Mathematics
| Piagetian Level | ||||
|---|---|---|---|---|
| Schemata | 2A, 2A/B: Early/Mid Concrete Operations | 2B: Mature Concrete Operations | 2B*: Concrete Generalisation | 3: Formal Operations |
| Properties of Numbers - ordination |
Add and subtract small number of objects, solve whole number problems, order and count to 100. Work out sequence backwards or forwards. Mentally add/subtract two single-digit numbers, the decades. | Know where 1,000 comes in number sequence. Can give change from £1 coin. Use addition and subtraction to find differences between numbers up to 1,000; can handle units of 10s and 100s. |
Understand use of negative numbers in practical context. Mental addition and subtraction of two-digit numbers, or on paper any addition or subtraction of whole numbers. Enter money in f./p on calculator. | Use negative numbers in multiplication. Make estimates in calculations. |
| Properties of Numbers - cardination |
Read/write to 100, then to 1,000. Count accurately 100 objects. | Read/write multiples of 100 up to 10,000. Adding zero does not change value. | Whole number approximation of numbers with one decimal place and interpolate 10ths. Read or enter any number to 1 million. Read scale marked in 100ths, labelled in 10ths. | |
| Multiplication and Division | Can double numbers up to 10. Practically knows half a set of objects. Identify halves and quarters. | Know and can use multiplication tables. Solve problems using multiplication and division. Use multiples 2, 5, 10. Practical quartering. Two inverses. | wo-digit x one digit. Multiplication commutative. 0.5 = 50% =1. Identify fractions, percentages. |
Mapping fractions onto 0-100 scale. Equivalence of fractions and decimals. Scaling as operation preserving multiplicative relations. |
| Expressions and Equations | Devise repeating patterns. Recognise use of symbol. (Can solve 3 + G = 4 for G.) | Can generate and continue a number pattern. Without enumerating intersection, can see problem of dividing pupils into those with pets and those with blue eyes. Provide many values of G & E which fit G + E = 9. | Use formulae to express words. Sets leading to bracketing. Can find member in each area of intersecting Venn. Make general statement about series (e.g. 1, 2, 4, 8 ... 'This doubles each time"). | Can evaluate simple expression, e.g. wage = hours x rate/hour. Solve one-step linear equations, e.g. C x 32 = 192. |
| Functions | Sort or order a set of objects using one criterion. | Sort on two criteria, two-stage ordering, use of 2 x 2 sorting table. | Three or more successive stages of ordering. 'Function machines, interrogate and interpret data in computer database. | Linear functions as generalised number. Express a simple function symbolically. |
| Shape and Dimensions | Identify 2D shapes. Estimate a metre by eye. | Identify common 3D shapes. Construct rectangle given dimensions. Sort 2D shapes in different ways. Read scales in units and 10ths. | Meaning of five major kinds of triangles. Order and estimate angles, know area units, understand volume. | Construct triangles, angles in triangle sum 180°. Areas of more complex figures. Use of 271 relationship in circles. |