Nt1310 Unit 2 Theoretical Framework

2 THEORETICAL FRAMEWORK

We learn more by looking for the answer to a question and not finding it than we do from learning the answer itself.” - Lloyd Alexander [3].
This quote encapsulates the essence of changing the focus of the mathematics tutorial to action rather than just solving and writing either alone or in small groups. Although, students do not always appreciate the journey to a solution, particularly an incorrect one, this navigation is where learning is occurring. Aspects of Bloom's Taxonomy of Cognitive Levels and Gardner’s Multiple Intelligences are utilised in the conducting of this new format tutorial session.
2.1 Bloom's Taxonomy
Bloom identified three domains of learning: Cognitive, Affective and Psycho-Motor (though
he never completed this level) [4]. These groupings and all sublevels broadly cover mental, emotional and physical skills respectively. The cognitive domain has been revised by Bloom’s student, Lorin Anderson which included renaming domains, slight rearrangement and introduced new levels [5]. The cognitive domain is associated with thought processes whereas affective domain is concerned with emotional processes of learning and psycho-motor domain with the physical aspect of applying these skills.

2.1.1 Cognitive Domain
The following table 1 gives a progression of stages of mental development starting from recalling facts through to generating new ideas elaborated from patterns of diverse elements. It is presumed that one level must be achieved before moving to next level. These categories are attributes or the learning outcomes quoted by educators as part of the unit outline and evaluated on the whole in the form of timed written assessments. This domain is most used in education, table 1 lists the levels in ascending order of complexity with brief examples.

2.1.2 Affective Domain
This domain deals with feelings, attitudes, and emotions and may be one of the reasons why it receives less attention than the cognitive domain. This area is hard to evaluate, however, it is more important for students to quantify and identify for their learning benefit. Educators may encourage and promote progression through levels by careful observations and timely interventions. Table 2 lists the five levels from the lowest skill of listening and awareness through to proposing, revising and qualifying problem-solving, a much needed skill in mathematics.

2.1.3 Psycho-Motor Domain
This domain is more about the physical skills which starts with simple repetition of exercises continues to recognising and modifying in fresh situations and finishing when the skill becomes second-nature and reached a level where there is confidence to impart knowledge to others. Table 3 lists the categories in ascending levels. Although included in Bloom’s taxonomy, Bloom did not complete this domain and there are several revised versions with some slight disagreements so table 3 will use the version by Ravindrakumar Dave, who was reportedly a student of Bloom.

2.2 Gardner’s Theory of Multiple Intelligences

Gardner put forward a theory that intelligence could be categorised into subgroups rather than just one intelligence [7]. Their origin was not in education, but rather psychology sometimes educators mistakenly consider the intelligences to be learning styles. It is best to use Gardner’s own words to state his perceived difference between style and intelligence: ‘In actuality, style and intelligence are fundamentally different psychological constructs.
Style refers to the customary way in which an individual approaches a range of materials—for example, a playful or a planful style. Intelligence refers to the computational power of a mental system: for example, a person whose linguistic intelligence is strong is able readily to compute information that involves Language. Whether a person likes to use a certain intelligence is not the same as strength in an intelligence: I might love music but have only mediocre musical intelligence’ [8].
Each person has their own personal framework of intelligences, some stronger than others, and a mixture of these intelligences may be accessed during learning not just one type. Educators can make use of these intelligences by creating an environment of material suitable to several intelligences. So if movement, images, writing, and numbers are combined to teach a topic, this will lay a path for students to learn according to their strengths of several intelligences. So the teaching experience, although delivered to all in the same manner, will be received and processed uniquely by each student. The following Table 4 lists the current intelligences according to Gardner though as the world changes so maybe digital or technological intelligences may be added
[9].

3 ORGANISATION OF MATHEMATICS TUTORIAL ACTIVITY
This tutorial style, as stated before, is not new and has been around for over thirty years originating from La Trobe University Melbourne Australia and hence is, often, referred to as the ‘La Trobe Method’ [10]. Over the years, there have been modifications but the basic concept of the students being active in the class remains the same.
3.1 First Introductory Tutorial of Semester
Students arrive at the classroom, find a seat, wait for the tutorial to start, sometimes even have pens and paper ready before the tutor has uttered any words or written on the board. Students are not informed about the change of style so there is no perceived extra pressure yet! It is often forgotten the role that very simple gestures play in the act of learning, a simple smile crafts a friendly learning environment to encourage a mathematical community class spirit of ‘we can help each other’. This will bring some comfort to students since vocalising ideas and strategies to others is unusual in this discipline where the anticipated activity is sitting and working through examples. Standing at whiteboards appears to be an invasion of the teachers’ territory, students may feel anxious about expected knowledge base and communication skills. Keeping the explanation and instructions as short and simple as possible means that students commence the activity before nerves and self-consciousness can manifest. The emphasis is directed at ‘the doing’ of the questions cooperatively using any technique rather than the completion in the quickest time. This will set the mood of optimism and enjoyment, a small but effective signal at the beginning of this new venture. Time flies when you are having fun and the aim is to kick-start students’ learning effortlessly without them even noticing. ‘We learn more by looking for the answer to a question and not finding it than we do from learning the answer itself.” - Lloyd Alexander [3]. Hopefully, the true value of this quote will resonate through the semester.
3.2 The Administration of Tutorial
The tutorial is administered in the following manner from students’ perspective:
• Students arrive and settle into groups of three or four
• Collect worksheets, colour whiteboard markers, erasers, and magnets
• Magnets hold the worksheets to the board so students can see problems easily
• Students tackle the tasks by talking using lecture notes, smartphones etc…
• Different colours indicate individual students’ work
• Students are welcome to take photos of their work
• Return all materials and clean whiteboards
• Collect worked solutions
For the educators, the tutorial looks like this:
• Check that all materials are near the boards
• Chat to students who arrive early mainly to remember names
• Move from group to group, giving encouragement to start, whilst recording attendance
• Observe the dynamics of students both as individuals and as a group
• Listen to the talk and discern individual workings on the whiteboard
• Keep the momentum of discussion though only required for the first few weeks
• Relate personal understandings, learning and teaching experiences
• Travel the room ensuring interaction with each student
• Reminiscing previous exercises, giving feedback
• Suddenly time has gone and tutorial is over
Ideally, the tutorial fills the room with conversations, boards covered with worked solutions and groups look at other boards to see other solutions. This does happen often only due to good preparation and guidance in the first few weeks. Some students will take time to adjust to the new style and then participate more fully as they become familiar with people, style and subject though they can choose to sit at the table and complete the tasks.
4 THE TUTORIAL AND GARDNER’S THEORY OF MULTIPLE INTELLIGENCES
The written test, some permitting students’ handwritten notes as an aid, can only convey a certain amount and type of knowledge. The knowledge behind the written solution is unclear as the student could have a complete understanding of the concept, or the permitted notes may contain easy to follow instructions to complete the question or by chance the similar worked solution so mere transcription is required. Students through conversation and questions can more easily convey the essence of their knowledge. Howard Gardner [11] commented on standardised tests:
I have become one of the most insistent critics of such tests, feeling that, whatever they successfully assess, they miss much; that they often fail to pick up the most important human capacities and attributes; they favor the glib and the conventional rather than the profound or the creative; and that people who do not understand these instruments attribute to them much more merit than they actually warrant. [11]
No matter how well educators design an examination paper, it is not always possible to determine whether the student knows the concept through the answering of the allotted question for that concept.
This style of tutorial gives students the chance to explain not only using words but with movement and gestures. Spatial-visual and Bodily-kinesthetic intelligences are used to aid both students’ learning and assessment. Mathematics is a subject where one is often asked to think beyond what one can see and it is this abstraction many find difficult to explain in written assessments. It is frustrating for student and educator alike, when an examination question either cannot be started or is misunderstood, so few or no marks can be awarded. Both know that this is not a true reflection of the understanding of the topic.
Students have the chance to perform and explore some of the more difficult abstract topics to visualise such as intersection of planes. This gives a concrete impression of an abstract idea and is more memorable and meaningful because students are demonstrating for themselves rather than watching.
Assessments in the semester consist of written and verbal contributions, the latter is more holistically evaluated rather than merely listening to one answer to a question. Students’ questions, no matter how simple, reveal where they are in their understanding of a topic and can be used as stepping stones for all in the group. Sometimes a member of the group solves the problem rather than the tutor, and then, at other times, it provokes more questions and explanations beyond the actual question itself. These interactions are noted by the tutor and is used as an aid during the verbal assessments. Students are given the opportunity of improving Verbal-linguistic intelligence through student-tutor and student-student communications.