Can goal-free maths problems help achieve exam success?

The best example of this is probably the sliding tile puzzle where the goal is to slide the tiles around the space to create a picture or to get the numbers in the correct order. Whilst many can solve the puzzle, most won’t be able to recall the steps they took in order to solve the puzzle and the act of solving one puzzle won’t necessarily help in solving future sliding puzzles.

What makes a good goal-free problem?

Goal-free problems work best when there’s a relatively limited amount of information that students can generate from the given information. Not so open-ended that students can continue ad-infinitum, but able to develop and adapt information. The problem prompts the practice of applying specific knowledge and skills when approaching similar goal-specific problems in the future. Often goal-free problems are adapted from goal-specific problems simply by removing the goal.

AQA’s new goal-free problem papers

Recognising the importance of this method of teaching, AQA asked me to help develop a resource to support your classroom teaching.

The goal-free problem papers were developed by identifying which specific goal questions would work well as a goal-free question, with each paper matching a topic in the curriculum.”

There’ll be 10 papers in all, one foundation and one higher tier for each topic area, including algebra, geometry and measures, number, ratio, proportion and rates of change, probability and statistics. Each resource will have teacher notes as a separate document that explains the purpose of each question and how students might explore them.

Using goal-free papers in your lessons

Students can work on the problem either individually or in small groups, and give feedback to the wider class. You can then look at the goal-specific version of the problem and examine whether the problem has already been solved, or if the necessary information has been generated to be able to solve it. Then importantly what you did in order to generate that information. If the problem hasn’t been solved completely, then you can examine how many marks have been produced and what is left to do to achieve full marks.

An alternative approach is based on the hit game show 'Pointless™'. Typically in small groups, students will work on the same problem. Then challenge them to find a 'Pointless' piece of (mathematically relevant) information from the problem. Groups then take turns to feedback something they have generated from the problem and score points based on how many other groups have generated the same information. The goal is to score as few points as possible. This sort of activity works well for graphical or diagrammatic problems, such as statistical representations or angle diagrams.

Another possibility is to ask groups to suggest what the goal-specific problem might have been prior to its adaptation to a goal-free problem. Potential questions can then be ranked in order of difficulty and attempted by different groups. We can then share either the goal-specific problem and compare it to the problems students have generated, or another similar goal-free problem for students to then approach in a similar way.

When to use goal-free papers?

Students should already be in the position where they’ve been taught the necessary mathematical knowledge to solve the original goal-specific problem. This will generally mean that the use of goal-free problems will occur after a period of instruction or after other activities that pupils have used to make sense of a mathematical idea.

They’re most commonly used in two different phases, either towards the end of a unit or as part of general revision prior to summative assessments. Typically, if a problem has the capacity to prompt students to bring different areas of maths together, it is marked as suitable for general revision. Whereas if a problem is likely to be focussed on a specific area of maths, then it’s marked as suitable for that area. Of course, every maths curriculum is different and so teachers will have to make their own decisions as to when to use each problem based on the mathematical knowledge they want students to apply.

Access goal-free papers on All About Maths

You can find the new goal-free papers on All About Maths. If you’ve not already registered, please do so to access these and all the teaching resources available to support you in preparing your students for their exams. I hope you find them useful.