This is a question we get asked quite often, and it’s a tricky one.
To start, let’s reframe the question. For us, the key is not about having answers or even model answers for investigations; rather, it is about how you as a teacher can best assess children’s mathematical investigative learning in order to:
What follows are some ideas and resources that will help you do investigations in such a way that children gain as much benefit as possible while allowing you to assess the outcomes for these purposes.
First – remember that investigations are wonderful!
It is wonderful that you are doing investigations with your class, taking on the challenge of developing mathematical thinking and deep conceptual understanding. Investigations have massive benefits for children’s fluency with and enjoyment of mathematics.
But, we know it’s not easy.
With investigations, it’s not the final ‘answer’ that children produce that is the most useful object of assessment. The process that the children went through in getting to that ‘answer’ is far more important. Assess children as they work through the investigation, as well as when looking at their results, according to how much they are demonstrating investigative skills, such as:
For some worked examples of how to look for these qualities, see Hamilton’s annotated children’s investigation examples (links below). You will see how quick notes on post-its can help you capture the nature of the problem-solving process that your children went through in doing their investigation. If you record observations as you circulate the classroom - working with and questioning children - you will have done lots of assessment before you open a book.
Assessing investigations is not like assessing a sheet of sums. There isn’t a fixed end point. Consider for a moment, how you assess extended writing. You do not assess the final value of the piece of writing as a finished piece of writing, but how well the child has met particular writing objectives.
One of the things that you may do when you assess writing that could help you here, is giving the children, and yourself, a particular focus. Perhaps, early in the term - tell them that you will be looking particularly at how well they get started on the problem. A bit further along in the year - tell them you will be looking at their recording and organisation. Further down the line - you could highlight the ways they spot and write about patterns. Towards the end of the year- you might look at their explanation, reasoning and generalisation. This gives you concrete skills to assess when looking at their books.
You can assess children's processes in a variety of contexts:
You can give children a framework to self-assess their 'solutions' to a problem as they progress. Instruct them to work out how they would
This gives you a focussed way to ask for their contributions one-to-one, in small groups or in a plenary session.
After the children have had time to work on their investigation, you should bring them back together for a plenary session. This is fundamental to consolidating their learning. You can choose a child that you know found an interesting solution, or ask for volunteers, and ask them to explain their ‘solution’ to the class. Another child can then ask questions about it. You can ask further probing questions, like ‘What happens if I choose zero?’, or ‘Does it still work if I begin at a different starting point?’, or whatever suits the particular investigation. This will allow you to assess both the child answering and their peers. By drawing their reasoning out orally, you will help the child conceptualise her/his work and also help other children see how it could be done. Then you can ask for different solutions or approaches.
You don’t always need to do a full set of written feedback on investigations. Sometimes, the plenary session will be enough, or some general feedback to the class after you have looked at the books. Or, after you look at the books, ask several children who took different approaches to explain what they did in the next session. But sometimes you will want to write more full feedback, focusing on the particular skill you highlighted in advance (see 3 above). If you have spent a whole maths session on an investigation, then use your marking time to feedback in more detail.
Hamilton have prepared a number of sample investigation slide presentations. You can use them to teach the sample investigations, or as a model to help you find ways to structure your investigation lessons.
For some units, the In-depth investigation provided is a link to NRICH activities. The 'Teacher Resources' tab on the left side of the activity on the NRICH website often includes analysis that can help with the marking of those activities.
Download the key stage specific sets of maths ‘challenges’ written by the DfE many moons ago. These are not full-on problem-solving investigations, but they do share some of their qualities. They encourage children to see numbers in real contexts, and to explore various solutions to problems. There is an element of play and number exploration that will help children solve more open-ended investigations. However, they are not fully open-ended, and therefore they do come with answers.
You can find these challenges at the bottom of our problem-solving investigations page.