the main vine, not the loose one.
Kei hopu tōu ringa kei te aka tāepa, engari
kia mau te aka matua
Thoughts on Teaching and Learning of Mathematics
Lesson #4 • Revised 9/1/20
Number and Algebra
- to be clear about the purpose of learning
Number and Algebra
Number is everything. In the beginning there was one!
This is my sense of humour acting but Anthony
Harradine challenged me one day by asking what is the most
important mathematical concept and I answered "The number 1;
without one there is nothing. From "1" we create mathematics." I
reminded him of this conversation when we met again at NZAMT #13
CONF in Lower Hutt 2019. It was good to catch up - relationships!
- to be aware that what is taught is not necessarily
what is being learned
There is a curious article on Reality in New Scientist (2016)
which suggests that at the root of everything, on the quantum
scale of size, is a binary number system which might underwrite
the universe. Quarks are Up and Down, Left and Right, Spin or
No-spin so all in pairs or in binary. We could also solve every
problem using counting if we had enough time. Luckily there are
some curious properties of number that make being numerate a
much more rewarding enterprise than simply counting. We all
count things however and often do not notice when we do so. An
example is counting loose change (coins). Some golfers have
trouble counting their stokes but that might be for another
Algebra allows us to generalise patterns and across all
mathematics to solve and to prove. Algebra is a language that
transcends geographical borders, age and time, gender and race and
mathematics itself. The language of algebra is adapted and
invented as new ideas are generated. New technology demands new
language and algebra provides the opportunity. Algebra as we know
it today had its roots in ancient Babylonia. The Arabian
influences gave us zero and along with Indian help the numerals.
Great mathematicians seem to pop up in all races and places and
Here is an interesting resource by Anthony Harradine that puts a
different spin on learning algebra. Algebra
Families - His copyright message covers this use.
Algebra has traditionally been a "sorting gate" for students
progressing to senior school courses in mathematics and some
schools persist in this silly requirement as though mastering
elementary algebra is some indicator of future success. I feel
this ludicrous requirement is more about teachers protecting the
false sanctity of their own knowledge and power. The "I know
something you do not" syndrome. This false power is sourced
in a lack of big picture knowledge and personal capability and
should be noticed and exposed. A better "sorting gate" would be
Deep understanding of a concept is about identifying key
understandings that form the "arch of sense and knowing.
I like to explore deep understanding with teachers and students
and ask them to try and write a statement that captures the
concept being discussed. This helps to develop big picture mental
models and position the teacher in a journey rather than at a
Anyone can blindly stumble through a forest but to comprehend the
way plants interact to form a forest and create complexity
requires a different approach. Mathematics is like that as well.
Deep understanding is about knowing the subject but also knowing
about learning. This reminds me of a book
written by Liping Mah in which she says "Know how
but also know why." I would recommend this book to help
understand the need for understanding!
Here is a useful
resource on counting and what happens next. Worth reading.
We learn number by counting objects in the first instance and
progress through operations using an increasingly complex
description and understanding of number and numbers. But at any
stage of development the same deep understanding unfolds...
The Deep Understanding of
With a little pondering it should be easy to see how we use
strategies like tidy numbers in addition, factors in
multiplication, parts in proportion, place value in all of these
to make a problem more accessible, to operate and to then
reconstruct the numbers to solve a problem.
Being enabled to take numbers apart, operate on them, and
reassemble them to solve problems.
A simple example is 18 x 5. Here we could see the 18 as 10 +
8 and then operate 10x5 and add 8x5 to get 50 + 40 which is 90.
This is using the Distributive Law and is a multiplicative
strategy or an indicator that a student is operating at early NZC
L4 (being multiplicative). We could also see the 18 as 2 x 9 (or
as 9 x 2) and then write 18x5 = 9 x 2 x 5 and see the answer as 9
x 10 which is 90. This uses the factors of 18 and is a more
complex multiplicative thinking. My goal for all students in
early years is to get them into this multiplicative thinking
Likewise in Algebra it is possible to make a deep understanding
I notice patterning is a very early event for all living
creatures. For example, it takes a few minutes for a new
born human baby to get a reaction to a cry and a few hours
to generalise that this is a good way to control need of warmth,
hunger and comfort. Who is controlling who? Who has become a
generalist in this case? Another example is when my dog
knows when it is walk time, food time, and time to play. The
magician (brain) is at work! Noticing patterns and making
generalisations from these is a very "life engaging and aware"
thing to do. Patterning is nature.
Generalisations of patterning logically follow and this
leads to proof and bold statements of truth in mathematics.
Finance and politics have generalisations as do food and sport. If
you could not generalise about your surroundings you would wake to
a new world everyday and that would be an amazingly scary
place and a reality for some old and all young people.
Generalising is about being alive, engaged and aware. Imagine the
issues of not being able to generalise about rooms in a house
where you sleep and having to explore the place to find the toilet
in the middle of the night. Generalising that the room you seek is
in the same place as it was before you went to sleep is a very
useful thinking ability.
The Deep Understanding of
have always been been a favoured way to communicate for a
very long time. It is natural to draw sketches in the
sand just as did the original designers of the Landrover (http://www.landroverusa.com/our-story/index.html)
in the 1940's and making their visuals become over 3,000,000
vehicles called Series 1, Series 2, Series 3 and Defender. The
Australian Aborigine used visual pictures and recorded 50,000+
years of habitation (First
Footprints also a TV documentary). The NZ Maori had a visual
history through carving.
Being enabled to notice, find and record
patterns and relationships, use visual interpretations, to
make generalisations and make sense of the world in solving
How many other projects and plans began the same way? A square
number is represented by a square and this also exposes other
properties all connected to geometry. Adding two equal sides and a
one makes a new square. Visualisation is a powerful method
and many famous mathematicians used geometry to help develop and
communicate their understanding.
This square of happy faces shows how adding two equal sides
and one makes a new square.
An odd number is made up from 2 equal parts and one as well, in
this example 3+3+1. We generalise that all square numbers are an
odd number away from the next. We could generalise that
every odd number sits between two square numbers.
By squaring all the odd numbers we discover an infinite series of
Pythagorian Triples. EG 9^2 = 81 so (81-1)/2 = 40, 9^2 + 40^2 =
41^2. 9, 40, 41 are a Pythagorean Triple. There are an
infinite number of odd numbers so there are an infinite number of
such triples and we have just made another generalisation and
actually a proof.
Different shading shows the difference between consecutive squares
are the odd numbers. The sum of n odd numbers forms the square
number n^2. Here we see 1 + 3 + 5 + 9 = 16 or 4^2. This is another
pattern and a generalisation. More patterns happen in one and
The triangle numbers form a staircase and making these important
numbers with interlink blocks shows how two consecutive
triangle numbers always form a square. This is also a generalisation
and an important one. I claimed to a group of teachers at a
workshop, in 2003 "if you cannot draw
a picture of what you are teaching do not teach it." Since
that day I have been trying to prove this statement wrong. Visual
mathematics and models are very very powerful.
Here are two representations of triangle
numbers. I strive to have students generalise the sum of these
numbers to n as n(n=1)/2. This can be done by putting two of
the same Triangle numbers together and looking at the
rectangle that is made. EG 2xT3 numbers form a 3x4 rectangle
or 3(3+1) rectangle. Pulling the two parts apart represent
division by 2 and so we get the formula. Proof.
More about Visual Models
Self Test your Visual
Skills and Mathematical Understanding with Fractions
(a) Draw a model explaining 3/4 (Answer will show the 3, the 4
and the relationship to the whole 1)
(b) Draw a model explaining 3/4 divided by 2 (Answer will show
dividing and the answer 3/8, showing the new 3 and the new 8]
(c) Draw a model explaining 3/4 divided by 1/2. (Answer will
show all numbers, the answer 1 and 1/2, and divided).
(Warning...the "one" changes!)
I have used this self test with teachers to help explain why
students find "derision" so hard. Lewis Carroll was absolutely
correct having the Mock Turtle describe "the four branches of
arithmetic as ambition, distraction, uglification and derision". Always
have a "growth mindset" about drawing and encourage students to
draw ideas. In the self test above, first two (a) and (b)
are relatively easy but the third (c) requires the re-unitisation
of what the whole or the "one" is. Think of (c) as "How many 1/2's
are there in 3/4?" or "How many 0.5m lengths are there in 0.75m of
cord?" and you may well see sense. Lipping Ma explained this to me
in her book (ref above).
Division, or the word I prefer "derision" has two meanings. We can
look at 6 divided by 2 as "How many groups of 2 are there in 6" or
"if I split 6 into 2 groups, how many would there be in each
group?". The context of the problem determines which meaning
applies. The answer remains the same! Students will pick
the interpretation needed once they understand the problem so my
advice is to allow this to happen naturally. Derision is hard!
One of the issues to successful learning of mathematics is
"experience". Students with a broad experience of many different
life adventures have little trouble constructing meaning. In too
many classes in 2018 I asked students to list all the life
experiences such as catching a fish, visiting the city, traveling
to a nearby town, riding on a train, flying in an aircraft,
hitting a golf ball that they have experienced. Alas I think I
stumbled upon a major barrier to learning. Most students do not
have broad life experiences and so are not able to make
connections to problems such as "I am
riding on a 1km long train travelling at 6o km/hr. It enters a
2km long tunnel. How long is the train in the tunnel? "
The student questions begin..."Can a train be that long? How
long is 1km? Where do you sit on a train? Is the tunnel bent?
What is a tunnel? Is it dark in the tunnel? How do you make a
tunnel? What if the train hits the tunnel?". Without
contextual knowledge students struggle to make sense of the
context and actually the problem presented is not the most
important problem. Remedy - take students on context
broadening experiences and advise parents to do the same. More
on this in a later chapter.
Why do number and algebra often become a barrier for
people learning mathematics? This is a wicked problem actually.
See NZCER http://www.nzcer.org.nz/nzcerpress/key-competencies-future
for an explanation of "wicked problem". Mathematics is a complex
and highly connected multitude of ideas and representations. It is
a language and is continually evolving for new situations, or
being discovered as Paul Erdōs would claim (https://en.wikipedia.org/wiki/Paul_Erdős#cite_note-23)
from "The Book". Everyone should read about Paul. He said "Coffee
turned thoughts into mathematics." he also traveled from place to
place overstaying his welcome for his entire life.
Number begins with thinking, continues with thinking, and does not
end without thinking. Young children develop a COUNT thinking
which parallels their view of the world of "Me", "I", "touch",
"in the mouth" and all this within a few centimeters of the little
face. Eye contact is the single most important development
tool a mother can give her child. It is all about "Me". All
information travels through this visual and causes brain
development. Add the strong negative feedback loop and when
"touch" causes "pain" and the brain very rapidly learns a lesson.
Neglect visuals in early life and the brain becomes sluggish. Why
would it need to develop? Stimulate and grow. Gain experiences.
Play safe but explore!
Counting is what I call a zero dimensional concept and has no
direction. It does not connect or link. It happens.
Learning to count is important but be aware that a 2 year old
reciting numbers to 100 is not evidence of counting. Counting
is being aware of the quantity involved and matching
objects to words.
I asked my 7 year old grand-daughter "How many blocks in the
staircase (see below) and she immediately asked for a calculator
and proceeded to add 1 + 2 + 3 + 4 + ...+20. Her strategy was
sound and her method modern and persevering, after a while, she
confidently announced "289". We asked Siri...who said "210".
Repeating the quest on a calculator rather than the iPhone app she
confirmed 210. She suggested the iPhone was not a good calculator!
I rearranged the 1 to the 19, the 2 to 18 and quite quickly she
finished the pattern. "Ahh" she said "it is 10 lots of 20 and 10
more. 10x20 +10 which she did with a calculator and of course
confirmed the 210 answer. I rearranged them to 21 groups of 10 for
her. So here we see in my 7 year old a counted solution and not a
lot of developed base ten ideas yet. That will happen however and
very soon. We had fun making the staircase, photographed it on her
suggestion and sent this to Mummy and her teacher.
The barrier to progression from
counting is the brilliant "place value" concept with
it's canon "ten" rule. The Romans missed this one. This
causes students to ask "what comes after 9, 99, 999?" Learning to
control the digits and see the multiplicative pattern of re-use,
each time with a more important (place) value takes time and make
no assumption that anyone understands this system until
they can explain it to you. Take a look at the Power of Ten
to convince yourself of the Power of Ten. Only 43 such powers
contain all we currently measure in the Universe. My 17 Year Old
Physics students would ponder and walk the "power of ten wall" in
CHECK POINT CHARLIE TIME
Do you understand and appreciate place value?
• Write the number "eleven thousand, eleven hundred and
• If I rearrange the digits of a number and subtract
the smaller from the larger, why is the result always
divisible by 9?
answers and see what understandings you expose.
Adding or Combining in a Linear way.
Following the development of counting strategies such as counting
in twos and the ideas of place value we soon see that adding the
place values gives the correct answer and we can now know the sum
of every combination of every number. We can develop a strategy or
an algebraic pattern to do that for us. Hence basic facts and
adding strategies like "tidy numbers", the common algorithm and a
calculator. There are some fascinating insights that can be
discovered in adding.
Additive thinking is however one dimensional or linear thinking.
More complex than counting and more connected but quite straight
2 + 3 to a "counter" is different problem than 2 + 4 but to an
"adder" is "just one less" or "just one more" depending on how you
ask the question. The two problems are actually connected.
It is this connected thinking and its development that separates
"a counter" from an "an adder". The connection shows a more
complex form of thinking and this shows development from NZC Level
1 to NZC Level 2. With place value ideas becoming established the
progression is assured.
We learn to read big numbers and solve problems like "How many
legs on 3 dogs" by adding 4 + 4 + 4. We see addition as the way to
solve most problems and for many people this additive way becomes
a final platform. Most people in any civilisation are
predominantly "additive". Sad but true. Sad because we allow this
happen and true because I notice it all the time in every place I
venture. Why is this the case?
My view is that too many teachers are only additive thinkers
in mathematics. I say "in mathematics" because that is where I am
noticing the sad fact. It is highly likely that this form of
thinking is across all learning areas (and life) for those
teachers and I am trying to get evidence to confirm this
prediction. The brain goes with the person so why would thinking
in other contexts be more, or less, complex?
Dilly dally and continue to revise addition at your student's
peril. My advice:- just like counting, once you have learned
to add get out of there and and explore multiplication. No
one needs 50 ways to add. You can learn more about adding when the
numbers become fractional and negative.
CHECK POINT CHARLIE TIME
More on Number!
What strategies do you use to solve these problems?
Add 997 to 5.
What is 299 + 37?
Add up 1 + 2 + 3 + 4 + 5 + 6 + ... 998 + 999.
The NZ Numeracy Project made an error here, in my view, and
allowed primary teachers who are predominantly additive thinkers
(my observational data and there is a research paper on the
nzmaths site) in number to learn, promote and develop many
different and horribly inefficient ways to add. This
hampers mathematical development by pausing too long and causing
boredom and reducing need. There are additive curiosities that can
be explored but depart and leave the "Fifty Ways to Add Your
Answer" to Simon and Garfunkel ["50 Ways to
leave your Lover", a great song]. When you discover a new
set of numbers like integers, decimals or fractions,
re-explore addition as need arises.
My First Target of Learning Maths,
Multiplication is, in
my head, two dimensional thinking. The person with
this ability holds the idea of 3 groups and the idea of 4 in each
group "at the same time" and this makes 12. It is "How many legs
on three dogs?" or "I have 4 bags each with three lemons, how many
all together?" Either way the answer is 12 and with the context
supplying the meaning of the 12.
An array of 3 rows and 4 columns is the visual model of 3 x 4 = 12
and explains every part of that equation. We see the "3", the "4",
the "12", the "=" and that the "x" multiplication is rectangular.
Multiplication is explained by the shape, a rectangle, that the
array forms. Every multiplication problem between two numbers
is rectangular. Getting students into multiplication ASAP
should be every teacher's goal. Once there we can do a lot more,
move on, explore. Become multiplicative!
"Every student is to become multiplicative by the end of Year
10" is a worthy "First Target" for all junior students and
please state this clearly and boldly in every classroom. I was
criticised for this comment but after explanation to my critics
that it is not a final goal and that for new to school Year 9
students only about 15% are in fact multiplicative, I am
usually approved. Typically, and by the end of Year 10 this data
point has become about 75%. It is indeed a worthy initial goal. In
one school last year (Feb 2016) the Y9 cohort began at 11%
multiplicative and one year later were measured to be 60% (Feb
2017). That is acceleration and we deliberately caused it to
happen. In 2018 I helped improve a whole cohort of 220 students by
twice expectation (100% acceleration) and nearly 60% became
genuine multiplicative thinkers. These students opened many doors
by doing so. [More on data - Measuring and Monitoring in a
Mooovin On, as the cow said!
But what about fractions and decimals? These ideas need some
knowledge and here there are a couple of axioms in mathematics
that are almost never exposed but are vitally important. [Referred
to in an earlier chapter]. We need also to count, add,
subtract, divide and multiply this fractional and decimal world.
Decimals are sometimes called decimals fractions.
But first the axioms.
One can be anything I choose
one to be.
When we combine or compare
anything in mathematics we do so using the same size bits.
An axiom is a self evident truth. That "one can be anything I
choose one to be" is not at all obvious. In the task above
of dividing 3/4 by 1/2 the "one" being referred to at first to
establish the fractions changed mysteriously to another "one" (the
half) and never once did "your hands leave your arms!". We should
identify "the one" in all problems and normalise this
The naming of the axioms above, by myself, happened because
like the Laws of Thermodynamics the most important and more
fundamental Zeroth Law was discovered after the others! I
think that "one can be anything I choose one to be" is most
Harradine of ACEMS asked the question "What is the most
important concept in Mathematics?" I said "one". You can explore
his fascinating website to find other answers and resources.
The combining and comparing axiom likewise is not obvious and is
exposed when we ask the sum of 1/2 and 1/3. Most young students
first say 2/5 adding everything in sight and most older
students, in the western world, pull out an algorithm and say
5/6ths but have no idea why. These two methods are the same
boat from my point of view, both in equal peril!
By understanding that the bits being added need to be the same
size and using number knowledge of common multiples we see 3/6 and
2/6 which can now make 5/6. There is numerator (the count) and
denominator(the part) knowledge needed here as well.
The visual shows the 1/2 being renamed into 6ths and likewise
for the 1/3. Now that the "ones" are the same size and we
are talking about "the same size bits" we can combine them and see
5/6ths as the answer. Do not underestimate the complexity of the
knowledge that underpins the learning to "add fractions".
CHECK POINT CHARLIE TIME
The NZ Numeracy Project (http://www.nzmaths.co.nz)
labelled operating with fractions and decimals, using rates and
ratios as proportional thinking. You can not be a proportional
thinker operating on factions without being a multiplicative
thinker either so this earlier goal is even more vital. One
researcher in the Rational
Number Project reported only 15% of the planet's human
population are capable of this complex form of thinking. I am a
bit skeptical of that figure but certainly it is not as high as
How well do you understand these two key theorems?
• Draw two different models of 1.
• What does half a small pie and half a big pie add up to?
• When can 1/2 + 1/3 be 2/5?
• What is 2/3 of 3/4? (See visual, and explain the
This researcher was Sue Lamon and at a Numeracy Conference in
Auckland said "You do not become a proportional thinker by
growing older". I love that blunt message.
She was adamant that, as the poster above says, deliberate acts of
teaching are needed to have people become proportional thinkers.
She challenged us with the problem "There is a tank of
200 fish. 99% of the fish are blue and the rest red. What
has to happen to make the tank have 98% blue fish?" It is
not that hard but just deceptive. How can a small %change cause
such a large number change or vice versa?
To me proportional thinking is many dimensional. I wrote a
.ppt for a
conference workshop on this topic to help explain my
thinking. It is fun.
I was driving to a workshop one morning and the radio announcer on
ZM came up with math problem competition called Farmer Brown.
This became another power point and I recorded many solutions here
The proportional problem about
hot and cold water filling a bath is worthy of attention as
well. The hot tap fills a bath in 60 minutes, the cold tap
in 30 minutes, if I turn both taps on full how long to get
half a bath of warm water?
(a) 60 minutes (b) 30
minutes (c) 90 minutes (d)
45 minutes (e) 24 minutes
(f) 12 minutes (g) something
else (h) who cares (i) all the above1
I usually add (g), (h) and (i) to multi-choice.
There is a classic proportional problem on Harvard's collection of
math in movies. http://www.math.harvard.edu/~knill/mathmovies/swf/league.html.
The problem is explained in the movie with a variety of solutions.
If Joe can paint a house in 3 hours and Sam can paint the same
house in 5 hours, how long would it take if they work together?
The Math Movie website, above, is full of curiosities,
impossibilities, great theorems and ideas all in a mathematics
context. Some right and some wrong!
The counting, adding, multiplying and proportional stages of
number thinking development can be paralleled to Bloom's and
others Taxonomy's. From recall to application. The SOLO
taxonomy is another and is explained here. What is
happening is an increasing complex way of connecting ideas and
processes in thinking. It happens in all learning areas of course
however I think we can measure this quite nicely using number.
More later on this when you discover the LOMAS test
( another Chapter or page in this book).
The brain, or the magician as I sometimes refer to it, is a
complex entity, a wicked problem! See http://www.brainwave.org.nz
to read about some of the developments in what we know about the
brain in the last few years. For good health ...exercise your
brain...learn to do Cryptic Crosswords! Here is a good link
MORE ABOUT ALGEBRA
Algebra at this level is the generalisation of number and number
patterns. We are all very good at generalising and do not realise
this. We learn from a few minutes of age that if crying makes
food. It works and seems to go on working. When I wake every
morning I already know what the room looks like and how to find
essential rooms. If I could not generalise the world would be a
very very scary place. It would be a new maze every step and turn
and highly confusing.
Learning to generalise and making this explicit is important to
learning mathematics. Learning how to record these generalizations
and manipulate them to make new understandings or generalizations
is called algebra manipulation. Often algebra is confused with new
procedures for a new problem and tedious examples where I make the
same mistakes over and over again. Learning early on that
algebra is the generalisation of number is a magical illustrator
of algebraic manipulation.
An example is the Distributive Rule where 2x(3+4) = (2x3)+(2x4).
This is generalised as a.(b+c)=(a.b)+(a.c) I have used "." to
represent "x" or the binary operation we call multiplication. The
number version illustrates that the left side is equal to the
right side and mathematical truth is preserved. It is a leap of
faith to extend this to all numbers but we prove this works with
Real Numbers and then check every time we find a new application.
Thoughts on HOMEWORK
Do we need homework? What do I learn by doing heaps of problems?
Am I not better off by pondering what I learned today? What do I
learn if I keep making the same mistake. Like reading books,
homework should be easily accessed, 95% of words recognisable in
reading and likewise in maths. So why do it?
I think it better to have students leave the classroom
pondering mathematics. What is infinity? What is past that
place? What is 0/0? [Ask Siri!] Sometimes students need to
practice or memorize methods and facts and then there is a
case for homework. I want to see students playing
games and interacting, arguing and explaining, planning and
building, learning to participate, collaborate and contribute.
That is the real and better learning.
There is an abundance of resources for learning number and algebra
in the form of text books, worksheets and on the WWW web.
The NZ Numeracy Project is the largest mathematics site in NZ and
is located at http://www.nzmaths.co.nz.
We do not have a resource issue but we do have a use of
resource issue. Investigate NRICH, youcubed.org, Otago Maths, Starter
of the Day, Cut the
Knot, NLVM, MAV, AMC...this
list is long and growing daily.
Noticing what students are doing when solving problems, noticing
their use of mathematical language and curiosity, involving them
in the daily discourse, having them reflect on their learning,
using number and algebra to develop those more important traits of
perseverance, collaboration, creative thinking, being critical and
reflective, developing and using the language of mathematics and
connecting to other learning areas is what learning is all about.
See Team and Problem Based Learning in Chapter 11.
Thinking like a mathematician and that statement draws this
chapter to a close and links to that main vine again.
Key features of any year level is monitoring the thinking and
skills of each student. Knowing the broad stages of number
development and monitoring this progression is a vital action
for any teacher of mathematical thinking. Being multiplicative
predicts future success in mathematics and a broad range of
occupations. Being multiplicative opens the door of
proportional thinking. The more mathematics you know the more
mathematics you will use. Place value knowledge underpins all
number and this is also a strongly multiplicative idea. Be
• How would you describe the Deep Understanding of Number
and the Deep Understanding of Algebra
• Describe how you develop number and algebra thinking for a
typical Year 7, 8, 9 or 10 student.
• Find out how you rate on the "Being numerate" scale.
• How do you develop visual representations? Do you draw
pictures of problems?
• What are your favorite on line resources?