Math Curriculum and Pedagogy

 

An important goal of math module in elementary stage is to eliminate the fear of math and dealing with numbers that is found in so many children. The module is designed to not only make children skilled in basic arithmetic operations, but also feel comfortable in handling numbers and their application in real life.

 

The math module attempts to help children use numbers not only to count but also see relationship among numbers as elements of an ordered set. While understanding, for example, that 4 is smaller than 5 by one and larger than 2 by 2, or that it comes in between 3 and 5, the essentials of addition or subtraction are introduced/ internalized. The children are taught to not only perform basic arithmetic operations, but also use alternate counting/ computational technique/ algorithms, and to cross-check the answer. They begin to understand the relationship among different operations, e.g. that the addition and subtraction are opposite; multiplication is repetitive addition; and the division is repetitive subtraction. Special emphasis is placed on understanding the place value, how it is used to represent larger numbers, and how this is operationalized in various standard procedures used in mathematical computations.

 

The sequencing of lessons, time devoted to each section/ stage, teaching guidelines, sequencing and composition of worksheets are all important elements in achieving above objectives. The worksheets and activities are so designed that besides getting adequate repeat practice, the children learn to substitute the lazy habit of pattern formation with exact computations. Exercises are set out in a way that requires children to examine alternate hypotheses before choosing the correct one, and to find pattern under a jumbled set. Children are enabled to estimate the size of a set of objects, with the number scheme providing exact measure of the estimate. Similarly, children are enabled to convert problems/ questions in daily life into questions involving arithmetic operations.

Examples of pedagogical approach employed.

1 Counting by using objects:  The child is to pick up a number card and correct number of counters according to the number spoken by the teacher. The child is also asked to add or reduce one counter from the set and choose corresponding number card.

2 Shop keepers and Buyers: Children are formed in pairs. One is to act as the shopkeepers and other buyers. Byers give number card to the shop keepers and shopkeepers gives correct number of counters to the buyers.

3 Which number is after and before me: Speak a number and ask child to show a number card either after/ before number of that number and put correct number of counters or Ask child to pick up number card and put correct counters accordingly to the number spoken by the teacher. Ask them to find either after/ before number of that number and put correct  number card.

4 How I can make: Ask child to put correct number of counters accordingly to the number   spoken by the teacher. Teacher speaks a number that is + or – 3 of that number .Ask child to  pick up that number card and ask them to make this number using correct number of counters

5 Sort and Count different object and link with number card: Ask child to sort and count the different object kept on a box and match it with number card

6 Sequencing: Ask children to speak numerals in different sequences ex. 1,2……,10 or 1,3,5…9 or 2,4,6……10 . The child who will make mistake will be out.

7 Comparing any two number using objects:  Ask child to pick up two number cards and put correct counters accordingly to the numbers spoken by the teacher. Ask them to find out smaller and larger sets through one to one correspondence. Ask to find out how many less in smaller sets than larger sets and the same way how many more in larger sets than a smaller set

8 Which number is smaller / larger number than me: Ask child to pick up number card according to the number spoken by the teacher. Ask them to find smaller/ larger number to that number

9 Comparing greatest and smallest number using objects:  Ask child to pick up four number cards and put correct counters accordingly to the numbers spoken by the teacher. Ask them to find out greatest and least number through one to one correspondence.

10 Arrange numbers in ascending and descending order: Ask child to pick up four number cards and put correct counters accordingly to the numbers spoken by the teacher. Ask them to arrange numbers in ascending or descending order through one to one correspondence.

11 Understand Zero using counters: Ask child to pick up number card and put correct counters accordingly to the number spoken by the teacher. Ask them to reduce counters accordingly to the number spoken by the teacher. This will be repeated till no counters will left with children and ask them to show correct number card.

12 Understand Zero using counters: Ask child to pick up number card and put correct counters accordingly to the number spoken by the teacher. Ask them to reduce the same number of counters all together and ask them to show correct number card.

13 Counting using fingers/ finger cuts: Ask child to pick up number card and show correct number of fingers/ finger cuts according to the number spoken by the teacher.

Understand place value:

14 Grouping in to units and tens:  Ask child to pick up number card and put correct number of tens bundles and loose counters accordingly to the number spoken by the teacher. Add or reduce one counter from it and put correct number

15 Dice game:

A (Using single dice): Ask child to throw the die and collect from the box a number of counters corresponding to the numeral thrown. As soon as a child has collected 10 counters he can exchange them for a bundle of ten. The winner is the first child to collect maximum bundles.

B (Using tens and unit dice): Ask child to throw two die and collect from the box a bundles    of tens and counters corresponding to the numeral thrown.

16 Secret number using arrays: Teacher will decide a number. Child locate the secret number first by its tens digit, then by its units digit referring to their arrays while making their guesses.

Addition and Subtraction:

17 Add or Subtract using counters: Ask child to pick up number card and put correct counters accordingly to the number spoken by the teacher. Ask child to pick up another number card and add/ reduce correct counters accordingly to the second number spoken by the teacher. Show number sentence using number card and symbol card.

18 Add or Subtract using number line: Ask child to pick up number card and go ahead on number line accordingly to the number spoken by the teacher. Ask child to pick up another number card spoken by the teacher. and go forward or backward on the number line depending on operation involved. Show number sentence using number card and symbol card

19 Add or Subtract using fingers/ finger cuts: Ask child to pick up number card and count fingers / finger cuts accordingly to the number spoken by the teacher. Ask child to pick up another number card and add/ reduce finger/ finger cuts accordingly to the second number spoken by the teacher. Show number sentence using number card and symbol card.

20 Dice game: Ask child to make a grid in book with number up to 10. Each child throws the die twice and can either add or subtract these two numbers using counters and then they have to cut the resulting number in their grid .The first child to cut maximum grid is the winner.

21 Guess how many: Teacher puts counters and children will count. She ask them to close their eyes and removes some counters and hides them in her closed hand. Then they open their eyes and look at the remainder and find how many counters the teacher is hiding in her hand .Then again count the counters together to verify. Write number sentence on black board .This will continue for all possible pairing of a given number.

22 Addition and Subtraction in daily life: Teacher puts counters and child will count It. She ask them to close their eyes and add/ removes some counters. Then they open their eyes and look at the counters and decides which operation involved. Ask them to show number sentence using number card and symbol card. Ask them to make a word problem using number sentence.

Multiplication

23 Equal grouping: Distribute the counters equally and found out how many counters in a single group and how many groups made.

24 Multiplication as repeated addition:

Key Concepts linked to Math Skills acquisition up to grade 2

  1. Memorize number sounds up to 100.
  2. Memorize sequence, both forward and backward) of 100 numbers.
  3. Writing of 100 number names.
  4. Counting,
    1. one-to one matching
    2. non-repetition
    3. non-exclusion
    4. bounding the set
  5. Properties of numbers:
    1. Which is more or less in a pair
    2. How much more/ less
    3. Highest and smallest in a set of n number (compare n! relationships)
  6. Representation of numbers in decadal order
    1. Place value
    2. Use of only ten symbols to write all possible numbers
  7. Addition/ subtraction of two numbers
    1. Count on/ down
    2. Finger based procedures
  8. Addition of a number column.
  9. Memorize the results of any two single digit additions.  (45 results)
  10. Memorize the results of two single digit subtraction (45 results?)
  11. There is a lot of remembering in early math.
  12. Some exercises on geometric shapes recognition and differentiation
  13. Some exercises on what is common in patterns/.

 

Math curriculum in middle-school stage:

An important feature of Gyan Shala math curriculum is its pace, which has been kept different from CBSE curriculum, to better suit Gyan shala children, so they attain terminal competencies of high school on par with excellent CBSE schools, in spite of their unique limitations and handicaps. This is illustrated in the following indicated pace of coverage.

CBSE Grade V (To be fully almost completed by 3rd or 4th grades completion in GS)

  1. Story of normal life and math concepts in that.(Gr.2,3,4)
  2. Shapes and Angles: In real life, (3/4)
  3. Finding squares in a shape (Foundation of area) (3/4)
  4. Parts and wholes (4)
  5. Similarities and patterns (2/3/4)
  6. Multiples and factors (4/5)
  7. Tracing patterns (3/4)
  8. Mapping your way (4)
  9. Boxes and stretches (4)
  10. Tenth and hundredths (4/5)
  11. Area and boundary, triangles, rectangle exercises, using graph paper (4/5)
  12. Smart Charts, graphs, bar charts, column-rows, table (3/4)
  13. Way to multiply and divide: Uses in life (3/4/5)
  14. How big/ heavy; volume/ density, estimation (3/4)

CBSE Grade VI (All to be completed in GS before the 5th grade)

  1. Knowing our numbers: Write large nos., estimate – nearest hundred/ rounding off, Use brackets, roman (4/5)
  2. Whole nos: Operations with and patterns in whole nos., (4/5)
  3. Play with numbers: Factors, divisibility by 2,5,10, (5)
  4. Basic geometrical ideas, angles, shapes,  (5/6)
  5. Understanding basic shapes: Polygons (4/5)
  6. Integers (Number line, negative nos. operations: Add/ subtract) (5/6)
  7. Fractions Graphic expression, Addition-subtraction rule, LCM, MCP, (4/5)
  8. Decimals (5)
  9. Data Handling: Table, bar chart, (3/4/5)
  10. Measurements: Perimeter, standard unit (4/5),
  11. Algebra: Mathematical pattern (4/5)
  12. Ratio and proportion: Interpretation (5/6)
  13. Symmetry (5)
  14. Practical geometry: Draw/ measure line, angles, shapes, using compass, protector,(4/5)

CBSE Grade VII (To be completed by grades 5/6)

  1. All operations on Integers, both positive and negative, number-line, commutative (a+b = b+a)/ associative {a+(b+c) = (a+b)+c} properties of addition & multiplications. (5/6)
  2. Fractions and decimals: All operations on fractions by integers and fractions. Graphical representation of fractions and operations (4/5/6).
  3. Data Handling: Tabular, Range, mean, mode, median, bar-chart, intro. To probability (5/6)
  4. Simple Equations: Formulate and solve simple equation:, 1 variable. (6)
  5. Lines and Angles: Complementary, Supplementary, Adjacent, opposite, (6) transversal, with parallel or non parallel lines (6)
  6. Triangles and properties: Equilateral & Isosceles, Right hand, Pythagoras theorem.(6)
  7. Congruence of Triangles: Identical (6)
  8. Comparing quantities: %, interest computations, (6)
  9. Rational Numbers =whole /+integer (Natural +0= Whole+ negatives=integers) (5/6)
  10. Practical Geometry (5/6)
  11. Perimeter/ area, (Triangle, Polygon, circle), conversion of unit (6)
  12. Algebraic expressions, formulation/ interpretation (6/7)
  13.  Exponents and powers (6/7)
  14. Symmetry, tessellations, (6/7)
  15. Visualize Solid Shapes, cuboids, cylinder, cube, sphere, pyramid, cone, Volume (6/7)

CBSE Grade VIII (All of this should be over by Grade 7, and some earlier too)

  1. Rational Numbers: Basic property of numbers and operations, definition?? (No)
  2. Linear Equations in one variables: Much too simple (grade 5/6)
  3. Understanding Quadrilaterals: Define Polygon, and use triangle based simple geometry problems (gr. 5-6-7)
  4. Practical Geometry: Relatively simple triangle shape linked problems (Gr. 6-7).
  5. Data Handling: Tables, graphs, Pie charts, Introduce chance, (Gr. 5-6-7)
  6. Square and Square Roots: Simple Arithmetic (7)
  7. Cube and Cube Roots: Simple Arithmetic (7)
  8. Comparing quantities: Use of %, Interest computations (7)
  9. Algebraic expression and Identities: Simple Algebraic manipulation, no  formulation/ generalization (6/7)
  10. Visualizing Solid Shapes: Nomenclature and two dimensional Z-section/ shadow of solid objects (7)
  11. Menu ration: Area, volume formulae of various shapes/ objects(6/7)
  12. Exponents and Powers: (7)
  13. Direct and Inverse Proportion (7)
  14. Factorization (5/6/7)
  15. Introduction to Graphs (6/7)
  16. Playing with numbers (4-7)

Grade 9 (All of this should be over by 8, while some topics of grade 10, particularly geometry would be covered by 8th grade)

  1. Number systems: Gr. 7
  2. Polynomials, Gr. 8,
  3. Coordinate Geometry, Gr. 7/8
  4. Linear Equations in two variables, Gr. 7/8
  5. Introduction to Euclid Geometry Gr. 6/7/8
  6. Lines and Angles Gr. 6
  7. Triangles Gr. 7
  8. Quadrilaterals Gr8.
  9. Areas of Parallelograms and Triangles, Gr. 8.
  10. Circles, Gr. 7/8
  11. Constructions, Gr. 7,
  12. Heron’s formulae, Gr. 8, advanced,
  13. Surface areas and Volumes, cuboids, cylinders, cone, sphere (9)
  14. Statistics (8-9)
  15. Probability: Introduction (7/8)
  16. Profs. In Math: Gr. 8, focus on geometry (7/8/9)
  17. Mathematical modeling Gr. 8 to start Programming, flow charting (9)

Gr. X (Most of the following should be started/ completed in grade 9, with grade X, focusing on more advanced applications, complex problems, more of trigonometry, coordinate geometry, Quadratic equations, probability, Excel based data-analysis, Matrix (?), Mathematical proofs & Programming)

  1. Real Numbers, Euclid’s division lema and theorems of arithmetic
  2. Polynomial, geometric meaning, division of polynomials
  3. Pair of linear Equation for 2 variables, graphical method of solution
  4. Quadratic equations
  5. Arithmetic progression (Gr. 8)
  6. Triangles- similar, Pythogoras theorem (Gr. 8)
  7. Coordinate Geometry (9,10)
  8. Introduction to Trigonometry (9/10)
  9. Some applications of Trigonometry (10)
  10. Circles-Tangent to circle (9/10)
  11. Constructions (9)
  12. Areas related to circle, segment, sector (9)
  13. Surface, area and volume of combination of solids (9/10)
  14. Statistics, Mean, mode, median, cumulative frequency distribution (8th)
  15. Probability (9/10)
  16. Proof in Math (7-10)
  17. Mathematic modeling (7-10)