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Explain the meaning of and differences between problem-solving and investigation in a mathematics context and identify ways in which mathematical modelling and investigations can be applied to the teaching of KS 2, 3 and 4 mathematics. In your essay, you should:
Describe the characteristics of these approaches - what makes an activity an investigation or problem-solving?
Discuss the rationale and possible benefits for teaching and learning of the use of problem solving and investigative approaches.
Discuss the specific benefits and drawbacks of these approaches to inclusive education.
Explore ways that students' work in investigations and problem-solving could be assessed.
Provide one investigation or problem for each of KS2, 3 and 4 and indicate why each is suitable for the Key Stage. You should include at least one investigation and one problem, and clearly identify which of your examples are of each type.
List some organisational considerations for the classroom of your investigations/problems and some possible solutions.
Refer to published material on investigation and problem-solving.
Orton (1996, p25) defines problem solving as a "process in which the learner combines previously learned elements of knowledge, rules, techniques, skill and concepts to provide the definite solution to a situation not encountered before." Whilst Askew and Wiliam (1995) state that success in problem solving requires "both specific content knowledge and general skills." In short, problem solving is a closed and focused work based problem and requires the use of previous knowledge and routine to give the solution to a specific problem or challenge.
Johnson and Rising (1967) believe that the most significant learning in every maths class is the learning to solve problems. They give the following reasons:
"Process whereby we learn new concepts
Problems may be a meaningful way to practice computations
By soling problems we learn to transfer concepts and skills to new situations
Problem solving is a means of stimulating intellectual curiosity
New knowledge is discovered through problem solving"
After successfully solving a problem, a learner has learned something new and potentially vital for their education such as the understanding of a subject in order to solve a number of similar problems. It is for exactly the same reasons that Gagne (1985) shares Johnson and Rising's opinion that problem solving is the 'highest form of learning'
The problem-solving process can be described as a journey from meeting a problem for the first time to finding a solution, communicating it and evaluating the route. According to Orton (1996) there are three distinct categories of problem within this topic, these are called Routine, Environmental and Process problems, all with a similar structure and will be outlined below.
Within the first category, routine problems, the student is required draw on their past experience and previous knowledge to understand the problem. These particular types of problem are designed to allow the learner to practice and develop number facts and mathematics techniques which they have recently been taught. Polya (1962) and Wickelgren (1974) are both advocators of the notion that it is possible to train ourselves to become competent problem solvers through the practice of other such problems, however, Gagne (1985) argues that giving learners this type of problem does not make better problem solvers per se, although agrees that they would become proficient at those types of 'problem'.
Environmental problems add more context to problem solving mathematics for the learners, generally being set in 'real life' situations. Because of the added context, environmental problems used correctly can be a very important and powerful tool used in mathematics education as it requires students to use informal as well as formal knowledge to get to their solution. Orton (1996, p 28) states that "the interaction between these two aspects of learning provides the most favourable conditions for the development of a student's mathematics. Frequently mathematics arises which is new to the student. 'Old' knowledge is consequently reorganised and restructured into an expanded and aggregated body of 'new' knowledge." This is to say that pupils working from this type of problem have the opportunity to develop links between mathematics and their life broadening their understanding of that particular topic of mathematics being taught. However, there are some problems with this type of problem for both learners and teachers alike. For example, the questions would attempt to draw on the learner's own real life experience something which may not be shared with their peers so a teacher must find topics and environments that relate to all learners in their class. Johnson and Rising (1967, p109) support this, stating "for most students it is not necessary to take problems from the immediate environment. Often, students are less interested in grocery bills than in cannibals and missionaries, less interested in volumes of oil tanks than in walks through Koenigsburg." Another problem faced by teachers when using such work is that once problems are written down in a text book or worksheet it then becomes just a word problem in the eyes of the learner subsequently losing its sense of reality (Orton, 1996).
Finally, Process problems are more centred on the mathematics rather than context of the problem. This particular problem is similar to Routine Problems in that it requires learners to reflect upon the processes and strategies they have used to solve other unrelated problems in order to successfully negotiate through the method of solving the problem.
It is evident that these types of problems are similar in terms of what is required of the learner to be able to 'problem solve'. According to nRich website the process of problem solving can be split into five steps:
Comprehension: this stage is about making sense of the problem by using strategies such as retelling, identifying relevant information and creating mental images. This can be helped by encouraging students to re-read the problem several times and record in some way what they understand the problem to be about (for example by drawing a picture or making notes).
Representation: this stage is about "homing in" on what the problem is actually asking solvers to investigate.
Can they represent the situation mathematically?
What is it that they are trying to find?
What do they think the answer might be (conjecturing and hypothesising)?
What might they need to find out before they can get started?
Central to this stage is identifying what is unknown and what needs finding.
Planning and Analysis: Having understood what the problem is about and established what needs finding, this stage is about planning a pathway to the solution. It is within this process that you might encourage pupils to think about whether they have seen something similar before and what strategies they adopted then. This will help them to identify appropriate methods and tools. Particular knowledge and skills gaps that need addressing may become evident at this stage.
Execution: During the execution phase, pupils might identify further related problems they wish to investigate. They will need to consider how they will keep track of what they have done and how they will communicate their findings. This will lead on to interpreting results and drawing conclusions.
Evaluation: Pupils can learn as much from reflecting on and evaluating what they have done as they can from the process of solving the problem itself. During this phase pupils should be expected to reflect on the effectiveness of their approach as well as other people's approaches, justify their conclusions and assess their own learning. Evaluation may also lead to thinking about other questions that could now be investigated. (http://nrich.maths.org/5569)
Garofola and Lester (1985) have suggested that learners are largely unaware of the processes discussed above and therefore it is the teacher's responsibility to address this issue. However, Polya (1965) states that care must be taken so that efforts to teach students "how to think" in mathematics do not get transformed into teaching "what to think" or "what to do", as this will inhibit the pupils thinking processes and lock in some potentially untapped potential.
Assessment in problem solving can be tricky if the pupils are not aware of a mark scheme or if they aren't aware of the information they are expected to include.
A way to do this is to either include a mark scheme for a set of example questions so that the pupils know how they are supposed to answer, lay out their answer, and what they are required to show.
If the children are not asked to follow any order as to how their work is presented then it instantly becomes twice as much work for the teacher to mark.
The OTRN website defines an investigation as "a situation originating in mathematics or the real world which lends itself to inquiry". Inquiry in this case would be the presence of a practical element or a physical activity which the pupil must actively takes part in in order to answer a question. Without the learner's taking part in the investigation it would be impossible for them to find the answer. Orton (1996) expands on OTRN's definition and adds that if an activity does not have a specific goal then that would be an investigation and gives the example of a topic such as 'exploring square numbers'. The fact that no clear goals are given makes this an 'open investigation'.
Because the nature of investigations is to give learners the opportunity to get 'hands on' with the task, they are given the freedom to determine their own goals within the task. This, however, could lead to problems for the learners as Orton (1996, p32) states that whilst some of the paths they take can "lead to exciting and novel revelations...others quickly lead nowhere with few conclusions reached." This would mean that learners have to understand and come to terms with the frustrations and disappointments as well as the satisfaction and feelings of accomplishment when they explore new mathematics. It is, therefore, the teacher's responsibility to manage the learners' expectations with regards to these investigations. With these types of questions being 'slow burners' the rewards for correct answers do not come as quickly to the learner as they do from the shorter, more directed and focussed problem solving.
As well as this the pupils also have the opportunity to apply their own hypothesis to an investigation, whereas in problem solving there is no hypothesis to prove or disprove, only answers to find, through a choice of method.
It allows pupils to study the situation at hand and figure out the best techniques to use in doing so.
During investigation work assessment becomes slightly trickier as pupils are expected to show their OWN methods towards finding the answers, rather than following a set number of steps or instructions. This is why it could be a good idea to instruct the pupils to create a presentation of their results (in groups?) to the rest of the class. Then the presentation can be marked and notes made, followed by the teacher simply having to check that this work was actually done and not copied from other places.
Specific Assessment Criteria
70 - 79
60 - 69
50 - 59
40 - 49
30 - 39
0 - 29
Identification and discussion of the processes of investigation and problem solving
Processes are accurately and succinctly identified and differentiated with references to a wide variety of academic sources.
Processes are accurately and succinctly identified and differentiated with references to a variety of academic sources.
Processes are accurately identified and differentiated with references to academic sources.
Processes are accurately identified and differentiated with some references to academic sources.
Processes are identified and differentiated with limited references to academic sources.
Processes are defined but not well identified or differentiated. Few if any relevant sources are cited.
Processes are simply defined with cursory discussion and no reference to academic sources.
Discussion of the benefits for teaching and learning
A well argued case is made for the inclusion of these approaches in the classroom, using a variety of examples and citing the use of the approaches in real classrooms.
A well argued case is made for the inclusion of these approaches in the classroom, using a variety of examples and practical ideas.
A well argued case is made for the inclusion of these approaches in the classroom, using a variety of examples.
A good case is made for the inclusion of these approaches in the classroom, using some examples.
A case is made for the inclusion of these approaches in the classroom, using a few examples.
Some arguments are put forward for the benefits of the approaches, but they lack coherence and fail to make good use of examples.
A few benefits are listed, but arguments are unclear and examples absent or irrelevant.
Discussion of the benefits and drawbacks for inclusive education
A balanced and well sourced discussion of the benefits and drawbacks of the approaches, with consideration given to all the major aspects of inclusive education.
A balanced and well sourced discussion of the benefits and drawbacks of the approaches, with consideration given to many aspects of inclusive education.
A balanced and well sourced discussion of the benefits and drawbacks of the approaches, with consideration given to several aspects of inclusive education.
A balanced discussion of the benefits and drawbacks of the approaches, with consideration given to several aspects of inclusive education.
A generally balanced discussion of the benefits and drawbacks of the approaches, with consideration given to a few aspects of inclusive education.
Benefits and drawbacks to inclusive education are discussed, but there is a lack of balance and only one or two aspects of inclusive education are considered.
Discussion is cursory, unbalanced and focuses almost exclusively on one aspect of inclusive education.
Discussion of methods of assessing open ended tasks
A variety of practical and useful methods for assessment are clearly delineated and exemplified.
Several practical and useful methods for assessment are clearly delineated and exemplified.
Some useful methods for assessment are clearly delineated and exemplified.
A few useful methods for assessment are delineated and exemplified.
One or two methods of assessment are outlined with some examples.
Consideration is given to assessment, but it is of little practical value and lacks clear examples.
Some cursory and ill-formed ideas about assessment are put forward. Examples are irrelevant or absent.
Quality and suitability of the examples given (identification as investigation or problem, suitability for Key Stage)
Examples are clear, well identified and each is suitable for its Key Stage. They are usefully referenced throughout the essay and serve to illustrate the points made.
Examples are clear, well identified and each is suitable for its Key Stage. They are usefully referenced throughout the essay.
Examples are clear, well identified and each is broadly suitable for its Key Stage. They are referenced throughout the essay.
Examples are clear, generally well identified and each is broadly suitable for its Key Stage. They are referenced in places in the essay.
Examples are clear, generally well identified and each is broadly suitable for its Key Stage. They are barely referenced in the essay.
Examples are misidentified or unsuitable for their Key Stages. They are not referred to elsewhere in the essay.
Examples are of poor quality, misidentified and unsuitable for their Key Stages. They are not referred to elsewhere in the essay.
Discussion of organisational issues
Several clear and practical ideas are outlined for the use of these approaches in schools.
Clear and practical ideas are outlined for the use of these approaches in schools.
Some good ideas are outlined for the use of these approaches in schools.
Potential problems and some solutions are clearly outlined.
A few potential problems are outlined, some with ideas for solutions.
Problems are outlined, but potential solutions are impractical.
Major potential problems are not discussed, with focus put on minor matters.