Do Hamilton provide answers to investigations?
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:
- give children feedback,
- inform your future planning for the class, and
- report to your maths coordinator, headteacher, etc.
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.
First – remember that investigations are wonderful!
It is wonderful that you are doing investigations with your class. It’s fantastic that there are teachers taking on the challenge of developing mathematical thinking and deep conceptual understanding in their children. Investigations have massive benefits for children’s fluency with and enjoyment of numbers and mathematics.
But, we know it’s not easy.
2. Assess the process not the result
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 have demonstrated investigative skills, such as:
- mathematical thinking,
- creativity, and
- application of skills.
For some worked examples of how to look for these qualities, see Hamilton’s annotated children’s investigation examples. 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.
3. Focus on an aspect of the investigation, not the total result
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 which could help you here, is to give 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 - tell them you will be looking particularly at their recording and organisation. Further down the line - you could spotlight the ways they spot and write about patterns. Towards the end of the term - you might look at their explanation, reasoning and generalisation. This gives you a concrete skill to assess when looking at their books.
4. Three opportunities to assess investigations:
A very useful framework for helping children self-assess is to ask them to:
- convince themselves, then
- convince a friend, then
- convince you (the teacher) or ‘the Headteacher’ or ‘the World’.
Tell the children that once they think they have ‘solved’ the problem, they need to work out how to convince these different audiences that they have a strong answer. They can do this:
- with you one-to-one while circulating around the classroom as the children work
- in small groups working with other children (and yourself or another adult if one is available)
- in a plenary session
After the children have had time to work on their investigation, bring them back together for a plenary session. This is fundamental to consolidating their learning. You can choose a child who you know has 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 then 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.
5. Written work
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.
6. More help on offer
Hamilton have put together a number of different pieces of advice about investigations:
- Listen to Professor Ruth Merttens talk about the value of doing investigations with children. They are not the icing on the cake, investigations are the heart of maths.
- Read Ruth’s Guidance sheet about how to run whole-class investigations.
- Read Nick Barwick’s Guidance about how to get young children off to a strong start with problem solving in Early Years.
- View Nick’s short video describing a Year 2 class’s investigation of ‘Mrs Multiple’s Cakes’.
- View Nick’s video talking through the impact of an investigative approach on a Year 5 class.
7. PowerPoint presentations
Hamilton have prepared a number of sample investigation presentations. You can use them to teach the sample investigations or as a model to help you find ways to structure your investigation lessons.
- Excellent Eights (in Y5 Multiplication - open Unit 3)
- Stars and Crosses (in Y6 Autumn Algebra - open Unit 1)
- Cycling Coordinates (in Y6 Autumn Shape - open Unit 2)
8. Challenges with answers
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.
Problem solving and reasoning investigations are now implemented within our new maths blocks: