Staff inset on concrete, pictorial and abstract teaching of maths

1. Guidance on Teaching Expectations
Teaching Mathematics using Concrete, Pictorial and Abstract methodologies

2. Find a seat next to somebody you normally
work with.

3. You have 3 minutes to work as a team to answer the
questions below!

4. What do we mean when we
say depth of learning?
How can this be achieved?
Core Teaching and Learning Principles
1 minute discussion with your partner

5. The concrete-pictorial-abstract approach suggests that there are three steps (or
representations) necessary for pupils to develop understanding of a concept.
Reinforcement is achieved by going back and forth between these representations.
According to the National Institute of Education, Bruner’s CPA sequence has been
shown to be particularly effective with students who have difficulties with
mathematics (Jordan, Miller, & Mercer, 1998; Sousa, 2008).
Use the Concrete – Pictorial – Abstract Approach

6. Concrete representation
This is the enactive stage as a student is first introduced to an idea or a skill by acting it out with real objects.
In division, for example, this might be done by separating apples into groups of red ones and green ones or by
sharing 12 biscuits amongst 6 children.
This is a 'hands on' component using real objects and it is the foundation for conceptual understanding.
Resources such as real objects and maths equipment such as Dienes equipment are important at this stage.
What concrete resources have you used when teaching maths?
What concepts have these resources helped to secure?

7. Pictorial Representation
This is the iconic stage when a student has sufficiently understood the hands-on experiences performed and
can now relate them to representations, such as a diagram or picture of the problem.
In the case of a addition, it would mean using pictures to develop a concept.
How have you used pictorial representations previously to
teach a mathematical concept?
How effective was this strategy?

8. Abstract Representation
This is the symbolic stage in which a student is now capable of representing problems by using mathematical
notation, for example: 12 ÷ 2 = 6.
This is the ultimate mode, as children find it the most mysterious of the three, however, teachers very
frequently only teach at this level, especially with children who are perceived to be further along in
mathematics.
How would teaching only the abstract limit the understanding of a concept?

10. YOUR TURN
Use the resources around you to express a calculation appropriate year group using:
• Concrete
• Pictorial
• Abstract

11. Provide mathematical settings, puzzles and questions
Mathematical Settings
Maths settings are objects and manipulatives that can be used for maths ‘play’. These can include traditional
maths objects (e.g. blocks, cubes, Cuisenaire rods, peg boards etc.) as well as natural objects (e.g. pine cones,
leaves etc.) or common household objects (e.g. string, laces, toys, dried beans etc.).
The aim of providing these ‘settings’ is to engage children with maths in a playful way. As well as simple,
mathematical play, this can take the form of completely free creation of mathematical questions or playful
solutions to open ended questions from teachers connected to topics in some way.
Consider your own maths cupboards
for a moment.
How accessible are the resources?
How often are these used?

12. Mathematical Puzzles
Open-ended puzzles can be adapted and extended easily to provide further challenges for pupils. The Nrich website
is an excellent source of these but once teachers become familiar with the principles behind problems and puzzles
they can design their own to broadly connect to areas of study or topics.

14. Using Blooms, can you come up with a
question for each of these headings to
develop a deeper understanding on a
single concept.

15. Mathematical Questions
Providing students with opportunities to design their own mathematical questions replicates what real
mathematicians do and conveys the message that maths is a subject that deals with problem setting and solving
rather than the memory exercise that most children associate with the subject.
QUICK TASK!
Create a word problem for your partner on fractions.
You need to represent the problem pictorially.
Make it real for them. Link it to something that they can relate to….not just pizza!

16. Develop and Support Mathematical Ways of Working
This is a crucial element of the teaching of maths. If the mathematical settings, puzzles and
questions above are the bones of the learning, these ways of working are the meat.
Research has shown that the following ways of working mirror the ways real mathematicians work
and distinguish children who are good at solving problems from those who are poor.
They should be revisited constantly throughout the year.
SOMETHING TO THING ABOUT:
What strategies have you implemented to develop and support
mathematical ways of working?

17. Planning Resources

18. Bar Modelling
Bar modelling provides students with a powerful, but simple visual model that they can draw upon and
use to solve problems.
What these visual models give you is an entry point when teaching a topic that all students seem able to
grasp. It presents the concept in its rawest, simplest form without the distraction of lots of words or
mathematical notation. The diagrams don’t replace the eventual algorithmic methods, but they provide
an entry point where students seem to understand what it is they are trying to solve; something that
often gets clouded when algorithms are presented to early on.
In primary education in Singapore, maths teachers follow a Concrete-Pictorial-Abstract (CPA)
sequence when teaching maths topics. They start with real world, tangible representations, move onto
showing the problem using a pictorial diagram before then introducing the abstract algorithms and
notation.

20. GAP TASK:
Teach a series of maths lesson using CPA and Bar Modelling .
Take phots and bring these with you to the next inset to share.