Make Math Real

8/14/2019 Make Math Real

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2007

Khalid Abozaid Morssey

[MAKE MATH REAL]A method for teaching math in primary school using helper objects

Cairo Governorate

El Sahel Educational Zon

Othman Ebn Affan ELS

Mathematics Departme


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MMR (Make Math Real)

Prologue:

As a teacher of maths Ill start by numbers, 15 years ago I was a student in faculty

of education in Ain Shams University, and since that time, Im searching for a method to

make maths real especially in my primary school. But for the good luck when I was in a

mission in Scotland 2004, I found a maths teacher called Richard Dunne. This old man

had the same problem and we made some workshops to find a method that Making

Maths Make Sense and at last was the great idea to use something looks like the first

man used to count calculus but must be familiar to our pupils so what about the soft

drink CUPS? Yes thats great, so my method is to use the CUPS. And because doing

something is the best way to learn it, we will act Mathematics operations using these

cups.

INTRODUCTION:

To make math real is the aim of this project by acting math concepts and

operations or by translate each math word into an action, so we have to usematerials that help our pupils to act math and make it real. The pupils will act math

in front of the class using concrete materials then using cards or what so called

pictorial materials, and at last using symbolic materials.

Ill concentrate on the big ideas of math which taught in primary schools as a

beginning.

The Big Ideas of Math1) Addition2) Subtraction

3) Multiplication4) Division5) Equals

Make Math Real (MMR) materials are designed to ensure that mathematics can

be taught so that it actually makes sense for all pupils. Teaching mathematics inan accurate way will ensure improve skill. It does require some study on yourpart but this work will be fully satisfied. The return is not only in the confidenceand ability of pupils; it is also in your recovering the complete enjoyment ofteaching.

Make Math Real (MMR)does not introduce new content (it is not newMath). Nor does it require expensive equipment. Its great value is in the fact thatit is based strictly on a powerful theory of instruction. The theory of instructionis widely applicable and similar materials have been designed for other schoolsubjects. In particular, it recognizes, develops and applies the difference betweenscientific conceptsand everyday concepts.

Khalid Abozaid


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In MMRThe plastic ( or paper) cups provide an educational object, this beingsomething that is used on purpose and only for the purpose of teachingbecause its use best represents the nature of what we want pupils to do. It isnot intended to be interesting. It does not offer any variety. You will see that

it is something for pupils to think with; not to think about. When pupils havedifficulty in recalling something they can return to objects that are veryfamiliar to them.

In MMRif you look at the first five Big Ideas + = below, you will seethat they are introduced in relation to two tables at the front of the class: theResources Table or Math Box (to the left) and the Math Table (to the right).Cups on the Resources Table or in Math Box (to the left) are moved to theMath Table (to the right).

Note that the Resources Table and the Math Table are themselves educational

objects.

Make Math Real (MMR):Acting the First Five Big Ideas.+ Add Get ready to get some more Pupil moves to the Resources Table and holds hands

aloft

- Take

away

Get ready to take some away Pupil moves to Math Table and holds hands aloft

Times Do the same thing lots of times Pupil picks up the same quantity andrepeats the previous action

Divided

by

Look at it and wonder about Pupil moves to the Math Table and

pointedly looks at it while scratching thehead

= Equals Look at the Math Table and

count

Pupil looks at the Math Table and

ostentatiously counts

This is my Resources Table: This is the Math Table:It is where I keep all my cups. This is where we do math.


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In MMR the language is very important.

Note in particular that looking at, say, 3 + 2 5, when the teacher says Readwhat it says, pupils respond with Three add two equals five. But when theteacher says Read what it means the pupils say Three cups add two cupsequals five cups.

In MMRwe need to be clear about the helper Units(CUPS).Look at reading what it means for each of these:3 + 2 5 Three cups add two cups equals five cups3 2 1 Three cups take away two cups equals one cup3 4 12 Three cups times four equals twelve cups6 2 3 Six cups divided by two cups equals three

This is because the educational object of cups is used to provide the highlyabstract equation 3 + 2 5 with a unit. But the unit is entirely imaginary;it is artificial; it is there because it enables learners to make sense of

something (3 + 2 5) that is otherwise so abstract it virtually defeatsunderstanding. The cups, introduced for educational reasons into numericalequations, are to be thought of as the Helper unit.

Let me return to the Big Ideas. Look first at the basic 3 + 2 5. In establishingfor educational reasons that we think of cups, the figures (3, 2 and 5) are

thought of, by the teacher, as nouns. But the + and the , being instructionsrequiring an action, are to be thought of as imperative verbs. And, of course,each of is an imperative verb. Therefore, 3 + 2 = 5 is thoughtof as:Noun imperativeverb noun imperative verb noun3 + 2 = 5

With this recognition in your mind, and the consequent ability to representsymbolism to learners in an active and visual way, you are well on the way to

successful teaching. Notice also that simply writing, say, on the board willdrive all your pupils to raise their hands in the general direction of theResources Table (math box). The symbol, and each of the symbols, has ameaning and some speech (eg Get ready to get some more) that can be

enacted in the absence of any surrounding numbers. In MMR the symbol does not require a context! It is therefore open to general use. It can be aseasily used with large numbers, fractions, decimals, negative numbers andalgebraic symbols as it can with small numbers. And that is the unique power


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of MMR. It teaches pure math to learners. It reverses the trend of trying toteach math always from a context. It is difficult to find any justification for

teaching math in context

The pattern of the appearance of the Helper Units for and in the last two,

is very different from + and

in the first two. That is why MMR refuses thepressure to teach multiplication as repeated addition. It is educationallyunsound because it entices learners to misunderstand the meaning of

multiplication. It is no good asserting to me that multiplication is repeatedaddition. Of course it is. But that is not the point. What I am talking about

here is how to design instructional methods that make math clear to pupils.What justification can there ever be for continuing with an approach that isdesigned to confuse?Multiplication must be in introduced separate from any mention of addition. Itis a Big Idea in its own right. Later, pupils will identify that repeated additiongives the same result as a multiplication: but that will be part of an on purpose

planned chance for symbolic reasoning. This act of reasoning (this act ofdiscovery; this act of creativity) can only take place if multiplication hasbeen introduced in its pure form. Equally, it is counter-productive tointroduce division as repeated subtraction because it is confusing for young

pupils and does not lend itself later to using factors. It is the logic of teachingdivision like (e.g. 3 = 6) and (7x 7 = x) that illustrates themethodology that pervades MMR. The Big Ideas + - x = are, remember, BigIdeas in their own right.


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Teaching the four operationsThe four operations ( +, - ,, ) are central concepts in teaching math. Theyare best taught in a way that applies unchanged to small numbers, large

numbers, vulgar fractions, decimal fractions, negative numbers, algebra andmuch else.

This avoids the multitude of rules generally taught in math In MMR, + - and = are given much more significance than is generally given. They are thefirst five Big Ideas designed to ensure rigorous math is accessible by all pupils.The manner in which they are taught is crucial to all subsequent learning.There is an essential feature that must be in the teachers mind in using theMMR Scripts to teach the first five Big Ideas. Each of the symbols + - = isto be thought of as an imperative verb; it is an instruction; it can be enacted

without reference to the symbol (usually a number) that follows it. When wesee 3 + 2 we do notsay3 [pause] add 2but 3 [pause] add [pause] 2.Each symbol is treated separately. It is for that reason the script below insiststhat + means get ready to get some more.You can see that 3 + 2 = 5, and all similar expressions, have the form:Noun; imperative verb; noun; imperative verb; noun. They have this formbecause cups are used as a proxy for the indeterminate object 3. The cupsenable you to think about the math.

This importantnoun; imperative verb; noun; imperative verb; noun form isuniversal. It is achieved by ensuring that novices without exceptionlearn that + means Get ready to get some more - means Get ready to take some away means Do the same thing lots of times means Look at it and wonder about = means Lookat the Math Table and count It is absolutely vital that each operation is thought of as an imperative verbprovoking a specific action without reference to the noun that follows. Inessence 3 + 2 = is thought of as what you start with; the type of job; the size

of the job; look at the math table; . This practiced set of actions ensures that

young learners inspect each symbol carefully (rather than assume any twonumbers on the page are inviting addition) and that this enables olderlearners to deal logically (rather than impulsively) with negative numbers,algebraic symbols etc.

Addition


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The basic script for learning the Big Idea of addition is shown below. It isfollowed by Detailed Notes on how the basic script underpins a full treatment

of addition at the elementary level.This is my Resources Table: This is the Math Table:

it is where I keep all my cups. this is where we do math.

Resources Table Math TableNeutral position

I need a helper. Stand in the middle here[places pupil in neutral position]. Myjob is to tell you what to do. You are the helper. Your job is to do what I tellyou. I want you toTeacher says Pupil does Comments

Put two cups on the math table picks up 2 cups, walks to the mathtable, places them on the mathtable and returns to the Neutral

position

Good

Now I want you toGet ready to get some more moves to the Resources Table andholds hands in the air, poisedabove the Resources Table, inpreparation for picking up cups

Good. You do notknow how much

to get until I tellyou. So wait

One cup . picks up 1 cup, walks to the mathtable and places it on the mathtable and returns to the Neutral

position

Good. Now listen

to what I wantyou to do.

I want you toLook at the MathTable and count the cups. Howmuch is there here?

Three cups. teacher uses bothhands tosurround the

cups to act theirmuchness, ie the

quantity


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This process is repeated many times so that helpers physically respond tothe spoken words: Start by putting [number]cups on the math table Get ready to get some more [Number]cups Look at the math table and count the cups [One, two, three, ] How m uch is there here? [Number]cups.Now I am going to pretend I have lost my voice: I cannot talk. My job is still totell you what to do. This time I am going to tell you what to do by writing.When I write [teacher writes the figure 2]Teacher writes Pupil does comments

2 Start by putting two cups on the m ath table teacher insertsplus sign2 + Get ready to get some more teacher insertsthe figure 12 + 1 pick upone cup and carry it to the math

tableteacher inserts

equals sign2 + 1 = Look at the math table and count the cups.count One, two, three

How much isthere here?

uses both hands to surround the cups to

act their muchness, say three cupsteacher inserts

the figure 32 + 1 = 3The numbers and signs have been written one by one (so that each is acted bythe helper) to build up the expression 2 + 1 = 3Learners need to understand expressions like 2 + 1 = 3 as a series ofinstructions that can be interpreted as physical actions and eventuallyvisualise as images of those physical actions.Detailed Notes1) The script above is illustrative. In fact, it is important to use longerexamples right from the start ( e.g. 3 + 1 + 2 + 2 = 8) so that pupils get usedto reading the symbols and responding to them (rather than assume thatthere will always be just two numbers before the arrival of the equals sign).

2) When 3 + 1 + 2 + 2 = 8 (for instance) has been acted, pupils need to betaught to respond to Read what it says by saying Three add one add two addtwo equals eightand to Read what it means by saying Three cups add onecup add two cups add two cups equals eight cups.


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3) When the teacher writes 3 + 1 + 2 + 2 = 8 (this is called a Math Story)pupils must be taught to act the Real Story (i.e. to act it with the cups). So

they must respond to:I will write the Math Story. You act the Real Story.

4) When the teacher acts the Real Story (i.e. acts it with the cups) pupilsmust be taught to write 3 + 1 + 2 + 2 = 8 (this is called a Math Story).Detailed attention must be given to ensuring they form their numbers andsymbols correctly. So they must respond toI will act the Real Story. You write the Math Story.5) The whole point of 3) and 4) above is to create an instinctive relationshipbetween the Math Story and the Real Story.Practice with examples like these:3 + 1 + 2 = 6 5 + 1 + 0 + 1 = 7 1 + 4 = 5 0 + 0 + 0 =0 2 + + = 3 + 1 + = 1 1 + 2 =4


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SubtractionI need a helper. Stand in the middle here[places pupil in neutral position]. My

job is to tell you what to do. You are the helper. Your job is to do what I tellyou. I want you toTeacher says Pupil does comments

Put three cups on the math table. picks up 3 cups, walks to the mathtable, places them on the Math

Table and returns to the Neutralposition

Good, I wantyou to

Get ready to get take away moves to the Math Table, holdshands in the air, poised above themath table, in preparation for

picking up cups

Good. You do

not know howmuch to get

until I tell you.

So wait

One cup picks up 1 cup, walks to theResources Table and places it inResources Table

Good. Now

listen to what Iwant you to do.I want you to

Look at the math table and count thecups.How much is there here?

Two cups. Good.

It is an essential part of the script for the teacher to indicate all the cups on themath table and to say How much is there here? to elicit the precise answertwo cups and not simply two.This process is repeated many times so that helpers physically respond tothe spoken words: Start by putting [number]cups on the math table Get ready to take some away [Number]cups Look at the math table and count the cups [One, two, three, ] How m uch is there here? [Number]cups.


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Now I am going to pretend I have lost my voice: I cannot talk. I am still theboss so I still have to tell you what to do. This time I am going to tell you what

to do by writing. When I write [teacher writes the figure 3]Teacher writs Pupil does comments3 puts three cups on the math

table teacher insertsminus sign3 - Get ready to take away teacher inserts minus thefigure 13 1 pick upone cup and carry it

to the Resources Tableteacher inserts equals sign

3 1 = Look at the math table andcount the cups. count One,two

How much is there here?

two cups teacher inserts the figure 23 1 = 2The numbers and signs have been written one by one (so that each is acted bythe helper) to build up the expression 3 1 = 2Learners need to understand expressions like 3 1 = 2 as a series ofinstructions that can be interpreted as physical actions and eventuallyvisualised as images of those physical actions.Although subtraction is introduced with this short example, most practicetakes place with long strings that include both plus and minus signs so thatpupils learn to carefully read each symbol and interpret it physically.Detailed Notes1) It is important to use long examples right from the start ( e.g. 3 - 1 + 2 - 1 =3) so that pupils get used to reading the symbols and responding to them(rather than assume that all figures are connected by plus signs!).2) When 3 - 1 + 2 - 1 = 3 (for instance) has been acted, pupils need to betaught to respond to Read what it says by saying Three take away one addtwo take away one equals threeand to Read what it means by saying Three cups take away one cup add twocups take away one cup equals three cups.3) When the teacher writes 3 - 1 + 2 - 1 = 3 (this is called a Math Story)pupils must be taught to act the Real Story (i.e. to act it with the cups). So they must respond to:

I will write the Math Story. You act the Real Story.4) When the teacher acts the Real Story (i.e. acts it with the cups) pupilsmust be taught to write 3 - 1 + 2 - 1 = 3 (this is called a Math Story). Detailedattention must be given to ensuring they form their numbers and symbols


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correctly. So they must respond toI will act the Real Story. You write theMath Story.

5) The whole point of 3) and 4) above is to create an instinctive relationshipbetween the Math Story and the Real Story.

Practice with examples like these:5 3 = 2 3 + 1 2 = 2 5 1 0 + 1 = 5 1 1 + 4 = 40 + 0 0 = 0 2 + - = 2 1 - 1 + = 2 - 1 = 1MultiplicationI need a helper. Stand in the middle here[places pupil in neutral position]. My

job is to tellyou what to do. You are the helper. Your job is to do what I tell you.i want youto

Teacher says Pupil does Comments

Put two cups on themath table.

picks up 2 cups, walks tothe math table, placesthem on the math table

andreturns to the Neutralposition

Good. I watched you dothat. I saw you walk tothe math

table with two cups. Iloved what you did

Do the same thing lots oftimes.

Ensure that at this pointthe helper moves to theResources Table, picks

up two cups and iswalking to the math

table before you say

I want you to do itfourtimes altogether

walk to the math tablefour times altogether

You have walked to themath table four timesaltogether. You werecarrying the samething each time.

Look at the math tableand count the cups.

How much is there here?Eight cups. Good

This process is repeated many times so that helpers physically respond tothe spokenwords: Start by putting [number]cups on the math table


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Do the same thing lots of times [Number] Look at the math table and count the cups [Number]cups.Now I am going to pretend I have lost my voice: I cannot talk. My job is still totell you what to do. This time I am going to tell you what to do by writing.

Teacher writs Pupil does Comments2 putting two cups on the mathtable2 Do the same thing lots of times2 4 Do the same thing 4 of times2 4 = Look at the math table and

count the cups. count One,two, three, four, five, six,seven, eight

How much is there

here?

2 4 = 8 eight cupsThe numbers and signs have been written one by one (so that each is acted bythe helper) to build up the expression 2 4 = 8Learners need to understand expressions like 2 4 = 8 as a series ofinstructions that can be interpreted as physical actions and eventually

visualised as images of those physical actions.Notice that the first number is what you start with: it is cups. The numberafter the multiplication sign is how many times you walk to the math tablealtogetherDetailed Notes1) It is important for pupils to get used to basic examples before moving tolonger ones (like those below).2) When 3 4 = 12 (for instance) has been acted, pupils need to be taught toRead what itsays [Three times four equals twelve] and to Read what it

means [Three cups times four equals twelve cups]. The form of thelanguage, the places where cups is spoken, is absolutely crucial!!3) When the teacher writes 3 4 = 12 (this is called a Math story) pupils mustbe taught to act the Real story (i.e. to act it with the cups). So they mustrespond to: I will write the Math Story. You act the Real Story.4) When the teacher acts the Real Story (i.e. acts it with the cups) pupilsmust be taught to write 3 4 = 12 (this is called a Math Story). Detailed


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attention must be given to ensuring they form their numbers and symbolscorrectly. It is extremely important for pupils be made to see that having got

as far as 3 they know by looking at the writing that the next thing theteacher will do is already known. The teacher will pick up three cups

because it says so! Itis the three cups that I loved. You know that. It says so.What you do not know is how many times I will walk. Do not write. Wait! Nowyou can see I have finished walking you can see how many times I walked.Four! So write it. So they must respond to: I will write the Math Story.You act the Real Story.5) The whole point of 3) and 4) above is to create an instinctive relationshipbetween the Math Story and the Real Story. but they must be taughtcarefully to interpret the symbols.6) After practice with examples containing only a times and an equals sign,move to longer examples that use plus and minus signs as well.

7) Of course, pupils do not always respond only to the symbols written one byone. I general, they see something like 3 4 = and put three cups on the MathTable and then, knowing there is a multiplication sign, move immediately

from the Math Table to the Resources Table to do the same thing. At this stage,they are not pausing at the Neutral position after putting the first three cupson the Math Table. It is this non-pausing that distinguishes multiplicationfrom addition and, pedagogically, we want novices to make that distinction.8) MMMS makes the distinction absolutely deliberately. Do not mentionanything aboutmultiplication being repeated addition. It is not! Look for

instance at7 2 3 = 1 to appreciate the meticulous rigour ofMMMS. We putseven cups on the Math Table; get ready to take away; move to the ResourcesTable with two cups; then do the same thing (i.e. move to the Math Table withtwo cups) lots of times. This is certainly not repeated addition.Practise with examples like these:3 4 = 12 2 5 = 10 1 5 = 5 0 4 = 02 4 + 1 3 = 11 2 4 - 1 3 = 5 5 + 2 3 = 11 7 2 3 = 1Do not use any rules. Teach the immutable logic of the (symbolic) language.

For instance, in 7 2 3 = 1 I start by putting seven cups on the math table; Iget ready to take away. I then walk to the Resources Table with 2 cups. It isthat (walking to the math table with two cups) that I loved. So I do it threetimes altogether. That is why the correct answer (the only correct answer) isone cup. When dealing with expressions like 2 4 + 1 3 = 11 ensure it isunderstood as: Put two cups on the table. Do the same thing lots of times.Walk to the math table four times altogether. Get ready to get some more. One.


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I watched you walk to the math table with one cup. I loved it. Do the samething [WALK TO THE MATHS TABLE WITH ONE CUP!!!] lots of times. Do it

three times altogether. You can now see 8 cups here and 3 cups here. You cansee 11 cups.


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Preparation for teaching DivisionBefore showing the actions for division you need to establish the importantidea of

Same Value; Different Appearance by using a version of this script:

Is this the same as this ?

Yes it is (because there is still six cups). No it is not (because these are on thetable and these are piled up). So the answer is: yes it is; no it is not; yes it is;

no it is not; yes it is; no itis not; yes it is; no it is not.The answer that mathematicians give is Same Value: Different Appearance.This is, of course, an important aspect of equality (i.e. the equals sign).Although it is true that 3 = 3 it is boring (same value; same appearance). On

the other hand 2 + 1 = 3 is interesting (Same Value: Different Appearance)The phrase and the idea ofSame Value: Different Appearance is in fact usedmuch earlier than this (in Reception) but it needs specific attention inrelation to division

DivisionI need a helper. Stand in the middle here[places pupil in neutral position]. My

job is to tellyou what to do. You are the helper. Your job is to do what I tell you. I want youto

Teacher says Pupil does comments

put six cups on the mathtable

moves to the ResourcesTable, picks up 6 cups,

walks to the math table,places them on the math

table and moves to theNeutral position

Look at it and wonderabout

stands at the math tableand scratches her headto act a mock confusion


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piles of two cups makes piles of two cups

Count the piles! Howmany piles of two cupscan you see?

Three

This process is repeated many times so that helpers physically respond tothe spokenwords: Start by putting [number]cups on the math table Look at it [the math table]and wonder about piles of [Number]cupsyou can see Count the piles of two cups How m any piles can you see? [number]Now I am going to pretend I have lost my voice: I cannot talk. My job is still totell you what to do. This time I am going to tell you what to do by writing.

Teacher writs Pupil does comments6 put six cups on the mathtable

teacher inserts thedivided by sign6 Look at it and wonder

aboutpiles of teacher inserts the figure 2

6 2 Make piles of two cups teacher inserts the equalssign6 2 = Look at the math table andcount the pilessay three

6 2 = 3The numbers and signs have been written one by one (so that each is acted bythe helper) to build up the expression 6 2 = 3Learners need to understand expressions like 6 2 = 3 as a series ofinstructions that can be interpreted as physical actions and eventually

visualised as images of those physical actions.Learners need to understand expressions like 6 2 = 3 as a series of

instructions that can be interpreted as physical actions and eventuallyvisualised as images of those physical actions. Notice that the first number iswhat you start with: it is cups. The number after the division sign is the much-ness in each pile (e.g. two cups). The number after the equals sign is an


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indication not of much-ness but of many-ness it is a pure number responseto How many piles of [for instance] two cups can you see?Detailed Notes1) It is important for pupils to work extensively with basic examples.

2) When 6 2 = 3 (for instance) has been acted, pupils need to be taught toRead what itsays [Six divided by two equals three] and to Read what itmeans [Six cups divided by two cups equals three] The form of the language,the places where cups is spoken, is absolutely crucial!!3) When the teacher writes 6 2 = 3 (this is called a Math Story) pupils mustbe taught to act the Real Story (i.e. to act it with the cups). So they mustrespond to: I will write the Math Story. You act the Real Story.4) When the teacher acts the Real Story (i.e. acts it with the cups) pupilsmust be taught to write 6 2 = 3 (this is called a Math Story). Detailedattention must be given to ensuring they form their numbers and symbols

correctly. It is extremely important for pupils be made to see they start byputting six cups on the math table; they then look and wonder; and theywonder about piles oftwo cups. They then count how many piles of cups theycan see and the answer is three (a pure number). So they must respond

to: I will act the Real Story. You write the Math Story.5) The whole point of 3) and 4) above is to create an instinctive relationshipbetween the math story and the real story. but they must be taught carefullyto interpret the symbols.Practise with examples like these:6 2 = 3 6 3 = 2 6 6 = 1 6 1 = 63 = 6 1 = 2 = 1 = 2


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Make Math Real