Cling to the main vine, not the
Kei hopu tōu ringa ki
te aka tāepa, engari kia mau ki te aka matua
Thoughts on Teaching and Learning of Mathematics
Monitoring Numeracy #14
This has become a focus along with Problem Based
Learning for me. It is great importance to a teacher to know
the student level and progression.
I will try and explain.
When we measure what Curriculum Level a student is operating in a
subject or strand we are actually measuring their complexity of
thinking. The thinking is wrapped in a developing cloud of
knowledge, skills, attitudes, experiences, situation and
upbringing. We all start out pretty much the same but pretty soon
after we develop differently. All the time we are learning. Our
thinking develops, is challenged, changes and adapts to our
situation. This is the complexity we measure.
I measure Number. I could measure any of the strands and probably
would prefer Geometry or Measurement but I measure Number.
Students before getting to Year 9 get a lot of number so some are
quite sophisticated in their thinking in Number. This measure also
reflects their other strand thinking, and in different learning
Areas with surprising agreement.
I use mainly the LOMAS test developed
by Dr P Hughes and Dr G Lomas. The original way of determining a
Stage in the Numeracy Project required what was called a NumPa
Interview taking about 20 minutes. Secondary teachers preferred a
faster way so Peter and Grigor generated and tested and selected
some very clever questions. The test will quite rapidly, maybe 20
minutes for a student, or 60 minutes for a class. I developed a
fast way to mark and also how to use the test in a formative
manner. There are many other test options such as PAT and eAsTTLe
Students are measured as being at Curriculum Level 1, 2, 3, 4, 5
or 6. I have noticed that a couple of tests will suffice to
establish a broad indication and successive tests confirm that
measure and help establish a trend.
Here is a cohort of about 200 Year 10
students. The HOD has tracked these students religiously from
entry at Year 9. We are looking at 7 terms of data. There are
no Level 1 Counting students at entry. The red bar is Level 2
and overtime has an annoying persistence as new students are
enrolled and existing students continue to be reluctant. The
grey bar is Level 3 and starts quite high at 30% but does
reduce to only 5%. These bars are dynamic and there are
students moving in and moving out so they tend to persist. The
Yellow bar, my Multiplicative favorite is a good example of
dynamic movement of data and remains around 30% over this
time. The 20% in 10T3 are almost certainly none of the 35% in
9T1. The Blue and Green bars are teh students who have become
proficient with fractions and decimals and are quite logical,
strategic, critical and communicative students. From only 10%
at entry they have grown to over 60% of the cohort. This means
60% of the following year NCEA candidates will easily gain
Merit and Excellence. I asked if they should sit NCEA Level 1
at all and just study all year.
When a student starts learning another strand like
geometry, or another subject like English, science or PE
there head with their brain walks along with them. They do not
miraculously become brighter, dimmer, more astute or less
connected when they do this. They carry there hard earned
complexity of thinking with them. They argue just as much in any
subject. They deduce and connect in the same ways. They take their
thinking to the sports field. The thinking is the measure. I am
actually measuring the complexity of student thinking when I give
them a test, any test. We, as teachers, just need to realise this
and use the information. We do not need to test everything or
every week or use many different tests. We need to know and use
I often see a lot of data in a school in the form of tests,
standardised tests, national tests, common tests, unit tests,
exams, assessments and when I ask if the teacher has used the
information I often get a "no" answer. The data from PAT and
e-AsTTLe just get recorded and then ignored. No use is made of the
information to improve thinking or inform learning. It should be
and there is a lot of researched indicators within the tests that
help to do this. But this is another problem.
FACT - You are actually measuring thinking ability, which
is why SOLO is on the background on this page.
I will describe now at each level in terms of NUMBER. This we
learned and described in the Numeracy Project
1999 to 2012. Overseas reseach projects confirmed these broad
Level 1 - In my words, a very
lonely place. This is the world of counting. I have 3 cats and 2
dogs, how many animals altogether? The fingers come out and we
arrive at 5. What is I have 3 cats and 3 dogs? The fingers come
out and we arrive at 6. 1, 2, 3, 4, 5, 6. There is no connection
to any previous problem. The fact that a door opened and a dog
entered are two separate events. An elephant might have entered.
Students at this level are plagued by no place value knowledge.
How the number system works is a mystery. Every number has a name
and none are connected to any other. I often hear parents boast
claiming "My 3 year old knows all his numbers 20". The child can
recite without error all names to these 100 numbers but they have
no idea how the numbers connect and relate to one another, Just
unconnected names. In reading we see the finger touching the
words, one by one. We learn terms like counting on and skip
L1 Key Features - No or not much connected thought. It is the
ever regenerating, Generation Me. A counted world.
Level 2 - Here there are some
connections and the student will see 2+3 and 2+4 as "one
more". They will start to understand placevalue or the
valueof the number changes depending on where it is. They will
know that 12 is a bundle of 10 and 2 more. The ten rule in place
value is starting to develop. Why we only need 0, 1, 2, 3, 4, 5,
6, 7, 8, 9 to make numbers is starting to make sense. The image
shows a rather sweet tool but the numbers shown are way to big for
this level. In Level 2, 20 is a big number!
When presented with 136 + 45 the student will just fold and walk
away. The numbers are too big. In reading we see a more smooth
approach and more comprehension joining the words that are being
read. Little words and little stories. Developing rapidly as
language develops as well. These students are on the way.
Key Idea - Some connections in thought. Understanding of simple
Place Value. Groups of 10 developing.
Level 3 - Here there are some very
good connections but it is all linear or straight line thinking.
This , then this. It is the world of logic. A lot of our
world works at this level. Xcel solutions are often additive.
Accountants doing the books, engineers planning a project,
programmers designing a program, buying an item at a shop, what we
do when we get up, what the dog does every day, is all linear.
Students can add reasonable big numbers at this level and will
master the efficient addition algorithm. They can double, triple,
know a few basic multiplication facts and will derive the rest.
"What is 4 x 6?" is answered with "2x6 is 12 so double that is
24." A student will choose addition to solve problems as first
choice, every time. The methods are often inefficient and time
consuming. "What is total of all the numbers from 0 to 100?" 1 + 2
+ 3 + 4 + ...+ 98 +99 +100? The answer, if developed, will be by
Key Idea - Linear straight line thinking. The world is an added
place. Addition rules. I have declared an War on Addition.
Level 4 - This is my first target
for Year 7,8,9 and 10 students. This is where a student develops
the ability to think of two things at once. This is what
multiplication is all about. What is 3 x4? 12. The student sees 3
groups of 4 at the same time. What is 4 x 6? 24. No hesitation. No
issue the facts are known and used.
What shape is a square number? What is a factor, multiple or
prime. What shape is multiplication? Many connections. The student
at this level will also be able to give you a reason for their
answer. This thinking is more advanced. More complex. More
informed by better knowledge and more creative due to better
problem solving experience and language.
Students here might not know 2/7 + 3/5 but are ready to find out.
They may not be able to to figure if how to divide 288 in a ratio
of 4:3. yet.
Key idea - Many connections, reasons are clear and conversation
is structured. Reading is with meaning. Good comprehension. My
Level 5 - The highly organised and
connected world of proportional thinking is where students will
not only answer but improve the question. These students debate
and enjoy the mental challenge. They can operate with all
operations on all fractions and decimals, percentages, rates and
This thinking is where we want people to operate most of the time
in our technology driven, busy world. This is the plan. I saw a
billboard on a trip to Auckland that said "Bepatientandwaitfora
gapinthetraffic". I knew a proportional and creative student
designed that idea.
How we figure out climate change, idiotic premiers of countries,
electric cars and batteries, nitrates in our rivers, our own
survival will be the realm of this thinker. We need as many as we
can make. It is a learned event. Sue Lamon of the Rational
Number Project once said "You do not become a proportional
thinker by growing older!"
Key idea - Highly connected reasoned students. Very clear
and powerful thinkers.
Level 6 - is where I depart from
using Number as a way to measure complexity of thinking and
instead use student argument and debate. These students can now
specialise or further develop mathematics from a self motivated
perspective. They are fun to teach. They learn quickly. These are
the Merit and Excellence students.
It may be of no surprise to know that only about 15% of students
entering Year 9 in NZ Secondary Schools are at Level 4 or higher.
They rest are mainly at Level 3 and Level 2, in that order. This
level has remained unchanged since it was first measured in 1999
despite the Numeracy Project, National Standards, ALIM and ideas
including my own. I wonder why?
A Curiosity as Alice's Cat Suggested
It may be of curious interest to know that the current adult
population is mainly at Level 3 with many professionals (engineers
and lawyers) at Level 4 and a few at Level 5 or better. Society
runs on addition. Look at your Tax Return and see if you can find
anything harder than addition. Who amongst you can see that the
sum of 0 to 100 is simply (101 x 100)/2 or 5050? If I travel to
town at 30kn/hr and home at 20 km/hr, who can see my average speed
is not 25km/hr? Who does cryptic crosswords?
Worse. The polls in an election are predicted by many different
and usually correct methods. How linear and predicable is this?
Propaganda and as Trump promotes "the false media" know that teh
population is controllable or at least can be influenced. The
news-media know this. They only promote what they want to promote
and they know darn well what is going to happen. Oh well, perhaps
this is the result of not improving thinking skills in our
The Cohort view.
Hence Lesson 14
This shows all 230 students or so and the
gains they have made over nearly two years at school, Year 9
and most of Year 10. The expected progress is 1 curriculum
level in this time. Nearly 70 students have been accelerated.
Why is this?
This cohort is going to be very successful in
this school. Afgter entering Year 9 at an extraordinary 50%,
nearly 3x normal, they have risen in a bumpy ride to nearly
85% and are certainly heading to exceed that target. Great to
see. The dip at the beginning of Year 9 and Year 10 is the
leaking of knowledge during our NZ Summer. The improvement
over this time is only OK. The real data indicates no cohort
acceleration but a definite improvement. They only gained .89
as a cohort when expectation was 1. I wonder why? Good data is
not always the best.
The good news is that over 80% of this cohort will easily cope
with our NCEA Year 11 courses. The issue for teachers and the
school is how do we extend the better students so that we can
maximise their efforts. No situation is ever easy or best.
Improvement is always possible. Again, I wonder why many of
these students will sit NCEAL1. Perhaps their time is better
spent expanding and deepening their mathematical curiosity.
Why is monitoring Number so important?
Can this be done in other ways?
How do you see thinking developing?
What messages does teh SOLO Taxonomy have for teachers?
Download the Lomas test and instructions and do the tests
Trial this test on some tame students.
Look at all the questions and ponder what they are seeking.
Circle three questions that are incorrect and use these to inform