Explain this
Thinkers
How the publisher describes it:
“‘Thinkers’ is a collection of activities to provoke mathematical thinking. The ideas are suitable for learners from KS2 to AS/A2 level. The book contains sixteen different contexts for exemplifying the general and generalising the particular which are processes at the heart of ‘doing mathematics’. There are examples of the way in which teachers can use the techniques for any topic in mathematics.”
Review by Alison Parish
In brief:
This book certainly does provide thought provoking ideas for questions, even for those who already use different questioning styles in the classroom. I particularly like the way the beginning of the booklet poses questions to the reader and encourages them to think. It certainly made me think about how I was questioning my pupils and how I could improve.
“Recommended reading and implementing!”
Thinkers How does one get pupils to think! Many pupils seem to look but not see. To some thinking appears to be a painful experience. Teaching secondary pupils one often becomes frustrated at the expectation of some pupils that teachers are there to do the thinking, and to tell them what to do in simple steps. How does one get all pupils to apply and build on the knowledge they already have to new problems, situations or ideas? Perhaps within these pages there are some clues!
The book begins by discussing whether there is a need to teach generalisation. They suggest that by asking learners to generate examples for themselves they will ‘provide the foundations for recognition, articulation and appreciation of generality’ and to look at variation. The book suggests sixteen types of question, tasks or prompts to develop the learner’s ability to create and question in mathematics by generalising from examples.
The book then works through these sixteen types of activities, with a longer example of how they might be applied to shape and space. It shows how they can be used with pupils from KS2 to AS/A2 level, activities being made easier or harder according to the desired response.
Activities themselves are not limited to one curriculum area, e.g. number, but are illustrated so as to show how they may encompass all aspects. They do not need to be worked in any order so can be ‘re-used’ at any time. Illustration is also made of how a small change in wording can produce quite different responses from students, do we as teachers think about how we word our questions or about how a slight variation in the presentation of a task might affect the outcome?
I liked the idea of introducing a ‘really peculiar example’ before asking for a general example, or asking for easy to hard examples. This encourages pupils to think outside the ‘safe’ answer and to start thinking constructively about the task. Too many of them are unprepared for the ‘impossible’ task such as those illustrated on page 17 (e.g. a triangle with sides 3, 4, and 8), but it is problems like these that really promote reasoning about what is and what is not possible. This ties in with ‘truth’ statements, are there constraints which should be put with the generalisation? There are also examples of some of the more complex questions (page 27) and, by asking students to construct their own examples, help them to unravel the techniques required for a satisfactory solution. Many students do have problems with familiar questions dressed-up to appear to be more complex than they really are.
Now more emphasis is being placed on the development of proof the section on using ‘thinkers’ in shape and space (page 28) is particularly useful. Through carrying out the activities the pupils acquire a sense of ownership and through their discussion reinforce the concepts they come across. In textbooks the instructions often result in a routine task of several similar examples, but the difficulty come when transfer of knowledge is required to solve a problem in a slightly different context and this is not recognised by the student. By looking further than the straight exercise and having to think about the consequences, a truer understanding will evolve and, hopefully, greater transfer of knowledge. Development of thinking skills, questioning, generalising will help pupils to raise their level of achievement, skills that are transferable and useful in situations other than just the Mathematics classroom! These activities will certainly help towards this.
The booklet does not leave the reader ‘high and dry’ as to what to do with the results of the activities. Various suggestions are made as to how to move forward. With the tasks being suitable for individual, group or whole class work they can be used at any point within lessons and not just as a starter or plenary. Classroom management can therefore be up to the teacher and take into account their own confidence at dealing with the possibility of a variety of outcomes.
This book certainly does provide thought provoking ideas for questions, even for those who already use different questioning styles in the classroom. I particularly like the way the beginning of the booklet poses questions to the reader and encourages them to think. It certainly made me think about how I was questioning my pupils and how I could improve. It lives up to its title. The school I trialled these had a ‘textbook’ culture, so pupils were not used to thinking. Indeed one pupil indignantly told me that it was my job to tell her what to do!
Recommended reading and implementing!
