Sampling and building relationships

Have you ever seen bullmastiffs taking a random sample? What about a pug?  Well… you’re in for a treat!

The backstory

I introduce sampling to my Year 12 students with a lesson progression adapted from Lindsay Smith (here).  The day before this lesson, a brilliant homework task was assigned – students had to find a silly dog picture to add to our wall – the story behind the wall is a completely different post that I’ll try to get written one day!


The next day we introduce sampling… the short version goes something like this:

  • Hand out class rolls to pairs of students
  • Ask them to come up with five different ways of selecting five students
  • Then share back the different ways they come up with – generally the basic methods are covered such as simple random sampling, stratified sampling (one from each group of desks or x boys and y girls (are girls are more likely to do homework?… hmmm…. you can see where this would lead to nice class discussion), convenience sampling, self-selected sampling 
  • I then go formally through the different sampling methods, and we take a sample using each method from the class, with replacement obviously.
  • If a student is selected and they have done their homework they get a lolly…. if they get selected and they haven’t done their homework then there is a “punishment” as deemed fit by the class.

The “punishment” is an opportunity to have some fun with the class, with good humour – we have a range of options for them to choose from that the class decides before we start.  This year we included star jumps, press ups, and drawing a horse (more on that in the promised later post).  Much laughter ensured this year as Louis, who of course hadn’t done his homework, was picked four out of the five times! He choose to draw each time, along with quite a few others, and my whiteboard ended up covered in a variety of creatures.  Here’s two that I managed to keep for a while amongst the rest of my busy back whiteboard

dog & horse

The new exciting part…

The one sampling method that was fresh for me this year, that my students had never come up with before, was to get my dogs to select a sample – I think maybe I’ve shared too much about them with my classes!  Always up for a challenge, and an excuse to do silly things with my dogs, I roped in the help of my husband that evening and here’s the results:

Cornelius sampling

daisy sample
Daisy sampling


Tess sampling part 1


Tess sampling part 2

As you can see… the student names were spread out in a circle in my garage, and what you might not quite be able to see, is that there is a small dog treat on every piece.  The dogs were spun to get a little dizzy (not harmed in anyway!) to aim for a “random” start point.

Were the dogs random?  I don’t know about the big dogs – yes, the start point was random, but they just went into hoover-mode from there so it depended on whether the names were also randomly spread in the circle (I have to admit, I don’t think they were). But… I’ve found a skill that a blind, deaf, losing-her-sense-of-smell, old-lady pug has!  I think Tess was quite random!  It took her ages to find each piece and make her selection – the clips above are just a small portion of the 2 minutes it took her to choose five students.

Why this was so cool? – other than having dogs sample of course – it gave me an opportunity to continue to reinforce the positive relationship I have with my class: they enjoyed seeing my dogs, I enjoyed being able to share a little of my dogs with them and involving them in class – we often, as teachers, seem to work really hard to remember to check in with students after their big sports event, competition, work experience, camp, music, new job, new sibling, new dog…. and its nice having this interest and care reciprocated.

The reward for being selected by Cornelius, Daisy or Tess…. a picture of them to keep!  Here’s Mya’s book – who was lucky enough to be selected by both Daisy and Tess:



The power of sorry

It feels like a long time since I’ve written anything here – I meant to write up this post at the start of the year, but now, with term 2 starting tomorrow, I’m finally getting there.  Oh well, better late than never (sigh).

Reflections on the start of year…

In our faculty session at the start of the year I showed Rita Pierson’s Ted Talk “Every kid needs a champion” (again – I showed it to the faculty last year and I’ll probably show it next year too 🙂   )  If you haven’t seen it I’d highly recommend you go and watch it now.  Its here.  One of her suggestions in building relationships with students is to say sorry.

One of the other things we discussed as a faculty that morning was bringing consistency into some of our behaviour expectations of classes, including introducing The RED Box in each class.


The idea for The RED box came originally from Sam (thanks Sam!) last year, a few of us trialed it successfully and hence the roll out to full faculty this year.  It is for cell phones (and other distractions) in class.

My method for using it is as follows: when I’m ready to start the lesson, I say to the whole class something like “okay, lets get started – now is the times for phones to go away”.  They know this means in bags or pockets.  Then, if there is a phone out during class, I just quietly walk over to the student and hold out the box for them to put their phone in, I don’t stop what I’m doing, its just a really low-key non-confrontational way of dealing with the ongoing distraction of cell phones in class.   Also note that in the bottom of each box, there are red cards – if a student argues about the phone at all, they get the choice of putting it in the box or getting a red card – simple!  I’ve not had to give out a red card yet, but have threatened a few times when they start the “I was just checking the time…” arguments.

This method works really well – students know what to expect, know how the system works and I don’t make a big deal over it with them.  As an aside – I’ve moved back to using my basket that I used last year – I found the lid on the box annoying.  A second aside – yes, I sometimes forget to give the “phones away now” signal, so when I see a phone out I just say something like “thanks Nigel for reminding me that its time to put phones away , phones away now please”.


Right – that’s the brief background.  Day 1 in my MAS201 class (Year 12 Statistics, last period after a full day, first day back at school for the year) – my goal was to establish positive relationships while making sure routines and expectations were clear.  Yes, as part of this, I introduced The RED Box to my class and made a really big deal over the “phones away now” signal.

A few minutes later, I spot a phone out… yes, at the back of the room… yes, from a J-name boy! (what is that??!).  I went over, and held out the box, just like we’d discussed.  He started arguing about how he was checking the time, of course, I mentioned the red cards and J-name reluctantly placed his phone in The RED Box.

A wee while later – The RED Box vibrated from the front of the room.  J-name calls out “can I get that, it’ll be my Mum?”.  Unfortunately, I made a classic teacher-mistake of discussing this with J-name across the class, and after a bit of too-ing and fro-ing I ended the debate with “please see me after class”.  (thiank goodness, now at the start of term 2, I can’t remember all the details!)

When J-name came to collect his phone from me at the end of class, our debate continued re using his phone in class, the appropriateness of answering it, even if he knows it was his mother, how come his mother was calling in class time etc etc.  It wasn’t pleasant, nor a moment that I felt I handled well at all.  J-name is a big personality in class, I knew I needed to get him onside, I was pretty sure I’d blown it – I worried and mulled over it all night before realising that I needed to channel Rita.

So the next day… the first thing I did as J-name had come into class and sat down (yes, at the back again) was to go over to him and apologise for the way I’d handled things yesterday – I told him that I’d worried about it all night, and I really wanted to just start fresh today.  I was really impressed with the reaction from J-name.  He also apologised and said that he had been a dick (yes, his word) and yes, that we should start fresh from then on.  Whow! How to turn things around Dr D…

But hang on – J-name’s phone was out again near the end of that period, I made a judgement in the moment to ignore it that time aiming to build that positive relationship.  Next day, I mentioned to J-name at the start of class that I’d seen him with his phone out again yesterday and that I was disappointed.  I’d have to get The RED Box out if it happened again – it did, but no argument at all this time (and I remembered to ask him to turn it off before it went in); when he collected it I suggested that he was welcome to put his phone in The RED Box at the start of the period when he came in as a preventative if he wanted to.

J-name hasn’t done this, I’ve had his phone (and plenty others) in The RED Box over the term but no grumbling about it at all from him and I think we have a pretty good relationship now, and lots of good work is being done.

Moral of the story – apologise, be humble, keep working on building the relationships but keep your expectations high and keep working towards them in a way that supports all the other things you are trying to do.  Never give up on your students.

From Rita: Teaching and learning should bring joy. How powerful would our world be if we had kids who were not afraid to take risks, who were not afraid to think, and who had a champion? Every child deserves a champion, an adult who will never give up on them, who understands the power of connection, and insists that they become the best that they can possibly be.  Is this job tough? You betcha. Oh God, you betcha. But it is not impossible. We can do this. We’re educators. We’re born to make a difference.


PART 3 – COMPARING TWO GROUPS Developing big ideas with sample-to-population inferences …

This is the third part of a series of posts on sample-to-population inferences, and progressively developing students understandings.

Comparing two groups:

Is what we are seeing when we compare the features of our samples of our two groups (in particular the median/mean) MORE than just sampling variation?

Our way of judging this difference gets more sophisticated as we move up our curriculum.  We essentially have a four-stage progression with details found HERE for NZ Curriculum Levels 5 to 7.  The fourth stage introduces formal bootstrapping methods for constructing confidence intervals.

STAGE 1: Curriculum Level 5 guide (Year 10 – 11ish) if the median of one group is outside the middle 50% of the other group then  you can make the call that “group 1 tends to be bigger than group 2 back in the population” (for samples of size 20 – 40).

boxI usually convince students that this works in two ways.  Firstly, by using the lesson progression developed by Pfankuuch et al (here are the key resources, but make sure you check out the workshop 1 material too.  This wiki also has all the resources and a summary together nicely (thanks Pip!).  The lesson progression uses a class investigation where students to take multiple samples from Karekare college, firstly comparing boys vs girls heights and secondly comparing times to school for students busing or walking.

Students CAN make the call that students who bus tend to take longer to travel to school than students who walk back in Karekare College – this is seen visually by the consistent shift between the multiple samples.  


This idea is reinforced with arm-waving again.  It is important this time to make sure you have bling on each arm (in this case watches) to represent the medians.  You can see below that we have the situation where the two groups are so far apart that although the middle 50%s move (each arm) with repeated sampling, the medians (watches) are consistently outside the middle 50% (arms) of the other group (for at least one group).  Please note that this one is a bit tricky to pull off – our arms aren’t quite designed right for this sort of functionality…

giphy (5).gif

Students CANNOT make the call that boys tend to be taller than girls back in Karekare College – visually the multiple samples don’t show the same consistency.  That is, there isn’t enough of a shift between the samples of the two groups to say whether boys tend to be taller than girls or if girls tend to be taller than boys back in the population.


With similar arm-waving ideas as previously – this time with repeated sampling we show that if the two group in the original sample are close together,  the middle 50% (arms) and medians (watches) may stay in the same positions or may switch round with repeated sampling due to sampling variation – so we can’t actually tell from this sample what’s happening with the difference between the two groups back in the population.

giphy (6)

STAGE 2: Curriculum level 6 guide (Year 11ish):  if the distance between the medians is greater than a proportion of the “overall visible spread” (OVS) then we can make the claim that B tends to be bigger than A back in the population.  The proportional relationship between the medians and OVS is linked to both sample size and spread.

L6 box

The idea that sampling variation is influenced both by sample size, and spread of the population is an important idea to start developing with students.  We work as a class to create multiple samples of different sizes (as discussed in Part 1) and to explore the link between the difference in medians and the size of the overall visable spread.  Resources are available on Census@School here.

STAGE 3: Curriculum level 7 guide (Year 12):  if the informal confidence intervals for the medians do not overlap then we can be pretty sure that the median of B is bigger than the median of A back in the population.  


Students take to this guide quite naturally and just run with it if the work has been put into developing their understanding of informal confidence intervals, why we need them and how we incorporate sampling variation measures.  We need to make sure we emphasise the slight difference in the conclusion we are drawing – this time it is directly about comparing population medians.

Again, I reinforce ideas of sampling variation and confidence interval construction with hand-waving.  When our groups are close together, with a sample size of n then the confidence intervals are THIS wide and with repeated sampling they will “jiggle” round quite a bit due to sampling variation.  [insert suitable sound effect at *].  If we increase the sample size then the confidence intervals will get narrower*, AND there will be less “jiggle” so we can be pretty sure that back in the population we will see the differences in the medians of the two groups.

giphy (7)

STAGE 4: Curriculum level 8 (Year 13):  Students complete their sample-to-population inference journey with us in their last year of high school by meeting a formal re-sampling method, namely bootstrapping, to create confidence intervals.

Further details on teaching bootstrapping for understanding can be found here  or here  (original roll-out teacher professional development). 

Part 1 – Sampling variation

Part 2 – Confidence intervals

PART 2 – CONFIDENCE INTERVALS Developing big ideas with sample-to-population inferences …

This is the second part of a series of posts on sample-to-population inferences, and progressively developing students understandings.

The need for a confidence interval (NZ Curriculum Level 7/Year 12-ish onwards)…

The next step in developing students’ appreciation of sampling variation is for them to understand that making a sample-to-population inference where they give a point estimate of the population parameter is setting them up for failure – they will always be wrong.  Instead they need to use an interval of plausible/believable values within which the population parameter is most likely to fall between, based on our best estimate of the size of the sampling variation.

My lesson progression, outlined briefly below in relation to developing the Level 7 informal confidence interval, is adapted from others’ work including Pip Arnold (Pip’s work) and Lindsay Smith (Lindsay’s work).

  1. Introduction to/ recap of sampling methods including how and why we sample.
    I set a silly homework assignment the period before this lesson, then get the students to brainstorms all different ways we could select five students to check their homework.  Generally students come up with our standard sampling methods needed (eg simple random, systematic, stratified, convenience…), occasionally with a little prompting from me. We then “check homework” which always involves students receiving a lolly if they get selected.  This does lead into interesting conversations on RANDOM (see my previous blog on this!) and our perception of RANDOM.  It does pay to save a few lollies for the end for any students that haven’t been selected!
  2. The need for confidence intervals – meet the Kiwis… 
    Students work in pairs each take four samples (n=15) from the kiwi population and use their samples (after creating dot plots, box plots and summary statistics) to finish the following sentence “From my sample data I estimate that the median weight for all New Zealand kiwis is….“.   Usually students who have worked together will give the same/similar answers but we get a variety of answers across the class.  Sometimes we even get a student or two who give an interval for their answer – send them to stand in the back corner of the room immediately!  The big reveal, of course, comes with the teacher-prompt of “but your answers are all different – who is right?“.  Generally your more confident personalities will claim that they’re right, while the quiet students sit quietly looking concerned.  Pad this out as long as you need to to make your point that “they are all wrong“.  How can they know EXACTLY what the population median will be?
    NOTE: Exactly how you deal with this will depend on the relationship you have with your class – it may be better to soften this statement to “you might be right, you might be wrong – in the real world you would never know…”.  I enjoy stirring my class a little and the reactions I get are always lots of fun!
    We now bring back into the room the students sent to the back of the class and praise them for being three steps ahead of the rest and putting an interval around where they thought the population median would be.
    We could use our medians from our repeated sampling (collect the class results on a big graph) and read off where most of the medians lie to give us a good idea of the interval for the population median.  This works fine in the classroom where we have easy access to the population to actually complete repeated samples, but in reality we want methods that we’re happy working with when we only have one sample.
  3. Increased sample sizes = decreased sampling variation. Therefore our interval should get narrower as our sample size increases.  See PART 1 for how I reinforce this with students.  Hand gestures are very useful again here: If your confidence interval is THIS wide for this sample size – what happens when you INCREASE your sample size? – make sure you include a sound effect as your confidence interval gets narrower with the increased sample size!
  4. Increased spread in the population = increased sampling variation.  Therefore our interval should get wider as our spread increases.  I use the simple situation (from Pip) of comparing student heights for new furniture in an intermediate school (Year 7 & 8) or a middle school (Year 7, 8, 9 & 10).  We ask which teacher is likely to get a closer estimate of the students heights?  
    Bring back your hand visuals again:  If your confidence interval is THIS wide for this spread – what happens when you INCREASE your spread? – make sure you include a sound effect as your confidence interval gets wider with the increased spread!   Of course, both here and for increased sample size we can start showing the ICI visually with a (red) line on our box plot, centered about our median.
  5. Formula for informal confidence interval (ICI).  I introduce the suggested formula for students to use, and we check that it meets our requirements.  Yes, the bit we add on or subtract does get smaller when we increase the sample size (as we’re dividing by a bigger number); yes, the bit we add on or subtract does get bigger when the spread (interquartile range IQR) increases (as we’re multiplying by a bigger number)
  6. Checking our informal confidence interval formula works most of the time.  We have class set of 100 different samples from the kiwi population (here if you want it) where students calculate 5 different ICIs themselves, then we collect these to check whether they captured the population median or not (collection sheet is here).  I am very clear to reinforce with students that we are working in TEACHING WORLD so we can do exactly what we’re doing – testing our ideas to convince ourselves they work.
  7. Reinforcing the ICI has been developed and tested by people who know what they’re doing, ready for us to use this year in class.  It is a step along our sample-to-population inference journey which culminates next year with the introduction of formal methods for constructing confidence intervals.
  8. EVERY TIME!!!!! you construct a confidence interval you should be interpreting it (even if it’s just in your head!).  This is almost a mantra in my class.  Students need to continually remind themselves what the point of creating the confidence interval is, and by interpreting it carefully we are doing this.  Remember, every confidence interval interpretation should include the statistics, the population, “pretty sure” (or equivalent indication of uncertainty), the variable, numbers and units.  For example “we’re pretty sure that the population median height of all Year 12 boys in New Zealand is somewhere between 173cm and 182cm“.
    Other formative assessment questions such as “Would it be believable that the median height of Year 12 boys in New Zealand is 185cm? Why? Why not?” are also super-important to check students’ understanding.  And of course slipping in annoying questions such as “Why do we use confidence intervals?”

Part 1 – Sampling variation

Part 3 – Comparing two groups

PART 1 – SAMPLING VARIATION Developing big ideas with sample-to-population inferences …

This is a post that’s been rattling round in my brain for months now, I’m finally getting it down in words.  Firstly, a huge shout-out to Anna Martin (@annamartinnz ) for all the help, suggestions, proofing,… she’s given so kindly to make sure this all makes sense! 

Right this has all got soooo long that I’ve split it into three parts (1) Sampling variation (2) introducing confidence intervals and (3) comparing two groups

Our NZ Stats curriculum is awesome and has developed a clear learning progression for developing (and assessing) students’ sample-to-population inferential understandings.  This progression is based on work from Maxine Pfannkuch, Chris Wild, Pip Arnold and others (for example Pfannkuch, M. J., Wild, C. J., & Parsonage, R. (2012). A conceptual pathway to confidence intervals. ZDM – The International Journal of Mathematics Education, 44 (7), 899-911; Arnold, P., Pfannkuch, M., Wild, C. J., Regan, M., & Budgett, S. (2011). Enhancing students’ inferential reasoning: From hands-on to “movies”. Journal of Statistics Education, 19 (2), 1-32) and is about having clear visual cue to develop and reinforce student understanding.  We are looking at what we can say about what is happening for two groups back in the population, based on samples, generally focusing on the median and mean. 

I’ve popped a summary together here of how I view the learning progressions from Year 10 to Year 13, including new ideas at each level, what needs reinforcing, and “watch-fors” (Year 10, Level 1 Multivariate data 91035, Level 2 Inference 91264, Level 3 Formal Inference 91582).  Please let me know if it’s helpful, ask for clarification, play spot the errors etc.  But… after teaching these progressions again this year, across all three levels (and still going!) there are some common themes and sticking points for students (and teachers).  Here’s my thinking on it.

Sampling variation: The variation in a sample statistic from sample to sample.

This is a BIG IDEA concept that we start developing in Year 10 with our students, repeat and build upon for the next three years.  Initially I look at this in a one sample situation with students.  Key teaching activities and tools are listed below (with links where possible):

  • Students in class each take a random sample from the same population (bag of data cards each) of size n (usually about 30).  After a dot plot, box plot and calculating summary statistics (usually median), they then use their sample to answer the investigative (summary) question – for example “What is the median height of doozers from Fraggle Rock?”.  Students notice that everyone’s samples are similar but different – they SEE sampling variation.  Collect summary statistics (usually lower quartile, median, upper quartile) from each sample into a class graph.  Students can SEE sampling variation in the summary statistics.  Links for these activities can be found here (Yr10 Karakare College population), here (Yr12, Kiwis population), here (Yr13, Pugs-in-Costumes-on-the-internet populations or Doozers).

These are some samples from Karekare College – our next step would be to create box plots above the dot plots. 

  • Chris Wild has animations here which track the median and middle 50% for repeated samples, reinforcing what you have done by hand.boxes_1samp_mem_30_600
  • iNZight had dynamic plots collecting the distribution in means or medians from repeated sampling

giphy (4).gif

  • Finally (and most often!) I do a lot of hand-waving…. firstly the simple situation of one sample where my hand is capturing sample median from repeated sampling (that is – its tracking the equivalent of the blue line in Wild’s animations above)

giphy (1).gif

The impact of sample size (NZ Curriculum Level 6 (Year 11-ish) onwards)…

The activities above can all be adapted to compare sampling variation with different sample sizes.  This year I also had students using the Kiwis population in Excel, combined with a random number to take larger samples.  The excel file is here if you want it (full lesson details are here near the bottom of the page), and the pictures of my class collection of repeated samples of different sizes are pictured below.  In this picture, the middle 50% (box) is shown, with the green section being from the lower quartile to the median, the median is in pink and the blue section is from the median to the upper quartile. This clearly shows that sampling variation gets less when you have larger sample sizes.  Please note that there are 700 kiwis in our population so when you are sampling 400 it is a large proportion of the population.

Hand-waving sampling variation with a change in sample size:

  • Take your left hand, have it track the change in sample medians for repeated sampling of size 30.  Got it?  Okay, now…
  • Take your right hand (don’t stop that left hand!), and have it track the change in sample medians for repeated sampling of size 300…
  • Okay – you’re laughing (or your students are) as it’s quite tricky to have your hands waving differently but… the point is (really easily, visually, laughably) reinforced that with larger samples the sampling variation is reduced.

giphy (3).gif

Part 2 – Confidence intervals

Part 3 – Comparing two groups

Random random random…


Start of year madness has developed and I’m just a wee bit late with my Week Four post of the 2016 #MTBoS blogging initiative – I’m working on the theory of better late than never…

I found this blog post last year from Bob Lochel (@bobloch) – we’ve now run it successfully with quite a few of our Year 13 classes, but I ran into a slightly different scenario with my awesome MAS302 students last week (MAS302 at Cashmere High is our second tier Statistics course).

In a nutshell, the activity requires students to generate a string of 50 coin flips while the teacher is out of the room (either randomly using the RandInt function on their calculator or making them up) .  They place their results sheet on the whiteboard.  The teacher then comes back in after an allotted five minutes (it’s enough time to make a cup of coffee and catch up on a few things, trust me) and sorts the students’ results sheets into “guts” (as in the students made the results up) or “random” using hints such as how many strings of heads or tails in a row, or whether they alternate a lot at the start etc.  See Bob’s post for a much better and more detailed description!

I really like using this activity right at the start of the year as, not only does it allow a discussion around what “random” looks like,  and how tough it is for us to be random, but it lets your students know that you trust them right from go.  My class did this activity in our third lesson of the year.  I purposefully let the students know that I trusted them to do what they needed to do, and walked out of the door with confidence.

I did set things up a little differently from normal this year after speaking to another teacher.  Students don’t know if they will be generating numbers at random or making them up until I’ve left the room (I just have a big pile of instruction sheets mixed up for them to come and collect one) so I showed everyone how to generate the required random numbers on their calculators IN CASE THEY GOT THAT SHEET.  I left the room with a “make sure you read your instructions carefully”.  When I returned, I did really well finding the “random” sheets, but got all but one of the “guts” sheets wrong – when we’d sorted the sheets into the right groups I gave my class a funny look – all except one group had randomly generated their numbers!  I knew I’d made sure there was about half-half of each sheet (another cunning trick to sway things in my favour) so knew something was up.  We discovered that lots of students in my class aren’t the best at reading written instructions, and figured that because I’d showed them how to use their calculators then they should be doing that…. hmm…. best to know this now I suppose… and yes, I’ve teased them about it every day this week!

We did manage to wrap up with a conversation about how its really hard to be random, and looked at the graphs from our senior student survey of our “randomly” picked numbers between 1 and 10, and between 1 and 100.  Here they are:


Some students knew that most people pick 7 when choosing a number between 1 and 10, and interestingly, when you look at the year level, you can see that many more Year 12 students have chosen 7 than Year 13s –  in discussion in class we thought it was likely to be because a lot of Year 12s didn’t know about picking 7, whereas lots of Year 13s did (after conversations like this one in class!).  Everyone in my classes who knew about 7 being picked most often made sure they picked another number. (try here for a short explanation of the 7 thing).


Yes, the second clear mode here is at 69… my comment with this one in discussion with classes was: “what would it have looked like in Year 10?”  I’ve colour-coded it here for fun, and it’s just like my classes suspected it would be (boys are in red) – I refused to do this in class.

Random came up again in my MAS302 class this week – we were running an independent group experiment, very well designed (cough, cough) where our treatment was forks or chopsticks and our response variable was the time taken to get 20 M&Ms into your mouth, one at a time, using your utensil.  We were all slightly scared about how long it would take us with chopsticks and decided we wouldn’t put a time limit on things unless we really had to (like if the bell was about to ring…).  In my superb teacher-planning pre class I hadn’t remembered to organise how I would randomly allocate to the two groups.  Luckily I found a box of dice in the draw – I gave everyone a dice and we decided that an even number put them in the fork treatment group and an odd number the chopsticks group.  We then rolled our dice all at once.  We got more forks than chopsticks and there was some great discussion around whether the difference was so big that I should suspect people cheating…  (I got a fork that time, thank goodness, with a time of 23 seconds, quite respectable).  I made this excel demo quickly for the next day to continue the discussion (see below).  If you’re like me, and not the best at chopsticks then my big hint is to make sure there is no requirement in the experimental design that says “utensils must be used in the proper way”.  I’ve got away with that twice now, using the chopsticks to scoop up M&Ms.  It makes for a great discussion in the debrief of the experiment about things we maybe should have considered.



Better questions for statistical thinking

MTBoS blogging initiative, week 3!  This week’s topic is better questions.


Kiri Dillon (@DillonK_Chch) and I have spent some time looking at ways to develop our students statistical insight – that is, how to encourage them to be deeper statistical thinkers.

Briefly what I mean…  Here’s a sample of weekly income by gender.

surf incomesurf income summary

Most of our students have got reasonably good at describing what they SEE in the samples, the median weekly income of these males is about $300 higher than the median weekly income of these females etc.  But what questions or prompts can we give students to encourage them to think beyond just what they are seeing?  Why might we be seeing this big difference in gender? Were we expecting this? What other variables might help explain this?  What else might be going on? …  Which might lead them to exploring things like:

surf income 2

which shows that the hours worked per week is also important.

And the best questions we have found to get the students thinking further are the simple ones…

  • Why?

  • So what?

And then just repeating them to students (you know, like your annoying five year old that just wants to know more and more and more and more…)  It works surprisingly well.

My Favourite* starter


*Favourite = New Zealand spelling!!  Sorry to anyone else reading it, it’ll just have to look wrong to you…

I’ve decided to write about STRIKE-MERE for my post for Week 2 #MTBoS 2016 blogging initiative – my favourite starter that I incorporated into my class routines last year – every Tuesday.

Firstly, a big thanks to Jane Gray who shared this with me 🙂

How to play: Students pick and write down four numbers from 1 – 6, repeats allowed.  I then roll my dice (big and soft with a really annoying bell in it) four times.  Students win if they have the same number as rolled, in the same position as I rolled it.
The prizes are as follows:

  • STRIKE ONE (one number matched) = high five from me (or fist bump upon request)
  • STRIKE TWO (two numbers matched) = one sticker
  • STRIKE THREE (three numbers matched) = one thing from my treat box (stocked with a variety of stationery, lollipops, silly toys from the party section at the cheap shop – sticky-men were very popular last year, no idea why…)
  • STRIKE FOUR (all numbers matched, in the right order) = one day playing games, at a time agreed by myself and the winner (to avoid the run up to high stakes assessment, or when the winner will be away)

For example: if a student picks 4 5 1 4 as their numbers, and I roll 3 5 6 4, then they win Strike 2 (for the 5 in the second position and the 4 in the last position) – make sense?

Why I like this game so much: This is a fast, quick game (takes about five minutes at the start of class).  Students all seem to enjoy it a lot.  You can work out the probabilities and expected number of periods playing games (about once every two years, on average but don’t tell…) if students seem inclined to head down this direction but (and this is the part I love the most) … I noticed last year, in the process of wandering round the class to see who had won, I checked in with every single student – it might have been a quick commiseration with them that they didn’t win, a smile and a high five when they did, a “don’t let sticky-man distract you” comment when they picked him from the treat box, or a very loud full class cheer when a STRIKE FOUR was rolled – I connected with every single student in the first few minutes in class.  And it was easy, natural and in good humour (NZ-spelling again)

Things I do to run STRIKE-MERE smoothly in class:

  • Students need to write their numbers down on paper quickly when they arrive at class (or get their more organised neighbour to write them down for them).  There is a bit of trust going on here – and I let my students know this.
  • Late students to class get to roll the dice (their lateness is acknowledged, but not in a bad way – those things can be dealt with later individually if needed)

That’s it…. and my favourite thing 🙂

First lessons – Year 12 Statistics


The start of the year is super-important in all my classes – it’s the time when I try to set up the culture, climate and expectations that I want to foster all year.

Here’s what I’ve got planned for my Year 12 Statistics class for the first couple of weeks.

Firstly though – a bit of background…

I’ve joined the #MTBoS 2016 blogging initiative.  Let’s see how long I last!  Timing wise – its a bit trickier over here in New Zealand – exploreMTBoS
my first day of classes for the new year isn’t until Tuesday 2 February, so I’ll miss the 31 Jan date for blogging about one of my classes… I’m running a start of year morning with my staff which will be done in time for me to write about.

My faculty has 15 Maths & Stats teachers, including 3 new teachers to the department starting this year: one first year teacher, one second year teacher, one experienced science teacher who hasn’t taught maths or stats since her teacher training.  We also have in the mix another second year teacher, and one returning from maternity leave, alongside a wealth of experience.  I willingly share my resources, ideas and lessons with all my staff – writing this blog is my way of making it easier to record what happens in my classes for me and for them to refer to.

Right – Year 12 Statistics … I have six lessons in the first two weeks where I hope to (1) get to know my students and develop the class culture and (2) to cover some activities on RANDOM (that’ll have to be another blog post – see here for a talk last year with Grant Ritchie)


Intros & Admin – we are focusing on being deliberate with our actions to help foster positive relationships among our class members – and this is a really tough question – you know that you get on well with your students, they like you and get good results in your class, but what is it you ACTUALLY DO to foster this culture?  What do you do that other teachers could try? Not all students know each other in our senior classes, so it’s important that we spend some time introducing ourselves to each other.  I share a little about myself (usually introduce them to my dogs) and ask them to share something about themselves that no one else in the class knows about them.  As students are doing this, I record who is sitting where and get started on learning names as quickly as I can.  I’ll then go through basic course administration etc.

Weekly Warm-ups thanks to Sarah Hagan (@mathequalslove) I am using these for a second year.  My plan for 2016 is Good things Monday, Strikemere Tuesday, Taboo Thursday & Figure it out Friday (or possibly Friday Funnies – yet to be decided…).  I don’t see this class on a Wednesday. I’ll explain these more in another blog (that’s at least two more I’ve committed to already!)

Relationship building fun task – whether I complete the pipe cleaner building challenge in groups (from Boston NCTM and ###’s workshop – use the pipe cleaners to build the tallest tower in 10 minutes, after 5 minutes silent work with a quirky story for why, after another few minutes, one are only with a bizarre reason…) or just make shape pictures (an idea that came from Angela Brett, give groups a pile of shapes – they have sixty seconds to make the best space ship or car or crazy animal or whatever you come up with – the prize… high fives of course!) will depend on how much time I have left.

Day 2

Senior Student Survey – Students will complete our senior student survey (copy here if you want to look/borrow/adapt etc).  This will collect a variety of information that we may or may not use for further stats classes.  I’ll share a link when its completed.  I have managed to include a couple of experiments, alongside standard demographics and perceptions of themselves as maths/stats learners,  and more quirky things such as student ratings on things like their intelligence, how scary they find pictures, how awful some noises are, how old they think people are, what age they perceive as “old” and “really old”.

Perseverance (thanks Annie @mfAnnie for introducing me to Game about Squares) – Students will play this game, and then I will use Annie’s post as my guide to leading a discussion on how the way they tackled the game challenges can be transferred into developing their statistical thinking processes and being successful in our course.
A big shout out here to Nathaniel (@nhighstein) for his blog post on building culture in his classes, with specific reference to Annie’s Game about squares – I’ll share the results of the survey if I do one in the same way Nathaniel has.  This may flow into Day 3 a little depending on how the survey goes.

Day 3

Random – Marshmallows… The lesson plan is here.  It covers three big ideas around “random” – what does random look like?; making connections between statistical distributions and probability distributions; and connecting theoretical probability models with experimental situations.



Day 4 & Day 5

Mean – MAD – SAD activity from Christine Franklin and the GAISE Report work.  Resources from Chris are here and here, and my powerpoint I’ve adapted is here.

The sequence of learning will take students through Levels A, B and C of the GAISE framework from viewing the mean as a “fair share” value and looking at variation from “fair”, through to the formal quantity of mean and variation from the mean using SAD (sum of the absolute deviations).  We plan to introduce the Levels A & B progressions to our younger students, but this year’s Year 12 students will not have met these concepts.  I’m hoping that by going through the background development, they will have a better understanding of standard deviation as that is introduced.

Day 6

Still to be confirmed, but likely to either be a lead into our next topic – Experiments or finishing off work things from Day 1 – 5 that we didn’t quite get to!