Catering For Individual Differences

Catering for Individual Differences PGDE (Full Time) Mathematics Major

Arthur Lee Dec, 2009

Catering for Learner Differences: Teaching Package on S1-5 Mathematics Learners vary tremendously in their family background, parental expectation towards their performance, cognition, learning sequences, motivation towards learnings, their own perception on performance in mathematics and their role in the learning process. These factors constitute the cause and nature of learner differences. They are variables for each learner and interact in a complex way. They affect teaching and learning activities and the quality of learning. Mathematics Section, Education Department (2001). Catering for Learner Differences: Teaching Package on S1–5 Mathematics. Hong Kong: Education Department. http://cd1.edb.hkedcity.net/cd/maths/en/ref_res/MATERIAL/ld_e/LD_e%20index.htm

Strategies, Central Curriculum The secondary mathematics syllabus is structured in three different parts: 1. the Foundation Part 2. the Whole Syllabus 3. the Enrichment topics

At the primary level, there is a choice of Enrichment topics on top of the core syllabus.

The syllabus contents are considered with an allowance of about 10% of the normal school time allocated to mathematics as spare periods.

Mathematics Section, Education Department (2001). Catering for Learner Differences: Teaching Package on S1–5 Mathematics. Hong Kong: Education Department. http://cd1.edb.hkedcity.net/cd/maths/en/ref_res/MATERIAL/ld_e/LD_e%20index.htm

Strategies, Central Curriculum an example in the learning unit "Congruence and Similarity" in the MSS dimension

The objectives with asterisk (**) are exemplars of enrichment topics. The objectives underlined are considered as non-foundation part of the syllabus. Mathematics Section, Education Department (2001). Catering for Learner Differences: Teaching Package on S1–5 Mathematics. Hong Kong: Education Department. http://cd1.edb.hkedcity.net/cd/maths/en/ref_res/MATERIAL/ld_e/LD_e%20index.htm

Strategies, School level ◆ Decide the aims and targets of the whole school mathematics curriculum and at each Key Stage. ◆ Adopt organizational arrangements such as providing additional lessons to certain students and ability grouping strategies like streaming, split class, withdrawal and cross-level subject setting. (Appropriate measures of flexible grouping would help reducing labeling effect.) ◆ Appropriately select the depth of treatment of the learning units that lie outside the Foundation Part of the Syllabus as the common core learning contents for all students. Mathematics Section, Education Department (2001). Catering for Learner Differences: Teaching Package on S1–5 Mathematics. Hong Kong: Education Department. http://cd1.edb.hkedcity.net/cd/maths/en/ref_res/MATERIAL/ld_e/LD_e%20index.htm

Strategies, School level ◆ Arrange the learning units in a logical sequence for each year level. This arrangement should take into consideration ◇ cognitive development and abilities of students; ◇ affective elements of students; ◇ learning objectives of the learning units and their inter-relation; ◇ the inter-relation of mathematical learning at different year levels; ◇ the resources (e.g. no. of periods) available to mathematics learning at different year levels. Mathematics Section, Education Department (2001). Catering for Learner Differences: Teaching Package on S1–5 Mathematics. Hong Kong: Education Department. http://cd1.edb.hkedcity.net/cd/maths/en/ref_res/MATERIAL/ld_e/LD_e%20index.htm

Strategies, School level ◆ Choose an appropriate textbook and adapt or produce instructional materials. ◆ Design a wide variety of informal and non-formal learning activities such as statistical projects, weekly questions posted in the mathematics bulletin boards, mathematics books reading scheme, poster design using transformation of shapes, mathematics camp, Mathematics Olympiad, etc. ◆ Set up assessment policies that allow the method of recording and reporting to encourage continuous effort of students and to provide feedback for teaching and learning.

Mathematics Section, Education Department (2001). Catering for Learner Differences: Teaching Package on S1–5 Mathematics. Hong Kong: Education Department. http://cd1.edb.hkedcity.net/cd/maths/en/ref_res/MATERIAL/ld_e/LD_e%20index.htm

Strategies, Classroom level ◆ Diagnosis of students' needs and differences ◇ e.g. by gathering information about their interests, strengths and weakness. Note also that results from HK Attainment Test, Basic Competence Assessment, class-tests and/or examinations are useful information. Your own observations in the classroom, their classwork and homework provide even more immediate impressions.

Mathematics Section, Education Department (2001). Catering for Learner Differences: Teaching Package on S1–5 Mathematics. Hong Kong: Education Department. http://cd1.edb.hkedcity.net/cd/maths/en/ref_res/MATERIAL/ld_e/LD_e%20index.htm

Strategies, Classroom level ◆ Variation in level of difficulties and contents covered ◇ select, adapt or design materials at appropriate level ◇ give less able students greater sense of satisfaction and hence greater confidence ◇ give more able students challenges to cultivate as well as to sustain their interest in mathematics

Mathematics Section, Education Department (2001). Catering for Learner Differences: Teaching Package on S1–5 Mathematics. Hong Kong: Education Department. http://cd1.edb.hkedcity.net/cd/maths/en/ref_res/MATERIAL/ld_e/LD_e%20index.htm

Strategies, Classroom level ◆ Variation in questioning techniques ◆ Variation in clues provided in tasks ◆ Variation in approaches in introducing concepts ◆ Variation in using computer packages (e.g. dynamic objects, spreadsheets, simulation) ◆ Variation in Peer Learning (e.g. various classroom organization like group work)

Mathematics Section, Education Department (2001). Catering for Learner Differences: Teaching Package on S1–5 Mathematics. Hong Kong: Education Department. http://cd1.edb.hkedcity.net/cd/maths/en/ref_res/MATERIAL/ld_e/LD_e%20index.htm

Strategies, Classroom level ◆ Variation in assessment items: Assessment that would cover various aspects of understanding and achievements. In particular, when assessment (ranging from classwork, quizzes, to tests) does not need to completely match any other "standard" examinations, it should encourage a broader scope of comprehension and a wider spectrum of understanding. ◆ Arousing Learning Motivation (varieties of tasks and activities including competitions, games, group discussion, and something extracurricular) Mathematics Section, Education Department (2001). Catering for Learner Differences: Teaching Package on S1–5 Mathematics. Hong Kong: Education Department. http://cd1.edb.hkedcity.net/cd/maths/en/ref_res/MATERIAL/ld_e/LD_e%20index.htm

Closer look at some classroom level strategies Variation in Questioning Techniques ... teachers can ask simple and straightforward questions to less able students and comparatively more challenging questions to more able ones ... Variation in Clues provided in Tasks ... for the more able students, teachers ask open-ended questions and provide fewer hints in the process of solving problems ... Your comments? Mathematics Section, Education Department (2001). Catering for Learner Differences: Teaching Package on S1–5 Mathematics. Hong Kong: Education Department. http://cd1.edb.hkedcity.net/cd/maths/en/ref_res/MATERIAL/ld_e/LD_e%20index.htm

One thing that people have in common is that they are all different.

F. Marton & S. Booth, 1997 Learning and Awareness

Catering for Individual Differences--Building on Variation another way http://iediis4.ied.edu.hk/cidv/ seeing individual differences What is 'Learning Study' http://iediis4.ied.edu.hk/cidv/learning/e_learn.htm

Theoretical Framework http://iediis4.ied.edu.hk/cidv/intro/e_intro_m3.htm

Some Beliefs on Catering for Individual Differences Difference in learning outcomes is caused by: ◆ difference in ability ◆ difference in motivation ◆ difference in teaching arrangement ◆ different ways of seeing the object of learning Lo, M. L., Pong, W. Y. & Chik, P. P. M. (2005) For each and everyone. Catering for individual differences through Learning Studies Hong Kong , The HKU Press.

Some Beliefs on Catering for Individual Differences The range of ability among normal children should not hinder students from learning what is intended in the school curriculum. Therefore in catering for individual differences, the focus is not on the variations in abilities. Rather, the focus is on the variations in the learning outcomes (what students actually learnt). For every worthwhile learning outcome that we can identify, there are also some critical aspects that can be identified and communicated. In order to help every student master these learning outcomes, teachers should be clear about the learning outcomes they wish to achieve in each lesson and the critical aspects that students must grasp. Catering for individual differences: helping every (normal) child to learn what is worthwhile, essential and reasonable for them to learn, given the school curriculum, and irrespective of their ability. What prevents students from learning an object of learning in school is not primarily due to their lack of ability, but mainly due to the incomplete ways of seeing that they acquired of the object of learning. Lo, M. L., Pong, W. Y. & Chik, P. P. M. (2005) For each and everyone. Catering for individual differences through Learning Studies Hong Kong , The HKU Press.

What is 'Learning Study' The main aim of the project is to establish an infrastructure in schools to facilitate teachers' professional development by learning from each others, from pupils' feedback and from the use of the theory of variation, thus improving the quality of teaching and learning. To realise this objective, we consider the 'Lesson Study Model' which is widely adopted in Japanese schools as a good method. Participant teachers form subject-based groups, who among themselves and with the university team meet regularly to carry out a number of 'research lessons'. For each research lesson, they discuss the objects of learning and its critical features. Together they develop ways to structure the lesson, taking into account the pupils' varied understandings of the subject matter in identifying the objects of learning. Some lessons are video-taped to facilitate the review work afterwards and serve as inputs for another round of study. Within such a model, we have also introduced an important element which is the use of the theory of variation (see Theoretical framework). Therefore, we describe our way of conducting the project as 'Learning Study' so as to distinguish it from the Japanese lesson study on the one hand; and on the other hand, to highlight our point of departure--how pupils understand what is to be learnt. http://iediis4.ied.edu.hk/cidv/learning/e_learn.htm

Theoretical Framework The basic idea of this project is to make use of the variation between pupils' different abilities and ways of understanding to actually decrease this variation. Such idea is derived from a learning theory which concerns variation and learners' structure of awareness (Marton and Booth, 1997; Bowden and Marton, 1998). The following briefly outlines the three aspects of variation which we have drawn from the theory of learning and variation to develop strategies to cope with individual learning differences. Variation in terms of pupils' understanding of what is taught Variation in teachers' ways of dealing with particular topics Variation as a teaching method

http://iediis4.ied.edu.hk/cidv/intro/e_intro_m3.htm

Variation in terms of pupils' understanding of what is taught A popular view about children's differences in learning is that they have different general abilities or aptitudes, and hence there are "stronger" and "weaker" pupils. Another popular view is that children have their own ways of thinking. As a result, if there are forty pupils in the classroom, there will be forty ways of understanding. We look at this differently, not because these two viewpoints do not carry any truth, but because they do not provide a good point of departure for addressing the issue. In contrast, we wish to focus on the 'object of learning', by which we mean the knowledge and skills that we hope the pupils will develop; we wish to focus on what is taught and how it is made sense of by the pupils. Our point of departure is that children understand what they are supposed to learn in a limited number of different ways. Our research shows that teachers who pay close attention to such differences (or variation) are better able to bring about meaningful learning for their pupils. Children learn better not only because they become more focused on the object of learning, but also because they are exposed to the different ways their classmates deal with or understand the same content.

http://iediis4.ied.edu.hk/cidv/intro/e_intro_m3.htm

Variation in teachers' ways of dealing with particular topics

Teachers have daily encounters with pupils, and from these they build up a bank of knowledge about the different ways pupils deal with particular concepts or phenomena, as well as a working knowledge of how to handle these differences. This knowledge is so powerful and becomes part of their daily teaching that sometimes it is unnoticed by the teachers themselves. We view such knowledge as extremely valuable. By knowing in advance the prior knowledge and understandings of the pupils, we can be more effective in helping pupils to learn what is intended. Therefore, instead of letting this knowledge remain at the back of the teacher's mind, it should be identified, sharpened, and systematically reflected upon, and above all, shared with other teachers.

http://iediis4.ied.edu.hk/cidv/intro/e_intro_m3.htm

Variation as a teaching method When we notice that some pupils have difficulties with their learning, it means that these pupils have not grasped the critical features of what has to be learnt. To cater for individual differences, the teacher should identify these critical features and help pupils to focus on them. This can be done by means of variation, i.e. using examples, non-examples, multiple representations, etc to give prominence to what is and what is not critical to the understanding of a particular object. For instance, the concept of having the same digit added on to itself for a number of times is critical to the understanding of multiplication, whereas the recitation of multiplication table without explanation is not. In our everyday experience, we cannot focus on everything at the same time. While some are taken for granted, some others are held in focal awareness. Features that are taken for granted or in the background are only discerned when they vary (Bowden & Marton, 1998, Marton & Booth, 1997). For example, a bird in the tree may not be noticed until it flies away and its movement catches the eye of the observer. Seen from this light, what is varied and what remains unchanged during the lesson is of decisive importance in determining how effective the lesson is. http://iediis4.ied.edu.hk/cidv/intro/e_intro_m3.htm

Learning Studies in Mathematics

http://iediis4.ied.edu.hk/cidv/e_front.asp?Open=2

P.4 Mathematics lesson on 'Perimeter and Area' L e a r nin g Stu d y 5: P. 4 M a t h e m a t i c s l e s s o n o n ‘ P e r i m e t e r a n d A r e a ’ Allen Leung Duration The second research period was from 19th January to 22nd June 2001, during which thirteen meetings were held after school hours. Stage I: Incubation of ideas During the first meeting, teachers promptly determined that they would like to do a research lesson on ‘area and perimeter’. The researchers were rather surprised by their decision, as ‘area and perimeter’ appeared to be quite a simple concept. However, the teachers were able to show the researchers that the students had difficulties in understanding this concept for the following reasons: Students’ difficulties/ misconceptions demonstrated in learning the topic The teachers noticed that students often mixed up the formulas for area and perimeter, as well as their measuring units. Moreover, most students were able to solve problems on ‘area’ or ‘perimeter’ easily, but they had great difficulties answering questions that involved both concepts, such as questions in the Hong Kong Attainment Test related to ‘area and perimeter’ on which most students scored poorly. Students at P.5 and P.6 appeared to share similar problems as well. Therefore, the teachers were eager to help students to solve this problem as early as possible, hence giving students a good foundation for future studies. In the next meeting, discussions were centred on identifying students’ differences in understanding, as well as their common misconceptions concerning the topic, which both the researchers and the teachers agreed to be very crucial for mapping out the teaching plan. Students’ differences in understanding ‘area and perimeter’ (a) Students with no concepts of ‘area and perimeter’: These were students who appeared to have not even a superficial 1

P.4 Mathematics lesson on 'Perimeter and Area'

Students' difficulties / misconceptions (general) The teachers noticed that students often mixed up the formulas for area and perimeter, as well as their measuring units. Moreover, most students were able to solve problems on ʻareaʼ or ʻperimeterʼ easily, but they had great difficulties answering questions that involved both concepts, such as questions in the Hong Kong Attainment Test related to ʻarea and perimeterʼ on which most students scored poorly. Students at P.5 and P.6 appeared to share similar problems as well.

http://iediis4.ied.edu.hk/cidv/e_front.asp?Open=2

P.4 Mathematics lesson on 'Perimeter and Area'

Students' difficulties / misconceptions Students thought that the perimeter of a rectangle should be equal to the length times the width even though they knew that this was the formula for area. Students tended to believe that the longer the perimeter, the larger the area of a rectangle becomes; and as the shape of the rectangle changes, so does the area. Some students also believed that if the perimeter of a rectangle is doubled, the area would also be doubled. In other words, they thought that the perimeter and the area usually increase at the same rate. http://iediis4.ied.edu.hk/cidv/e_front.asp?Open=2

P.4 Mathematics lesson on 'Perimeter and Area'

A Gap in Curriculum ... 'area' and 'perimeter' were never taught together in the curriculum, rather, they were treated as two different topics. ... teachers were thinking of teaching these two concepts together in the hope of providing students with an opportunity to learn through comparison and contradiction. Moreover, they believe that 'area and perimeter' should be taught along with other related topics, such as the characteristics of polygons.

http://iediis4.ied.edu.hk/cidv/e_front.asp?Open=2

P.4 Mathematics lesson on 'Perimeter and Area'

Development of Lesson Plan In the subsequent meetings, teachers came to realize that the reason for their studentsʼ difficulty in grasping the basic concepts of perimeter and area might lie in the fact that these concepts were usually 'definedʼ by the teachers via formulae rather than by using studentsʼ intuitions of space and measurement. Some teachers pointed out that students often confused the two formulae by using them interchangeably. This might indicate that the concepts of perimeter and area are purely symbolic for students, missing out the primitive geometrical elements. http://iediis4.ied.edu.hk/cidv/e_front.asp?Open=2

P.4 Mathematics lesson on 'Perimeter and Area'

Development of Lesson Plan Discover pattern(s) for rectangles with the same perimeter, different areas: ◆ The school is planning to build a rectangular fishpond with a fixed amount of wiring to surround its shape. What dimensions would give the largest fishpond? ◆ Students will be grouped and given pieces of unit-square to construct their rectangular fishponds. They will be asked to count the number of unit-square used. http://iediis4.ied.edu.hk/cidv/e_front.asp?Open=2

P.4 Mathematics lesson on 'Perimeter and Area'

Development of Lesson Plan Discover pattern(s) for rectangles with the same area, different perimeters: ◆ What is the least amount of wiring needed (hence the most economical) to surround a fishpond with a given area? ◆ Students will be grouped and given pieces of unit-square to construct their rectangular fishponds. They will be asked to count the number of units that give the perimeter. http://iediis4.ied.edu.hk/cidv/e_front.asp?Open=2

Assessment for Learning (Secondary Mathematics) The Open-ended Questions Examples of open-ended questions and samples of students' work can be found in this booklet. It is prepared by the Mathematics Education section of EDB and distributed to schools in 2003. Besides assessment, how can this type of questions be used in our teaching.

Assessment for Learning (Secondary Mathematics) The Open-ended Questions

Assessment for Learning (Secondary Mathematics) The Open-ended Questions

Assessment for Learning (Secondary Mathematics) The Open-ended Questions

(b) their relations are twice of x plus 1 Assessment for Learning (Secondary Mathematics) The Open-ended Questions

(b) for each pair of numbers, their difference is a multiple of 5

Assessment for Learning (Secondary Mathematics) The Open-ended Questions

Assessment for Learning (Secondary Mathematics) The Open-ended Questions

Low attaining students can think mathematically Failure in mathematics ... can be a result of affective issues, disrupted education and specific learning difficulties. ... can also be due to lack of development in the ways of thinking about mathematics which come naturally to those who, hence, succeed. Often, however, support for weaker students focuses on rules, techniques and procedures - sometimes called the 'basics' of mathematics.

Watson, A. (2005) In Houssart, J., Roaf, C., & Watson, A. (2005). Supporting mathematical thinking. London: David Fulton. (Chapter 2)

Low attaining students can think mathematically Emphasis might be on recall and application, yet these learners have not remembered and may not recognise situations as familiar ones in which to apply their knowledge. Little emphasis is generally given to helping them develop ways of thinking which may improve future learning. Little help is given them to construct complex understanding which provides the context for recall and application of procedures. Little attention is given to building on pupils' existing understanding and mental images. Currently, the materials provided to schools for those 'falling behind' largely fit this description (that is, rules, techniques and procedures), though of course they may be imaginatively used by teachers to create more challenging lessons. Watson, A. (2005) In Houssart, J., Roaf, C., & Watson, A. (2005). Supporting mathematical thinking. London: David Fulton. (Chapter 2)

Watson, A. (2006) ... offer a more accessible and functional language about doing mathematics, and show that learners in a very disadvantaged group were all able to demonstrate mathematical ways of thinking. This contrasts with 'normal' practices in which low-attaining learners are taught simplified mathematics. ... argue for a new mindset based on proficiencies of thinking rather than deficiencies of knowledge; thinking abilities ought to be nurtured, rather than left to atrophy while focusing on mundane content. Nurturing mathematical thinking is a job requiring skills and techniques which come from a structural understanding of mathematics.

Watson, A. (2006). Raising achievement in secondary mathematics. Maidenhead: Open University Press. p.102

'Boring' lessons Observation of many mathematics lessons aimed at low-attaining learners ... confirms that many such lessons frequently deal with simplified mathematics, broken down into step-by-step processes, often in short chunks, or packed with practical features such as colouring in, cutting out, tidying up and so on. Typical arguments for this approach are persuasive and commonplace. For example, it is said that learners who cannot concentrate for long periods need frequent changes of task; they grow bored if you do not change the topic every lesson; they need activity which uses their energy because many are so-called 'kinaesthetic' learners; they need the quick success which comes from getting things right easily; and so on. The irony of these arguments is that if you follow these guidelines low attainment is the inevitable results, as well as the reason. It is simply impossible to learn mathematics if one is constantly changing topic, or task, or doing related but irrelevant tasks, or only doing the easy bits, or being praised for trivial performance. Watson, A. (2006). Raising achievement in secondary mathematics. Maidenhead: Open University Press. p.103

'Boring' lessons A problem with a fragmented, mechanistic approach to teaching mathematics is that learners who find mathematics hard are thus often taught in ways which make it hardest for them to learn it. Simultaneously, students who get stuck at the lower levels of the National Curriculum in secondary school have to churn through content requiring a high level of accuracy and technical recall, while peers are doing work which is much more interesting and in which technical inaccuracies such as minor algebraic mistakes, dropped negative signs, and forgotten multiples are tolerated as less important than overall conceptual understanding.

Watson, A. (2006). Raising achievement in secondary mathematics. Maidenhead: Open University Press. p.103