Welcome to the second of five

videos that have been developed in a collaboration

between the University of Melbourne and the

Victorian Department of Education and Training. This is the second video

and it follows exactly on from the first video

where we talked about core number competence. This now moves into primary

and secondary maths development and some

of the difficulties. So learned symbolic number

then is the first stage from the basic number

abilities we talked about earlier, and in addition to the basic

core systems for exact small number and approximate

large numbers, students then develop knowledge

about number words, the counting sequence

written in number symbols and also other arithmetic

and mathematical symbols, the operations of addition

subtraction and so on, and then they are ready to go

into the whole world of mathematics, and by the time students

start school many have mastered some of these skills, but as I emphasized in video 1

we’re going to talk just about the typical development here. There will be a number

of students who haven’t achieved all of these:

counting to 20, ordering numbers up to nine, identified numerals 1

to 9, comparing symbolic quantities up to 9. Now

counting to 20 is actually, for students speaking English, not as easy as it is in

some other languages because we have

a different word for every number up to 20, and so there’s a lot of memory

needed there to get to get to 20. Everything becomes regular at

20, 21, 22, 31, 32 and so on, but because of this irregularity

up to 20, we do find that some students

are not as competent at the counting when they

start school and that was visible

in the second last line of the previous video where

we saw that there was a set of students

who were pre-schoolers but they had poor

counting skills, but quite advanced other

mathematical skills and a number of them can

place a mark on a line to represent a number and some

simple non symbolic additions and we saw that in

the preschoolers from the previous video. So how do students

then progress? Well, once they’ve mastered counting

they can quite easily progress to simple addition

because counting is just the basis of the development

of arithmetic skills for most students. So adding to numerosities

is the equivalent to combining the number of objects from two

different and distinct sets, and it might look like this. So a student could count

the four animals in the left and then move on to the three

fruit by going five six seven, or they could say four and three

and then they count their fingers but most

would just count from one. Students make use of their

counting skills in the early stages of learning

arithmetic and the number words have both a sequence, that is to say they’re

in the ordinal sequence and also a cardinal meaning

and it is that cardinal meaning that can present

difficulties for many students particularly

those with maths learning difficulties and dyscalculia, and there are three main

stages in the development of these addition

counting strategies. The first of those

is counting all, so for three plus five

students will count one two three and then they’ll count one

two three four five, often using fingers, and then they will

count all of them. That is time consuming and also

can be error prone. Some students come to realise

that you don’t really need to stop by counting

the first at the end you could start at three

so they start with three and then they go

four five six seven eight. So using finger counting, the student will no longer

count out the first set but start with the word

three and count on. Some students recognise

that it’s quicker and also they’ll be less prone

to error if they choose the larger of the two addends

to add to the smaller. So this time they would select

the larger number five and then carry on with six

seven eight. Now that more sophisticated

the strategy depends on knowing that five

is larger than three and we’ve already mentioned

that that is a fundamental problem in the core

number ability, the number sense for students

with dyscalculia. So early number difficulties

then. Well, simple subtraction can also be

successfully solved with counting. We look at this image we can

say cross out two pies to find out how many are left

and they cross out two and count how many are left

or cross out seven leaves to find out how many are left. Students don’t have difficulty

with this kind of problem but once we remove the concrete

materials or the pictures it’s not always as easy

as you might imagine. We often would say, it’s just the reverse

of addition, but this requires the student

to understand which particular aspect

of addition is being reversed and particularly

for a student who has been doing addition by counting all, reversing that process

is not easy for them. So sometimes the language

we use can can affect the ability for a student

to understand what we mean and we talk about this much more

in the fifth video on intervention. So in a normal learning

progression then, students learn to multiply

and divide after adding and subtracting, and multiplication can also be

successfully negotiated by successive additions, either by counting individual

objects or by counting objects and sets

and combining them. So here we have full plates

with five cupcakes on each and a student could easily

start at one and count all of those cakes one or two at

a time and a student with MLD may well do this. More numerically advanced

student though, might count the cakes

on the first plate: one two three four five, and notice the same number

of cakes on every plate. So they then go 5 10 15 20. What’s significant about this is

that both students will produce the answer 20 and any

underlying specific number difficulties experienced

by the first student will not be apparent

because they have both answered 20. So parents and teachers often

comment that specific number difficulties in maths are not

picked up as early as say reading difficulties

and dyslexia. So why is this given

the evidence presented in the first video

that pre-school children are already showing differences

in basic core number abilities? Well in part it relates directly

to the child’s progression in learning arithmetic. Most children are able

to count quite competently up to 10 as they

enter formal schooling, even though some may count more

slowly or be more error prone. Maths learning difficulties

don’t really become evident until children need

to manipulate numbers in a way that depends

on symbolic to non symbolic relationships. So they need to start

to understand the meaning of the symbol and that is the

point at which we start to pick up

the children have number difficulties because if

they don’t understand the mapping

of that number symbol onto a non symbolic array. So counters or dots

“that” presents difficulties with working

with numbers, and it happens later

than you might think. The problem is there early

but revealing it comes later than you think, because of this ability

just to count. So when do they become

more obvious? Well usually it comes to light

when we move from physical objects into symbolic

arithmetic. So the student who is able

to add this without any difficulty,

counting the objects, is then faced with it like that, and if they don’t understand

what the numeral 4 represents in terms of that left hand set

and the numeral 3 represents in terms of a set

of fruit there, then they find that sort

of problem difficult and to successfully perform

this addition the student needs to understand the relationship

between the symbols and then on symbolic form. Now we’re going to pick

this up again in Video 5 when we talk about

intervention strategies for students with MLD

and dyscalculia, but from words to symbols

is a really important step

in primary school. So typically students will

acquire counting words and use them for counting

before they learn the numerals 1 2 and 3 and so on. Once the student has learned

the symbols, the numerals for counting words, there’s an important

developmental step which is needed that of

associating the non symbolic form of the number

and the word and the symbol. So they need to know

that that array of dots is represented by a symbol 5

and the word 5 and in video 3 we’re going to talk about

where in the brain that association happens

or where this neurological deficit is for dyscalculia

students and this mapping of symbols and words onto non

symbolic arrays is thought to be fundamental to number

sense and number knowledge and it is showing

that it results in this neurological deficit

for students with dyscalculia, and without this mapping, symbols and words have no

meaning or any relationship with magnitude

for these students. So apart from the mapping

difficulties described earlier the transition from words

to symbols can also cause problems for many students

as words in all languages use a name

value system, that is to say the powers of ten

are given special names: ten, hundred, thousand, million.

Similarly, in English multiples of tens

are given special names 10 20 30 40 and so on but the numerals

by contrast use a place value system so that the same

numeric symbol can mean different things

in different places. So the numeral seven

could represent 70 if it’s in the tens column

or seventy thousand if it’s in the tens

of thousands column and this transition from name

value for place value causes difficulty for many students

especially early learners and can result in them

proforming 102 and writing it as 1002, so they’ve written one hundred

and then they’ve written a two and that is an important

indicator of the sort of error patterns

that students make when they haven’t understood

place value, and we’re going to discuss

these types of areas more in video four

because they’re highly informative about the way

students understand the base 10 system of numbers

and we will talk about them in Video 5 in the assessment. So base 10 knowledge then relies

on understanding relationships between the decimal reference.

So what do I mean by that? Our system is built

up on base 10, so we have 10 units to make

it 10. We have 10 tens to make one

hundred and ten hundreds to make a thousand

and those images there: the red block is a thousand

little cubes, the yellow cubes are individual,

the greens are tens, the blues are one. Or we can have them in this sort

of form but these are really useful manipulatives

for students because they can be used to perform calculations, they can represent numbers

using them and it can help them understand place value

which is an area which is problematic

particularly for MLD students. Writing numbers is the next

stage in understanding numbers. Early in primary school students

learn to write words and symbols

representing magnitude but it isn’t always that easy. If you look on this diagram

and I’ll highlight this one here I can see here that the

student was probably asked to write down two thousand

five hundred and sixty nine and what they’ve done

is written down exactly that two thousand five

hundred and sixty nine without understanding

place value here. The student has probably been

asked to write down one hundred and twenty nine

and this student has made exactly the same error, this student has made

the same error as Kieran on this task and Felicity

has made the same error as well although she’s put twenty

thousand in her first attempt at it, so you can see that place

value is problematic and when we’re looking at

reading and writing numbers and the student’s ability

to read and write in numbers this is actually our data. Five to six year old students

reading numbers correctly and you can see here

that they are fairly accurate on eight and 10.

They become less accurate, they they are accurate

on one hundred and they regain accuracy to a

degree at two hundred and one thousand, but you can see that they

are highly accurate at small

numbers and 100, but when we get to numbers

like one hundred and twenty nine the accuracy

drops off fairly quickly. Similarly when they’re

asked to write numbers, the same idea, the students are just asked

to write down the numbers as they’re read and you can

see a high degree of accuracy on the numbers 10 and less, and again one hundred

but you can also see that we have some poorer

performance when we get into the larger numbers. So this idea of place value

then is quite critical. It is something which we used

as an indicator in our clinical

research as well. The ability to read

and write numbers, particularly to read numbers

is predictive of other other issues and reading and writing

numbers is known as transcoding. Now some people might suggest

that in fact this is an issue of attention or tracking

but in fact those sorts of things don’t

predict these differences, but interestingly the core

number abilities do, so short subitising span

does also predict performance on these

sorts of tasks. So what about older students? So once students have mastered

counting and simple arithmetic problems they move

into a more formal calculation. Problems are presented

in symbolic form exclusively and set out with the

correct place values. So addition problems might

look like this and these are problems where you don’t

have to do any carrying, each of the columns can

be added up and they come to a number less than 10. Similarly with subtraction

you can subtract the number and they will all subtract. You don’t need to borrow any

numbers from the next column along. Students are then introduced

to addition and subtraction where they do require carrying, if the total of one column

exceeds nine or borrowing if the number being

subtracted in any column is the larger, so they

might look like this, and in this case of addition

you need to carry if you do six plus five you’re

going to need to carry a one across to the next column, and in fact that will also

make that column exceed 9, and so and so on

and with subtraction you need to borrow from the

adjacent column. Now this requires a real

understanding of the base 10 system in order to understand

how to reorganize a number to make it possible

to do these additions and subtractions

subtraction particularly, and so some of the sorts

of known errors that we find are these, sometimes the student will

just take the smaller number from the larger number. Whether they’re above or below

so you can see here that that is a student

who simply just subtracted the smaller

from the larger regardless. They also have this problem

with borrowing from zero they know that they need

to cross it out and make a nine but they then don’t

in turn take one of the hundreds. Now we’re going to leave

this discussion now but when we get to a video 5

where we look at interventions, we cover this issue in some

detail how to assist students with this kind of of

a subtraction but also I want to just emphasize

that students with maths learning difficulties

and dyscalculia aren’t at base one for these sorts of problems

so they have already got into difficulties

because these sorts of problems don’t lend

themselves into the normal counting procedures

in arithmetic and that’s why it’s so imperative that we can

intervene to bring them up to a different level

so that they have the entering competence

for these sorts of tasks. Now word problems is something

which students often groan about but we do

need to just have a look at them. We need to also recognize

that word problems and number fact knowledge and arithmetic, they’re all overlapping

competencies so word problems aren’t something on their own, they are simply testing number

effect knowledge and also ability to manipulate numbers. So it’s well known that word

problems can present a number of difficulties to students. Here’s an example

of a word problem: Lucy has three fewer apples

than Julie, Lucy has two apples, how many

apples does Julie have? And if you had 14 squares

of chocolate and five chocolate fingers how

many do you have in total. I think they mean how many

objects do you have in total, and translating words

into symbols is not straightforward and often

students complain that we leave word

problems to the last five minutes of a lesson, they’re often almost

like an afterthought, but they’re actually

a critical part of understanding

because mathematical relationships can be

expressed in multiple representations including

diagrams and graphs which we’ve already seen

in these presentations, verbal representations

in written and spoken language as here,

and symbolic representations, numbers and letters, and often what students

are doing with word problems is representing

the mathematical relation which is now in a written

language into numbers and letters. Interpreting translating

switching between representations contribute

to building this conceptual understanding of mathematical

relationships but that can be difficult

if you haven’t got a basic understanding of number

and the relationships of number then trying to take

them from different, translating from different

formats, can be very difficult

for a student and for some students,

maybe many in fact, maths may not be about sense

making, in fact, Alan Schaeffer once said

that “students leave their logical sense making at

the door when they enter a maths classroom” and here

is an example of it. John wants to make wooden

bookcases that are two metres wide. He has two five

metre long boards. How 2m too long boards

can he cut from them? And 70 percent of 11 year

olds answer 5 and they’ve done that because they’ve said

well he’s got 2 x 5m boards that’s 10 metres of wood

so he can cut 5 from them without realising that you can

only actually get two from each of the… you’re going to have

some leftovers. So why do students have such

a hard time with maths problems? Well one important difficulty

is that students tend to think linearly

in a step by step manner and try to make the numbers

in the text match each other in the same order. So for example Jane

had twenty five pens and she gave away fifteen.

How many does she have now? This translates in a word order:

twenty five minus fifteen. So that sort of problem is no

difficulty for a student, but what if it doesn’t follow

this step by step recipe? So if we have this, after giving away some

cards Jane now has 17 cards left of her original 30. How many cards did

she give away? This time none of these

calculations can give you the answer:

17 minus 30, 17 plus 30, or 17 times 30. So the numbers are going

to be in a different order for the calculation and that

is where errors occur. Now we’re going to cover

intervention strategies for where problem solving

difficulty in video 5 and suggest a model

for intervention based on something called interactive

assessment which I’ll describe in more detail at the time

to determine the exact nature of the student’s understanding

and difficulty and then beyond the basic

four rules of arithmetic: addition, subtraction,

multiplication, and division, two areas of maths

are well-known to present difficulty to older

primary school students: fractions and algebra. Now clearly I’m not going

to cover the entire curriculum of primary

and secondary education in video 2 because it would take

far too long. So I selected two areas

that are known difficulties for all many students and also

those with maths learning difficulties to focus

on because they are often the building blocks for other

sorts of progression. So fractions are first

introduced very early in primary school in the form

of students being introduced to the concept of one-half, they are taught that one-half

represents one of two equal parts of the whole, and most students don’t find

this difficult. Young children are also used

to dividing a pizza between two siblings or even some

toys or fruit equally beside between themselves and a friend. Now this highlights already

differences in fractions. Fractions can mean two things

you could have a group of objects being divided

or you could have a single object being divided, and this distinction starts

to impact students understanding of fractions

fairly early. So it is the case many students

never master fractions, so we’re not just focusing here

on MLD students or dyscalculic

students, there are many who never

master fractions. So for example and this

was a study done in the US, when asked whether 12

thirteenths plus seven eighths was closest to nineteen

or twenty one, only 24 percent of year eight

students answered correctly, and that test was given

about 40 years ago, and then there was work

by innumerable teachers mathematics coaches,

researchers, government commissions, in all

of the intervening years, and then the study was repeated

and this time 27 percent got it correct. So it is still very hard

for students to manipulate those sorts of fractions

and these sorts of difficulties are not

limited to just fraction estimation problems

and they don’t end at Year 8 either. So on standard fraction

addition, subtraction, and division problems

with equal denominators, so three fifths plus four

fifths and unequal denominators three fifths plus

two thirds, year six and year eight

students tend to answer correctly only about 50 percent

of the time but fractions are foundational to many

more advanced areas of maths and science. Grade 5 student fraction

knowledge predict secondary school students’ algebra

learning and overall maths achievement even after

we control for whole number knowledge, student IQ,

and family education and income. So this was an interesting

image I found, the reason I was interested

in it was because we often focus fraction learning

to numbers less than 1 and students then are faced

with fractions greater than one and they’re not sure

what to do but this image I was pleased

to see that they’ve actually included five on four there:

an improper fraction, which helps to make

students understand that fractions also

go beyond the whole. What is the problem though, why do students find

fractions difficult? Well there are three

types of difficulty that you need to overcome

when you’re learning fractions. First of all the notation

that’s used, understanding the relation, A over B, is much more difficult

than understanding just a simple quantity like a number. The complex relations

between fractions, arithmetic and whole

number arithmetic. Remember that students

would have done a number of years on whole number

arithmentic before they reached fractions

and then suddenly there seems to be something

different happening. The complex relations

among different fraction arithmetic operations, so things like multiplication

and division, for example, multiplying fractions involves

applying the whole number operation independently

to the numerator and the denominator. So that looks like it’s exactly

the same as whole numbers arithmetic: three sevenths times two

sevenths you multiply three by two seven by seven

and you get six forty-ninths but then if you try

and do the same thing in addition: three sevenths plus

two sevenths, it doesn’t work. So why do we need equal

denominators to add and subtract fractions but not

to multiply and divide them? Why do we invert and multiply

to solve fractions division problems? Why do we invert the fraction

in the denominator rather than the one in the numerator? Now if I had much more time

I would show you the proof of why we need to invert

the denominator rather the numerator, but that will take

much too much time, but if you are interested

in contacting me later, I’m happy to show you a proof

for that which students do find somewhat compelling

for understanding why we do it that way. Fractions are not

always concrete. Although we may start

a fraction instruction with objects perhaps paper

folding plastic fraction pieces. We too often move quickly pass

these into real world models, into paper and pencil

computation and even drawings of fractions sometimes can fail

to be concrete enough for some students with limited number

sense or MLD and dyscalculic students really need to be

cutting up paper and using manipulatives

very long after those that are normal

progression students. So this physical act of cutting,

folding, manipulating it’s so important

in understanding fractions and then of course

to make matters a bit more complicated in fractions, when we multiply whole numbers

we can make groups of objects: three times six can be

represented as three piles of six. So do students really

understand what happens when we multiply fractions

and for this slide I wanted to find an image

that would express this and it was an image

presented to students to help them understand

multiplication of fractions, and this is what the image

looked like, and to be honest I’m not sure

that that is a particularly helpful image for understanding

multiplying fractions but I also think

it characterizes exactly what the difficulties

are when we’re trying to explain multiplication

of fractions to students. Fraction notation

and terminology can also be challenging. We use words like numerator

and denominator and sometimes we forget the students may not

have internalized those terms and have most likely forgotten

what the words represent, they’re unusual terms. They’re not something

that they’re meeting in their environment normally, we need to continually

reinforce what we mean by numerator and denominator.

Words like equivalence, improper, reduce also add

to the confusion. So let’s have a look at

this image below and try and imagine it through the lens

of a struggling mathematician. So there’s the image, both of the yellow fractions

represent the same thing but they’re written in two

different ways. One, if written carelessly

could end up looking two one hundred and ten. The other is easy to read

but difficult to make using technology and I will concur

with that as I’ve attempted to always use the horizontal

line version of fractions in these in these

presentations using a technology equation editor

and it isn’t as easy as just putting a forward slash.

How about the purple ones then? Students need to understand

that both of those are the same value:

one and seven eighths… 15 on eight… they are exactly

the same. Some students will

read that one is seven eighths as 17 over eight, even understanding that the one

is a whole and the seven eighths

represents part of another whole, can be

confusing for many. We need to be constantly

assessing for this type of misunderstanding

and students also need to understand that one

whole is the same as when it’s notated

with the same numerator and denominator. So we can’t assume

that understanding is in place and in the intervention Video 5

we will talk about how we can instantiate some of these

fundamentals of fractions to aid

with progress in fraction understanding

for all students numbers. Fractions are just

numbers actually. Do we help students

to understand that they are real numbers

and even parts of numbers? So when we have a drawing

of three eighths it means we have less than the whole

object or even less than half an object. Also fractions allow us to show

amounts between whole numbers. In the whole number range, there isn’t a number between 3

and 4. If we’re talking about fractions there

is a fraction between 3 and 4. An important stage

in understanding where fractions fit in our

number understanding. So the drawings we used

to show fractions are merely representations of numbers

from a number line and maybe putting fractions along

the number line may assist in students understanding

the relationship between fractions and whole numbers.

Now another area of difficulty, and this is for older students

mostly, is algebra. Now the thing about algebra

is we use letters to represent unknown quantities and these

letters we call variables and terms that have the same

letter of variable are called like terms

and they can be combined, and if they have

different letters or variables they are unlike

terms and they can’t be combined. So if we look at

this image: orange juice and orange

juice can’t be easily combined into any

meaningful sum, but if you’ve got two sets

of apples you can combine them because they are alike, and similarly you can combine Xs

a set of Xs with another set of Xs because they

are the same variable. In some ways the problems

experienced in learning algebra have commonalities

with those discussed previously

for fractions. Some of the difficulties

that students express in processing fractions result

from incorrect application of arithmetic processes. So they’ve had years of working

with arithmetic and then they start

applying those same procedures into fractions, that exact same approach

is used in algebra and also it results in some problems. So it could be argued though

that learning algebra is a little like learning

a new language. Language in itself is a

combination of symbols such as the alphabet to express

and communicate a coherent idea, a message. Algebra similarly uses a new

set of symbolic notation to communicate

mathematical expressions, introducing alphabetical

letters to represent variables in a calculation. So this is different

from arithmetic which is about absolute

computation of specific numbers. In algebra, students have to visualize

multiple possible answers for a given

algebraic expression, and I’m going to discuss four

possible components that might hinder students’ understanding

when they’re learning algebra. So I would refer to these

as potential roadblocks in algebra. So the first is this use

of symbols to replace numbers with algebra. Students have to visualize

multiple possible answers for a given algebraic

expression whereas twenty four plus seven equals

thirty one used to be the brand

of calculation students would work with every

day in school, they now have to contend

with this new approach to a mathematical expression

where we have 24a plus seven 7b equals two

hundred and forty seven and a and b can represent

entirely new dimension in the calculation, and this sort of step

into understanding how to introduce these variables

can be a roadblock for students, so they need to understand

that a and b can be a range of numbers and it

could be integers or rational numbers

like fractions, it could be irrational

numbers or complex numbers when we are getting

into the higher levels of secondary education. So where arithmetic uses real

numbers to find an absolute answer, I’m using the word real numbers

there to include fractions, algebra uses symbolic notation

to represent a wide range of possible solutions. Students find that difficult

because they are used to being asked for an answer, they’re used to getting

a correct numerical answer. Now they are being asked

to really produce something that they would regard

as somewhat ambiguous. The next area which we

would describe as a potential roadblock

is difficulties understanding the equals sign. So after many years

of arithmetic students come to interpret the equals

sign as an invitation to perform a calculation. So three plus four equals

tells them that they’re going to add three and four

and it’s going to give or yield or produce or lead to. So it’s the end

of the expression and in order to to interpret

an equals sign like that we are going

from left to right. So we’ve got something on the

left because an equal sign and it leads to something

on the right and we see it as one directional.

So this equals sign, which is actually a sign

of equivalence or balance, and many algebraic equations

use it to describe a relationship between

and the mathematical characteristics

of the variables, in arithmetic the equals sign

is one directional usually from left to right and can be

interpreted as gives or yields. So to solve the arithmetic

equation 3x plus four equals nineteen. The student would first

subtract four from nineteen reversing an operation

or unwinding and then divide by three

reversing the operation or unwinding. They’re often taught

that in order to solve algebraic problems of this sort

they start with the X and they say

what happened to the X first and what happened first was it

was multiplied by three, what happened next, four was added and the answer

was nineteen and that’s why they use this reversing process. Now these operations

are reversed to the solution of this equation this equation

is actually arithmetic it might be look

like an algebraic equation because it contains an X

but actually it’s arithmetic because you

can apply arithmetic procedures to it. So could you though apply

the same procedure to three G past seven

equals four G minus two? The answer is no you couldn’t, because this time the equal

sign is an equivalent sign. It is balancing the two

expressions on the left and right hand side

of the operator. So this balancing requires

you to actually solve the problem in several stages

by performing the same operation on both sides. So algebraic problem solving

with equations is different from arithmetic because the

equal sign means something different. Although in fact the equal sign

has always been an operator of equivalence, simply that it has not been

interpreted that way in arithmetic, and then we come to algebra

word problems which of course are even more complicated

than arithmetic problems because we now have some

unknown values that we’ve got to accommodate and you need

to designate those with some sort of a letter

and then build up the mathematical relationships

in the problem eventually to produce a symbolic equation.

For example, Ellie has four times as many 20

cent pieces as 50 cent pieces in her piggy bank. She has a total of $10.40

in her piggy bank. How many of each type

of coin does she have? So apart from trial and error,

how might you solve this? Well you might say let’s

let the number of 50 cent pieces be X or any other letter. Now we know that there

are four times as many 20 cent pieces as there

are 50 cent pieces. So there’ll be 4x 20 cent

pieces and if we add those together by making

the number of cents, so remember 50x and 80x

because there are four times, 4x there, the total of that is 130

X cents and now we’re going to divide $10.40 by 130x

and get that x is eight. That is a lot of steps

in problem solving in order to get the answer

to that problem, and we would then say

Ellie has eight 50 cent pieces and 32 20 cent pieces. So simply constructing

an equation to represent this algebra word problem

requires many steps and there are many places for a student

to go wrong along the way and it is often hard to find out

which stage in the problem solving process the student was

unable to complete accurately. So in Video 5 we’re going

to pick this up again in this discussion

of interactive assessment which is an intervention

strategy that assists all students to achieve success

in many of these sorts of aspects of maths

by breaking it down into these steps and also

providing support for the student when they

go wrong. So why does algebra matter? Well algebra is critically

important because it is often viewed as a gatekeeper to high

level mathematics and it’s a required course

for virtually every post secondary school program. When students successfully

make the transition from concrete arithmetic

to symbolic language of algebra, they develop abstract

reasoning skills necessary to excel in maths and science. So the take home message

for this video is that we’ve focused on the development

of number beyond basic number and pre-school number abilities, through primary

and secondary school, to the more advanced

maths competencies involved in fractions

and algebra. It’s important to note that the

sorts of known errors students make in advanced

arithmetic fractions and algebra are not confined

to MLD students many students who don’t have specific maths

many difficulties can experience problems

in these areas, however, the basic number

deficits characterizing MLD students impact significantly

on more advanced arithmetic processing fractions

and algebra. For example, a lack of understanding

of the base 10 system, which is commonly associated

with MLD students results in their inability

to successfully add multi digit numbers involving

carrying or subtract multi digit numbers

involving borrowing. This is going to be addressed

as I’ve repeatedly said in much more detail in the video

5 when we’re going to look at intervention

strategies for MLD students and students

with dyscalculia in these more advanced mathematical domains.