In maths classes teachers often approach differentiation by setting different levels of work for learners at different levels of ‘ability’. However, this approach may fail to develop the types of mathematical thinking needed most for students’ futures.
Here’s why, and how, ‘rich tasks’ can help.
Teachers are often being asked to differentiate teaching and learning to ensure they are catering for a broad range of student abilities, interests and readiness to learn.
In mathematics, this means simultaneously teaching new content to all students, providing ample opportunity for students to master content, supporting students experiencing difficulties with mathematical understanding and ensuring experiences in higher levels that extend and enrich more capable learners.
It can be exhausting to think about, especially when we know that in our classrooms we have such a broad range of students. As teachers, how can we manage this in real-world classrooms, without driving ourselves to the edge of exhaustion by micro-managing separate tasks for each and every student?
Back in 2016, the World Economic Forum predicted that within five years, over one-third of skills that were considered important in the workforce will have changed. They argued that by 2020, artificial intelligence, advanced robotics, autonomous transport, new materials, biotechnology and genomics will have transformed the kind of skills required in the workforce and in the way we live. The three skills that topped the WEF’s ‘list of 21st Century Thinking Skills’ were, in order, complex problem solving, critical thinking and creativity (Grey, A. 2016).
As educators we can’t ignore these realities. We know we need to be preparing as many of our students as possible for the imperatives of life in the 21st century, developing their problem solving, reasoning and creativity proficiencies, especially in mathematics. These skills underpin so many roles in the new economy. However, it’s also important that at the same time we are catering for students who are having difficulty understanding concepts at much more basic levels.
The standard approach to differentiation in mathematics has been to prepare multiple ‘levels’ of content set to different standards, and/or to make available a variety of learning resources, individualised for each learner in the classroom. When teachers are responsible for classes of around 30 students, often within the context of engagement or behavioural issues, this can be time consuming and professionally draining.
Some digital or online learning providers claim to have solved this by developing software that allows students to follow an ‘individualised’ program of learning in mathematics. Such programs use adaptive pathways which respond according to students’ progressive successes and failures against set tasks. Smart, huh?
However, the problem with many such online or digitally based mathematics learning solutions is the same as that facing the traditional ‘textbook only’ based approaches to mathematics that have plagued maths learning since the 1950s. Typical ‘drill and answer’ exercises and closed worded problems – such as those typically used in textbooks and on online mathematics websites – usually focus students’ learning only on developing understanding and fluency in mathematics.
These two math proficiencies – the skill of grasping a maths concept (understanding) and of using it flexibly and efficiently (fluency) are of course necessary – but by no means sufficient. Tasks in maths that develop only understanding and fluency for students tend to suggest a single linear pathway to ‘working out’ and a single correct answer.
More importantly, they neglect the problem solving and reasoning proficiencies both prescribed within the Australian mathematics curriculum framework and required for humans to cope with life and work in the 21st century.
Problem solving, reasoning and creativity: ‘rich’ mathematical thinking
In mathematical learning, generally ‘closed’ (single answer/single method) tasks tend to stimulate basic conceptual understanding and develop concept fluency. However, ‘open-ended’ types of problems (more than one correct answer, multiple pathways for working out and justifying) have greater potential for stimulating higher order mathematical thinking, that is, creative problem solving and complex reasoning capacities.
This is partly because such tasks involve a search for patterns and relationships between elements in the problem. Students must ‘play around with’ different variables in order to generate different solution pathways, use trial and error (much the same as for problems faced in the real world) and explore a range of methods. They must compare the efficiency and accuracy of solution pathways and use reasoning to adapt and apply previously learned concepts to new situations.
Of course, students who have not understood and/or who are not fluent in math concepts will struggle to solve problems and to apply mathematical reasoning. The key issue for teachers is thus how we effectively and efficiently cater for these less fluent students without compromising opportunities for more fluent students to extend and enrich their mathematical thinking.
‘Rich’ or ‘Low Threshold, High Ceiling’ (LTHC) tasks
Above we mentioned the desirability of open ended maths tasks, or tasks which provide scope for more capable students to move toward more sophisticated thinking skills.
As an extension to this, ‘rich tasks’ – or ‘low threshold, high ceiling’ tasks in mathematics are structured so that all students can make a start to the problem, even if needing support. The ‘low threshold’ of such tasks reinforces understanding and fluency of a given concept and allow less confident learners to experience some level of success.
At the same time, however, further levels or iterations of the same tasks are designed to engage more independent learners in deep or complex problem solving and reasoning. The ‘high ceiling’ in these tasks provide plenty of opportunity for the participants to have a go at much more challenging maths, albeit within the same concept, topic or skill area.
According to Lynne McClure (2011), “A LTHC mathematical activity is one in which pretty well everyone in the group can begin, and then work on at their own level of engagement, but which has lots of possibilities for the participants to do much more challenging mathematics.”
She goes on to explain that rich tasks can go some way toward resolving the dilemma of more efficiently differentiating for a diverse group of learners; a single task designed with low thresholds and high ceilings can provide for the whole class. Teachers often believe that the only way to challenge learners is to offer them different content at a higher grade level. However, in rich mathematical tasks the content itself remains quite simple but the level of thinking required – such as non-linear problem solving and mathematical reasoning – can become very complex (McClure 2011, p.2).
Rich or LTHC tasks allow teachers to set one problem for all students, provide some explicit whole-class instruction, and then respond to individual and small group strengths and needs as they arise during the problem. An important role for the teacher during such learning is to cultivate a problem solving ‘culture of iteration’ whereby students learn to push themselves into levels of the task that are difficult or ‘problematic’ for them.
Having most students persevering on harder levels of the task (for them) will also often free up the teacher to work more intensively with students struggling with basic concepts. Establishing collaborative problem solving guidelines such as time for quiet thinking and reflection, ‘ask three before you ask me’, pausing periodically to discuss the approaches of different students in various parts of the task (not asking for ‘answers’!) and analysing errors as well as ‘correct answers’, will further facilitate the freeing of teacher time to be spent with less confident individuals.
Developing LTHC or ‘Rich’ tasks in Mathematics
Click to here to download an exemplar LTHC (rich) task: ‘catering-canapes’ (approximately Grade 5 or 6 level).
To write your own rich tasks in maths in a grade and topic suited to your own classroom (which is always much more fun than using someone else’s!), the following development guidelines might prove useful.
1. Start with a closed version of a problem within a given topic or concept. Grade level textbooks and standardised tests (such as past NAPLAN papers) are often a good source of closed mathematics problems.
2. ‘Open up’ the problem by removing or adapting parameters, to allow for a range of solutions. As an example, a closed problem might read:
‘Sarah takes 4 hours and 55 minutes to complete a 200-page novel, while Derek takes 5 hours and 12 minutes to read the same novel. Assuming they are each reading at the same speeds as they were for the 200 page novel, how much faster is Sarah than Derek, in seconds, if they both read a 300-page novel?’
To open up this question up, we could remove the parameters in the second sentence of the question, such that it instead reads:
‘…Assuming they are each reading at the same speeds as they were for the 200 page novel, compare the times taken by Sarah and Derek to read one of the novels selected from a book shelf either in your classroom or your library.’
3. Push further by adding mathematical or procedural ‘problem solving’ complexity. For example, in the example above, we could add a third reader (with a different reading speed), select a range of different length novels to read and/or ask students to produce line graphs that compare the reading speeds of students against novels at different lengths.
They key concern here is to ensure a few different iterations of the same type of problem, each of which add increasingly complex levels of problem solving for students in the class.
4. Introduce a requirement for students to demonstrate reasoning and justification for a version or versions of the problem. Ways in which to do this might be:
- have students challenge one another by setting their own versions of the task (they will need to have reasoned out their own version and to have ‘worked backwards’ in order to do this);
- have students compare several different methods of working out and write reasons for which they think one is ‘better’ than another; or
- have students design a model, experiment or product that applies the concept or topic in the real world. (An example of the above might be to design an experiment that compares any students’ reading speeds and draws conclusions from this experiment that could be useful to students when choosing the type and length of novel to read).
Designing and using rich tasks in mathematics can be a rewarding and motivating experience for teachers as they watch their students engage in problem solving and demonstrate thinking in ways that traditional ‘text book’ questions rarely allow.
They can also provide a new and rich source of assessment information as teachers gain new insights into how their students are working mathematically.
– Marcus Garrett
EduGains (2015), ‘Knowing and Responding to Learners in Mathematics’. Online resource available from http://www.edugains.ca/newsite/di/knowing_responding_to_learners.html. Accessed 17/01/18. Ontario Ministry of Education : Ontario, Canada.
Grey, A. (2016), ‘The 10 skills you need to thrive in the fourth industrial revolution’. Article published by World Economic Forum, 19 January, 2016. URL: https://www.weforum.org/agenda/2016/01/the-10-skills-you-need-to-thrive-in-the-fourth-industrial-revolution/.
Herter, R. (2015), ‘Growth Mindset for Math – Mistakes’ (Youtube). URL: https://www.youtube.com/watch?v=LrgpKjiQbQw. Accessed 8/11/2017
McDonald, S. and Watson, A. (2012) What’s in a task? Generating mathematically rich activity. A report commissioned by the United Kingdom Qualifications and Curriculum Development Agency (now the United Kingdom Standards and Testing Agency). London : United Kingdom. Online report – URL: http://xtec.cat/centres/a8005072/articles/rich.pdf. Accessed 17/05/18.
McLeod, S. A. (2012). Zone of proximal development. Retrieved from www.simplypsychology.org/Zone-of-Proximal-Development.html
McClure, L. (2011), ‘Using Low Threshold High Ceiling Tasks in Ordinary Primary Classrooms’. Online article on nrich.maths.org. URL: http://nrich.maths.org/7701. Accessed 10/1/2018. NRich Maths, Cambridge University : United Kingdom.
Motter, A. (Date uncertain), ‘George Polya’. Online article, in ‘math.wichita.edu’. URL: http://www.math.wichita.edu/history/men/polya.html. Accessed 10/04/16. Wichita State University : Kansas, United States.
New Zealand Ministry of Education (2010 ? – 2017), ‘Problem Solving’ online resource page. URL https://nzmaths.co.nz/level-5-problems. Accessed 3/11/2017. New Zealand Ministry of Education : Dunedin, NZ.
NRICH Mathematics (2018), ‘What Was In the Box’. URL: https://nrich.maths.org/7819. Accessed 17/01/18. Cambridge University, United Kingdom.
NSW Department of Education (2014), ‘Newman’s Error Analysis’. On Numeracy Skills Framework support website. URL: http://numeracyskills.com.au/newman-s-error-analysis. Accessed 23 October, 2016. Government of NSW : Sydney, Australia.