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Thoughts on Teaching and Learning of Mathematics
#6 • Revised 4/1/19 Probability and Statistics
- to connect probability
with all parts of mathematics - to know how
statistics makes sense of very difficult problems
- to understand "random" as a concept
Ahh.. chance! Perchance! What a wonderfully misleading,
often abused but valuable concept. Einstein said "God does
not play with dice!" challenging claims that "quantum chance"
rules the world and which it turns out it does. Currently accepted
quantum theories abound with probabilistic ideas and deductions
suggesting God might well have "played with dice'. Experimental
observation supports the randomness proposed within the theory.
Here is the summation to infinity, over all space and time, of the
square of the probability density of a particle in one dimension.
I admire the simplicity and complexity in this statement. How
something so obvious as "it has to be somewhere" is captured by
something so complex and all to do with probability.
The Deep Understanding of Probability Being enabled to measure and use random events
to make sense of the world around us to solve problems.
Probability is a natural part of our language and the concepts are
components of all human endeavour. We use expressions like "on the
balance of probability", "beyond all reasonable doubt",
"possible", "probable", "almost certain", "always", "never", "in
all likelihood", "hardly ever", "sometimes" to name just a few.
The list is large and when asking people of the numeric
probability that might be assigned to each of these expressions
the variation is wide. I asked a court judge of what probability
might he assign to "beyond all reasonable doubt". His answer was
anything over 50%. That frightened me. I had my mind set around
80% or more.
Task 1 for Years 2 to Year 100 Make a list of all the probability terms you can think of,
encounter or might remember, and then order them from least to
most likely. Try and assign a number to describe the
probability. Compare. Warning!...there will be disagreement.
I have a task for Y 7 to 10 all prepared ready for use on the nzmaths.co.nz
website called https://nzmaths.co.nz/resource/number-probability-1
For young people, playing with dice, gaming, and learning
the language of probability is vital. Roll the dice! Play cards.
Play Monopoly. Normalise and use "Paper, Scissors, Rock to sort
issues", and understand the unpredictability of randomness.
Destroy the misconception that if I throw everything but a 6 on 20
or so throws the next must be a six. In random, we always
underestimate the long run.
Marion Steele showed me this experiment with great glee: "Imagine
you are a coin and write down the result of throwing yourself 100
times, ten rows of ten outcomes. Now do the experiment and record
your outcomes of tossing a coin 100 times. Do not mark which is
which. I will come around and tell you which is your guess and
which is your coin toss result." It is very easy, and fast, to
tell the difference between experiment and our perception of what
happens. I challenge you to repeat this task and see if you
can sort how obvious it is to distinguish the difference and be a
"mindreader" in your math class. This task connects to the develop
of "relationship" and "students not realising they are learning".
We need to learn to use probability to select a "random sample"
and know why this is a good way to sample. We must learn to use
probabilistic language. We need to know what number might be
assigned to "no chance" and "will always happen".
To establish 1/6 and 1/10 and how different these two
probabilities are use a six sided die (plural dice) and a 10 sided
0-9 die. The task for students is to guess a number and then try
and toss that number. It happens quite often on a six sided die
but on the 10 sided is decidedly harder. I made up a sheet for my
local GP to use with heart risk patients who have little
understanding of the improvement in outcomes when moving from a
1/10 (10%) risk to a 1/6(17%) risk. It does not sound much but I
know which group I want to be in. This is a valuable exercise, try
Interwoven in the ideas underpinning probability are proportional
thinking and fractions. These are difficult concepts and it is
wise to stick to the language development and playing games
until fraction knowledge is secure. We do use fractions to
represent probability and we operate on these fractions in normal
ways as well but all probability is between 0 and, unlike
The PPDAC Statistical Inquiry Cycle or be a Data
Detective applies to probability, of course.
P - Ask a Question - "I wonder if I can test a coin to see
if it is fair?" P- Make a Plan - I will toss the coin 200 times and make a
bar graph of the results. If it is fair it will be even, or
symmetrical. D - Design and do the experiment - I toss the coin 200
times recording all outcomes. A - Analyse the outcomes - I draw a bar graph of the
results. I also put the results into iNZight and get significance
bars. That is interesting. C - Make sense and a conclusion - This experiment suggests
the coin appears to be fair. The two parts are almost the same
size, Heads show 95, Tails 105, and the iNZight software error
bars overlap so no difference can be supported. The coin appears
to be fair...(answer the question!).
Try figuring out the probability of randomly breaking a stick in
two places and being able to make a triangle with the three
pieces. This is not something that is obvious and a great problem
for statistics and randomness to answer.
What is the likelihood of Kane Williamson being bowled out for a
duck in the next cricket match? What is the likelihood of
President Trump seeing out his first presidential term? Well, he
made it to 2019 despite losing a few decisions and is now in a rut
demanding a wall with Mexico. I wait!
Playing games using dice like Yahzee, or my variation Numahtzee,
Monopoly, Battleships, Dice Games, Black
Jack, Paper-Scissors-Rock are all essential "hands-on" learning.
The folly of gambling can be learned at this level. More advanced
students can simulate LOTTO draws and make up sample tickets using
XCEL to learn how ridiculous winning anything in Lotto can be. The
experience of games is very useful.
The obvious is not always apparent in probability and a wide
experience of problems is important to experience this bizarre and
highly variable idea. In the "Breaking a Stick" problem mentioned
above the geometrical solution is easiest to comprehend. In the
Birthday Problem of "How many people do you need in a group to
have a 50% or better chance that two of the group share a common
birthday" is easier to answer when you ask the opposite question.
How likely I am to win the 2019 Match Play Golf Competition with
the 9 Hole Men's Group is an even more obscure problem despite me
winning it in 2018. Probability infests all problems!
The Deep Understanding of Statistics Being enabled to collect and use data to make
sense of the world around us and to solve problems.
Statistics has undergone major change in NZ and I
acknowledgement the NZ Statistics Association for their collective
wisdom, guidance and hard work over quite a long time. It took
many meetings and emails by many people to settle on ways of
reinventing statistics for the modern world during 1998 to 2008
period when the new NZC was being written. John Tukey
influenced the future with this work and created the dot plot,
and, the box and whisker diagram now so common in NZ schools.
The computer has brought about a revolution in the management and
analysis of data and it is up to us to be good interpreters and
make valid conclusions. Chris Wild's team in Auckland University
caused the Data Explorer on the Census@Schools website and the
easy to use iNZight Statistics Software. The computer and data
storage, data collection and Google have changed the way we
perceive the world. Even the Russians appear to have influenced
the Trump election and again, I wait to see what happens.
The Statistical Inquiry Cycle and what is now called PPDAC guide
or should guide all investigations in statistics and probability.
The past fascination with mean, median and mode has become
"measures of middle". Now "the eye's have it" and "shape,
middle, spread and oddities" of dot plots and box and
whisker graphs rule the reasoning. Add "WWW" for each of
these four features as "What am I talking about", "Where is
it" and "What does it mean" and we have a worthy guide to
become a Data Detective.
Statistical Questions. Pip Arnold completed a PhD on this topic.
Get the question right and you will be much more likely to be able
to answer it. One of the three types of stats questions is
"comparative". I use the frame of "Variable"; "2 Groups";
"> or <", "Population" in explaining and guiding
students to a good question. Pip was on the button with her
research. Without a clear and defined, answerable question anyone
is pretty much in the dark in Statistics.
Task In the question "I wonder if in NZ Year 9 boys carry
heavier bags than Year 9 girls?" identify all parts using my
PPDAC. This is actually a version of the inquiry cycle
commonly used in science. P = Problem, P = Plan, D = Data, A =
Analysis and C = Conclusion.
A statistical inquiry begins with a query or Problem. "I
wonder if in NZ Year 9 boys carry heavier bags than Year 9 girls?"
I might wonder this because I have a heavy bag usually as a Year 9
student and I see that most boys are the same. There might be a
case here for getting some storage at school for some of my gear
My Plan is to go to the last Census@Schools website
survey and get random sample Data of 30 boys and 30 girls
and the bag weights they reported. Then I will make a dot plot of
these two groups and a box and whisker graph using the data
Explorer. Thirty (30) worked when I used this in my class work. My
These comments apply to different Year levels and show the
difference in thinking at these levels.
Year 6/7/8 - The boys box is more to the right (or higher) than
the girls so this suggests that typically boys carry heavier bags.
Year 9/10 - Boys bags are typically heavier since half of the boys
bags are heavier than 3/4 of the girls bags. The median is 5kg and
spread upwards to over 9 kg. The girls upper quartile is about
4.5kg and spread lower or lighter to about 1kg.
Year 11 - The gap between the medians is large compared to the
overall spread of the two boxes. The difference in the medians is
5-3.2 = 1.8kg and the overall middle spread is 5.5 - 2.2 = 3.3kg.
This means the higher boys median is likely to be reflected in the
whole population so we can say "Yes; boy's bags are likely to be
heavier than girl's bags".
Year 12 becomes more technical again as use of the informal
interval ± 1.5*IQR/√n is established. The results here support the
claim that boys bags are heavier.
Year 13 is about "bootstrapping" and testing for difference. Again
this analysis supports the claim.
Means, medians and modes as measures of the middle have been
replaced by the concept of "typical". The Census@Schools
website is the GOTO site for all information. The "dot
plot" and the "box and whisker plot" of sampled data is everything
for Y7 to 11. The concepts of survey and sampling prioritised.
software is found by asking the internet "iNZight software".
Other online software like NZGRAPHER work on
iPads and Cellphones. InZight Lite is a smart 21st C software
WARNING! Do not let
younger students (Year 6,7,8 or even 9 and 10) make use of
software to make dot plots and box and whisker graphs before they
can tell you how to make these things by hand and explain all
features. Show them too soon and they will never draw one again
and miss out on that "learn by doing" experience. "Learning
travels up your arms"and I told many students this view.
Learn by doing!
Students should not be introduced to the Box and Whisker until
they are multiplicative and appreciate that 1/4 of the data
populates each part despite the whiskers and boxes in the middle
50% being different sizes. The Box and Whisker is a brilliant tool
for displaying information and comparing information. It is loaded
with deep understandings and connections. In the diagram is LQ,
Middle 50%, median, UQ, IQR, Range, Min, Max and with teh Dot Plot
we add Shape, Spread, MIddle and Oddities. These "simple" graphs
contain everything except the context!
Here is a diagnostic test to help know if students are ready to
move on. TASK Which section A, B, C or D of this box and whisker
has the most data?
Students who choose the largest D or even A
or even B are of course all incorrect but do so because
larger is bigger and more. This is the world of
the additive thinker; "What happened yesterday will happen
tomorrow", "More is better", Bigger is better", "One
think dominates thinking and connections do not exist past
simple obvious links".
Students need to experience and make meaning for them selves
to get to the correct answer of "All sections have the
same number of data! Reason - In this case there is
28 dots so each section has 7 and this can be seen. Section
A is Minimum to LQ or 25%, Section B is LQ to Median or 50%,
Section C is Median to UQ or 75% and Section D is UQ to
Maximum. (Clear connections showing Multiplicative and
Proportional thinking). The IQR is the width of the Box and
is a measure of spread. This is a trick question! (Critical
thinking)" You would be very happy for a student to present
Everyone should study statistics for as long as they can and all
pathways for all students from school should include Year 12
Statistics knowledge and skills. Today's world has a data
infestation and "making sense of data" is now more vital
than ever. Computers can deal with millions of data points and
every time you log on to the internet or use a credit card you are
and monitored. Know what is happening. Go to the GeoNet website or EBOP
buoy website to see how much data is being collected for
scientific purposes. How much data is Nasa collecting from space
probes? The answer is trillions of Gbytes daily. Our task is to "make
sense of the data."
A good grasp of the language of statistics is vital. I describe
statistical language as "floppy" as it includes phrases
like "tends to", "the data suggests" rather than exactness of
mathematics like "x = 3" and "length of the pipe is 3.4±.1cm". We
can not say that the mean weight of a trout in Lake Taupo is
2.34kg because it is a population statistic that we can never know
exactly. We can only use a sample to get some interval in which
the mean or median of the population is likely to be. That is
Statistics is the place for literacy development in mathematics
using words. In Mathematics we also have literacy but it is better
described as subject specific literacy. Setting out a proof in a
logical way so others can read and comprehend the reasoning is
clever subject specific literacy. Solving an equation in a problem
in a logical way likewise. Constructing a pentagon using a compass
and ruler is also literacy in mathematics. Literacy in mathematics
looks most like literacy in another subject when we have word
lists and essays on math topics.
Statistics however requires a strong writing and reading ability.
It is a great place for STEM students to develop the skills of
writing and reading, and speaking and listening. It is a bit
meaningless to say the mean weight of trout in lake Taupo in
1.24kg. This says nothing of the population except that one
statistic, the mean. It is better to use the word typical and add
an interval to get "The typical trout in Lake Taupo is between
1.1kg and 1.75kg. This is the spread of the middle 50% or IQR, the
Inter-quartile Range. There are smaller and larger fish."
Statistics and probability walk hand in hand with the concept
of randomness. Why is a random sample better? Why do
we use a random sample? Are there other ways?
Random sampling causes all those questions we did not
ask of the data to be evenly likely to turn up in any
sample and so minimise bias and false inference.
This is hardly ever explained. In the example of asking how many
text messages a group of male and female students send the
previous day to try and answer the question of "Do female students
send more texts than male students?" notice we did not ask a few
other questions such as;
- is this your first phone?
- Is this a new phone?
- have you got a new partner?
- is your Mum ill?
- has your friend just won Lotto?
- and about a million or so other questions that I have thought
of and the million or so other questions that I did not even
By randomly selecting students we also spread all this questions
and answers randomly across all the data so that when we select
the two groups of Male and Female the question that we did ask has
more validity and is not biased. The comparative inference we then
make has more validity.
Here is another issue and usually teachers just tell students "Select
a sample of size 30". Good heavens
and good gracious and OMG - WHY? Telling destroys
learning! It prevents the understanding from students making their
own meaning. Yes, it might be faster and teachers will always cry
"I do not have time to explore this idea and wait for students
to make sense of it all". I respond... "You do not have time
not to allow the students to explore and make sense of
these deep and meaningful understandings. This is the real
learning!" Like all thinks, there is a time to tell, but it 'ain't
IN 2016 I ran a study group of several senior Statistics teachers
and coached them to make inquiries into the Teaching and Learning
of Statistics. We were learning how to "Inquire" amongst other
things. Not long after the group formed I was aware no one
actually could explain "How big should a sample be?" and provide a
cool learning experience and reasons. Item #2 in this link is the
.ppt I developed as a result and have used many times since. Sample Size
Powerpoint. I used the iNZight software to select samples of
different sizes and measured the computer screen to get a spread
estimate. This was graphed and modelled. The task is a great
student learning experience and they should all take the time to
do this and then try and answer the question of how big a sample
There are a couple of new learnings that came out of this for me
as well. I was reminded of the old adage by statisticians that
"quadrupling the sample size halves the spread (or 1/√n)". I also
noticed computers could handle large amounts of data easily, now
making a sample a bit irrelevant.
Is Statistics Mathematics?
Yes. Statistics deals with the floppy questions our world
presents. It determines standard error and ±limits
and why. It allows Google to make better software and retailers to
target products to consumers. "Making sense of data" is
the primary mantra of Statistics. The reason I said "yes" to this
question is that mathematics is inside statistics doing the
median, LQ, IQR calculations, running the software and drawing the
graphs. It runs the internet and data gathering systems, the
telemetry systems, the sensors and satellites. Mathematics runs
the atomic clock that can now measure the daily variation in the
Earth's rotation rate! It is why our cellphones work! Mathematics
is deep within Statistics, just like all the STEM subjects.
Hence Lesson #6
Making sense of data is statistics. The PPDAC cycle is vital
with hands-on sampling to learn about variation and what you can
say about a population. This is an excellent subject to help
students learn to write to convey meaning. Making sense of
probability and using random, chance, outcomes is just as vital.
Both areas of knowledge, Staistics and Probability,
applied to problems becomes the learning.
What are the key competencies opportunities that present
themselves for learning probability and statistics.
How are these part of your main vine of learning?
List all the probability words you can and order these from
never to always.
Investigate and master one computer based stats programme, such
as iNZight or NZGrapher. There are others.
Look up Taupo Trout Resources on my website and use these in
junior programmes. Find Kiwi Kapers on Census at Schools.
What are the three types of statistical questions.