least liked activities were formal breathing exercises (‘pranayama breathing’), with only half the cohort enjoying this activity. The ‘human camera’ activity also stood out for dividing student opinions, combining both high approval ratings with a high percentage of explicit dislikes. This is also reflected in the raised stress levels that nearly one third of the students reported at the end of the day. The ‘pass the cup’ activity was interrupted by a fire drill and therefore had to be excluded from comparative analysis (last line in Table 1). This example can serve as an indicator for gauging the overall sensitivity in student responses, with only half of the participants reporting feeling happy, clear and alert, and engaged.

Student responses to individual mindful activities on exit slips

Table 1. Student responses to individual mindful activities on exit slips. The responses are normalised as percentage points for each activity and grouped into the three domains of student socio-emotional wellbeing, student executive functioning and mental health, and student engagement and enjoyment. Individual activities are colour-coded into four categories relating to health literacy (blue), sensory awareness (red), movement awareness (yellow), and attention and emotion regulation (green). The activity that was interrupted by a fire drill is listed in grey. Data discussed in the text above are emphasised in bold letters.

The participant responses to individual activities can be aggregated into corresponding thematic modules. The total number of responses varies significantly between the four modules. The two health literacy practices combine only fifty-three responses. In contrast, the four sensory awareness practices required to cover tactile, visual, auditory and gustatory/olfactory mindfulness experiences produced a total of ninety-four combined responses (Appendix C). Consequently, participant responses for each of the four modules were added, normalised and expressed as percentages to enable data comparisons. The summary statistics were plotted as a column chart to highlight patterns in the data (Fig.1).

Exit slip responses summary statistics

Fig. 1. Exit slip responses summary statistics plotted as a column chart. Participant responses are grouped into four categories relating to health literacy (blue), sensory awareness (red), movement awareness (yellow), and attention and emotion regulation practices (green). Responses across groups are normalised as percentage points to allow for comparison.

Movement awareness practices proved to be most beneficial in regards to students’ socio-emotional (83%) and mental health (73%), closely followed by attention and emotion regulation practices. In stark contrast to these positive outcomes, the movement awareness practices received the least ‘likes’ (77%) and most ‘dislikes’ by the students (7%).

While sensory awareness practices such as ‘mindful eating’ made most participants feel ‘happy’ (79%) and were liked (83%), they exercised a comparably limited positive impact on mental health (66%). Sensory awareness practices also registered the highest percentages in participants feeling ‘sad or angry’ (5%) and/or ‘stressed and confused’ (12%). The differences in percentage points is small and needs to be interpreted with caution, considering the low and variable number of participants. However, it is safe to conclude that a small number of students did not respond well to sensory awareness practices compared to the health literacy educational practices in the first week that serve as a control.

The attention and emotion regulation practices came second best in all categories in terms of participants feeling ‘happy’ (80%), ‘clear and alert’ (72%) and engaged (81%). In contrast to the sensory awareness practices, significantly less students were isolated in negative emotions (1%), stressed (7%) or disengaged (1%).

The cogenerative dialogue at the end of the unit helped to interpret and explain the quantitative data (Appendix D). Students were asked which mindfulness practices they enjoyed most, least, and why, and answered with a range of activities and reasons. For example, the ‘mindful eating’ activity involving a single jelly bean was enjoyed by some students for the intense experiences of smelling, flavour and having to regulate their urges. Others found this temptation annoying and hard to suppress. Similarly, some students enjoyed stretching in postural yoga, while others were uncomfortable and experienced a lack of balance. Likewise, the ‘human camera’ activity was mentioned both in the context of positive experience of trusting other people, and a discomfort resulting from lack of control, balance and sense of direction. When asked to highlight something interesting they learned over the course of the program, the single most disliked mindfulness activity of ‘pranayama breathing’ was surprisingly repeatedly highlighted by the students as a good and practical way to calm down. The mindfulness practices that students wanted to practice more often combine all the aspects found in a traditional yoga program, including postural work (asana), breathing exercises (pranayama), sense withdrawal (‘yoga nidra’) and meditation. Finally, it was remarkable that during the dialogue circle students repeatedly referred to terms that highlight positive socio-emotional skills such as “you can trust other people [ . . . ] not just your friends”, and “calming down can help you a lot instead of just fighting back”.

Discussion of findings and implications

This study supports a growing body of research that reports on the positive effects of mindfulness interventions on socio-emotional wellbeing, executive functioning and mental health (Brown & Ryan, 2003; Gu et al., 2015; Van Leeuwen et al., 2012), including mindfulness programs delivered in school settings (e.g. Zenner et al., 2014). A comparison of participant responses between the control activities at the beginning of the mindfulness program (e.g. ‘mindfulness jars’) and all subsequent modules shows that students’ self-reported socio-emotional wellbeing was raised from ‘okay’ to ‘happy’ by an average of 10%. Corresponding student mental health improvements are even more pronounced, increasing from ‘okay’ to ‘clear and alert’ by nearly 20%. This finding is particular interesting in a school context, where students’ executive functioning play a critical role in their ability to learn (Flook et al., 2010).

In alignment with similar studies (i.e. Arthurson, 2015), students overwhelmingly (77-83%) enjoyed participating in the mindfulness activities and remained fully engaged throughout the four weeks. Unexpectedly, some sensory awareness practices resulted in mixed student experiences. Sensory awareness practices that required either the exertion of self control (i.e. ‘mindful eating’) or the relinquishing of control (i.e. ‘human camera’) were enjoyed by some students but disliked by others, while registering an overall increase in students feeling socio-emotionally or mentally stressed. This observation is at odds with other studies that discuss sensory awareness activities as student favourites (Arthurson, 2015). In this context, it is important to note that two weeks later students recalled experiences they did not enjoy, such as fighting with temptation and trusting others, as important things they have learned. Similarly, the formal breathing activity was least enjoyed according to the exit slips data, but most emphasised by students in the dialogue circle as helpful and something they want to practice more. This suggests that the students realise the value in learning practical coping strategies, even if their initial experiences are not always pleasant. At the same time, the contrasting student experiences highlight the importance of providing children and young adolescents with many different entry points into mindfulness practices (Thompson & Gauntlett-Gilbert, 2008).

The formal practices that proved most popular and effective in improving overall student wellbeing are those commonly integrated in traditional yoga practice. This could simply reflect the positive cumulative effect of sustained mindfulness practice throughout a mindfulness program designed to culminate in these more formal practices (Hooker & Fodor, 2008). However, it also highlights the potential of yoga to inform and complement mindfulness intervention programs in schools. Integrated yoga sessions, rather than separate mindfulness mini-lessons might prove to be an even more effective format. Research into the integration of formal yoga practices into mindfulness intervention programs is preliminary (Salmon, Lush, Jablonski, & Sephton, 2009) as well as research into yoga-based intervention programs in schools (Khalsa & Butzer, 2016; Serwacki & Cook-Cottone, 2012). Any program that requires teachers to combine multiple aspects of mindfulness practices including postural yoga would also require a certain level of professional training, understanding and depth, and a personal yoga and meditation practice (Arthurson, 2017). As a consequence of a growing interest in teaching yoga and mindfulness to children in Australia, the peak body for Yoga in Australia recently released new standards for teaching yoga to children that ensure developmentally appropriate, evidence-based, inclusive, and safe practices that are taught in registered postgraduate teacher courses (Lewis, 2015). Following the reflective phase of the action research (Milton-Brkich et al., 2010), the teaching researcher will revise subsequent mindfulness school intervention programs and pursue more formal training in teaching yoga to children.

Conclusion

This study supports previous published research that reports on the positive effects of mindfulness intervention school programs on students’ emotional wellbeing and executive functioning. A significant and consistent increase in students’ self-reported state of socio-emotional wellbeing (10%) and mental health (20%) was recorded. Student responses to daily questionnaires and in the cogenerative dialogue at the end of the program suggest that the Year 5 cohort benefited from the full range of mindfulness practices. Formal movement awareness practices and attention and emotion regulation practices provided the best outcomes in improving both students’ socio-emotional wellbeing and executive functioning. The most effective single mindfulness activities include traditional yoga practices such as postural yoga (asana), body scan (yoga nidra) and meditation. This suggests further scope to explore and develop more explicit links between the health and physical education dimensions in the AC:HPE curriculum through integrating mindfulness- and yoga-based school intervention programs. Student engagement remained high throughout the program. A detailed analysis of the data shows that some students can find certain sensory awareness practices and formal breathing practices challenging. However, student responses in the cogenerative dialogue explicitly emphasised the value of these practices as effective coping strategies, with repeated requests for additional practice in future. More research involving longer term programs would be required to establish the effectiveness of mindfulness practices that are less liked but overall valued by a student cohort.

Action research always comes with a number of limitations in terms of transferability of results. The informative value and significance of the study is limited by the small number of participants (24-28) and the short quantitative data acquisition interval (twelve mindfulness activities over four weeks). Qualitative data were collected in a single cogenerative dialogue at the end of the program where every student was given only one opportunity to respond to the program. The students’ experiences of, and responses to individual mindfulness activities would also most certainly have been influenced by external factors, such as their playground experiences during preceding breaks.

With these limitations in mind, the consistent self-reported increases in the student cohort’s socio-emotional wellbeing and mental health emphasises the positive role that mindfulness intervention programs can play in schools. The recorded increase in students’ wellbeing and in particular executive functioning is likely to translate into improved academic learning and reduce the need for reactive behaviour management. This research hopes to inform and encourage school leadership and teachers to develop and implement mindfulness programs across Australian schools that address the national Health and Physical Education syllabus and are responsive to their specific school communities, year levels, and individual students’ needs.

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  • Hanh, T. N. (2011). Planting seeds: Practicing mindfulness with children. Berkeley, California: Parallax Press.
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  • Huang, A. H., Yen, D. C., & Zhang, X. (2008). Exploring the potential effects of emoticons. Information & Management, 45(7), 466-473.
  • Ivtzan, I., & Hart, R. (2016). Mindfulness scholarship and interventions: A review. In A. L. Baltzell (Ed.) Mindfulness and performance. Cambridge, United Kingdom: Cambridge University Press, pp. 3-28.
  • Kabat-Zinn, J. (2004). Wherever you go, there you are: Mindfulness meditation in everyday life. United Kingdom: Little, Brown Book Group.
  • Kaiser Greenland, S. (2016). Mindful Games: Sharing Mindfulness and Meditation with Children, Teens, and Families. Shambhala Publications.
  • Khalsa, S. B. S., & Butzer, B. (2016). Yoga in school settings: a research review. Annals of the New York Academy of Sciences, 1373 (June 2016), 45-55.
  • Kidsmatter. (2013). Mindfulness made easy. Getting in early for Good Mental Health, 5(August). Retrieved from http://www.kidsmatter.edu.au/health-and-community/enewsletter/mindfulness-made-easy.
  • Lambert, M. (2012). A beginner’s guide to doing your education research project. Sage, London.
  • Langer, E. J., & Moldoveanu, M. (2000). The construct of mindfulness. Journal of Social Issues, 56(1), 1-9.
  • Lewis, R. (2015). New Standards for Teaching Yoga to Children. Yoga Australia. Retrieved from https://www.yogaaustralia.org.au/new-standards-for-teaching-yoga-to-children/
  • Meditation Capsules. (n.d.). Meditation capsules for schools. School programs for staff and students, Meditation Capsules. Retrieved from http://www.meditationcapsules.com/school-programs.
  • Milton-Brkich, K. L., Shumbera, K., & Beran, B. (2010). Action research. Science and Children, 47(9), 47.
  • Queensland Government. (2006). Education (General Provisions) Act 2006 (Qld). Queensland Legislation, Office of the Queensland Parliamentary Counsel. Retrieved from
  • https://www.legislation.qld.gov.au/LEGISLTN/CURRENT/E/EducGenPrA06.pdfQueensland Studies Authority. (2010). Dialogue circles. Aboriginal and Torres Strait Islander Perspectives — Resources. Retrieved from https://www.qcaa.qld.edu.au/downloads/approach2/indigenous_res_dialogue_circ.pdf
  • Rix, G., & Bernay, R. (2015). A study of the effects of mindfulness in five primary schools in New Zealand. New Zealand Journal of Teachers’ Work, 11(2), 201-220.
  • Roth, W. M., Robin, K., & Zimmermann, A. (2002). Coteaching/cogenerative dialoguing: Learning environments research as classroom praxis. Learning Environments Research, 5(1), 1-28.
  • Salmon, P., Lush, E., Jablonski, M., & Sephton, S. E. (2009). Yoga and mindfulness: Clinical aspects of an ancient mind/body practice. Cognitive and Behavioral Practice, 16(1), 59-72.
  • Serwacki, M., & Cook-Cottone, C. (2012). Yoga in the schools: a systematic review of the literature. International Journal of Yoga Therapy, 22(1), 101-110.
  • Smiling Mind. (2016). Establishing an evidence base for the Smiling Mind Education Program, Smiling Mind. Retrieved from https://smilingmind.com.au/media/1194/research-project-report.pdf.
  • Smiling Mind. (2017). Smiling Mind (version 3.1.4), iTunes app store. Retrieved March 23, 2017, from https://itunes.apple.com/au/app/smiling-mind/id560442518?mt=8.
  • Snoeren, M. M., Niessen, T. J., & Abma, T. A. (2012). Engagement enacted: Essentials of initiating an action research project. Action Research, 10(2), 189-204.
  • Thompson, M., & Gauntlett-Gilbert, J. (2008). Mindfulness with children and adolescents: Effective clinical application. Clinical Child Psychology and Psychiatry, 13(3), 395-407.
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  • Van Leeuwen, S., Singer, W., & Melloni, L. (2012). Meditation increases the depth of information processing and improves the allocation of attention in space. Frontiers in Human Neuroscience, 6, 133.
  • Zenner, C., Herrnleben-Kurz, S., & Walach, H. (2014). Mindfulness-based interventions in schools—a systematic review and meta-analysis. Frontiers in Psychology, 5, 603.

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QAR – Exploring Question-Answer Relationships

Question-Answer Relationships (QAR) is an explicit reading comprehension instruction model for teaching students about reciprocal question-answer relationships. Most recently popularised by Sheena Cameron in her book Teaching Reading Comprehension Strategies (Cameron, 2009), the QAR model is based on a reading strategy developed by Taffy E. Raphael at the University of Illinois in the early 1980s (Raphael, 1982). The idea of QARs is to engage students in thinking about and discussing what they read, progressing from literal text evidence to text synthesis and inference.

In the original concept, students were taught three types of questions that can be asked about texts and ways to find answers. The students need to locate the response information, which shifts with the type of question from a particular sentence, to multiple passages across a text, to drawing on prior experiences and background knowledge about similar or related texts and genres.

Original QAR model by Taffy E. Raphael (1982, p. 188)

Original QAR model by Taffy E. Raphael (1982, p. 188)

Taffy Raphael called the three type of QAR questions:

Right There: Right There questions are answered by pointing to a literal reference. Students are required to “… find the words used to create the question and look at the other words in that sentence to find the answer” (1982, p.187). The students learn how to find information by looking for key words. I call these questions On the Line questions, because the answers generally require pointing to one line in the text.

Think & Search: Think & Search questions are answered by synthesising different passages of a text. Students are required to integrate literal information “… , so students must skim across text segments to find the question and response information” (1982, p.187). The students learn how to find and integrate information based on their prior knowledge about text structures and by applying reading strategies such as skim reading. I call these questions Across the Lines questions, because answers generally require pointing at multiple paragraphs (or lines) in the text.

On My Own: On My Own questions demand thinking beyond the text and “… require the student to determine what background knowledge can be applied to the question” (1982, p.187). The students learn to think beyond the provided textual information, to question texts and to develop critical literacy skills. I call these questions Beyond the Lines questions, because they can be answered without referring to the text at all.

Taffy Raphael subsequently expanded her original model to include Author & You questions (Raphael, 1986): Author & You questions require inference from the students, who learn that some answers “... must come from the readers’ own knowledge base, but only in connection with information presented by the author” (1986, 519). Students learn how two very different resources of information, those that must e acquired from the text and those that are based on their prior knowledge and understanding, must be considered to answer some questions. These questions support reading strategies such as inference. I call these questions Between the Lines questions, because answers require to “read between the lines” by integrating the author’s text and the readers’ existing knowledge.

The revised QAR taxonomy is divided into two major categories: In the Book with the answer resources Right There and Think & Search, and In My Head with the answer resources Author & Me and On My Own (Raphael, 1986).

Updated conceptual QAR model now including four types of question-answer relationships (Raphael, 1986, p.517)

Updated conceptual QAR model now including four types of question-answer relationships (Raphael, 1986, p.517)

The QAR model provides a practical literacy resource to develop three comprehension strategies: (1) locating information; (2) determining how text structures convey information and can be synthesised; and (3) determining when inferences are required or invited. Students learn that depending on the type of question, answers may involve both literal and referential comprehension, references to the text as well as their interpretations that rely on their personal knowledge bases.

It is instructive to align the QAR model with the Four Roles of a Reader model, originally developed by Peter Freebody and Allan Luke (1990). This model emphasises the four interrelated essential roles of a reader: (1) decoding text as a ‘code-breaker’; (2) developing semantic meaning as a ‘text-participant’; (3) developing functional meaning as a ‘text-user’; and (4) critically analysing text as a ‘text analyst’.

The Four Resources of Reading by Freebody & Luke (1990) as compiled by the Barefoot Literacy Project

Right There questions operate at the level of key words and Code Breaking. Think & Search questions require text comprehension strategies that build on knowledge about language and genre conventions required to Make Meaning. The Author & You questions asks readers to respond and connect to the texts, to bring in their prior knowledge and “communicate” with the text as Text Users. Finally, On My Own questions demand critical engagement from readers in the form of Text Analysis. Readers are asked to draw on other texts or use the text as a stepping stone into inquiries that go well beyond the read text.

Early QAR Resource packages, such as that developed by the Irish National Behaviour Support Service (NBSS) are using four quadrants that are very similar to the Four Roles of a Reader model.

QAR teaching resource by NBSS (p.4)

QAR teaching resource by NBSS (p.4).

Since, a plethora of teaching resources based on the QAR model have been developed, including such some useful tools that include example questions such as QAR bookmarks by Sheena Cameron (2009). Deb Lawrence from Queensland wrote a useful four-step instructional guide detailing ideas on how to introduce QAR in the classroom (Lawrence, 2015), including:

  1. Introducing the question types
  2. Teaching about clues for identifying the question types
  3. Modelling how to think aloud
  4. Teaching about text organisations

The last step is about unpacking literature genre convention and nicely links with a number of functional grammar-related content descriptors (see below).

QAR bookmark

Photo copy of QAR bookmark as part of the comprehensive digital resources developed by Sheenan Cameron (2009).

References:

  • Cameron, S. (2009). Teaching reading comprehension strategies. North Shore: Pearson.
  • Freebody, P., & Luke, A. (1990). Literacies programs: Debates and demands in cultural context. Prospect: Australian Journal of TESOL, 5(7), 7-16.
  • Lawrence, D. (2015). Why use Question-Answer Relationships (QAR) as a comprehension strategy?. Literacy Learning: The Middle Years, 23(1), i-vii.
  • Raphael, T. E. (1982). Question-answering strategies for children. The Reading Teacher, 36(2), 186-190.
  • Raphael, T. E. (1986). Teaching question answer relationships, revisited. The Reading Teacher, 39(6), 516-522.

Relevance to Australian Curriculum content descriptors:

  • Foundation: Use comprehension strategies to understand and discuss texts listened to, viewed or read independently (ACELY1650)
  • Year 1: Understand that the purposes texts serve shape their structure in predictable ways (ACELA1447)
  • Year 1: Use comprehension strategies to build literal and inferred meaning about key events, ideas and information in texts that they listen to, view and read by drawing on growing knowledge of context, text structures and language features (ACELY1660)
  • Year 2: Use comprehension strategies to build literal and inferred meaning and begin to analyse texts by drawing on growing knowledge of context, language and visual features and print and multimodal text structures (ACELY1670)
  • Year 3: Identify the point of view in a text and suggest alternative points of view (ACELY1675)
  • Year 3: Understand that paragraphs are a key organisational feature of written texts (ACELA1479)
  • Year 3: Use comprehension strategies to build literal and inferred meaning and begin to evaluate texts by drawing on a growing knowledge of context, text structures and language features (ACELY1680)
  • Year 4: Use comprehension strategies to build literal and inferred meaning to expand content knowledge, integrating and linking ideas and analysing and evaluating texts (ACELY1692)
  • Year 5: Understand how to move beyond making bare assertions and take account of differing perspectives and points of view (ACELA1502)
  • Year 5: Investigate how the organisation of texts into chapters, headings, subheadings, home pages and sub pages for online texts and according to chronology or topic can be used to predict content and assist navigation (ACELA1797)
  • Year 5: Understand how texts vary in purpose, structure and topic as well as the degree of formality (ACELA1504)
  • Year 5: Recognise that ideas in literary texts can be conveyed from different viewpoints, which can lead to different kinds of interpretations and responses (ACELT1610)

Book Creator for iPad literacy resource review

In 2008, the Australian Education Ministers declared a principal educational goal for young Australians is to be successful learners by developing “. . . the essential skills in literacy and numeracy and [becoming] creative and productive users of technology, [. . .], as a foundation for success in all learning areas” (Barr et al., 2008, p. 8). Consequently, the national English syllabus (AC:E) designed by the Australian Curriculum, Assessment and Reporting Authority (ACARA) defines literacy as “. . . the ability to read, view, listen to, speak, write and create texts for learning and communicating in and out of school” (ACARA, 2017a). These six receptive and productive language macroskills (Barrot, 2016) are emphasised across all key learning areas (KLA) with the general capability ‘Literacy’ as interrelated elements essential for comprehending and composing texts (ACARA, 2017b). The AC:E further draws attention to the social and multimodal nature of language learning (ACARA, 2017a). Consequently, the demands on twenty-first century literacy teaching and learning resources are different to those developed for the twentieth century industrial model of schooling (e.g. Seely Flint, Kitson, Lowe, & Shaw, 2014). Here, the educational app ‘Book Creator for iPad’ (Red Jumper Ltd., 2017a) is critically reviewed in context of the AC:E from multiple perspectives, including literacy development theories, language macroskills, the six guiding principles for teaching reading and writing in the twenty-first century (Seely Flint et al., 2014), and the four-resources model of reading (Luke & Freebody, 1997) as applied to multimodal texts (Serafini, 2012) and creative writing (Heffernan, Lewison, & Henkin, 2003). The author concludes that Book Creator for iPad is versatile and age-appropriate literacy resource that can be employed to teach and learn critical and multiliteracies in Australian primary schools, across KLAs including English.

Resource description

Book Creator educational app

Book Creator educational app

Book Creator is a best-selling educational software running on Windows, iOS and Android platforms, with a browser extension in development (Kemp, 2017). It was launched in 2011, with the current release version 5.0.2 available on the iTunes app store for iOS 9 and above. The iOS app is priced at AU$ 7.99 per licence, with a 50% discount offered for schools through the Apple’s Volume Purchase Programme (Red Jumper Ltd., 2017a).

Book Creator supported media formats

Book Creator supported media formats

The Book Creator app is designed for school-aged children to create and publish multimodal ebooks. The core functionality includes widgets that allow adding text, images, drawings, shapes, audio and video to virtual book pages. Individual pages and final ebooks can be read out aloud, supporting twenty-seven languages, including thirteen English speaking voices and four Australia dialects. In reading mode, spoken words can optionally be highlighted and the speech rate adjusted. Book Creator supports publishing ebooks in multiple formats, including ePub, PDF and as as a video file with spoken text. Ebooks can be saved locally, or in the cloud (e.g iCloud, Dropbox, Google Drive) to support access-controlled sharing of student work with parents and school community. Alternatively, ebooks can be locally shared in the classroom using the AirDrop iPad functionality. As a result, Book Creator supports distributed content creation, where multiple students can work collaboratively on individual chapters that can be combined at a later stage (Hallett, 2013).

 

Critical evaluation and discussion

Most educational literacy software is designed with a narrow focus on developing and practicing particular skills such as phonemic awareness (e.g. Oz Phonics (DSP Learning Pty LTd., 2015)), sight words and spelling (e.g. Reading Eggs, (Blake eLearning, 2016)). In contrast, Book Creator is designed to be open-ended and to be used in creative ways across various KLAs to support the development of critical and productive multiliteracies. The developers of Book Creator value creativity, collaboration, cross-curricular integration, and “app-smashing”, i.e. the ability to seamlessly integrate other apps as part of the workflow (Red Jumper Ltd., 2017b). The company also prioritises dialogue with educators by offering free webinars and comprehensive customer support.

Book Creator excels as a top-down literacy development resource. The core intention of the app is to support students in creating ebooks. Multimodal ebooks are a whole-language product. Book Creator supports socially-situated learning through purposeful collaboration and dialogue, editing, and publishing. The app can be used in inquiry-based teaching and learning across all KLAs, for example in activities involving journaling and reflection. Perhaps the greatest value as a literacy resource is the ease-of-use with which all receptive and productive language macroskills (Barrot, 2016) can be meaningfully and seamlessly integrated into a single authentic product. Books are not just written but created, by seamlessly integrating text with images, audio and video recordings. The productive language skills are even expanded into the often neglected aspect of publishing for audiences (Jaakkola, 2015). The student experience between reading, listening (or being read to), and viewing the story as a movie is fluent. The app can be used to explore intertextuality, the links between different texts, personal experiences and outside knowledge. The students are invited to construct meaning by linking multimodal sources and developing the three schema-building connections: text-to-text, text-to-self, and text-to-world (Fountas & Pinnell, 2006).

The app can also be employed for critical literacy development. For example, Book Creator can be used to construct comic books with social justice themes (Stone, 2017), perhaps making use of speech and thought bubbles to explore multiple perspectives.

Basic Book Creator shapes

With the ability to publish in ePub and video formats, Book Creator lends itself as a tool to express opinions and take social action. The four-resources model (Luke & Freebody, 1997) is perhaps the most widespread model based on critical literacy theory in Australian schools. Originally developed by Peter Freebody (1992), it emphasises the socio-cultural practices and four interrelated essential roles of the reader: (1) decoding text as a ‘code-breaker’; (2) making semantic meaning as a ‘text-participant’; (3) making functional meaning as a ‘text-user’; and finally (4) critically analysing the text. This model is aligned with the four language cueing systems (graphophonic, semantic, syntactic, pragmatic) of the whole-language approach (Seely Flint et al., 2014). Frank Serafini (2012) expanded the original print-based model to address multiliteracies. Accordingly, the literate reader-viewer of multimodal texts acts as a :(1) ‘navigator’; (2) ‘interpreter’; (3) ‘designer’, and; (4) ‘interrogator’. Book Creator is designed to develop all four interrelated skills, with a particular focus on productive language skills. Lee Heffernan and co-authors adapted the four-resources model of reading to a four-resources model of writing in a primary school context (Heffernan et al., 2003). This model is used to support students in better communicating ideas, improving text composition, drawing on background experiences to construct meaning, and becoming more explicit and reflective in the representations and positions argued in the text. The simplicity with which students can add their voices (i.e. record audio) and perspectives (i.e. record photos and videos) makes Book Creator a great tool to develop critical and creative writing that addresses all four resources.

Ultimately, Book Creator as literacy resource is not limited to any particular theory of literacy development. Some creative teachers have used the app to support bottom-up literacy development through activities such as multimodal vocabulary practice (e.g. Dodds, 2015).

Another approach towards evaluating Book Creator as literacy resource is to critically assess ways in which this app can be used to address the six guiding principles for teaching reading and writing in the twenty-first century (Seely Flint et al., 2014):

1) Literacy practices are socially and culturally constructed. Book Creator encourages social interaction by offering multiple ways of collaboration between students, teacher, parents and the school community (Hallett, 2013). Cultural and linguistic diversity in the classroom is supported by offering few limits in terms of languages and genre conventions. However, the default page flow from left to right does not support languages that use right-to-left scripts, such as Arabic or Urdu.

2) Literacy practices are purposeful. Writing and publishing books is a purposeful form of literacy practice, in particular if the task design is inclusive and responsive to the students’ lives, and encourages cross-disciplinary learning. The simplicity with which text can be integrated with photos, audio and video recordings provides numerous opportunities for students to express themselves, organise and document their learning through journaling, support design thinking and prototyping (Holland, 2017), even playing interactive learning games (Dodds, 2015). The app can be used to support and complementing reading activities, for example by creating audiobooks of class readers. The sharing and publishing functionality offers opportunities to create ebooks for both enjoyment and assessment.

3) Literacy practices contain ideologies and values. Book Creator supports a range of literacy practices in virtually limitless social and cultural contexts. On the iPad, the app is portable and can even be used outdoors in nature for many hours. Shapes, such as speech and thought bubbles superimposed on images, can be used to communicate perspectives (Baker, 2015) and allow individual book characters to speak and think for themselves, perhaps juxtaposed on facing pages.

4) Literacy practices are learned through inquiry. The starting point of a new Book Creator project is a blank canvas, which can be customised. All content needs to be developed, the ideal starting point for student inquiries. The app is explicitly designed to support students in the drafting, composing and publishing processes. At each stage, students can work individually or in groups, and share their work for discussions, assessment and reflection (Vasinda, Kander, & Redmond-Sanogo, 2015).

5) Literacy practices invite readers and writers to use their background knowledge and cultural understandings to make sense of texts. Book Creator supports multiple ability levels and prior experiences with texts through inbuilt scaffolding tools such as the read-aloud function. Options to adjust the speed and dialect of the voice, and the ability to highlight spoken words make this app a powerful tool for supporting students struggling with unfamiliar aspects and practices around literacy development, e.g. EAL/D students. Emerging writers will enjoy the ability to creatively express themselves through multiple media to complement their writing (Rowe & Miller, 2016).

6) Literacy practices expand to include everyday texts and multimodal texts. Book Creator supports any type of genre and register, and can complement literacy practices across multiple contexts and KLAs. Multimodal texts are the core function of the app, supporting written, visual, auditory and spatial modes in any possible combination.

While all this demonstrates that Book Creator can be applied to a wide range of literacy teaching and learning scenarios, one fundamental question remains: to what extent does the app transform literacy learning compared to traditional, non-technological alternatives such as scrapbooking? A practical framework to critically evaluate educational technology and software is the SAMR model by Ruben Puentedura (Romrell, Kidder, & Wood, 2014).

Accordingly, Book Creator is reviewed in terms of its ability to Substitute, Augment, Modify and Redefine literacy learning experiences compared to traditional scrapbooking. The answer to the question above depends on how the teacher and students are employing the app. Book Creator can be used to simply substitute paper-based story writing through activities that are limited to individual writing exercises, perhaps allowing students to include pre-selected images. However, once students make use of the camera and microphone on their iPads to include spoken words, photos and videos, Book Creator will augment scrapbooking by functionally improving the possibilities. In order to modify the traditional resource, the app will need to be used in unprecedented and novel ways. This is for example the case in the area of collaboration. Book Creator enables easy duplication and sharing of documents, instant contextual feedback through annotations, and process documentation for assessment (e.g. Sample, 2014). Finally, traditional scrapbooking is only truly redefined when the app is used in ways inconceivable without technology. Arguably, workflow integration between Book Creator, other apps and cloud services is the area that establishes Book Creator as a transformative literacy resource. Examples include the ability to import any student-generated content, such as stop-motion movies, student images in front of a customisable backgrounds, and the novel ways that content can be shared and published to reach new audiences (Sample, 2014).

Book Creator comes with a price tag. Although reasonable in comparison to other educational resources and technology, it will require a purchase plan that can limit its appeal for teachers that plan to use the app only for a single project. The software is also limited in terms of editing images, audio and video. Advanced editing functionality will require integration with other apps that often need to be downloaded. Other useful functionality, such as the ability to automatically save the history of drafts, and to protect shared documents with passwords requires integration with a cloud service. Finally, as with any educational software, there is a learning curve for teachers and students involved, especially for lower year levels. All this suggests that the appeal of Book Creator as a literacy resource will depend on the IT environment of the school, and the intention of the class teacher to use the app for multiple projects and across multiple KLAs.

Conclusion

Book Creator for iPad is a literacy resource with the potential to transform traditional writing activities. It designed to enable primary school children to create and share multimodal texts in the form of ebooks and videos. Book Creator can be compared to a digital scrapbook, or a white canvas that can be employed across a range of teaching and learning activities. While primarily useful in supporting top-down and critical literacy approaches, it can also make bottom-up skill development activities more engaging, and support emerging readers and EAL/D learners through scaffolding functionality like text-to-speech. Book Creator is a powerful resource to teach receptive and productive language macroskills. While supporting the creative integration of all forms of media, it remains rooted in the traditional format of a book, thereby emphasising the writing and reading modalities above all others. The app becomes a transformative resource if it is integrated into a broader app environment including cloud services. This aspect, as well as the initial purchase price and the learning curve involved for teachers and students to master the app, make Book Creator a more attractive literacy resource for the sustained use across multiple key learning areas, as opposed to a resources for a single teaching and learning activity.

References

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Teaching and learning fractions

Mastery of fractions is the foundation for many more advanced mathematical and logical reasoning skills, including proportional, probabilistic and algebraic thinking. The degree of early year fraction understanding often correlates with secondary school mathematical achievement (Siegler, Fazio, Bailey, & Zhou, 2013). At the same time, fractions present a wide range of teaching and learning challenges that have been the subject of educational research (Petit, Laird, Marsden, & Ebby, 2015).

In the first part of this post, issues surrounding the teaching and learning of common fractions are described and linked to teaching and learning strategies that can address these. In the second part, implications for the teaching and learning in diverse classrooms are investigated and addressed by the Universal Design for Learning (UDL) framework, with particular reference to opportunities that modern information and communication technology (ICT) can offer. Drawing on both parts, a logical sequence is developed detailing conceptual and procedural steps for teaching and learning the fraction equivalence concept.

Issues surrounding the teaching and learning of common fractions

In primary school, learners move from non-fractional, through early fractional and transitional strategies, to mastery in applying fractional knowledge to magnitude, unit fraction and benchmark reasoning, and in operations (OGAP, 2012). In the Australian Curriculum, teaching and learning of fractions starts in Year 1 with content descriptor ACMNA016recognise and describe one-half as one of two equal parts of a whole”, and it progresses to Year 6, where students are expected to have developed procedural fluency in all operations with fractions, decimals and percentages, with the capacity to solve authentic problems (ACARA, 2017).

Fractions, ratios and proportions are the most cognitively challenging concepts encountered in primary school mathematics (Booker, Bond, Sparrow, & Swan, 2015). For students, fractions often mark the transition from concrete to formal operational mathematical thinking (Siegler et al., 2013), where numbers do not anymore relate to whole objects, or the size, shape and arrangements of their parts, but instead to part-whole relationships between two quantities composed of equal parts of a whole (Pantziara & Philippou, 2012). One difficulty in expanding whole-number to rational-number thinking is that both share overlapping cerebral processing areas in the intraparietal sulcus of the prefrontal parietal cortex (Siegler et al., 2013). Additional difficulties are encountered with the notation system used to represent fractions (Brizuela, 2006). Explicit teaching of fraction notation is essential, since “one whole number written above another whole number, do not transparently communicate the meaning of fractions” (Gould, 2013. p.5). The relational action associated with the symbols is not an intrinsic property of the symbols. Learners first need to experience the symbols as an expression of the relational outcomes of actions that they have carried out or observed (Dörfler, 1991). Finally, there is the concept of changing units, where one whole can refer to both multiple objects or composite units within a single object; partition fractions or quantity fractions. Students need to learn to move between different representations, including multiple symbols referring to the same amount (Booker et al., 2015).

In teaching fractions, it is essential to explain and establish fraction terminology first, explicitly addressing language and conceptual misunderstandings that surround rational-number thinking. The links between terminology, symbology, notations and concepts such as whole-number and part-whole relationships must be established before moving on to mathematical operations involving fractions. Mastery requires that students develop both conceptual and procedural knowledge and understanding of fraction concepts (Pantziara & Philippou, 2012). Therefore, teachers need to value and at least initially prioritise deep conceptual understanding over automatic procedural skills (Booker et al., 2015).

Visual models are a central component in teaching fractions at all stages of conceptual development, rational-number thinking, procedural and operational problem solving (Petit et al., 2015). Provision of a variety of visual representations of identical fractions that differ in perceptual features, such as the location and shape of shaded areas (numerator), were demonstrated to be important in the development of a multi-dimensional understanding of fractions. However, it is important that teachers guide learners in developing the knowledge about how visual representations relate to the fraction concept (Rau, 2016).

There are three common visual fraction models: linear, area, and discrete. These can be taught using a variety of representations (e.g. rectangular and circular segments, arrays, object collections) and physical and virtual manipulatives. Recent research into cognitive numerical development highlights the importance of teaching students that fractions represent magnitudes that can be located on a number line. Number lines, where equal parts are defined by equal distance, can serve as a conceptual bridge between whole numbers, proper, improper and mixed fractions, decimals and percentages, and highlight the concepts of equivalence and continuous quantities of fractions (Booth & Newton, 2012; Siegler et al., 2013). Gould recommends focussing on the linear aspects of fraction models as the primary representation of fractions in younger years (2013). Nevertheless, traditional area models, where equal parts are defined by equal area, continue to play an important role in the conceptualisation of numerator and denominator, fraction division, the relationship between unit of measure and reference unit, and equivalence (Lamberg & Wiest, 2015; Booker et al., 2015). Discrete models or ‘set of objects’ arrays, where equal parts are defined by equal number of objects with countable sets and subsets of discrete entities, visualise the mapping of distinct countable sets onto numerators and denominators (Rapp, Bassok, DeWolf, & Holyoak, 2015) and help students to understand equipartitioning (Petit et al., 2015).

All three visual fraction models can be used in different learning modes, including group discussions (verbal, aural), kinesthetic activities, and even through music (Courey, Balogh, Siker, & Paik, 2012). Physical manipulatives are a valuable resource stimulating hands-on learning that can make abstract mathematical ideas more tangible (Petit et al., 2015). Access to a variety of representations and activities support students in building the foundations for solving complex questions and real problems that involve rational-number thinking which cannot be achieved by rote learning alone.

Learners need guidance and practice to expand their conceptual numerical understanding to include rational numbers (Petit et al., 2015). Procedural fluency and algorithmic operational problem-solving skills are best learned by moving back and forth between conceptual and procedural knowledge and practice. Individual students have different learning styles and learning preferences. Student diversity can be accommodated by empowering learners to make choices between different activities and task designs, including group, paired and individual work, different modalities and types of questions, resulting in increased motivation and persistence (Landrum & Landrum, 2016). A degree of choice of tasks, task sequence and stimulus can be introduced into the classroom through blended learning, where students engage part-time with online content and instructions using learning platforms such as Mathletics (see below). Blended learning also provides a degree of flexibility over time, place, path and pace, and can be implemented as station-rotation, flipped classroom, or flex model among others (Staker & Horn, 2012), depending on the opportunities and constraints of individual teaching and learning environments.

There is also a cultural dimension to how students learn mathematics in general and fractions in specific. Mathematics is a cultural construct with its own epistemology. It cannot simply be assumed to constitute a “universal language”. Indigenous Australian mathematician and head of the ‘Aboriginal & Torres Strait Islander Mathematics Alliance’ Chris Matthews developed a model for culturally-responsive mathematics that links students’ perceived reality with curriculum mathematics through a hermeneutic circle of abstraction and critical reflection based on practical problem-solving (Sarra, Matthews, Ewing, & Cooper, 2011).

It has long been argued that Indigenous Australian students prefer kinesthetic learning experiences with physical manipulatives, narrative learning, valuing group discussions and explicit guidance (Kitchenham, 2016). It is therefore important to link formal mathematical concepts to something concrete endowed with real meaning. In reference to the Maths as Storytelling (MAST) pedagogical approach (Queensland Studies Authority, 2011), the fraction concept could for example be learned by acting out, using groups of students to represent fractions in terms of varying parts of the class (e.g. boys vs girls), or perhaps more dynamically by connecting fractions with rhythm and dance (Campbell, 2014).

At the same time, it is important that students also learn that there are differences between everyday colloquial expressions and empirical understanding of fractions, such as in acts of sharing and distributing, and formal mathematical equivalents. Mathematical definitions are developed through theoretical or operative generalisation and abstraction and use symbols (verbal, iconic, geometric or algebraic) to describe the conditions or schemata of actions (Dörfler, 1991). Therefore, explicit teaching of the meaning behind the symbolic mathematical language through exposure to multiple representations and models is essential for student learning of mathematical concepts including rational-number concepts.

Providing a creative and active learning environment, offering choice and variation in learning activities, mathematical representations, and task and assessment modes, will foster student engagement and the development of a positive disposition to mathematics. Similar to the fraction understanding itself (Siegler et al., 2013), a positive mathematical self-belief is another key predictor of middle years students’ mathematics achievement (Dimarakis, Bobis, Way, & Anderson, 2014).

Implications for the teaching and learning in diverse classrooms

Australia is a multicultural country and home to the world’s oldest continuous cultures. Nearly half of the population have an overseas-born parent, 5% identify as Aboriginal and/or Torres Strait Islander, and 20% speak a language other than English at home (Australian Human Rights Commission, 2014; Australian Bureau of Statistics, 2016). This diversity translates to classrooms with diverse social, cultural, religious and linguistic approaches to learning (Shahaeian, 2014). The Australian-wide promotion of an inclusive education policy emphasises the right of students of all abilities to participate in all aspects of the mainstream education, adding an additional dimension of physical, sensory and intellectual diversity (Konza, 2008). According to the Australian Bureau of Statistics, 5% of all primary school-aged children have disabilities resulting in significant core-activity limitations and schooling restrictions (2012). At the other end of the ability spectrum are the 10% of gifted and talented students, often unidentified and significantly underachieving (Parliament of Victoria, Education and Training Committee, 2012).

It is therefore the legal, moral and professional obligation of teachers to embrace all learners in their diversity and make reasonable adjustments to facilitate their full participation towards achieving their best potential (Cologon, 2013; Poed, 2015). There are a number of models for responsive teaching that addresses all learning needs in diverse classrooms. The Universal Design for Learning (UDL) is a set of principles guiding teachers towards developing universally accessible learning environments and instructional practices (Flores, 2008). The fundamental idea is to make the curriculum delivery as accessible as possible to all students, limiting the need for additional modifications and individual support. The design focuses on providing equitable access to the curriculum by offering multiple means of representation, expression and action (Basham & Marino, 2013). Students are offered choice over curriculum content, learning activities and resources to best meet individual skill levels, learning preferences and interests. Assessments offer learners multiple ways of demonstrating acquired skills and knowledge. While UDL can cater for most students in the diverse classroom, preferential intervention and special provisions is given to small groups, including access to resources (e.g. teacher aide) materials (e.g. manipulatives) or equipments (e.g. calculator) for task completion, including additional time or accelerated curriculum, alternative input and response formats (Ashman, 2015). A third level of prevention and intervention offers short-term intensive and explicit instruction for individuals (Fuchs & Fuchs, 2001), for example explicit practice of mathematical terminology and symbols for new EAL/D students.

Utilisation of ICT, including augmented and alternative communication devices that can support students with physical impairments, has great potential to help addressing all individual learning needs in a diverse classroom (Blum & Parete, 2015). Modern teaching and learning devices such as the iPad have been designed with disabilities in mind and can be easily configured to support the visually, hearing and physically impaired (Apple Inc., 2016). The iPad provides quick and simple access to a wide range of mathematics apps. Preliminary research highlights the potential of using iPads in primary school Mathematics classrooms to motivate and engage students (Hilton, 2016). Mathematics teaching and learning software, such as Mathletics developed by the Australian company 3P Learning provides teachers with tools to custom-design learning sequences for any topic in alignment with the Australian Curriculum, even activities with year level and content descriptors, lesson plans and ebooks (3P Learning, 2016). Australian schools that use Mathletics are performing significantly better in NAPLAN numeracy tests irrespective of socio-economic and regional status (Stokes, 2015).

The reported positive outcomes for all students, including students with learning support needs as well as gifted and talented students, could be a result of the combination of design features in the software:

  • student-led design that encourages learning at individual pace and at multiple difficulty levels (easier, core, harder)
  • instant and encouraging feedback to learners highlighting mistakes and solutions without teacher intervention
  • powerful formative assessment capabilities allowing teachers to monitor student progress and to identify learning gaps
  • tools that allow teachers to develop individual student learning pathways
  • app and web-based access allows Mathletics to be used as a flipped classroom tool and assign individual homework
  • gamified character in modules including class, school and world challenges (LIVE Mathletics)

Apps can also provide virtual manipulatives that enable more creative work with objects. For fractions, the educational graphing calculator GeoGebra is discussed below for building fraction bar models (Cooper, 2014).

As powerful as some apps and technology can be, ICT should only complement the teaching and learning of mathematics side by side with explicit teaching and multi-modal activities that encourage verbal and written communication, group discussions and the use of physical manipulatives that encourage kinesthetic learning. Also, apps are not always designed in alignment with UDL and can include barriers for students with disabilities (Smith & Harvey, 2014). Particularly in intervention instruction, it is advised to make use of both virtual and physical manipulatives to teach fractions (Westenskow & Moyer-Packenham, 2016).

Teaching and learning steps for acquisition of the equivalence concept

Fraction equivalence is one of the most important mathematical ideas introduced in primary school and know to cause difficulties for many students (Pantziara & Philippou, 2012). The big idea behind teaching equivalent fractions is for students to understand that fractions of a given size can have an infinite number of different names and corresponding symbols, and to develop efficient procedures for finding equivalent fractions. Finding equivalent fractions enables students to compare, order and operate with fractions (Petit et al., 2015).

The curriculum is the starting point for the design of teaching and learning units by defining the learning objectives and expected outcomes for each year level. The Australian Curriculum (AC) follows a spiral-based approach that gradually builds mastery of skills and concepts by sequentially increasing the cognitive demands (Lupton, 2013). Equivalence is introduced in the AC v8.3 in Year 4, where students are expected to “recognise common equivalent fractions in familiar contexts and make connections between fraction and decimal notations up to two decimal places”. In Year 5, equivalence of fractions is not specifically addressed but students are expected to develop the capacity to “... order decimals and unit fractions and locate them on number lines. They add and subtract fractions with the same denominator”. The equivalence concept is expanded in Year 6, where students are expected to “connect fractions, decimals and percentages as different representations of the same number”, more specifically detailed in content descriptor ACMNA131Make connections between equivalent fractions, decimals and percentages”. Full mastery of equivalence of fractions is not expected until Year 8 (ACARA, 2017).

In the learning continuum encountered in diverse classrooms, it is critical to develop an understanding of the sequence of teaching and learning steps of mathematical concepts and establish prior understanding of conceptual knowledge and procedural skills in all students.

  1. Step One starts with diagnostic assessment to establish existing foundational knowledge of common fractions, notation conventions, the relation between fractions to whole numbers, including proper/improper fractions and mixed numbers. Explicit teaching and practice of terminology and revisiting previously learned concepts might be required to establish critical conceptual understanding without which any further learning would be only procedural and rely on rote learning.
  2. Step Two explores new concepts and terminology by making use of physical manipulatives and encouraging student discussion. One example would be having students folding paper rectangles that have been vertically subdivided into equal, partially-shaded parts lengthwise in two, three, four bars of equal thickness The shaded fraction remains the same while the total number of equal parts as outlined by the creases increases. Students count shaded and unshaded parts and discuss equivalence (Booker et al., 2015, p.184).
  3. Step Three elaborates and reinforces equivalence fractions through multiple representations working from the visual-concrete towards the symbolic-abstract. The activities help to develop procedural fluency, the accurate, efficient and flexible use of mathematical skills in renaming equivalent fractions (Petit et al., 2015). Fraction games, ideally focusing on equivalent fraction grouping, are employed using material (Booker et al., 2015) or online virtual resources (e.g. Math Playground Triplets). A “fractional clothesline” can be used to establish the magnitude of fractions, sort and locate equivalent fractions, improper fractions and mixed numbers (Heitschmidt, n.d.). This activity involves kinesthetic and visual learning, and can encourage verbal learning through student discussions. It also serves as a formative assessment tool. Number lines illustrate the big idea that equivalent fractions share the same value (Petit et al., 2015) and are highly recommended as a representation that can conceptually bridge whole-number and rational-number thinking (Booth & Newton, 2012; Gould, 2013).

Fraction clothesline example

  1. Step Four integrates the acquired procedural knowledge and conceptual knowledge by looking for patterns and developing rules, progressing from concrete presentations towards symbolic presentations and abstract algorithms. The focus is on finding the next, rather than any equivalent fraction, making use of “fraction bars” as graphical representations. Fraction bars can be build using Lego blocks and extended by educational dynamic mathematics software (Cooper, 2014). Alternatively, an innovative lesson sequence works with stacks of papers of different thickness (Brousseau, Brousseau, & Warfield, 2014).

Example for Lego fraction bars that can be used to investigate equivalent fractions.

  1. Step Five extends the learned knowledge and understanding of equivalent fractions to real-world scenarios. This includes investigating the relationships between alternative representations of fractions (e.g. decimals, percentages) in wide variety of cross-curriculum contexts (e.g. Science, Economics and Business, Music). At this stage, a summative assessment of learning is important to evaluate the achieved mastery of the concept.

Conclusion

Quality teaching is based on proficient subject-matter and pedagogical knowledge. Teachers need to understand the full spectrum of individual challenges and potential barriers that students can face with cognitively challenging mathematical concepts such as rational-number thinking. It is important to invest the time to allow students to gain deep conceptual understanding before moving on towards procedural fluency. This will require well-sequenced teaching and learning steps, supported by multiple representations, modes and questions, working from physical and visual towards more symbolic and abstract problem-solving activities. Both hands-on manipulatives and appropriate use of ICT can support the learning process, especially at both ends of the ability spectrum. Offering variety and choice will help to engage all learners and establish students’ confidence and positive dispositions towards mathematics.

References

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8 Aboriginal Ways of Learning – a review

The 8 Aboriginal Ways of Learning is a pedagogy framework for embedding Australian Indigenous perspectives into the classroom by emphasising Indigenous learning techniques across all subjects. The framework was developed by the James Cook University School of Indigenous Studies, in collaboration with the Western New South Wales Regional Aboriginal Education Team and DET staff in 2007-2009. It postulates that Indigenous perspectives and knowledges in the classroom are not about introducing “indigenised content”, but rather by practicing a pedagogy informed by “Indigenous processes” of knowledge transmission and identity.

The 8 way learning model explained at the Australian Indigenous College

Dr Karen Martin, a Noonuccal woman, NAIDOC Scholar of the Year 2008 and Associate Professor in the School of Education and Professional Studies at Griffith University unpacks how a culturally-informed pedagogy informed by means by speaking of ways of knowing, ways of doing, ways of being and ways of valuing”.

8 Aboriginal Ways of Learning pedagogical framework

The 8 Aboriginal Ways of Learning pedagogical framework highlighting the connections to axiology, ontology, epistemology and methodology.

The 8 Aboriginal Ways of Learning therefore is quite a unique approach linking pedagogy with with:

  • axiology – Indigenous ways of valuing, in particular particular cultural protocols, systems and processes
  • ontology – Indigenous ways of being, in particular cultural protocols of behaviour
  • epistemology -Indigenous ways of knowing, in particular cultural protocols such as initiation
  • methodology – Indigenous ways of doing, such knowledge transmission through storytelling

What makes the 8 Ways pedagogical framework so compelling in the Australian main school context is that it is informed by a large overlap between Indigenous and non-Indigenous learning processes. There is much common ground in the value to all students of teachers including:

Story sharing:

Story sharing

 

Great teachers are storytellers that can also teach through narratives and songs. Sharing stories is also an important tool to connect with each other.

Learning maps:

learning maps

The visualisation of pathways of knowledge is also called concept mapping. According to the latest Hattie Effect Size update, there is good evidence that the development of learning maps is among the most effective teaching practice.

Non-verbal learning:

non-verbal

Six out of the seven learning styles are non-verbal and include the kinesthetic and interpersonal approaches. In the words of the 8 Ways pedagogical framework, “we see, think, act, make and share without words“.

Symbols and images:

symbols and images

Providing visual cues, including symbols and colour-codes in learning routines can significantly help students with hearing impairment and social communication difficulties such as Autism Spectrum Disorder. This point addresses that learning is often visual and supported by objects, images, symbols, signs, art and metaphors to explain concepts and content.

Land links:

land links

Great teachers make teaching content relevant by connecting it to the world in which their students live. This includes teaching lessons about the local environmental, highlighting traditional knowledge and connection to the land, including climate, fauna and flora, as well as the history about a place. The aspect of land links is also an important factor for training students’ ability of acute observations as required in Science and often best taught in nature. Land linkst is arguably an important part of teaching Sustainability, one of three cross-curriculum priorities in the Australian Curriculum.

Non-linear:

non-linear

 

Non-linear thinking can be translated as Critical and Creative Thinking, a key general capability to be developed in the Australian Curriculum. This is because lateral thinking or what we also call “thinking outside the box” is the foundation of innovation. In the words of the 8 Ways pedagogical framework, “we put different ideas together and create new knowledge“.

Deconstruct and reconstruct:

deconstruct and reconstruct

On the one hand this pedagogy describes the gradual release of responsibility instructional framework, where teachers first unpack new knowledge with the students by means of modelling and scaffolding, then encourage shared and individual practice. It also emphasises the importance of holistic knowledge, always anchoring new content in prior student knowledge, by “work[ing] from wholes to parts“.

Community links:

community links

 

This pedagogy is an important aspect of social pedagogies that emphasise the importance of community engagement and authentic audiences to bolster student engagement. Arguably, community links goes one step further by applying learning for community benefit, or to paraphrase UNESCO to “empower disadvantaged communities through innovative education“.

The 8 Ways pedagogical framework is hardly radical, allows for broad practical applications in a wide range of local school contexts and does not prescribe any particular or commercial classroom materials and training requirements. It also sidesteps possible constraints in curriculum content choices, which prior to the introduction of the Australian Curriculum and in particular the cross-curriculum priority Aboriginal and Torres Strait Islander histories and cultures had the potential to constrain Indigenous education in mainstream schools and classes. It also recognises that every school community and every local Indigenous culture is different, and that one-size-fits-all prescriptions are problematic and limiting. Rather than being explicit about teaching content (e.g. Indigenous lesson units by commercial providers such as sharingculture.com), classroom and school management styles (e.g. Stronger Smarter developed by Chris Sara), particular teaching styles (e.g. Direct Instruction advocated by Noel Pearson), or conceptual frameworks for constructing individual teaching and learning episodes (e.g. Uncle Ernie’s framework), the 8 Aboriginal Ways of Learning approach promotes culturally sensitive and informed ways of teaching and learning practically anything in ways that benefit all students.

The challenges with such an open, inclusive, and non-commercial pedagogical framework are in professional adoption and meaningful translation and applications. The 8 Ways is only one of an ever growing number of culturally-informed pedagogical approaches advocated to Australian teachers, and while not necessarily in conflict with the others risks of being only superficially adopted and watered-down in practice to the point where it would make little difference to Indigenous students. Without any specific units, class material, applied recommendations in areas such as EAL/D, it will be easy for teachers to endorse it in theory but not in practice. This is even more likely within the non-commercial context of this framework, as it will not be actively marketed by consultants for professional development to schools.

The AITSL professional teacher standards 1.4 (strategies for teaching Aboriginal and Torres Strait Islander students) and 2.4 (understand and respect Aboriginal and Torres Strait Islander people to promote reconciliation between Indigenous and non-Indigenous Australians) frame much of the professional classroom practice in relation to Indigenous students, and teaching Indigenous perspectives and understanding. The 8 Ways approach has the potential to directly inform the teaching practice by offering a rich, culturally-informed framework to design teaching and learning episodes and activities. While not supporting this process with specific teaching material or recommendations for lessons on Indigenous people, culture, country/places which would inform teaching about the reconciliation process between Indigenous and non-Indigenous Australians, the 8 Ways framework supports meaningful cross-cultural dialogue and shared learning experiences.

Personally, I find the comprehensive nature and non-prescriptive approach of the eight interconnected pedagogies very appealing, because they can easily be to be applied across all curriculum areas. The are also an excellent starting point for discussing curriculum and pedagogy choices with the local Indigenous and non-Indigenous school community. This flexibility also ensures compatibility to work with any (future) version of the Australian Curriculum, across changing cross-curricular priorities, different whole-school approaches and communities, by offering pedagogical approaches that benefit all students and make real connections to local knowledges and practices.

Teaching and learning Maths: learning sequence catering for diversity

This post is addressing the Year 6 content strand ‘measurement and geometry’, substrand ‘using units of measurement’ and content descriptor ACMMG137solve problems involving the comparison of lengths and areas using appropriate units” (ACARA, 2017), which were discussed in the previous posts on Maths unit and lesson planning process, rubric construction, multiple representation of mathematical concepts, and using Math apps. The achievement standards are mapped to the proficiency strands and include:

  • students are to understand and describe properties of surface area and length,
  • develop fluency in measuring using metric units,
  • solve authentic problems, and
  • be able to explain shape transformations

A short learning sequence of comparison of lengths and areas – major steps

Booker et al. detail the conceptual and procedural steps required to master length and area (2015). Applied toACMMG137, these include three major steps:

  1. Perceiving and identifying the attributes ‘area’ and ‘length’
  2. Comparing and ordering areas and lengths (non-standard units => standard units)
  3. Measuring areas and lengths (non-standard units => standard units), including covering surfaces without leaving gaps

This sequence is introduced using multiple representations, progressing from hands-on experiences with manipulatives towards abstract logical thinking and transformation tasks (see examples).

Activities to aid the learning sequence

The steps are mapped to a range activities that cater for diverse classrooms in alignment with the framework of Universal Design of Learning (UDL) (Fuchs & Fuchs, 2001):

  • Students cut their own tangram puzzle (with or without template) and explore how small shapes can create larger shapes
  • Students order tangram shapes by area and perimeter and establish base units: smallest shape (small triangle) as area unit, side of small square and hypotenuse of small triangle as length units
  • Students colour tangram pieces and puzzle range of objects (with and without colour, line clues), exploring how larger geometric shapes can be covered by smaller and making statistical observations on the number of units within each shape and corresponding perimeter. Non-standard units are measured and used for calculations.

(The activities are detailed with examples in the post on multiple representations of mathematical concepts)

Adjustments for a child with learning difficulties

Student with very limited English knowledge (e.g. EAL/D beginning phase). ACARA provides detailed annotated content descriptors (ACARA, 2014). The language and cultural considerations are specifically addressed by keeping discussion relevant to the tasks, offering alternatives to ‘word problems’ in both activities and assessment (as highlighted in the rubric design). Teaching strategy considerations are followed by explicitly teaching the vocabulary, making explicit links between terminology, symbols and visual representations (e.g. by pausing explanatory movie and writing out and illustrating on the whiteboard using colours (e.g. area = blue, equal sides = green, hypotenuse = red, labelling the count of units). The EAL/D student is provided with opportunities to develop cognitive academic language proficiency through mixed-ability group work. All content knowledge can be demonstrated by the student using physical manipulatives, charts and algorithms.

Adjustments for a child with advanced abilities

Children with advanced abilities can only develop their potential if provisions are made to deliver a challenging, enriched and differentiated curriculum, and a supportive learning environment
(Gagné, 2015). Maker’s updated recommendations on the four dimensions of curriculum modifications (2005) are applied as follows:

  • Content – content is framed in an interdisciplinary way, using tangram that connects to Japanese culture and art
  • Process – design emphasises self-directed learning, choice, variety and discovery of underlying patterns by offering a range of tangram puzzle options at multiple levels of difficulty to be explored in abstract terms (i.e. sorting by ratio of area to perimeter)
  • Product – high-ability students are encouraged to work on expert puzzles and transform learned concept knowledge by designing their own tangrams with constraints (e.g. tangrams with identical perimeter, sequence reduced by one length unit, …) and present their products to the class
  • Environment -high-ability students are provided access to spreadsheet software (e.g. for statistical observations, to graph relationships between area and perimeter) and allowed time to work independently

References

Teaching and learning Maths: multiple representations of mathematical concepts

Multiple representations

The representation of mathematical concepts and objects plays an important discipline-specific role. Doing Maths relies on using representations for otherwise inaccessible mathematical objects. The concept of multiple representations (MR) has been introduced to teaching and learning of mathematics in the 1980’s (i.e. Janvier, 1987). Some primary school curricula (e.g. Germany) highlight MR as a key mathematical idea (Leitidee) (Walther, Heuvel-Panhuizen, Granzer, & Köller, 2012), while the Australian Curriculum (v8.2) includes specific references to some year-level proficiency standards (ACARA, 2016). This could reflect that different mathematical content domains apply particular kinds of representations (Dreher & Kuntze, 2015).

Benefits and difficulties

Research emphasises both the importance of MR to developing mathematical understanding and the difficulties that can be faced by learners (Ainsworth, 1999). Multiple representations can make all facets of mathematical objects visible. The ability to move between different representations is key to develop multi-faceted conceptual mathematical thinking and problem solving skills (Dreher & Kuntze, 2015). The difficulty with MR is that no single representation of a mathematical object is self-explanatory. Each representation requires understanding of how this representation is to be interpreted mathematically, and how it is connected to corresponding other representations of the object. These connections must be made explicit and require learning that engages higher cognitive levels. Interpreting individual representations, making connections between MR of corresponding mathematical objects, and changing between MR can present significant obstacle to learners (Ainsworth, 1999).

Sequencing the introduction of multiple representations

Booker, Bond, Sparrow & Swan (2015) highlight the importance of gradually sequencing the introduction of MR from the concrete to the abstract over time and identify the functions that MR can serve in developing mathematical understanding.

One such sequence is illustrated for content domain ‘geometry’ (compare ACMMG137) by applying the five ways of working (Battista, 2007).

Step 1: Visualisation of spatial arrangements – Students are provided with the following A4 template and are asked to cut out Tangram pieces along the blue lines and arrange them in one row by size.

A4 tangram template for students to cut out

Step 2: Development of verbal and written communication skills – Students are asked to discuss and describe their size order using explicitly taught concepts of ‘area’ and the small triangle as ‘1 unit’.

Tangram pieces sorted by size

Step 3: Symbolic representation through drawing and model making – Students are asked to colour their tangram pieces and puzzle the objects of projected image below (rotation, transformation)

Example colours for student tangrams

Step 4: Concrete and abstract logical thinking – Students are asked to create a column chart of the number of units (triangles) within each shape (colour). Students are allowed to cut one set of shapes into triangles (working in pairs).

Column chart depicting number of triangle units for each (coloured) tangram piece

Step 5: Application of geometrical concepts and knowledge – Students are asked to investigate how many different parallelograms they can form and the number of units required. Next, they measure and calculate the base unit and apply multiplication to calculate the areas.

Examples:

Smallest possible parallelogram consisting of 2 small triangle units

2 units, 2 x 8 cm2 = 16 cm2

Largest possible parallelogram consisting of 16 small triangle units

16 units, 16 x 8 cm2 = 128 cm2

References

  • Ainsworth, S. (1999). The functions of multiple representations. Computers & Education, 33(2), 131-152.
  • Australian Curriculum, Assessment and Reporting Authority. (2016). Home/ F-10 Curriculum/ Mathematics.
  • Booker, G., Bond, D., Sparrow, L., & Swan, P. (2015). Teaching primary mathematics. Fifth edition. Pearson Higher Education AU.
  • Battista, M. T. (2007). The development of geometric and spatial thinking. In Lester, F.K.Jr. (Eds) Second handbook of research on mathematics teaching and learning, Volume 2. National Council of Teachers of Mathematics, 843-908.
  • Dreher, A., & Kuntze, S. (2015). Teachers’ professional knowledge and noticing: The case of multiple representations in the mathematics classroom. Educational Studies in Mathematics, 88(1), 89-114.
  • Janvier, C. E. (1987). Problems of representation in the teaching and learning of mathematics. Centre Interdisciplinaire de Recherche sur l’Apprentissage et le Développement en Education, Université du Quebec, Montréal. Lawrence Erlbaum Associates.
  • Walther, G., Heuvel-Panhuizen, M. V. D., Granzer, D., & Köller, O. (2012). Bildungsstandards für die Grundschule: Mathematik konkret. Humboldt-Universität zu Berlin, Institut zur Qualitätsentwicklung im Bildungswesen.

Teaching and learning Maths: using Math apps

Benefits of apps to the Maths teaching and learning process

With the widespread introduction of mobile learning technology to Australian classrooms (i.e. iPads), an unprecedented development of educational software (apps) takes aim to complement traditional teaching. The potential benefits of apps need to be critically appraised for their pedagogical content, learning-area specific knowledge and technological requirements and ease of implementation (Handal, Campbell, Cavanagh, & Petocz, 2016). The emerging research suggests that the use of iPads in primary school Mathematics classrooms has great potential to develop and maintain positive student attitudes (Hilton, 2016) and support self-paced learning. However, research also points out that individual apps can have both supportive and inhibitive consequences on students’ learning performance and efficiency, depending on the student, prior instruction and the phase in the learning and teaching cycle (Moyer-Packenham, 2016).

Examples of three Math apps

  1. Mathletics by 3P Learning Australia, Sydney. Mathletics is the most widely used app in Australian primary schools with comprehensive modules that complement for the K-12 Maths curriculum. (see more detail below)

Screenshot of Live Mathletics challenge

  1. Khan Academy, Mountain View, California.
    Khan Academy started out as a content provider of free educational movies and since evolved into student-centred learning app with a strong focus on Maths, with recent initiatives towards more international curriculum alignments (Khan Academy, 2017).

Khan Academy Maths opening page

  1. LÜK-App by Westermann Gruppe, Braunschweig, Germany.
    German curriculum-aligned quality app with a unique gamified approach towards learning, including all areas of Maths covered in primary schools (no German knowledge required)
LÜK app Maths task example

LÜK app Maths task example

Detailed description of Mathletics

Mathletics software is developed in Sydney since 2004 and is marketing itself by stating that Australian schools that use Mathletics are performing significantly better in NAPLAN tests, irrespective of their socio-economic and regional status (Stokes, 2015). While running as an app, Mathletics is more of a comprehensive cloud-based educational platform offering school and class management tools, individual student learning pathways, global online competitions, and professional teacher training courses. The author has been using this app with his daughter throughout F-Year 3 and is particularly impressed with the pedagogical quality that went into the sequential buildup of mathematical concepts, the comprehensive content and close alignment with the Australian Curriculum (Australian Curriculum, Assessment and Reporting Authority, 2017), the quality of technological implementation and support. It is one of the few Math apps that combines declarative, procedural and conceptual knowledges (Larkin, 2015).

References

  • Australian Curriculum, Assessment and Reporting Authority. (2017). Home/ F-10 Curriculum/ Mathematics.
  • Handal, B., Campbell, C., Cavanagh, M., & Petocz, P. (2016). Characterising the perceived value of mathematics educational apps in preservice teachers. Mathematics Education Research Journal, 28(1), 199-221.
  • Hilton, A. (2016). Engaging Primary School Students in Mathematics: Can iPads Make a Difference?. International Journal of Science and Mathematics Education, 1-21. DOI 10.1007/s10763-016-9771-5
  • Khan Academy. (2017). An uncommon approach to the Common Core.
  • Larkin, K. (2015). “An App! An App! My Kingdom for An App”: An 18-Month Quest to Determine Whether Apps Support Mathematical Knowledge Building. In Digital Games and Mathematics Learning (pp. 251-276). Springer Netherlands.
  • Moyer-Packenham, P. S., Bullock, E. K., Shumway, J. F., Tucker, S. I., Watts, C. M., Westenskow, A., … & Jordan, K. (2016). The role of affordances in children’s learning performance and efficiency when using virtual manipulative mathematics touch-screen apps. Mathematics Education Research Journal, 28(1), 79-105.
  • Stokes, T. (2015). National Numeracy Study Mathletics and NAPLAN. 3P Learning Australia Pty Ltd.

Montessori grammar symbols and colours

In researching the conventions in functional grammar colour-coding, I came across Montessori’s grammar pedagogy. Montessori associated geometric shapes and colours with the building blocks of traditional grammar creating semiotically-rich classroom manipulatives.

In 1923, Russian painter Wassily Kandinsky created a questionnaire that asks participants to fill out three geometric shapes (triangle, square, circle) with one of three primary colours yellow, red and blue. While concerns can be raised about presenting the “correct” order of answers, the primary colour-geometry relationship that came out proved very influential and can be found in all of Kandinsky’s paintings.

Kandinsky’s questionnaire. 1923. Bauhaus Archiv, Berlin

The Museum of Modern Art (MoMA) recreated an online version of Kandinsky’s original questionnaire. This could be a great starting point and teaching and learning resource for getting student’s to think about colour-shape- and ultimately grammar associations.

Kandinsky’s student Monica Ullmann‐Broner took a step further in 1931 and associated additional “secondary” geometrical forms with secondary colours.

Monica Ullman-Broner’s secondary form-colour associations. 1931. Bauhaus Archiv, Berlin

Inspired by his contemporaries in Weimar, Maria Montessori designed objects to recontextualise formal grammar for pedagogic discourse. She came up with nine grammar symbols still used today in Montessori schools to represent “nine parts of speech” (noun, verb, adjective, adverb, article, pronoun, conjunction, preposition and interjection). Montessori further groups these into four “functional families” (‘noun family’ – noun, adjective, article, pronoun; ‘verb family’ -verb, adverb; ‘the servants’ – conjunction, preposition, and ‘special case’ – interjection). The system uses three variables: colour, shape and size. The resulting grammar symbols can be compiled into a chart. Differently-sized sets of grammar symbols can further be combined into advanced grammar symbols that facilitate semiotic discussion of more complex grammatical concepts.

Montessori ‘noun family’ grammar symbols (http://www.montessorialbum.com)

Montessori ‘verb family, the servants, and special case’ grammar symbols (http://www.montessorialbum.com)

Montessori advanced ‘noun family’ grammar symbols (http://www.montessorialbum.com)

Montessori special ‘verb family’ grammar symbols (http://www.montessorialbum.com)

Montessori advanced ‘verb family’ grammar symbols (http://www.montessorialbum.com)

  • Verbs are depicted in red as circles or spheres. The verb is considered to be the central word (latin verbum) and to depict movement, actions, like a ball or the planets.
  • Nouns are depicted in black-blue as triangles or pyramids. The noun is considered stable like a pyramid. The pronoun is purple because it links the noun (blue-black) with the verb (red) (see Feez, 2007, p.361).
  • Modifiers reflect the shape and colour of their “parent element”, but are lighter in colours (blue for adjectives, orange for adverbs) and smaller in size.
  • Conjunctions are depicted as pink rectangles. The shape is considered to symbolise a hyphen.
  • Prepositions are depicted as green crescents. The shape is considered to symbolises a bridge.
  • Interjections are the special case, combining circle and square in golden colour.

It is fascinating to realise how much didactic thought Maria Montessori put into recontextualising abstract formal grammar into pedagogical manipulatives. One might ask why SFL did not adopt these semiotic colours in their transitivity system, making Processes red, Participants blue, and Circumstances … well I guess yellow/golden because they make situations special? However, as Gunther Kress and Theo van Leeuwen conclude in their seminal analysis of colour as a semiotic mode: “Colour does what people do with it” (2002, p.350). So, perhaps it is then best to have children invest into their own grammar symbols and colours?

References:

  • Feez, S. (2007). Montessori’s mediation of meaning: a social semiotic perspective. Learning to read with grammatics. Unpublished PhD thesis. University of Sydney, 312-366.
  • Kress, G., & Van Leeuwen, T. (2002). Colour as a semiotic mode: notes for a grammar of colour. Visual communication, 1(3), 343-368.