Introduction to conceptual understanding
- 1. Introduction to Conceptual Understanding and the CPA Approach (Concrete, Pictorial, Abstract)
- 2. Children and adults can find maths difficult because it is abstract. The CPA approach helps children learn new ideas and build on their existing knowledge by introducing abstract concepts in a more familiar and tangible way. Concrete is the “doing” stage, using concrete objects to model problems. The CPA approach brings concepts to life by allowing children to experience and handle physical objects themselves. Every new abstract concept is learned first with a “concrete” or physical experience using cubes, bears, numicon, beads, counters etc. Pictorial is the “seeing” stage, using representations of the objects to model problems. This stage encourages children to make a mental connection between the physical object and abstract levels of understanding by drawing or looking at pictures, circles, diagrams or models which represent the objects in the problem. Abstract is the “symbolic” stage, where children are able to use abstract symbols to model problems. Once a child has demonstrated that they have a solid understanding of the “concrete” and “pictorial” representations of the problem, the teacher introduce the more “abstract” concept, such as mathematical symbols. Children are introduced to the concept at a symbolic level, using only numbers, notation, and mathematical symbols, for example +, –, x, / to indicate addition, multiplication, or division.
- 3. To ensure that all pupils: • become fluent in the fundamentals of mathematics, including through varied and frequent practice with increasingly complex problems over time, so that pupils have conceptual understanding and are able to recall and apply their knowledge rapidly and accurately to problems • reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language • can solve problems by applying their mathematics to a variety of routine and non-routine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions. Mathematics Programme of Study: KS1-4 September 2014
- 4. Key components of conceptual understanding •Mathematical vocabulary and language •Connections between old and new knowledge •Questioning to extract reasoning •Manipulatives/Resources •CPA Teaching Approach
- 5. • The following slides will show examples of manipulatives which aid the development of conceptual understanding.
- 9. Counters Base 10 Grid and Counters
- 10. Cuisenaire rods
- 11. Addition and Subtraction • Part + part = whole
- 12. Bead Strings What if the bead string is worth 1? …0.1?
- 13. Addition – Numicon, Straws, Arrow Cards
- 14. Partitioning using Diennes 47 + 76 = 7 + 6 = 13 40 + 70 = 110 123 47 40 7 47 + 76 = 40 + 70 = 110 7 + 6 = 13 110 + 13 = 123
- 15. Column method T U 6 7 + 2 4
- 16. The movement between concrete, visual and symbolic representations of a problem is often quite complex, and the connections between the different forms and the implications of using different representations need to be explored. NCETM, (2014)
- 17. “Research has shown that the optimal presentation sequence to teach new mathematical content is through the (CPA) approach.” Sousa, (2008) “students who use concrete materials develop more precise and more comprehensive mental representations, often show more motivation and on-task behaviour, understand mathematical ideas, and better apply these ideas to life situations” Hauser(2009) “I utilize the CPA approach ………………….. to foster a deeper understanding of mathematics so that students are gaining greater conceptual knowledge rather than procedural knowledge. Through this approach, students are experiencing and discovering mathematics rather than simply regurgitating.” Gujarati,(2013)
Editor's Notes
- #4: Aims of the new Mathematics National Curriculum for KS1-4 for 2014 NCETM - The Essence of Maths Teaching for Mastery, June 2016 NCETM Mathematics Textbook guidance, Jan 2015 Both these document stress that conceptual understanding should be at the heart of textbooks, schemes of work and all lessons.
- #5: Mathematical vocabulary and language Discuss the importance of Maths dictionaries/glossaries, maths literacy activities etc…. Connections between old and new knowledge Discuss Questioning to extract reasoning Discuss
- #7: Delegates to watch the ‘CPA’ video and then discuss. Concrete. The “doing” stage using concrete objects to model problems Representational. The “seeing” stage using representations of the objects to model problems Abstract. The “symbolic” stage using abstract symbols to model problems
- #8: This slide shows how the CRA/CPA approach can be used with the problem 4+5=9. CPA _Sorting Cards CPA Table _Template Work with a colleague and decide which card goes into which column of the table. Discuss with a colleague Which images clearly fitted into a specific column? Which images are you unfamiliar with? Were there any images which fitted into more than one column? Can you add any further examples to the table
- #11: Cuisenaire activity Pick a black, orange and lime green – find the relationship between these three – write 4 statements. Determine the need for correct vocabulary
- #13: Pringles Tin_Activities Beadstrings_Activities
- #17: The Venn diagram allows us to think about the connections between the different stages and how some of the resources/models/diagrams are either clearly categorised as representing one stage or the link between the stages which support cognitive development. CPA Venn Diagram_Template Using the cards from before, work with a colleague and decide where the cards go in the Venn diagram.