A strategy is different from an algorithm, which is a well-established sequence of predetermined steps that are executed in a particular order to carry out a commonly-required procedure. The evidence suggests that teachers should consider the following when developing these skills. 17 Select genuine problem-solving tasks that pupils do not have well-rehearsed, ready-made methods to solve. Sometimes problem-solving is taken to mean routine questions set in context, or word problems, designed to illustrate the use of a specific method. But if students are only required to carry out a given procedure or algorithm to arrive at the solution, it is not really problem solving; rather, it is just practising the procedure. Consider organising teaching so that problems with similar paperless structures and different contexts are presented together, and, likewise, that problems with the same context but different structures are presented together. Pupils need to experience identifying similar mathematics that underlies different situations, and also to identify and interrogate multiple relationships between variables in one situation. Teach pupils to use and compare different approaches. There are often multiple ways to approach a problem.
However, while using multiple representations can aid understanding, teachers should be aware that using too many representations at one time may cause confusion and hinder learning. 16 Problem solving generally refers to situations in which dream pupils do not have a readily-available method that they can use. Instead, they have to approach the problem flexibly and work out a solution for themselves. To succeed in this, pupils need to draw on a variety of problem-solving strategies which enable them to make sense of unfamiliar situations and tackle them intelligently. What is a problem-solving strategy? A problem-solving strategy is a general approach to solving a problem. The same general strategy can be applied to solving a variety of different problems. For example, a useful problem-solving strategy is to identify a simpler but related problem. Discussing the solution to the simpler problem can give insight into how the original, harder problem may be tackled and the underlying mathematical structure.
The evidence indicates that number lines are a particularly effective representation for teaching across both key stages 2 and 3, and that there is strong evidence to support the use of diagrams as a problem-solving strategy. The specific evidence regarding the use of representations more generally is weaker, however, it is likely that the points above regarding effective use of manipulatives apply to all other representations. Using a number line The teacher noticed that some pupils were incorrectly adding fractions by adding the numerators and the denominators. She gave the class this task: text Is this true? frac12 frac18 frac210 Some pupils noticed that: frac210 text is less than frac12 With the teachers help, the pupils represented the three fractions using a number line. This helped pupils to see that: frac12 text is equivalent to frac48 And then to work out that the answer is: frac58 The pupils then invented their own examples of incorrect and correct fraction additions using number lines to make sense. While in general the use of multiple representations appears to have a positive impact on attainment, more research is needed to inform teachers choices about which, and how many, representations to use when. 15 There is promising evidence that comparison and discussion of different representations can help pupils develop conceptual understanding. Teachers should purposefully select different representations of key mathematical ideas to discuss and compare with the aim of supporting pupils to develop more abstract, diagrammatic representations.
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The teacher said, Im going to use multilink cubes to see whether this will help us see why we always get a multiple. The teacher made want four sticks of 10 cubes each. So here. What does it look like if we remove 4 cubes? A pupil came to the front of the classroom and removed 4 cubes.
How else could you do it? Another pupil removed 4 cubes in a different way. A pupil said, Ah yes! If we take away one from each 10 then we are left with four. Another pupil said, And if we started with 70 uses wed have 10 sevens take away 1 seven is 9 sevens. The teacher wrote: and the pupils discussed what is going on here, before the teacher concluded with the generalisation: 10t t (10 1)t 9t What about other types of representation?
Manipulatives can be used across both key stages. The evidence suggests some key considerations: Ensure that there is a clear rationale for using a particular manipulative or representation to teach a specific mathematical concept. Manipulatives should be used to provide insights into increasingly sophisticated mathematics. Enable pupils to understand the links between the manipulatives and the mathematical ideas they represent. 12 This requires teachers to encourage pupils to link the materials (and the actions performed on or with them) to the mathematics of the situation, to appreciate the limitations of concrete materials, and to develop related mathematical images, representations, and symbols. 13 Try to avoid pupils becoming reliant on manipulatives to do a type of task or question.
A manipulative should enable a pupil to understand mathematics by illuminating the underlying general relationships, not just getting them to the right answer to a specific problem. 14 Manipulatives should act as a scaffold, which can be removed once independence is achieved. Before using a manipulative, it is important to consider how it can enable pupils to eventually do the maths without. When moving away from manipulatives, pupils may find it helpful to draw diagrams or imagine using the manipulatives. Manipulatives can be used to support pupils of all ages. The decision to remove a manipulative should be made in response to the pupils improved knowledge and understanding, not their age. Using manipulatives an example a teacher said, give me a two digit number ending. A pupil said, forty. The teacher said, Im going to subtract the tens digit from the number: 40-4 text gives me 36 The pupils tried this with other two-digit numbers ending in 0 and discovered that the result was always a multiple.
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They need to gpa be used purposefully and appropriately in order review to have an impact. 10, teachers should ensure that there is a clear rationale for using a particular manipulative or representation to teach a specific mathematical concept. The aim is to use manipulatives and representations to reveal mathematical structures and enable pupils to understand and use mathematics independently. What are manipulatives and representations? A manipulative is a physical object that pupils or teachers can touch and move which is used to support the teaching and learning of mathematics. Common manipulatives include cuisenaire rods and dienes blocks. The term representation refers to a particular form in which mathematics is presented. 11, examples of different representations include: two fractions represented on a number line; a quadratic function expressed algebraically or presented visually as a graph; and a probability distribution presented in a table or represented as a histogram. What does effective use of manipulatives look like?
In this situation, teachers could think about how a misconception might have arisen and explore with pupils the partial truth that it is built on and the circumstances where it no longer applies. 8, counter-examples can be effective in challenging pupils belief in a misconception. However, pupils may need time and teacher support to develop richer and more robust conceptions. Knowledge of the common errors and misconceptions in mathematics can be invaluable when designing and responding to assessment, as well as for predicting the difficulties learners are likely to encounter in advance. 9, life teachers with knowledge of the common misconceptions can plan lessons to address potential misconceptions before they arise, for example, by comparing examples to non-examples when teaching new concepts. A non-example is something that is not an example of the concept. Manipulatives and representations can be powerful tools for supporting pupils to engage with mathematical ideas. However, manipulatives and representations are just tools: how they are used is important.
marking. A summary of the evidence regarding different marking approaches and their impact on workload can be found here. Addressing misconceptions, a misconception is an understanding that leads to a systematic pattern of errors. Often misconceptions are formed when knowledge has been applied outside of the context in which it is useful. For example, the multiplication makes bigger, division makes smaller conception applies to positive, whole numbers greater than. However, when subsequent mathematical concepts appear (for example, numbers less than or equal to 1 this conception, extended beyond its useful context, becomes a mis conception. 7, it is important that misconceptions are uncovered and addressed rather than side-stepped or ignored. Pupils will often defend their misconceptions, especially if they are based on sound, albeit limited, ideas.
This information allows teachers to adapt their teaching so it builds on pupils existing knowledge, addresses their weaknesses, and focuses on the next steps that they need in order to make progress. Formal tests can be useful here, although assessment can also be based on evidence from low-stakes class assessments, informal observation of pupils, or discussions with them about mathematics. More guidance on how to conduct useful and accurate assessment is available in the eefs. Assessing and Monitoring Pupil Progress. Responding to assessment, teachers knowledge of pupils strengths and weaknesses should inform the planning of future lessons and the focus of targeted support (see recommendation 7). Teachers may also need to try a different approach if dream it appears that what they tried the first time did not work. Effective feedback will be an important element of teachers responses to assessment information. Consider the following characteristics of effective feedback: 4 be specific, accurate, and clear (for example, you are now factorising numbers efficiently, by taking out larger factors earlier on, rather than, your factorising is getting better give feedback sparingly so that it is meaningful (for example. Feedback needs to be efficient.
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Log in, home page, mickleton Primary and Nursery School, Broadmarston Lane, mickleton, Chipping Campden, Gloucestershire, gl55 6SJ. On the first page of this file there is a link to a bbc clip introducing Palm Sunday through animation. There are some key words on which you may need to discuss with the chn to check their understanding. Then there is some information and images about how Palm Sunday is celebrated. Hope it is useful! Read more, recommended Categories. Mathematical knowledge and understanding can be thought of as consisting of several components and it is quite possible for pupils to have strengths in one component and weaknesses in another. 3, it is therefore important that assessment homework is not just used to track pupils learning but also provides teachers with up-to-date and accurate information about the specifics of what pupils do and do not know.