A headmaster of a primary school asked us if we had some good maths tasks for the lower grades.
Reminding the expression:
‘Give someone a fish, and they’ll eat for a day. Give them a fishing rod, and they’ll eat for life.’
we decided to give some tools in order to be able to:
(1) change existing tasks into better tasks, and
(2) design challenging math tasks.
But what are criteria for a challenging math task? We refer to Prof. Jo Boaler and use her criteria and add some of our own ideas based on experience with realistic mathematics education (Freudenthal Institute) and other resources.
We strongly advocate Inquiry-based learning. Note that we do not write ‘teaching’, we write ‘learning‘, because the goal is not teaching, but learning. More about inquiry-based learning can be found in earlier blogposts: Research on inquiry-based learning and Inquiry-based learning in practice. Inquiry-based learning uses more open tasks. Closed tasks with more scaffolding are less exciting and challenging. If students get step-by-step instructions (cooking-book practices), they stop thinking.
Challenging maths tasks give students the opportunity to learn, think, explore, discuss, be creative and learn (about) different strategies and respresentations or visualisations in the process.
Jo Boaler (‘Mathematical Mindsets’, 2015) offers the following rules to open up an existing task, thus making it more challenging:
Can you open up the task to encourage multiple methods, pathways and representations?
Can up make it into and inquiry task?
Can you ask the problem before teaching?
Can you add a visual component?
Can you make it low floor and high ceiling?
Can you add the requirement to convince and reason?
A) If you take a simple task from a textbook such as 5 + 7 Ask the students how they came to their answer (before even asking what the answer is). This way you encourage pupils to think about their strategy, to express their strategy in words or visually, to use mathematical vocabulary, to see and learn different strategies from each other.
B) Take the simple task 24 x 3 Students have already learned different strategies, such as swop the numbers 3 x 12 which is probably easier for some. Or they have used the strategy to double one number and half the other number 12 x 6. Now ask the students why 24 x 3 is equal to 3 x 24. Can they prove that this is true? They could for example use rectangles on a grid.
C) Take a task like 28 : 4 Do not teach the way to check the answer, but first ask: How could you check your answer yourself?
D) Take the following task 1/4 x 5 . First offer visualisation and ask: How much is 1/4 of the 5 circles? Or instead of asking for an answer, ask: How could you divide the 5 circles in four?
E) Take the topic average. Do not teach the different averages, but let the pupils work on average without specifying what it is. They will have a basic notion of what average is. Let them for example measure their own height and register this for all students in class. Let them come up with the class average. Discuss which ‘average’ they think is best/fairest.
F) Take a task about measuring. This task deals already with understanding, namely of units of measurement. We can change the task so that it becomes ‘low floor, high ceiling’ (accessible and challenging for all).
- Estimate height or length and write this down using a unit of measurement.
- Measure the objects (except for e).
- If the students in your group came up with different results, discuss how this may have occured.
- Try to write the height/length in a different unit of measument.
Another task on measurement. Instead og asking What is the circumference of a given rectangle? ask: Draw different rectangles with a circumference of 12 cm.
G) See the task Beads on a string from FI-rekenweb. See also the blogpost Early Algebra on Beads on a string.