Design math tasks

In the previous blogpost on how to Change math task so that they become more challenging, we wrote about guidelines on how to change existing tasks. In this blogpost we discuss how to design new math tasks. 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.

The guidelines below are particularly useful when you design new, challenging tasks.

  1. Can you make it into an activity?

  2. Can you make it hands-on: use materials, equipment and tools?

  3. Can you make it into an experiment?

  4. Can you use problems from the real world?

  5. Can you integrate it with other subjects?

H) Fold an air-plane from and A4 piece of paper and measure whose air-plane gets farthest. Pupils work in groups. (They decide how many tries they are allowed, how they can measure in a fair way, how they can improve their paper plane etc.)

I) Use polydron squares. See the blogpost Cube 3D-2D and Make a cube and fold it out into a nett. How many different netts can you find?

J) Let the students measure the schools playground using ‘steps’ (or a rope). What is the shortest way to cross the playground? Ask them to draw the playground and their shortest route on cm2 grid paper.

K) Plan a trip from your home town to Oslo. Work out and compare different options.
When is the trip fastest? Which means of transport is cheapest? Depending of how much time you want to spend on the project, you can decide how much information you give: timetables, maps, pricelists.

L) The task in the blogpost on Balancing Act is both physics and maths. It offers students the opportunity to investigation, experiment, and to find the rule. The last step is generalisation and (early) algebra. Instead of using a computerprogram you can also use a real balance/scales or have the students construct their own.

weegschaal met schaal


Change maths tasks

A headmaster of a primary school asked us if we had some good maths tasks for the lower grades.

fishingrodReminding 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:

  1. Can you open up the task to encourage multiple methods, pathways and representations?

  2. Can up make it into and inquiry task?

  3. Can you ask the problem before teaching?

  4. Can you add a visual component?

  5. Can you make it low floor and high ceiling?

  6. 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?

vijf cirkels

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).

multi 5b measurement

  • 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.

Multi 5b circumference

G) See the task Beads on a string from FI-rekenweb. See also the blogpost Early Algebra on Beads on a string.

kralen applet



MOOC -Success

Young woman juggler is juggling balls.A good start in 2016 !

Second run of the Open Online Course Success- Unleash yourself  starting in January 2016.

The course is about learning good habits that lead to success. Participants are introduced to four main focus areas mindsetgoals and tasks – individually and as part of a team. The course runs for 9 weeks and anyone with internet connection can participate. Everyone can benefit from the course, especially students and young professionals.

Course Date: January 11, 2016 up to March 14, 2016
Registration is OPEN.


MOOC stands for Massive Open Online Course and was first developed by Stanford University in 2008. The aim of MOOC is to offer online courses for an unlimited number of students wordlwide and is published via open access on the web. 

Concept Cartoons

concept ice in waterConcept Cartoon is a relatively new approach to teaching, learning and assessment in science. Concept Cartoons were first developed and created by Brenda Keogh and Stuart Naylor in 1991. Concept Cartoons feature cartoon-style drawings showing different characters arguing about an everyday situation. They are designed to intrigue, to provoke to encourage discussion, and to stimulate scientific thinking. The problems or questions posed may not have a single “right answer”.

The characters in the Concept Cartoons offer the students a role model they can identify with. This encourages students to choose a character and thus discuss freely. It does not become too personal what the student expresses about the concept. The cartoons can be used with pupils from 6 to 14.

Concept Cartoons can be an introduction to a more practical and hands-on experiment, a summary after experimenting, or just a discussion in class.

concept cave dark light

More on Concept Cartoons-2 and Concept Cartoons_3.

Research on Inquiry-based learning

This post is related to the DaVinci2020 project in Norway. The project introduces and implements STEM education at two primary schools. An important goal is to apply Inquiry-based learning and Hands-on learning.
STEM (Science, Technology and Mathematics) and the interdisciplinary tasks within it, are not part of the official Norwegian curriculum.


The UK has had a strong focus on science education and introduced and implemented science education at primary school level in sixties and seventies. After 1996 many countries have included technique or technology in science education.

Projects in different countries:
  • UK – Nuffield Science 5-13, STC, FOSS, Insights
  • France – Insights 1996
  • Sweden – STC 1996 -> Naturvitenskap og Teknikk for Alle (NTA)
  • Berlin – STC 2004 (Science Technology and Children)
  • Netherlands – VTB, Talentenkracht
  • EU – Fibonacci project
The effect of Inquiry-Based Science Education – Results from research:

Pupils who received science education using inquiry based learning scored higher on science achievement tests than pulpils taught using the traditional approach. M.A. Selim & R.L. Shrigley, R. L. (1983). The group dynamics approach: A sociopsychological approach for testing the effect of discovery and expository teaching on the science achievement and attitude of young Egyptian students. Journal of Research in Science Teaching, 20(3), 213–224.

Attitude towards technology is more positive after girls participated 1-2 years in technology club. Self-confidence is higher. Attitude towards career in technology is more influenced by society as a whole. Creativity and design are important factors to appreciate technology. E.van Eck, M.Volman, 1999. Techniek, leuke hobby, saaie baan? Eindrapport evaluatie Technika 10 Plus. Kohnstamm Instituut/Vrije Universiteit.

Attitudes towards science increased and students were more interested in science careers. After four years the positive attitude towards science had decreased but was still considerably higher than with other students. H.L. Gibson, Ch. Chase (2002). Longitudal Impact of an Inquiry-based Science program on Middle School Students’Attitudes Toward Science.

Pupils from 7th and 8th grade participated in a year and a half program in project-based science education. Thereafter their scores on standard statewise test were 20% higher. They had a better understanding and better process skills. Higher scores were measured in all science subjects, not only in the areas that were covered by the program. Geier (2007). Standardized Test Outcomes for StudentsEngaged in Inquiry-Based Science Curricula in the Context of Urban Reform. Journal of Research in Science Education. Vol.45, no.8, pp. 922-939.

The positive effect from inquiry-based learning is higher when combined with hands-on learning, thus when learners manipulate and investigate by using artefacts and materials. D.D. Minner et al. (2009). Inquiry-based Science Instruction- What is it and does it matter? results from Research Sythesis Years 1984-2002. Wiley InterScience, online:

Students in the hands-on classes are more favorable to science and have a better understanding of the nature of science than students in textbook classes. SB.J. Foley and C. McPhee, 2008. Students’ Attitudes towards Science in Classes Using Hands-On or Textbook Based Curriculum. AERA.

Undervisning av Inquiry based learning er viktig og vanskelig og for å være effektivt trenger lærerne 80-160 timer etterutdanning. T. van Eijck, E van de Berg (2011), Effecten van nascholing Wetenschap en techniek in het Pimair Onderwijs in de regio Amsterdam, Tijdschrift voor Didactiek en Beta-wetenschappen. 28, nr. 2

Brainpower: Facebook friends

Every month DiScoro writes about resources that can be used in schools or about inspirational issues. See Services in the Menu for workshops, training etc.

Archimedes is  a task in a series of Brainpower questions and tasks. See also Brainpower: Milk packaging and ArchimedesThis type of questions and tasks require higher order thinking skillsmeaning that students have to apply several types of knowledge and skills. Higher order thinking skills involve critical thinking, problem solving, research, argumentation, discussion, evaluation, collaboration, judgement etc. The questions and tasks require brainpower and often several different strategies can lead to the solution.

We give you examples that you can use in the classroom. We appreciate your comments which you can write in as a comment. With your comments we are able to improve the tasks and the information about them.


Brainpower tasks require a different attitude and behaviour from both teacher and students. As a teacher you guide the students without disclosing answers, strategies or algorithms. You will rather guide the students with questions that encourage them to think in different ways, which help them to discuss further, or to visualise the problem in a creative way. Usually, students work on a task in small groups and for quite a while.

Download the PDF for the teacher: Facebookfriends.pdf

Brainpower tasks require a different attitude and behaviour from both teacher and students. As a teacher you guide the students without disclosing answers, strategies or algorithms. You will rather guide the students with questions that encourage them to think in different ways, which help them to discuss further, or to visualise the problem in a creative way. Usually, students work on a task in small groups and for quite a while.


Inquiry- Based Learning in Practice

We offer a course Inquiry-based learning in Practice. In this course you learn how to change tasks in such a way that learning improves. We use a strategywhich is based on several theories:

  1. Higher order thinking skills
  2. Bloom’s Taxonomy as defined by Bloom & Krathwohl (2002)
  3. 21st Century Skills as described by OECD (2009, 2012)
  4. Two projects by Dr. Sugata Mitra: A whole in the wall, and The Da Vinci project
  5. Lecture by Prof. Eric Mazur: Turning Lectures into Learning.

The strategy is a modification of RTTI, a widely used strategy and in-service-training for teachers in Dutch schools. We have  implemented the taxonomy of learning activities linked to tasks in at the educational publishing company in The Netherlands. Here we used the strategy in the design and publication of text books for mathematics as well as language and history.

Important is that we adher to the notion that the ultimate goals in education are:

(1) students acquire knowledge and skills that are long lasting. This is called Retention;

(2) students are able to apply knowledge and skills in new situations, in combination with other skills, across disciplines, and in daily life. This is called Transfer.

Retention and Transfer pyramids

Retention and Transfer pyramids

The strategy uses four levels of knowledge and four different levels of learning activities. The mastering of these levels of knowledge depends on the learning activities teachers offer. The strategy works towards teachers that are able to make or change tasks so that students learn at all levels of knowledge and work with all levels of learning activities. This way the students get the opportunity to reach the ultimate goals: rentention, of knowledge and transfer (deep learning)

Levels of knowledge:

  • Factual knowledge
  • Procedural knowledge
  • Conceptual knowledge
  • Higher knowledge

Learning activities:

  • Remember
  • Use
  • Apply
  • Integrate

We have made some tasks at the level of Conceptual and Higher knowledge combined with the learning activities Apply and Integrate. Examples of these tasks can be found in the Brainpower activities and in the HOT Math series (=inquiry-based learning activities).

Below you find the schema used to identify the nature of tasks within the curriculum.

Schema for identification, analysis and inventory of tasks