malaysia number framework

Early number and whole-number numeration Frameworks for teaching primary mathematics

MALAYSIA 10.4.03

INFORMATION ON EARLY NUMBER AND WHOLE-NUMBER NUMERATION

This module, early number and whole-number numeration, focuses on the early work in number, from prenumber to counting and number itself and onto numeration for small and large numbers. The sections to be covered (and the page numbers) include:

(i) prenumber (page 2) - attribute recognition, sorting and classifying, patterning, comparison and order, and 1-1 correspondence;

(ii) the concept of number (page 4) - counting, numerals, number principles, and counting stages;

(iii) numeration principles (page 6) – counting (odometer), separation (base or value, position), wholistic, structure (multiplicative relationship);

(iv) numeration processes (page 8) - number recognition, place value, seriation, equivalence, comparison and order, renaming, and rounding;

(v) sequencing (page 10) - concept of number, early grouping, 2 digit numbers, 3 digit numbers, and large numbers;

(vi) practice activities (page 11) - six directions (Rathmell triangle), & games; and

(vii) misconceptions (page 12) - unlearning and relearning, teens and zeros, base vs position, role of the one’s position, additivity, and division relationship.

Although this is the content that we have to cover in this module, we will also be looking at how to teach this material. Thus, the focus of the module is on the processes, principles and strategies that make up number, the materials that can most effectively develop these processes, principles and strategies, and the techniques and language which can be used most effectively with these materials.

Number is one of the mathematical ideas that has a central language component. A number, like 7, is associated with:

(i) a number of objects, say the number of days in a week;

(ii) a name, here “seven”; and

(iii) a symbol, here “7”.

Thus, the major teaching model utilised in this section of work is the Rathmell triangle which relates model (a representation of the objects) with language (name) and symbol in 6 relationships:

model --> language model --> symbol language --> symbol

language --> model symbol --> model symbol --> language

Note: Before we begin there is one point to make about number. The things we write down and say (e.g. 2, 3, "four") are not strictly numbers. They are numerals or words. The word "number" is reserved not for the label but for the idea, i.e. what two means, the "twoness" of things. And this "number" is a concept fairly unique to children and any learners in that it does not exist in reality. There is no real world example of "number" (like there is of "chair" or "cube"). It is totally a construct of our minds, something we put on the world. It is an example of how our minds structure the reality around us.

Tom Cooper, School of Mathematics, Science and Technology Education, QUT, 1999


PRENUMBER

To count requires the recognition of different things - the sorting of the objects being counted into those to be counted and those that have been counted, 1-1 correspondence between number names and objects, and the ordering of a path through the objects for counting.

Attributes

Number can not be understood unless children can recognise when things are the same and when they are different. Difference is the basis of number because number tells how many different things there are. Same is the opposite of different and is essential for understanding difference.

Hence, the first step to number is attribute recognition, that is, the ability to recognise likeness and difference. For example, recognising everything that has the colour red, or can be sat on, or rattles, and so on. This is well explained in Irons (1992a - READING 1).

The second step is matching objects on the basis of attributes. This focuses on the likeness of objects, matching those that are like.

Underlying this section is the continuous-discrete dichotomy. Some things are continuous (unbroken), like height. Other things are discrete (in parts), like pencils. To be counted, something must be discrete. One of the earliest learnings is detecting when there is a difference - when something continuous has a break, becomes discrete (e.g., a door in a wall, a change of colour). The basis of measurement is changing something continuous like mass into something discrete so that number may be applied to it. This is done by using a unit and determining how many of the unit equal the measure.

Sorting and classifying

From matching, activities move onto sorting and classifying. This means identifying attributes and sorting sets of objects into subsets by these attributes (e.g., sorting by colour, shape, size or number). The idea is that sorting by colour, size or shape will underpin sorting by number, and, therefore, identifying different numbers and giving one number to all collections with that number of objects.

The sequence of activities would be starting by sorting on one attribute, going on to sorting by two attributes, and then to three or more. Once again, there is reversing - asking children to sort objects, and asking children to identify the attributes used with objects that have already been sorted. Remember, also that sorting can be done in terms of no attribute (e.g., not red or not large).

Venn diagrams (see activity below) can be used with sorting. These are used to separate subsets when a set is being sorted in more than one way. They are overlapping circles or hoops.

It should be noted that sorting lays the basis for set language such as AND, OR or NOT. AND means that both attributes have to be present. OR means that only one attribute (or both) have to be present. NOT means that the attribute is not present.

Materials that can be used for sorting vary from informal to formal. Anything that has more than one attribute is useful (e.g., colour, size, shape, etc.). Informal objects are toys, house-hold material (clothes pegs, scissors, etc.), pencils, etc. Formal materials are unifix, pattern blocks, logic attribute blocks, etc.. Pictures can also be sorted by rough circles drawn around groups of objects.

ACTIVITY

Materials - logic attribute blocks, plastic rings

(1) Place two rings so they overlap. Label one ‘blue’ and the other ‘triangle’. In turn, children choose a logic attribute block and place it correctly with respect to the rings - blue objects in ring labelled blue, triangles in ring labelled triangle, blue triangles where the rings overlap, and all other objects outside of the rings.



(2) Place three rings so they overlap. Label one ‘red’, one ‘thick’ and the third ‘square’. In turn, children choose a logic attribute block and place it correctly with respect to the rings.

(3) Place three rings so they overlap. Label one ‘not larger’, one ‘circle’ and the third ‘not yellow’. In turn, children choose a logic attribute block and place it correctly with respect to the rings.

(4) Place two rings so they overlap.

(a) Label one ‘red’ and the other ‘rectangle’. In turn, children choose a logic attribute block and place it correctly with respect to the rings. The material in the overlapped part is ‘red AND rectangle’.

(b) Label one ‘yellow’ and the other ‘circle’. In turn, children choose a logic attribute block and place it correctly with respect to the rings. The material in both hoops is ‘yellow OR circle’.

(c) Remove one ring. Label this ring ‘blue’. Place the material. The material outside the ring in ‘NOT blue’.

Patterning

Patterns are rules that repeat. Patterns can be linear, two-dimensional and three dimensional. Finding and following patterns is the basis of mathematics. The idea is that patterns with other attributes (colour, size, shape) will precede and lay foundations for number patterns.

Children often confuse patterns with designs that have some form of structure, particularly if that structure is symmetrical. Students often believe that the next item in this pattern, A B B A B B A B B A B, is A because the pattern started with an A.

When undertaking patterning activities, it is important to both interpret or follow patterns (give the children a pattern and ask them complete it) and construct patterns (children make a pattern and teacher has to check that it has a rule that allows the next member to be determined. This interpretation-construction dichotomy is an example of reversing. When interpreting a pattern, it is easier for a learner to determine the next member on the end of a pattern than to determine a missing member in the middle of a pattern, so finding middle members should follow finding end members. When sequencing patterning activity, it is useful to begin with colour and shape patterns before moving on to number patterns. It is also useful to build patterns on one attribute before moving onto patterns made up of two or more attributes and before moving onto patterns with distracters. Patterning activities should occur throughout the primary years. In fact, patterning is one of the ways to introduce algebra.

Useful materials are unifix, pattern blocks and logic attribute blocks. Patterns can also be drawn and picture patterns can be used in this area.

There are three types of patterns in later years (and, I guess in early years).

(1) The first is when the pattern has a section that repeats, for example, 1, 2, 2, 1, 2, 2, 1, 2, 2, …

(2) The second is when the pattern is between consecutive members, for example, (a) adding 3 - 1, 4, 7, 10, 13, 16, ...; and (b) multiplying by 2 and adding one - 1, 3, 7, 15, 31, 63, ....

(2) The second is when the pattern is between the number of the member and the member, for example, (a) number plus 3 - 4, 5, 6, 7, 8, 9, 10, ...; and (b) number multiplied by 2 and adding one - 3, 5, 7, 9, 11, 13, ....



There are also “pattern” when a rule is used to change a list of numbers, for example, (a) adding 3 - 2 --> 5, 11 --> 14, 45 --> 48, ...; and (b) multiplying by 2 and adding one - 3 --> 7, 32 --> 65, 457 --> 915, ..… This rule can be called a “change rule” rather than a “pattern rule”.

In all these examples, it is important to remember that we must reverse - follow the pattern, and find the pattern.

Comparison and order

The idea here is that comparing objects for attributes such as height and mass leads into comparing numbers in terms of size of number. Comparing is looking at two objects and seeing which has the more of a particular attribute. Ordering is looking at more than two objects and putting them in sequence from the one that has the least of the attribute to that which has the most.

Comparing should precede ordering. Ordering should move begin with three objects. For three objects, the first activities should involve comparing two objects and then adding in a third object which is the smallest or the largest. Then, activities should move to comparing two objects and then adding in a third which is in between the first two. The important feature in the move from two objects (comparing) to three (ordering) is the development of the notion of ‘betweenness’ - that objects can have attributes which means that one is between the other two in terms of these attributes. Comparing means a lot of new language. In terms of length we have thick/thin, narrow/wide, short/tall, etc..

1-1 correspondence

One crucial thing in number is putting number names in 1-1 correspondence with objects - one number name for each object. The idea here is that this can be assisted by putting one set of objects in 1-1 correspondence with another set (e.g., one egg for each eggcup, one knife for each fork, one fish for each fishing line, etc.). The evidence for this particular approach is not strong. There appears to be evidence that putting two sets of objects in 1-1 correspondence with each other is more difficult than putting number names (words) in 1-1 correspondence with objects.

This activity can be done with 3D materials and with pictures. For pictures, the corresponding images are shown by a line drawn between them.

THE CONCEPT OF NUMBER

Counting

Counting involves the following:

(i) rote counting - the words in the right order;

(ii) rational counting - count how many and count out a subset;

(iii) cardination - “how many” (last number name gives this);

(iv) ordination - aspect of number which gives order;

(v) numerical comparison/order - using 1-1 to see which has more;

(vi) conservation - number is the same when nothing added/taken away;

(vii) 1-1 correspondence - number names to objects;

(viii) perception - pattern or path through objects;

(ix) separation - into those that have been counted and those yet to be counted;

(x) subbitisation - counting by “seeing” (numbers 2 to 5); and

(xi) adding (subtracting) - counting by subbitising and adding the subbitised groups.

Numerals



The Rathmell triangle is used to develop numerals. Objects are counted to give number as language and then a numeral is associated with this language and the objects (model).

OO OOO OOOO OOOOO OOOOOO OOOOOOOO“two” “three” “four” “five” “six” “seven” 2 3 4 5 6 7

Because, we have a base of 10 in our number system, we only have numerals for 0 to 9. The numeral for ten is one ten and zero ones or 10. The problem for us in our numbers is that our language is based on a base of 20 (“score”) and so we have language for 0 to 19 (“one”, “two”, ..., “nine”, “ten”, “eleven”, “twelve”, “thirteen”, ..., “nineteen”). The last ten number names should be “onety”, “onety-one”, “onety-two”, ..., “onety-nine”.

Sequencing involves the following order - 1-4, 0, 5-9, 0-9 sequence, 1 more/1 less, ordinal numbers (fifth, etc.), 10, counting on/back, separate subgroups, and subsets. Reinforcing numbers involves matching sets, relating objects, name and numeral, ordering numbers, subbitisation, and subsets.

Number principles

The number principles are:

(i) fixed order - number names follow a set order (“one”. “two”, “three”, ...);

(ii) 1-1 correspondence - each item tagged with one and only one number name;

(iii) cardination - last number name tells how many;

(iv) invariant order - items can be counted in any order (number is the same); and

(v) abstraction - any discrete items can be counted.

These number principles the counting information above are based on the work of Gellman & Gallistel (1978). This is the work upon which Education Queensland’s Early Numeracy Net is also based.

Counting stages

Steffe et al (1983) developed counting stages (these stages are used in early intervention programs in NSW and Victoria):

(i) perceptual - bounded pattern of counted items;

(ii) countable perceptual - perceptual coordinated with number name;

(iii) countable motor unit - motor act coordinated with number words;

(iv) number word pattern - pattern of number words uttered in sequence;

(v) composite unit - items are units but also combine to give number;

(vi) abstract unit - can be counted and counted on;

(vii) number more - counting can be repeated for an extension;

(viii) iterable unit - can count in terms of the number as well as count the number; and

(ix) measure - number can be interchanged with one as a counting unit.

NUMERATION PRINCIPLES

This section has no reading and all material is here. These principles result from a structural perspective on mathematics. This perspective believes that it is more important to develop these principles across all numbers than learn particular sized numbers. It also provides framework in which all whole umber and decimal numbers can be considered.



Counting

The starting point for numeration is counting. However, counting does not stop with 100. All place values count, as do decimals and fractions.

Odometer

Odometer is a name given to the principle which describes the nature of all place-value positions to count like the ones position. For example:

COUNTING COUNTING COUNTING BY ONES BY HUNDREDS BY THOUSANDS

25 47 567 335 76426 47 667 336 76427 47 767 337 76428 47 867 338 76429 47 967 339 76430 48 067 340 764

This odometer pattern is for counting back as well as counting on, for example:

32 65 218 872 45531 65 118 871 45530 65 018 870 45529 64 918 869 455

The best material for teaching this principle is a calculator. For example, enter 47 567 and then add 100 and keep pressing =.

Separation/Numerical partitioning

Separation or numerical partitioning (Van de Walle, 1990) is the part-whole part of numeration - it is seeing a number as its place value components. That is, 263 is 2 hundreds, 6 tens and 3 ones. Children need to partition flexibly - 25 can be 2 tens and 5 ones, but it also can be seen as 25 ones and 1 ten 15 ones.

Materials such as MAB and bundling sticks have the partitioning/separation built in. As soon as a number is represented with this material, it is numerically partitioned. Numeral expanders also expand or separate/partition numbers.

Base or value

Our numbers are based on ten and powers of ten. As we move along the place values to the left, the value of the numbers increases from one to ten to hundred to thousand and so on. This increase of value is based on grouping by tens - a hundred is ten tens, a thousand is ten hundreds, and so on - and a pattern of threes:

| * ones, tens and hundreds | * thousands, ten-thousands and hundred-thousandsV * millions, ten-millions and hundred-millions

Then, if we follow the USA number system, this pattern continues:

| * billions, ten-billions and hundred-billionsV * trillions, ten-trillions and hundred-trillions

However, if we follow the British system, the pattern continues in sixes:

| * millions, ten-millions and hundred-millions | * thousand millions, ten-thousand millions & hundred-thousand millionsV * billions, ten-billions and hundred-billions



That is, a billion is a thousand million in the USA system; while the billion is a million million in the British system,. We are tending to adopt the USA system.

To show value or base, we need to use a base material. The best of these include bundling sticks and MAB. Unifix can also be used as can pictures of bundling sticks and MAB. However, unifix tends to lead to the base vs position misconception (see below). Bundling sticks tend to be used before MAB because MAB have to be traded while bundling sticks can actually be regrouped and grouped by removing the rubber band or adding the rubber band respectively.

Position

In our number system, we do not have a special numeral for the bases. We only have numerals for the number of each base. (Thus, we only have numbers 0 to 9.) We show our base by the position of the digit with respect to the ones position. This position is called place value. Thus, the 2 in 254 is a hundred while the 2 in 324 is a ten. This also means that we need a “place-holder”. When we have 2 hundreds and 5 ones, the 2 must still be two positions to the left of the ones (the 5). This is done by using a zero, that is, 205. However, since our language does not require us to say this zero (we say “two-hundred and five” instead of “two-hundred and zeroty five”), children tend to have problems with numbers that have a zero in them.

Position is shown by position materials, such as place value charts (PVCs), abaci or chip trading (different coloured counters on PVCs). Position is a visual image, hence, we need to use the kinaesthetic sense in teaching. An effective way to do this is to move one’s hands across the positions as one says the number. This is a good approach regardless of year level. Base and position are effectively introduced by placing a base material on a position material (e.g., MAB on a PVC). The teacher has to make sure that the base vs position misconception (see later) does not occur. Numeral expanders and calculators are also useful.

Wholistic/Quantity

It is a weaknesses in some areas that MAB and bundling sticks emphasise separation – it means it is difficult to use them to teach the aggregation (e.g., 45+28=65+8=73) and wholistic (e.g., 45+28=45+30-2=75-2=73) strategies for computation. These require a wholistic emphasis – for example, the number 28 is one number which represents an actual single quantity.

The materials for quantity are different to those for separation - one has to consider numbers more wholistically - there is a need to use materials that emphasise position, for example, 99 or 100 boards and number lines

- drawn with intervals and numbers:

0 20 40 60 80 100 120 140

- drawn without numbers or intervals (open):

Structure/Multiplicative relationship

This is almost the most fundamental of the principles. It says that the place-value positions follow a multiplicative or exponential pattern where movement to the left is multiplying by 10 and movement to the right is dividing by 10. This pattern is bi-directional and continuous across the positions.

The pattern is evident in the base materials, for example, the ten in MAB (long) is one-tenth of the hundred in MAB (flat). It is not well understood by children, probably because of the way new place-value positions are introduced. We tend to introduce thousands from hundreds by bundling hundreds into groups of 10. We then say that 10 hundreds is one thousand. Thus, we move to the left by multiplying by 10. However, we tend not to reverse - show that a hundred is one-tenth of a



thousand. We have to do this so that children can see that dividing by 10 and moving to the right applies in whole numbers as well as decimals.

The materials to teach this relationship are calculators, digit cards and PVCs. As the calculator multiplies by 10 and divides by 10, the numerals/digits can be seen moving to the left and right respectively on the PVC. The Chinese or Japanese abacus is also useful here.

NUMERATION PROCESSES

Once again, this section has no reading. The processes in this section represent the straightforward skills a child needs in whole-number numeration. The principles separation and wholistic to which they most commonly align are given in square brackets.

Number recognition and place value [Separation]

This consists of reading and writing numbers and determining the place values of their digits. The difficulties in reading and writing are, of course, the teens and the zeros. The difficulties with place values seem to be restricted to larger numbers and the pattern of threes.

The reading and writing represent a reversing. A reversing which is the basis of an interesting place-value activity is to provide children with positions and require the number, for example,

3 tens, 2 hundreds and 5 ones ……..... 8 ones, 4 thousands and a ten ............

The material for this process is MAB and bundling sticks on PVC s, calculators, and numeral expanders. Calculators are an interesting and effective teaching resource for these processes (e.g., the game wipeout is an effective calculator activity). They can be used for reading and writing. A teacher can state the digits and the children can enter these on their calculator and read back the number, and a teacher can read the number and the children can enterthis number on the calculator and read back the buttons pressed.

These processes are based on relating model to language to symbol (the Rathmell triangle). The sequence of steps is as follows (MAB ten is represented by | and MAB unit by .):

| Tens | Ones Questions and directions: How many tens? | | | | | . . . . [Three] Write the number of tens in the tens | position. How many loose ones (or, if we | Tens | Ones disregard the tens, how many ones are left | 3 | 4 over)? [Four] Write the number of loose | ones in the ones position. (Note - there are | 3 tens and 4 ones 34 ones in the number.) Remove the lines in | the PVC to obtain the final symbols, V 34 stressing the role of the ones.

The aim of the questions and directions is to stress that the numeral for the left column has to be the number of tens and the numeral for the right column is the number of ‘left-over’ ones. That is, 3 MAB longs and 4 MAB units ( | | | . . . .) is 3 tens and 4 ones which is 34. It is important for the correct development of numerals from MAB, bundling sticks and PVC that 30 as 3 tens and zero ones. Difficulties can eventuate if three MAB longs (| | |) are labelled with the symbol 30.

Seriation, comparison and order [Wholistic]

Seriation is determining the number that is one before or one after (similar to odometer). Comparing is working out the larger/smaller while ordering is putting three or more numbers in sequence smallest to largest. Seriation is based on the odometer principle. Comparing and ordering are based on comparing in turn from largest place-value position to smallest. This is the difficulty for comparing and ordering. Words are read left to right. Numbers are read left to right but their place values are determined right to left. Now with comparison, we have to go back to looking at place values from left to right.



A calculator can be useful for seriation but so can an abacus and MAB on PVC. Comparing and ordering are assisted by the use of a PVC. MAB and bundling sticks can assist but so can a 99 board and a number line. The game dice-order is excellent at teaching that the left-most digits crucial in comparison.

Equivalence and renaming/regrouping [Separation]

The zero in between numbers is a place-holder and effects the value of the number. However a zero on the LHS of other digits does not change the number (e.g., 024 = 24).

Renaming/regrouping is equivalence in which numerical partitioning applies (e.g., 24 = 1 ten and 14 ones). Renaming we allocate to this equivalence when it is represented symbolically and regrouping when it involves material. Thus, 2 longs and 4 units traded to 1 long and 14 units is regrouping and 24 = 1 ten + 14 ones is renaming.

Although regrouping is a generic term, it also has a particular meaning. In whole numbers, when ones form tens or tens form hundreds this is commonly called grouping and when hundreds form tens and tens form ones, this is commonly called decomposition.

There are two types of renaming/regrouping: type 1 - 234 = 23 tens and 4 ones = 234 ones (changing complete place values); and type 2 - 234 = 1 hundred 11 tens and 24 ones (only some of a place-value position are changed).

The type 1 renaming/regrouping is best taught with the numeral expander. The type 2 requires MAB or bundling sticks. The game Blockbuster has an important learning role here.

Rounding and benchmarking [Wholistic]

Rounding is determining the ten, hundred, thousand, etc., that a number is closest to. Benchmarking is finding a ‘central’ number (e.g., 50, 300, 11 500, etc.) that the number is near and noting whether the number is less or more than the benchmark. The most effective material here is a numeral expander or 99 board and number line.

SEQUENCING

Concept of number

The concept of number is an amalgam of rational counting, counting principles and numeral understanding. It is the starting point for numeration and for operations. It needs a lot of work with many objects relating language and symbol to model and real world situation.

Early grouping

When number is first visited (in preschool and Year 1), children are encouraged to count one object at a time. This is to reinforce rote and rational counting. Then, in later Year 1, the children are encouraged to start to count in tens. To assist this, counting activities can be organised where the children count in groups of less than 10. These activities can provide background to numeration. They can help children learn:

(i) how to group with materials;

(ii) that groups go on left of ones and ones go on right of groups;

(iii) that groups of groups take another place to the left of groups;

(iv) that ones can be grouped to groups and groups can be decomposed to ones; and

(v) that the group is crucial to understanding number.

Thus, early grouping underlies base, position (place value) and renaming/regrouping.

2 digit numbers



To introduce 2 digit numbers, the recommended approach is to introduce tens and ones (counting by tens) and the use of a PVC before moving onto full name and full symbols:

| * tens with material and language | * tens and ones with material and language | * numbers like 46, 78, etc. with material, language, symbols | * numbers like 27, 34, 56, etc. with material, language, symbols | * numbers with a zero (e.g., 30) with material, language, symbolsV * teens with material, language and symbols

For translating to symbols, the sequence is:

| * 4 tens and 7 ones | | * Tens | Ones | 4 | 7 | | forty-seven |V * 47

The materials that appear most effective for 2 digit numbers are bundling sticks and PVC followed by MAB and PVC (base material and position material together).

3 digit numbers

Once again, the recommended approach is hundreds, tens and ones with material, then PVC, and finally symbols. The sequence is:

| * hundreds with material and language | * hundreds, tens and ones with material and language | * numbers like 468, 784, etc. with material, language, symbolsV * numbers with zeros and teens with material, language, symbols

Once again, the zeros and teens are left to the last.

Large numbers

Large numbers must be taught. There is still the need to use PVCs, numeral expanders and calculators (2D and pattern materials). There is still the need to put digits on PVCs and to use the kinaesthetic sense to drive home the pattern of threes. There is a special numeral expander for the pattern of threes.

A way of setting up stating a number like 346 786 254 is to cover, in turn, two of the threes 346, 786 and 254 and to read the three remaining digits as hundreds, tens and ones. Then apply millions to LHS three digits, thousands to middle three and ones to RHS three and read the number.

The sequence for introducing large numbers is:

| * thousands with material and language | * thousands, hundreds, tens and ones - material, language, symbols | * thousands etc. with zeros and teens - material, language, symbols | * pattern of threes & ten-thousands/hundred-thousands - lang/symbols | * millions as thousand thousands - language, symbolsV * pattern of threes past millions - language, symbols

There are some interesting calculator activities for large numbers. For example: (a) the game target; and (b) operating on large numbers, as below

43 256 785 953 + 28 858 459 268 = ............................. 121 654 829 002 - 67 878 392 117 = .............................



9 456 347 x 4 582 = .............................

PRACTICE ACTIVITIES

It is important to practice numbers until they become very familiar, almost automated. Then the numbers will be available in problems without removing thinking from the problems.

The six directions of the Rathmell triangle

The six directions of the Rathmell triangle below provide a basis for practice:

model ----> languagelanguage ----> modelmodel ----> symbollanguage ----> modellanguage ----> symbolsymbol ----> language

Games

There are a variety of game-like activities that are useful in practicing numeration skills. Some examples are:

(1) Blockbuster. Children each have a PVC. In turn, the children throw two dice and take MAB units to the value of the sum of the two dice. They regroup the units to tens if necessary. The first child to 100 wins. The teacher asks children their score and the ten above, asks if they think their next throw will get them to this ten, and, after the throw, asks if it has been large enough to do this. If it has, the teacher asks the child if the number is enough to get to the next ten, or past the next ten, and then how far past. The game can be played backwards from 100.

The game introduces place value, regrouping/renaming, building to 10s, and basic addition facts.

(2) Mix & match cards. For a variety of numbers, A4 sheets of coloured paper (all the same colour) are covered with different representations of the same number (with the same representations in the same positions on the A4 sheet). Representations are symbols, language and different models. Each of the A4 sheets is cut up differently into the various representations and these put in a box. Children have to assemble all the pieces that are associated with the same number - they can see that they are correct because the pieces fit together.

(3) Bingo. Three (or more) different coloured A4 sheets are divided into 3 by 4 small rectangles. Different representations of numbers are placed on each sheet (e.g., one sheet is numerals, one sheet is language, one sheet is models). The same numbers are in the same position on each sheet. Two sheets are cut up for playing cards. The third sheet is kept for a playing board. The cards are divided up amongst the children who, in turn, cover the board with the corresponding cards. Three in a row wins?

(4) Dominoes. Make up a set of dominoes with different representations of numbers on each side.

(5) Dice order. This has a playing board consisting of rows like:

[ ] [ ] less than [ ] [ ] ..................[ ] [ ] less than [ ] [ ] ..................[ ] [ ] less than [ ] [ ] ..................

A dice is thrown four times. After each throw, students place the number in one of the four squares. If, at the end of four throws, their LHS number is less than their RHS number, they



score the value of the digit in the tens position of the LHS number, otherwise zero. Game can have rows like above or:

[ ] [ ] [ ] less than [ ] [ ] [ ] ..................

[ ] [ ] less than [ ] [ ] less than [ ] [ ] ..................

(6) Wipeout. Students enter a number on their calculator. A digit is called and students have to wipe that digit to zero by one subtraction, for example,

2 345 wipe the 4 [subtract 40] 52 987 wipe the 2 [subtract 2 000]

Numbers can be wiped to zero, that is, all digits wiped; or only one digit wiped. Can also add to wipe digit, and do something to just change the digit (not necessarily wipe it).

(7) Target. Students given a starting number and a target number. The starting number is entered on the calculator with the multiplication sign. Now guesses are pressed along with equals until the target is reached. If the calculator is not cleared and multiplication is not pressed again, the calculator will continue to multiply the guess by the starting number. The idea is to reach the target by the least number of guesses. You need to develop a worksheet with spaces to record the too high and too low guesses.

MISCONCEPTIONS

Unlearning and relearning (changes in grouping - series of compromises)

The children are taught, first, to count in ones, then to count in tens exactly, and finally allowed to decompose to have more than ten ones. This requires unlearning and relearning. Without effort in these areas, problems can occur.

Teens and zeros

Seventeen should be “onety-seven”. It clashes with seventy and seventy-one because in seventeen the ones are said first. This causes misunderstanding.

Similarly, 30 is 3 tens and zero ones, but so often students see 30 as just thirty or 3 tens. Our language does not say the zero, so zeros also cause difficulties.

For example: two thousand and four 2 004

thirty thousand and fourteen 30 014

Base vs position (204 instead of 24)

In the representation below, three tens and four ones is shown by 3 MAB longs and 4 MAB units on PVC (base on position). The longs are ten times the size of the ones.

Therefore, the material will tend to show that there are 30 tens and 4 ones, that is, that the number is 304 not 34. This has to be prevented by questioning and directions that focus children’s thinking on the number of tens being the number put in the tens column.

Error: Tens | Ones | | | | . . . .

30 4

304

This difficulty is enhanced in the hundreds. In the example below, it is easy for students to think of the 5 flats as 500 giving 50034 for five-hundred and thirty four.

Error: Hundreds | Tens | Ones [ ] [ ] [ ] [ ] [ ] | | | | | . . . .

500 3 4



50034

Role of the one’s position

The position of digits gives their place value. This position should not be determined in relation to the RH digit but in relation to the ones. For example, two positions left of the ones is the hundreds. Failure to see the relation to ones gives rise to errors in decimal numbers, where the ones sets up both whole-number and fraction place values.

Additivity

When, for example, MAB are used to move from 10 to 100s, it is done by adding nine more 10s to the already present one 10. Thus students can come to believe that the relationship between 1 and 10 is +9, 10 and 100 is +90, 100 and 1000 is +900. Thus, they see the relationship in additive terms when it should be seen in multiplicative terms, that is, x10 as move to the left and /10 as move to the right. Structural materials such as digit cards and calculators have to be used with PVCs to introduce and reinforce this multiplicativity rather than separation materials such as MAB and bundling sticks.

Division relationship (reversing)

It is important to show that a hundred is ten tens and that a ten is one-tenth of a hundred. It is important that children realise that as digits move from left to right, their value decreases by a tenth. Without this, children will have problems with the continuous and bi-directional nature of the multiplicative relationship between place value positions in decimal numbers.

HM | TM | M | HTH | TTH | TH | H | T | O | | | | | | | 3 | 4 34

x10 | | | | | | 3 | 4 | 340x10 | | | | | 3 | 4 | | 3 400x10 | | | | 3 | 4 | | | 34 000x10 | | | 3 | 4 | | | | 340 000x10 | | 3 | 4 | | | | | 3 400 000x10 | 3 | 4 | | | | | | 34 000 000

/10 | | 3 | 4 | | | | | 3 400 000/10 | | | 3 | 4 | | | | 340 000/10 | | | | 3 | 4 | | | 34 000/10 | | | | | 3 | 4 | | 3 400/10 | | | | | | 3 | 4 | 340/10 | | | | | | | 3 | 4 34

The idea is to reverse, say, ten hundreds is one thousand to show that one hundred in one-tenth of a thousand.

REFERENCES

Gellman, R. & Gallistel, C.R. (1978). The child’s understanding of number. Cambridge, MA: Harvard University Press.

Steffe, L.P., von Glaserfeld, E., Richards, J., & Cobb, P. (1983). Children’s counting types: Philosophy, theory and application. New York: Prager Scientific.

Van de Walle, J. (1990). Concepts of numbers. In J.N. Payne (Ed.). Mathematics for the young child (pp. 63-87). Reston, VA: National Council of Teachers of Mathematics.


malaysia number framework

Documents

van de walle

logic attribute block

logic attribute blocks

teaching primary mathematics

putting number names

usa system

bundling sticks

a4 sheets