Developing Mathematical Intuition in a Knowledge-based World

1. Developing Mathematical Intuition in a Knowledge based world. Paul Cliffe

2. What is intuition? The ability to understand something instinctively, without the need for conscious reasoning. A thing that one knows or considers likely from instinctive feeling.  Are our students first instincts giving them appropriate mathematical responses? What causes these MISCONCEPTIONS?

3. What is the most important thing about NUMBERS? What are the important concepts that a student needs to understand their value? AND… 60=6x10 BUT… 162 =100+60+2 100 = 10x10 = 102 AND… BUT… BUT… AND… At the very least can be imagined as or as minus or below zero.

4. Different Number Representations • Every number representation involves one or more or the operations • + ADDITION • SUBRACTION • X MUTIPLICATION • ÷ (/) DIVISION

5. For starters: to create Mathematical intuition that can be relied upon, students must know what the operations are. What is addition? SO… What is division? What is multiplication? What is subtraction?

6. Your definition for WHAT IS … • A definition that describes how to do it • A definition for your students to remember • Something that makes sense

7. Dictionary Definition of … • ADDING • a) Join (something) to something else so as to increase the size, number, or amount. • b) Put in (an additional element, ingredient, etc.) • SUBTRACTING • a) the process taking one number or amount away from another. "subtraction of this figure from the total" • b) MATHEMATICS • the process of taking a matrix, vector, or other quantity away from another under specific rules to obtain the difference. • MUTIPLYING • a) Obtain from (a number) another which contains the first number a specified number of times. • b) Increase or cause to increase greatly in number or quantity. • DIVIDING • a) The action of separating something into parts or the process of being separated. • b) Disagree or cause to disagree.

8. My definition for WHAT IS… Moving DOWN (the number line) Repeatedly adding the same value How many times does the bottom add to make the top

9. These definitions produce the structure below for students to understand numbers and make calculations. SUBTRACTION: is moving down (the number line) 5 DIVISION: how many times the bottom adds to make the top

10. Knowledge and Understanding Does knowledge of something mean that a student understands it? I believe that understanding something means that you can JUSTIFY why it is true?

11. What does intuition feel like? Three quarters is bigger than a half and five tenths is one half? When we add something positive to three quarters the result must be bigger than one half. There must be something in what we are looking at that makes sense. (3/4 is bigger than ½) MEAN MEAN MEAN = Why did I have no clue that 5x24.5 was wrong? And yet I noticed that 23.17 was wrong? Students must have ideas that make sense to them and they must consciously decide to look for sensibility.

12. Where does intuition come from? Intuition is not something that forms in isolation. It is developed by incorporating knowledge and understanding from prior learning with the new experiences that are occurring. Our brain is constantly assessing and gathering information – some of which we do with conscious awareness and some is gathered completely unintentionally. The gathered information is stored as “patterns” of information. However, what starts as a correct mathematical idea can be used incorrectly and quickly become a misconception. CONSIDER: What has gone wrong and how to fix it? In each of the example above the student has confused the rules for adding, multiplying, dividing and powers. How do we develop correct mathematical intuition that does not become a misconception? To understand how you can benefit from intuition, we must first lay a FOUNDATION of where the knowledge comes from.

13. We have to start connecting mathematical ideas in a way that makes sense to our students.

14. Why are numbers so important? Solve the equation for and What is ? The ratio of the opposite side length of a right angled triangle divided by the hypotenuse length. What is What is What is ? How can two numbers multiply to make zero? What is? One of them has to be zero? When this becomes Either or Either or Because almost everything in mathematics stands for a number. Either or

15. Foundational Knowledge What knowledge is at the foundation of mathematical learning? This has to be knowledge that makes sense

16. WHY? What is the Area of a TRAPEZIUM The same as half a parallelogram What is the Area of a PARALLELOGRAM? WHY? AREA = base x height The same as a rectangle AREA = base x height What is the Area of a RECTANGLE? WHY? What does the height tell us about the area? What does the base tell us about the area? What is AREA? The number of squares on the surface

17. I have recently written a book called … It ain’t MATHEMATICS DOESN’T MAKE SENSE if it

18. Students need a deep well of Mathematical “Foundational Knowledge” that makes sense and that they can justify. • Number operations • 1) Adding: moving up the number-line (except for adding a negative number which is the same as subtracting.) • 2) Subtracting: moving down the number-line (except for subtracting a negative number which is the same as adding.) • 3) Multiplying: repeatedly adding the same value. • 4) Division: how many times the denominator adds to make the numerator • 5) Powers: repeatedly multiplying the same value. • 6) Not all ones are the same. (ie different units) • 7)You can only add or subtract similar things. • 8) When only multiplying numbers, you can do them in any order. • 9) BEMA is the order in which operations are done. • Geometry • 10) Angle: the amount of turn or rotation • 11) Opposite X angles on intersecting lines are equal • 12) Corresponding angles The are equal • 13) Isosceles Triangle: base angles are the same (due to symmetry) • 14) Similar shapes and Enlargement: - Have the same angles and the ratio of any two similar sides are the same. (TRIG) • 15) Transformation Geometry properties: -Reflection, Rotation Translation.(Lengths, areas and angles remain the same size) • 16) Co-ordinates : are always plotted on a cartesian plane from (0,0) • - Horizontally in the -direction and Vertically in the -direction • - The -intercept occurs when (ie no horizontal movement) • - The -intercept occurs when (ie no vertical movement)

19. Measurement • Perimeter: the distance around the outside of a 2D shape • (distance is the number of units of length: mm, cm, m, km, etc) • Area: the number of squares that fit on, or completely cover, the surface of a shape • Volume: the number of cubes that fit inside, or completely fill, a 3D shape. • Probability • Probability: is the number of favourable outcomes divided by the total number of possible outcomes. Is there a problem if students ROTE learn this “Foundational Knowledge”. Not if the Knowledge can be justified by them.

20. Developing Intuition (continued) 1) There must be a logical connection to the problem being solved. 2) Students must be consciously looking for sensibility when solving problems. THIS TAKES TIME but….. What else can be done…? 3) Allow students to use their knowledge and understanding to create new theory . WHERE does it come from? WHY does it work? (ie Discovery Learning) Multiplying two negative numbers. Subtracting a negative. Expanding quadratics (using FOIL?) Pythagoras’ Theorem. Area of a parallelogram. Angles in a triangle. We need to change our students expectations from “KNOWING” to “UNDERSTANDING If students learn these rules without understanding them, then an incorrectly remembered RULE makes just as much SENSE as one remembered correctly.

21. Why Discovery Learning? Discovery is ACTIVE learning compared to the PASSIVE approach of being told or shown what to do. Researching what others have done, on the internet or in a book is NOT Discovery learning. This will never work unless students have a foundation of knowledge they understand and have the confidence to use it. Students must know the difference between understanding something and just knowing the procedure for how to do it. Introducing Differentiation TASK: Investigate the gradients of the tangents on the graph of where can be any point between -1 and 3