Background – Why?
I’ve been using goal-free problems for the last 16 months in the classroom with some extremely encouraging results in pupil success and confidence when interpreting tables, charts and sentences. The approach encourages flexible thinking (Hasanah, R et al 2017)1, exploring different contexts for problems, improved recognition of similar problems and as a consequence, the language associated with it.
In the last 9 months, I’ve led a NCETM innovation with 8 fantastic London schools and their teachers. The project looked at mathematical vocabulary and deciphering word problems. Goal free being one of the many methods explored but all with some encouraging results in a short space of time in both KS1 and KS2.
The accompanying resources to this blog have been created out of need! It will now be my go-to when planning (rather than an internet search with the snipping tool). It’s been a strange journey, revisiting past papers with a purpose, going back to 2001 to present day. It seems for the first nine years there were an abundance of questions on soup, bird watching and cheese!
Sweller (1998)2 championed goal free problems; problems lose their specificity and they become non-specific, opened-up and flexible. Supporting knowledge acquisition and cognitive load. Craig Barton talks about this in more detail in his book ‘How I wish I’d taught Maths’ (2018)3 a must read for Primary and Secondary teachers alike.
Without a goal (often highlighted in light blue on a test paper), a word problem becomes an opened-up ground of possibilities. With just a picture, sentence, table or graph the goal is left to the children. They are the creators of what this will mean. Without a goal there is no overwhelming amount of text to distract the reader and increase cognitive load. It’s mathematical freedom.
A large number of problems children will see in published materials are goal led, particularly those presented in end of Key Stage tests.
Some years ago, at the start of my word problems research, I presented goal led problems in tables and charts to a Year 5 test group. I asked them to simply highlight the parts of the table/graph they would require to answer the question and make notes of any part they were not sure of.
What fascinated me in particular, was the highlighting. A large number of children were either unfocused or unable to identify information required by a question. Tables and posters were more problematic. My assumption at this stage could be we are au fait with teaching word problems using graphs and related tally/frequency charts as it’s a taught strand in Mathematics, Statistics and given emphasis as a result. Tables and posters stand alone in Mathematics and these, yielded the highest errors. Many of these reside in real life; price lists, hire price/time and price savings. They have a future purpose.
I first came across ‘goal-free’ through Pete Mattocks ‘Goal Free Problems’ website4. Some of these were perfect for Year 6 teaching, however, an approach across the Primary range was something the innovation group really wanted to consider. Clare Sealy’s article for Third Space Learning, ‘How I wish I Taught Primary Maths – Focusing Thinking and Goal Free Questions’5, exemplifies Primary examples.
‘….with the first type of question, concerns about the final goal can intrude upon their thinking. The pupil’s attention is split between thinking about the first step and thinking about the other steps that they need to take next. Therefore, even if they successfully answer the question, they may not have learnt anything generally that could be transferred to new kinds of problems.’
This is something many teachers can relate to, I often found my classes looking at step two of a problem before trying to answer step one, as mentioned, unable to decipher the ‘set-up’ information a sentence/table/chart would give. Take that away and you have no alternative but for deep exploration.
In KS1 I’ve started using them as a whiteboard enquiry, ‘what can you tell me about this …?’ The teacher acting as scribe for ideas discussed. In KS2, they can be discussed one at a time or 2-3 enlarged to A3 on different tables; children rotate and add comments to further explore under the visualiser or in a presentation.
Goal free provides an activity all children can access at their own ability. Their discussions can reveal so much about the learner as well as the metacognitive possibilities, ‘I noticed you decided to …. with this table, can you explain a little more about that?’
Care must be taken in teacher questioning; probing thinking without directing one line of thought over another. Ideas must come from the children. It’s all about paying attention, noticing.
A Goal Free Example
In this example (1), children may consider the total number of chocolates in both the large and small box, they might just look at the large or small box in isolation or consider how many more chocolates there were in the large box compared to the small box. Children may notice, the small contains half the amount of chocolates than the large. By clearing away the detail of the goal, there emerges some clarity and a chance for real exploration. The inclusivity of this sort of activity is clear.
We could end our journey there, return to the original problem (2), From experience, I know the children would always feel more fully equipped to solve. However, often, a third step (3) could be this; we could take away even more goals/information, open-up the opportunity for even further exploration, connection making and creativity.
Supposing there were no Ken? We start to think about the possibilities. What if we gave the boxes some dimensions? What if we priced them? What would be reasonable? What if there was a special offer on? All ideas must come from the children but you can see how easily one problem could take half your lesson (if explored properly), as in my last post, it’s quality over quantity. Slow learning.
If you’ve never tried goal free, like all new approaches, it must involve an introductory lesson. Children soon become familiar with the idea.
So, when to use it? If I were starting a unit on teaching perimeter and area, I’d start with the core knowledge and models, then I’d use goal free in the application to deepen understanding,
Clare Sealy’s article mentions its use,
‘to further cement knowledge, using them after you have taught pupils a principle. This will encourage focused thinking within the classroom.’
I have also used them as homework (at the same point in the learning) and got the class to discuss their findings as part of a lesson later that week.
Here’s some points to consider:
- You only need 2-3 a lesson or one for a starter you could keep revisiting throughout the week. Don’t be tempted to over plan. It’s all about the quality discussion and developing layers of understanding.
- Encourage annotation – What do you notice? In SATs we are told to write in the box; answer and working. A goal free activity is meant to be messy. I often paste a problem in the centre of a sheet of A3/A1 paper with a bunch of marker pens.
- In KS2, you could start with 3 goal free problems, rotated from table to table before a whole class discussion. Each table has the responsibility to identify something else the last didn’t.
- Younger children, may require shared/guided support with their observations whilst using goal free.
- Comparing two goal-free problems – same/different (as the problems in the free resources are sorted into categories, you could easily do this). What is the same? What is different?
- Goal free retains the existing context of the problem, however, what about this idea? Keep the question as similar as possible but change the context. Is it possible? Which context could you have? For the chocolate box example, it could be biscuits or seeds planted in big/small trays. A change of context can be just as challenging for some children.
- Can they add another step? Introduce another area of mathematics?
In the resource I’ve created below, there are over 200 KS2 and 100 KS1 goal free problems sorted into categories and a few extra ideas and sources of information too. The resources are hyperlinked to a GoogleDrive as the files are large. They are available as a PPT with/without links to the last three years of original questions (found below the slide) or PDF for each Key Stage.
They are also hyperlinked in the contents to provide quick access. Some of the slides have part of the problem, some just an image (different levels of goal free as mentioned).
- KS1 Goal Free Resources – https://drive.google.com/drive/folders/1DulErJO98jHHOWt_QcerF42JZ3hHwpo5?usp=sharing
- KS2 Goal Free Resources – https://drive.google.com/drive/folders/1pb4Q73N4bL3cedQqzv3agwDlX4o0eoIi?usp=sharing
If you are searching for more sources of goal free inspiration here’s some of my favourites:
- With younger children use images instead of written problems.
- Use images from the ‘real-world’. I often use photographs I’ve taken whilst out of supermarket offers and car parking charges.
- Infographics are a fantastic source of information; word problems can be sourced from them to be presented in a ‘goal free’ format.
- Bar Models – In this lesson example, I added children’s names to each row. In this ‘times as much’, multiplicative reasoning question. Discussing the possibilities for possible questions before similar and different contexts were applied.
- An all-time favourite book of mine is by ATM’s publication by Jill Mansergh & Margaret Jones (2007) ‘Thinking for Ourselves’6. There is a section in it about children problem solving from initial presentations like the one below.
There are many problems for you to choose from. Using different posters, questions posed are labelled ‘straightforward, challenging & more challenging’ or simply discuss what you see. Every time I have used this, a limited experience of the theatre prompts a whole discussion about seating and the price of tickets.
Image from ATM’s publication by Jill Mansergh & Margaret Jones (2007) ‘Thinking for Ourselves’. pp30
- Old, Science SATs papers with useful graphs and charts. For further cross-curricular sources, Geography and History have some good sources too.
Here’s where to find the sources I’ve mentioned and more:
- ATM – Mansergh, J & Jones, M (2007) ‘Thinking for Ourselves’ – https://www.atm.org.uk/Shop/Thinking-for-Ourselves/dis019
- Barton, C (2018) ‘How I’d wished I’d taught Maths’ pp161-165
- O’Brien, K Goal Free Word Problems (2018) – there’s a KS1 and KS2 set of PPTs to purchase – https://www.etsy.com/listing/628768034/ks2-and-ks1-goal-free-problems
- http://goalfreeproblems.blogspot.com/ – Mr Peter Mattock’s Website ‘Goal Free Problems’ has a range of Upper KS2-Secondary ‘A’ Level to choose from, including a selection of writing from other sources on the goal free approach.
- https://www.jaggersmaths.co.uk/goal-free-problems – Again the range is from Upper KS2-Secondary but still a useful source of problems.
- https://thirdspacelearning.com/blog/how-teach-primary-maths-goal-free-question/ – Clare Sealy’s Blog for Third Space Learning on using Goal Free Problems in the Primary Classroom
- Infographics Bank from Maths Shed – http://www.mathematicshed.com/infographics-shed.html
- More Infographics – http://www.infographicsshowcase.com/
- Information presented beautifully (KS2) – https://informationisbeautiful.net/
- http://ntimages.weebly.com/photos.html – Images for mathematical discussions.
- https://www.ncetm.org.uk/resources/50723 – NCETM ‘First, Then, Now’ images.
1 – Hasanah, R et al (2017) – ‘Can Goal Free Problems Facilitate Students Flexible Thinking?’- https://www.researchgate.net/profile/Endah_Retnowati/publication/318923144_Can_goal-free_problems_facilitating_students%27_flexible_thinking/links/5a820648a6fdcc6f3ead6fc3/Can-goal-free-problems-facilitating-students-flexible-thinking.pdf?origin=publication_detail – Accessed 3 June 2020
2 – Sweller, J., Merriënboer, J.J.G. and Pass, F.G.W.C. (1998) ‘Cognitive architecture and instructional design’, Educational Psychology Review 10 (3) pp.251-296
3 – Barton, C (2018) ‘How I’d wished I’d taught Maths’ pp161-165
4 – Goal Free Problems – http://goalfreeproblems.blogspot.com/
5 – Sealy, C (2020) Third Space Learning Article, ‘How I wish I Taught Primary Maths – Focusing Thinking and Goal Free Questions’ – https://thirdspacelearning.com/blog/how-teach-primary-maths-goal-free-question/
6 – ATM – Mansergh, J & Jones, M (2007) ‘Thinking for Ourselves’ – https://www.atm.org.uk/Shop/Thinking-for-Ourselves/dis019