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Anne Bayetto, Lecturer in Education, Flinders University
Classroom educators already know there is a wide range of student abilities within a year level and with this come significant planning and programming issues. Commenting on mathematics teaching, Elkins (2005) notes there has been a move away from the transmission model of content delivery for all, or what has been referred to as ‘you watch what I do, and then you do it’, to a focus on conceptual understanding that is supported by constructivist teaching approaches.
However, there is a dilemma for educators. On the one hand, they are told not to teach to the whole class and, on the other hand, that a completely individualized mathematics program, or even multiple groups, will likely pose organizational challenges. While there will always be situation specific factors to consider, research has highlighted fundamental principles that maximize mathematics learning for all students. It may be the intensity and amount of learning that needs to be different for students with learning difficulties. As the National Council of Teachers of Mathematics (NCTM) (2006) states, effective mathematics teaching requires understanding what students know and need to learn and then challenging and supporting them to learn it well.
A small number of students is identified as having a specific mathematical learning difficulty (dyscalculia) but there is a divergence of views about causes and identification (Hannell 2005). Interestingly, it has been suggested that very few students actually have a mathematical learning difficulty (Carnellor 2004, Westwood 2000). In fact Booker, Bond, Sparrow and Swan (2004) suggest that most students have a learned difficulty. That is, educators have created their mathematical problems. Regardless of the reason for students’ mathematical difficulties i.e. intrinsic or extrinsic factors, educators still need to get on and teach. Some examples of how effective and considered practice may prevent confusion and problems will now be discussed.
It is timely to preface these points by reiterating that learning occurs in settings that are supportive and caring. Students with learning difficulties may already have an external locus of control i.e. they believe they can’t improve their mathematical capacities. It is when they feel confident to have a go, make mistakes, discuss and question, that engagement and achievement will occur.
Sherman, Richardson and Yard (2005, p 3) remind us that mathematics instruction must provide many opportunities for concept building, relevant challenging questions, problem solving reasoning, and connections within the curriculum and real-world situations. Westwood (2000) also reminds us that the educator is the pivotal person in ensuring successful learning.
In order to work within curriculum guidelines while accommodating the diversity of students in their classrooms, educators need to be realistic and systematic in the way they structure their mathematics programme. The benefits of cross curricular teaching cannot be overemphasized. It could well be that use of an engaging, and age appropriate, theme is the way into developing conceptual knowledge and skills. For example, a topic such as patterns could have students exploring patterns not only in mathematics but also in Health and Physical Education (team games), Society and Environment (climate, history), Arts (dance), and Design and Technology (measurement processes used when designing and constructing). However, Tucker, Singleton and Weaver (2002, p 3) suggest that the primary criterion for judging an instructional activity is what are the pupils learning during the activity… [what is] the learning objective? Educators of students with learning difficulties must be quite clear about intended learning outcomes as they work toward closing the learning gap.
Westwood (2000) and Carnellor (2004) highlight the importance of educators using a judicious blend of constructivist and explicit teaching with ample guided practice/scaffolding toward independence.
Where does this leave one-to-one instruction and drill activities that have long been the mainstay of many mathematics remediation programs? Before any practice is undertaken, a secure understanding of underpinning concepts, where new learning is linked to previous learning, must be assured. If not, it may become a cycle of practise and forget, practise and forget. How often has one heard said, “I taught him/her and it’s already forgotten"? A response might be “How do you know he/she understood it in the first place?” Sherman, Richardson and Yard (2005) believe that students with learning difficulties are given tedious and boring activities to develop the basics. They go on to remind educators (p 3) that it is critical that the same content not be taught year after year, in almost the same manner of delivery. Students who did not “get it” the first time are not likely to “get it” the next several times it is taught in the usual manner. This may be where technology can provide a different way to develop conceptual understanding (More about this later). It has also been suggested that doing more of the same low level tasks not only narrows the curriculum but that it does not enable a student to show what they truly know and can do. Instead of watering down, educators must program-up and have ambitious but achievable goals.
So what might an educator do to acknowledge the wide diversity in a group and honour what students can do and need to do next? Following is a collection of key teaching issues in no particular order but all worthy of reflection.
1. Use brief, mini-lessons for specific skills with the whole class or targeted groups (Peterson, Hittie & Tamor 2002).
2. Provide opportunities to work alone and together. Learning mathematical concepts and skills is more than receiving it like a gift from an educator. While an educator can introduce new learning and lead students toward understanding, there is much benefit in moving beyond a whole class approach by using paired and group work. Problem solving can promote discussion between peers as they share strategies and justify processes and answers. It is a way of talking and moving into understanding. Heterogeneous grouping based on need is much preferred to the use of stable ability grouping where the ‘budgies’ group never break away to join the ‘doves’ or ‘eagles’ groups. A note of caution though, Westwood (2000, p13) reminds us that [in the] US, Britain and Australia, a significant distance between the highest and lowest achievers exists from the early years of schooling and the gap often increases-partly due to the emphasis on individual progression and group work rather than on whole-class teaching. Further elaboration about effective whole class approaches, based on the Trends in Mathematics and Science Study (TIMSS) can be found in Westwood’s book or at the TIMSS website (http://timss.bc.edu).
3. Use problem solving with divergent questions. Booker, Bond, Sparrow and Swan (2004, p 44) state that problem solving is a task or situation for which there is no immediate or obvious solution. Along with other writers they question whether what educators provide as problems are little more than algorithms with words around them. Authentic problems must pose a challenge that encourages strategic thinking and are a vehicle for development of concepts and skills (Westwood 2000, Sherman, Richardson and Yard 2005). It is also important to remember that there is more than one way to be right and there is more than one way to be wrong! A student’s sense of satisfaction at having developed a successful process for solving a problem must be warmly acknowledged rather than discounted as not being ‘the preferred way’.
4. Use concrete materials. For reasons unknown, many educators in Australia have moved away from use of concrete materials when students transition into the primary years. Sounds of “Do it in your head” have been heard across the country. However, researchers have a different view (Booker et. al. 2004, Carnellor 2004, Westwood 2000). It is materials that provide tangible ways to explore mathematical ideas and, for educators, they are a window into student thinking. Counters, Base 10 materials, interlocking blocks, real money (it’s never quite the same with plastic) etcetera provide materials to manipulate and talk about. One resource already in schools is an overhead projector. With the use of transparencies and pens, educators are able to model and think aloud as they tackle algorithms and problems. Additionally, there is a wide variety of materials that can be used on the overhead projector e.g. transparent counters, clocks, and calculators.
5. Confirm student understanding of mathematical language. Sherman, Richardson and Yard (2005) believe that students become confused about the meaning of words in mathematics lessons. For example, while an educator may be explaining the concept of 10 to the power of… the word power takes on a whole new meaning for many South Australian children that has (possibly) little to do with what is being explained. Westwood (2000, p18) believes that one of the main problems encountered by students…is translating between their own intuitive and concrete understanding of the real world and the language used to describe and quantify for mathematical purposes for school. Educators must build upon a student’s level of language, check for understanding and not assume that nods and smiles are indicating comprehension.
6. Have students keep math journals. While educators have used journal writing for many years as part of their literacy program, they can also be used as a resource for mathematics. Carnellor (2004, p 25) comments that children…need to be given as many opportunities to write about their tasks which allows them to incorporate the strategies they are happier using. By writing about their mathematics, students can firm up their thinking and explain on paper what they did and why. This process also allows the educator to see into a student’s thinking.
7. Play games. Historically, games have been used as a reward when the real work has been finished (early). Booker (2000) reminds us that games can be powerful teaching and learning tools to develop conceptual understandings. It is engagement with interesting and fun activities that can keep a young person practising a skill well beyond what they might tolerate if asked to do (another) worksheet. Simple and versatile, games such as those presented in Booker (2000) can provide important learning opportunities; student with student or student with adult. Card games are especially useful as they are cheap, portable, and socially acceptable for all ages.
8. Use technology. NCTM (2006) states that technology is essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances students' learning. Far from being just electronic downtime, the internet offers educators a huge range of research-based practices, interactive websites, resources, and lesson plans. Again, it is the engagement of otherwise reluctant students that shows us the internet can be a valuable teaching resource. A computer is a patient teacher and it is interesting to observe some students who will stay with a game well beyond the time they might usually stay with a pen and paper exercise. Far from being a solitary activity, pairs and small groups can use internet mathematic resources.
Computer software can often do what a textbook or worksheet will never be able to do, and that is to engage. While an educator will need to introduce a concept and ensure understanding, a software program can often provide varied practice to develop automaticity and fluency. Selection of programs must be judicious and meet the learner at their point of capacity. Some of the latest mathematics software is found in the SPELD SA shop.
It’s interesting that some educators believe the use of calculators is making students lazy and yet employers expect their staff to have an effortless capacity to use them. (What about the use of graphics calculators in year 12 exams?) Carnellor (2004, p 54) takes a more proactive view in relation to students with mathematical difficulties when she says that calculators may provide these students with opportunities to investigate the necessary mathematical ideas, without the concern of accuracy in the recording process. Swan (1996) has a very helpful book of teaching approaches for developing skills in using calculators.
9. To optimize learning for students who already have mathematical difficulties it is essential that educators have a robust pedagogical knowledge and positive attitude. Carnellor (2004, p 5) states that for many adults…mathematics generates unease and insecurity. ….These feelings probably originate from their own classroom experiences where mathematics consisted of drill, rules, and recipes, instead of understanding and application. This suggests that educators need to be quite clear about what mathematical concepts they feel comfortable teaching and where they need further learning. The writer would like to suggest that as she gets wiser (aka older) she has come to the realization that the older she gets, the less she knows. Professional learning through collegial sharing of practices, participation in workshops and self-selected reading must be part of every educator’s repertoire.
10. Choose published materials carefully. The belief that purchase of a packaged program/textbook will be the key to successful mathematics teaching and learning is too simplistic. While it may seem irresistible to put one’s faith in another’s compelling testimonials, it is highly unlikely that ONE of anything will meet the needs of all in a class. What happens if a few students can’t read the textbook? What about individuals who already know all of the concepts in the book and it’s the start of the year? Pincott (2004, p 147) suggests that a textbook approach in the curriculum is akin to rote learning tables i.e. carrying out the process often without the understanding.
11. NCTM (2006) reminds us that assessment should support the learning of important mathematics and furnish useful information to both teachers and students. By taking a strengths based approach and using a range of measures i.e. observation, questioning, work samples, diagnostic tests, student self assessment, and asking students to explain a process, educators can establish what is known, and therefore what needs to be known. The writer believes that having a student explain what they did (or would do, if they could do it!) provides rich information. Whatever process is used to gather evidence and data, it is to inform strategic and timely teaching and learning opportunities. Assessment must be for learning, and of learning.
Hastening students through a curriculum at the expense of understanding is short sighted and inefficient. (Recent research supported by the Department of Education, Science and Training (2004) suggested that Year Four was the time to think about a first introduction of algorithms!) Rather, educators need to work in a cycle of assess, plan, program, assess etcetera and meet the student at the point of their knowing and what they need to know next. As educators consider what they might do tomorrow, next week or next year, it is challenging to reflect on what effective practices should be maintained or taken up, and what practices have had their time and are best left behind. Students with learning difficulties are like all other students: they must be taught mathematics in a way that engages and dignifies them as learners.
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