The SITi M project

Teacher questionnaire

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1. What are your main priorities when teaching mathematics?

1......

2......

3......

4......

5......

Estimate the relative importance you currently give to each of the following.

Give each one a percentage, so that they add to 100%.

Purpose / %
  • fluency in recalling facts and performing routine skills

  • interpretations for concepts and representations

  • strategies for investigation and problem solving

  • awareness of the nature and values of the educational system
    (e.g. ‘study skills’, ‘how to revise’, ‘exam technique’)

  • appreciation of the power of mathematics in society

Linking our purposes to teaching method

Purposes of Mathematics Teaching
Fluency in recalling facts and performing skills / Interpretations for concepts and representations / Strategies for investigation and problem solving / Awareness of the nature and values of the educational system / Appreciation of the power of mathematics in society
• Memorizing facts, names and notations
• Practising algorithms and procedures for fluency and 'mastery' / • Constructing networks of relationships between mathematical concepts
• Interpreting and translating between representations of concepts
• Generating representations of concepts
• Discriminating between examples and non-examples of concepts / • Formulating situations and problems for investigation
• Constructing, sharing, refining, comparing strategies for exploration and solution
• Monitoring one's progress during problem solving and investigation
• Interpreting, evaluating solutions and communicating results / • Recognising different purposes of learning mathematics.
• Appreciating aspects of performance valued by the examination system
• Developing appropriate strategies for learning/
reviewing mathematics / • Creating and critiquing 'mathematical models' of situations
• Appreciating uses/abuses of mathematics in social contexts
• Appreciating mathematics as human creativity
(+ historical aspects)
• Using mathematics to gain power over problems in one's own life

This list presents us with an intimidating design challenge. Different purposes may be emphasised within a lesson or series of lessons. Thus, when considering the topic of number, an imaginative teacher may include a combination of:

  • exercises to develop fluency with multiplication or division algorithms (a skill focus);
  • discussions concerning the meaning of place value and its links with fractional notation (a concept focus);
  • 'rich' calculator-based, investigative activities or real problems to solve (a strategy focus);
  • discussions on the types of questions that might come up in the exam (for awareness of values of the system) and
  • discussions on the uses and abuses of percentages in the media (a social context focus);

2. How would you describe your current teaching practices?

For each row;
Tick one box to show how often the statement is true in your Maths lessons.
/ Almost never / Occasionally / Half the time / Most of the time / Almost always
1.Learners learn through doing exercises
2.Learners work on their own, consulting a neighbour from time to time.
3.Learners use only the methods I teach them
4.Learners start with easy questions and work up to harder questions.
5.Learners choose which questions they tackle.
6.I encourage learners to work more slowly
7.Learners compare different methods for doing questions.
8.I teach each topic from the beginning, assuming they know nothing.
9.I teach the whole class at once.
10.I try to cover everything in a topic
11.I draw links between topics and move back and forth between topics
12.I am surprised by the ideas that come up in a lesson
13.I avoid learners making mistakes by explaining things carefully first.
14.I tend to follow the textbook or worksheets closely
15.Learners learn through discussing their ideas
16.Learners work collaboratively in pairs or small groups
17.Learners invent their own methods
18.Learners work on substantial tasks that can be worked on at different levels.
19.I tell learners which questions to tackle
20.I find myself encouraging learners to work more quickly.
21.I only go through one method for doing each question
22.I find out which parts learners already understand and don’t teach those parts.
23.I teach each learner differently according to individual needs.
24.I only cover important ideas in a topic.
25.I tend to teach each topic separately.
26.I know exactly what maths the lesson will contain.
27.I encourage learners to make & discuss mistakes.
28.I jump between topics as the need arises

3. Who are you most like?

Read the following pen portraits of four teachers. These are based on genuine case studies. Underline the things which resonate most strongly with your own beliefs and experiences.
Which of these teachers are you most like? You may like to write your own pen portrait.

Ellen:

I think I'm a bit old-fashioned really. I tend to do a lot of teaching from the front. All the students that I have, like to be organised...They seem to sort of like to be told what to do. They think that if I'm not standing at the front doing something on the board that I'm not doing my job.

Ellen sees mathematics as a well-defined body of knowledge and procedures that needs to be covered. Her teaching involves structuring the work into a logical, linear sequence, then explaining and demonstrating results and procedures. Her students learn by listening to her explanations and then working individually using the methods and procedures she has demonstrated in her worked examples. Ellen is well organised and knows exactly what knowledge and methods will be learned before each lesson.

Ellen believes that the most effective way to 'cover' the syllabus is to 'deliver it' through clear explanation and repeated practice. She has carefully planned things so that nothing is left out. She starts each topic assuming that students know nothing and reteaches it from the beginning.

Ellen's classroom is organised into rows of tables, with students facing the front. She spends three quarters of the time giving explanations to the whole class, leaving the remaining time for individuals or pairs to work on practice exercises from a textbook. Her explanations are clearly delivered and include questions directed at individual students. She does not waste time following up errors and alternative answers. She feels under pressure from the syllabus - there is so much work to get through and so little time. Her lessons are taken at a brisk pace. Ellen encourages students to keep careful notes which will be comprehensible when they are re-read later on, nearer the examination. She encourages students to copy out examples from the board, or occasional worked examples from the textbook. She encourages students to do as many examples in each exercise as they can. They start with easy examples and gradually work up to harder examples.

During Ellen's explanations, students are quiet and attentive, though most are unforthcoming when asked questions. Ellen believes that most of the students who attend her class do not have the maturity to work independently and need the pressure and structure which she provides. She has a formal, but cheerful and positive relationship with students.

Chris:

I think the important thing is to try and get to know them as individuals and try to form some sort of personal relationship with them.. I'm not too hard on them. I try and understand their situation and maybe start them off again. So I work quite hard at that. It's nothing directly to do with the maths.

Chris sees mathematics is an interconnected network of ideas which he and his students must create and work on together. he claims that there are many links between topics and there is no 'best' way or order in which to teach the subject. He sees teaching as non-linear dialogue in which meanings and connections are explored and refined through genuine two-way discussion. This means that he often follows up ideas arising from the class and cannot always predict which direction the lesson will take beforehand. He sees learning as a collaborative activity based on discussion.
Chris allows students to work at their own pace. He is relaxed about syllabus coverage. He is willing to spend time discussing mistakes and errors when they arise, even if this takes considerable time. Chris teaches in a classroom containing desks arranged in a U shape. He considers this a flexible arrangement, allowing whole class discussion and also some small group work. He spends about a third of the time leading whole class activity; explaining work, giving examples and leading discussion. He allows the remaining two thirds for students to work individually, in pairs or in small groups while he goes round prompting and helping.

When using a textbook, Chris will suggest an exercise, but then allow students to select the questions they work on. He will define time limits for this. Chris occasionally offers students more open, exploratory tasks in which students can choose the direction they take and devise their own examples. Chris has a friendly, informal relationship with students. He appears relaxed and unhurried and is positive and encouraging in his manner.

Denise

Students have the idea that they should be doing lots and lots of problems and looking the answers up in the back of the book and ticking them.....Students are not happy to concentrate on whole class activity for more than a few minutes because they’re used to getting on their own

Denise, like Ellen, sees mathematics as a given body of knowledge and procedures which must be 'covered'. She sees teaching as structuring a series of exercises and problems that students work through individually while she goes round giving support. She gives much less emphasis to whole class demonstration and explanation than Ellen. Denise reasons that her students 'cannot concentrate for long' and so she keeps introductions short. Denise prefers to deal with students individually. Like Ellen, she sees learning as an activity based on listening and practising.

In Denise's classroom tables are arranged into 'blocks', but students work as individuals or in pairs. Lessons begin with Denise giving a short introduction and maybe one or two worked examples. A few questions are directed at students. This is followed by an extended period when students work on exercises from a textbook or worksheet. She stops the class from time to time to through questions she thinks are particularly difficult. Denise spends about a quarter of her time explaining work, giving examples and asking questions while the remainder is spent helping students as they work through exercises.

While students are working, Denise adopts a directive style. She spends only brief periods with individual students, telling them where they have gone wrong and showing them how to put errors right. If a student cannot do a question, she shows them how to do it. She does not spend much time with individuals as others in the class also need to be kept 'on task'.

Students do not have to complete every exercise from the beginning. They are encouraged to 'miss out' questions which they feel are too easy. Denise feels that many in her class are disinterested in maths and she has to work hard to 'keep on top of them'. She sometimes battles against the noise when trying to attract their attention so that she can explain something. Denise is a sympathetic and positive teacher, but feels that the odds are stacked against her.

Alan

There is a pressure to get through the worksheets... if students sense they are not getting their fair share, that is they are not doing all the sheets then resentment creeps in. The sheets are all at the back of the classroom so they can check

Alan's practices are not consistent with his own beliefs about learning. While he claims to share Chris' beliefs, he feels constrained and frustrated by the scheme of work. He believes that students learn best when their own ideas are discussed and investigated, but there is little evidence of this in his classroom.

Alan's students sit in rows and tend to work alone or with a partner. His lessons are dominated by a worksheet scheme.

Overall, Alan spends about a quarter of the time in trying to engage the class in whole class discussion and three quarters helping individuals as they work through the worksheet scheme.

Students seem quiet, passive and undemanding. When problems arise, Alan stops the class and works through specific questions on the board. He does this as interactively as he can, though students reply monosyllabically to most of his questions. Most of his interventions are unplanned and stimulated by a particular difficulty arising from a worksheet.

In spite of the security and apparent 'completeness' offered by the school scheme of work, Alan feels unhappy with it as it is not consistent with his own philosophy of teaching and learning.

3. What are your current views on Mathematics, Learning and Teaching?

Give each statement a %, so that the sum of the three % in each section is 100.
If you wish, you may add your own personal statements underneath.

Mathematics is..


  • a given body of knowledge and standard procedures. A set of universal truths and rules which need to be conveyed to students.

  • a creative subject in which the teacher should take a facilitating role, allowing students to create their own concepts and methods.

  • an interconnected body of ideas which the teacher and the student create together through discussion.

......

......



Learning is..

  • an individual activity based on watching, listening and imitating until fluency is attained.
  • an individual activity based on practical exploration and reflection.

  • an interpersonal activity in which students are challenged and arrive at understanding through discussion.

......

Teaching is..



  • structuring a linear curriculum for the students; giving verbal explanations and checking that these have been understood through practice questions; correcting misunderstandings when students fail to 'grasp' what is taught.
  • assessing when a student is ready to learn; providing a stimulating environment to facilitate exploration; and avoiding misunderstandings by the careful sequencing of experiences.

  • a non-linear dialogue between teacher and students in which meanings and connections are explored verbally. Misunderstandings are made explicit and worked on.

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