Learning Difficulties in Numeracy – Numeracy and the Curriculum (2/5)

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.

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