Examples can illustrate standard techniques, or standard ways of presenting an argument. The tasks may include comparisons, or an invitation for the student to come up with their own examples, as a way of getting the feel for the scope of an idea or technique. These may be followed by problems or questions for students to think about that may aim to deepen understanding or highlight links to other areas.
These problems are designed to give students an opportunity to notice ideas or concepts for themselves by building their knowledge through a particular problem. Either the situation, or the questions asked, will be carefully focused in order to guide the students in their learning.
An opportunity to rehearse and use well defined procedures and notations with the goal of using these without effortful thought. These might be a single problem or a collection of problems that have other aspects to them, such as classifying statements, but where the underlying aim is for students to practise and develop fluency of a technique.
Package of problems
This is a set of problems that have been designed and that should be thought of as a single entity. It might be that the problems build on each other, so that by the end they are requiring more sophistication but in a way that is accessible to students who have worked through the whole set. Or it might be that by working on all of the problems, the student is naturally prompted to explore some underlying structure or to make a generalisation.
Many ways problem
Students sometimes seize the first idea that comes to mind, when it might not necessarily be the most effective. These problems draw students’ attention to the thought that there might be several ways to tackle a problem or to represent an idea. For example, they may encourage students to switch between graphical and algebraic representations, or they might illustrate different approaches that can be taken to solve a problem, and offer opportunities to compare them.
A task that gives students an opportunity to engage with material that may otherwise be out of their grasp. It will offer structure that should allow students to make progress on more challenging or unfamiliar tasks. For example, smaller subtasks could draw on student knowledge to take them step-by-step through something more complex, or it could be a ‘proof sorter’ activity that enables students to understand the components of a proof and their ordering.
Problem requiring decisions
Students are often used to problems being posed in such a way that they have all the information that they require in order to start, and no more. Problems (especially from the real world) are very often not like this, and so resources of this type will give students the opportunity to develop the skills needed to deal with this. Some problems might not contain enough information, so students may need to decide on classifications, make assumptions or approximations, or do some research in order to move forward. Some problems might contain too much data, so that part of the challenge is to identify the useful information.
Food for thought
When students are familiar with concepts and ideas they often benefit from exploring them further to improve their understanding. These problems aim to allow this further exploration, and for example, might bring different techniques together, highlight interesting or unusual cases, or probe the definition of mathematical terms.
A more open task, which might start with an interesting context, or some initial questions to explore. Students will have to decide how to investigate the ideas or may be asked to pose and explore their own questions.
Go and think about it...
A problem for which students (probably) have the required mathematical knowledge, but where the challenge is identifying how to get started, what tools might help, and how to apply the relevant mathematical knowledge. These problems may not contain solutions and therefore might be something students tackle for a challenge, perhaps at home, as they are often more suited for individual consideration than classroom collaboration.
A resource that puts the mathematics into context, perhaps by offering a historical perspective, describing the mathematicians who worked on it, linking it to areas of current research, or illustrating how it leads on to further topics (e.g. at university level). They may be articles or videos, and they will often draw on many different areas of mathematics, including those that might be unfamiliar to the student. Many of them will contain questions for the students to consider whilst reading to help them further engage with the topic being presented.
These are questions designed to test students’ understanding of one or more topics and to exercise their problem-solving skills. In many cases they can also be used as a classroom resource to help teach concepts and methods. They are mostly drawn from past examination questions and have been chosen as ones that are interesting in nature and require non-routine thinking. The hints and solutions are designed to explain the reasoning and highlight connections as well as giving the answer. In many cases, alternative methods or solutions are presented.
More detailed notes on using Review questions in the classroom can be found here.
Examination questions have been reproduced verbatim wherever possible though page layout and part numbering may vary from the original. Some older questions involving imperial units have been modified to use appropriate metric ones instead. Where we have added explanatory notes, these appear in italics in square brackets.