Cling to the main vine, not the loose one.
Kei hopu tōu ringa kei te aka tāepa, engari kia mau te aka matua

123Thoughts on Teaching and Learning of Mathematicsrand dice

Lesson #6
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! Perhance! 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. Currently accepted quantum theories abound with probabilistic ideas and deductions suggesting God might well have "played with dice'. Experimental observation supports the randomness 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 (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 estimate, measure and use chance events to make sense of the world around us and 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 website called

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 teh next must be a six. In random, we always underestimate the  long run.

Task 2
M 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. 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.

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".

10 die6 dieTask 3
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 1/6 (17%) risk to a 1/10(10%) risk. It does not sound much but I know which group I want to be in. This is a valuable exercise, try it.

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 fractions.

Task 4
The PPDAC statistical inquiry cycle or be a Data Detective applies to probability, of course. See below for more details.
P - Ask a Question - "I wonder 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 pie graph of the results. If it is fair it will be even, or symmetrical.
D - Do the experiment - I toss the coin 200 times.
A - Analyse the outcomes - I draw a pie graph of the results. I can also put the results into iNZight and get significance bars.
C - Make sense and a conclusion - This coin is pretty close to being fair. The two parts are almost the same size. Heads show 95, Tails 105. The iNZight error bars overlap so no difference can be supported. The coin appears to be fair...answer the question!

Task 5
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 an obvious calculation and an inquiry is needed.

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?

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 can be.

The obvious is not always apparent in probability and a wide experience of problems is important to experience this bizarre 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 easy when you ask the opposite.

The Deep Understanding of Statistics
Being enabled to collect, use 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. 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 his 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, 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 Statistical Inquiry Cycle or what is now called PPDAC guides 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. My colleague 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.

In the question "I wonder if in NZ Year 9 boys carry heavier bags than Year 9 girls?" identify all parts using my frame.

. 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 sorted out! 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. My results are:-

sample analysis

Year 6/7/8 - The boys box is more to the right or higher than the girls so I think 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 teh 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.
Year 13 is about "bootstrapping" and testing for difference.

In statistics "The Eyes have it!" the
        eys have it

Means, medians and modes as measures of the middle have been replaced by "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.

 iNZight software is found by asking the internet "iNZight software". Other online software like NZGRAPHER work on iPads and Cellphones. INZ Lite is a smart 21st C software solution.

WARNING! Do not let younger students 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"; well, I told many students this. Students should not be introduced to the box and whisker until they are multiplicative and appreciate that 1/4 of the data is in each part despite the lines and areas being different sizes. 

Here is a diagnostic test to help know if students are ready to move on.
Which section of this box and whisker container the most data?

quarter of BW

Everyone should study statistics for as long as they can and all pathways for all students from school should include Year 12 or 13 Statistics knowledge. 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 being measured. 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?

BIG 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. 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 statistical inference.

Statistics and probability walk hand in hand with the concept random. Random sampling causes all those questions we did not ask in the data to be evenly likely to turn up in any sample and so eliminate bias and false inference. The size of a random sample is typically 30 to 50 for a school situation and I made a .ppt to help answer this question. You can use this at any level and should get a class to take random samples of different sizes as an exercise. Best done on computer!

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 applied to problems becomes the learning.

Teacher TASK
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.


This is to help look around my pages. I have tried to make it consistent in all chapters.
1. Intro and Relationships, L#1
2. The Main Vine, L#2
3. Beginning a Year, L#3
4. Number and Algebra, L#4
5. Geometry and Measurement L#5
6. Probability and Statistics L#6
7. Problem Solving L#7
8. Investigations L#8
9. Visual Mathematics L#9
10. Assessment and Learning L#10
11. Team and Problem Based Learning L#11
12. Engagement L#12