The Class-size debate – it matters to teachers

Class size matters to teachers

Class size is a perennial question in education. What is the ideal size for a school class? Teachers would like smaller classes, to improve learning. There is evidence of a small positive effect size due to reducing class size from meta-analysis published in John Hattie’s Visible Learning. But it makes sense, teachers argue – fewer children in the class means more opportunities for one-to-one interactions with the teacher. It makes for easier crowd control, less noise and less stress for teachers and pupils. And in these days of National Standards, it makes the assessment load more realistic.

Educational Research is difficult

I’d just like to point out that educational research is difficult. One of my favourite readings on educational research is an opinion piece by David Berliner, Educational Research: The hardest science of all,  where he explains the challenge of educational research. It was written in response to a call by the US Government for evidence-based practices in education. Berliner reminds us of how many different factors contribute to learning. And measuring learning is itself an inexact science. At one point he asks: “It may be stretching a little, but imagine that Newton’s third law worked well in both the northern and southern hemispheres—except of course in Italy or New Zealand—and that the explanatory basis for that law was different in the two hemispheres. Such complexity would drive a physicist crazy, but it is a part of the day-to-day world of the educational researcher.”

Ask the teachers

So with this in mind, I decided to ask the experts. I asked NZ primary school teachers who are just gearing up for the 2017 school year. These teachers were invited via a Facebook group to participate in a very short poll using a Google Form. There were just eight questions – the year level they teach, the minimum, maximum and ideal size for a class at that level, how many children they are expecting in their class this year and how long they have been teaching. The actual wording for the question about ideal class size was: “In your opinion what is the ideal class size that will lead to good learning outcomes for the year level given above?” There were also two open-ended questions about how they had chosen their numbers, and what factors they think contribute to the decision on class-size.

Every time I do something like this, I underestimate how long the analysis will take. There were only eight questions, thought I. How hard can that be…. sigh. But in the interests of reporting back to the teachers as quickly as possible, I will summarise the numeric data, and deal with all the words later.

Early results

There were about 200 useable responses. There was a wide range of experience within the teachers. A third of the teachers had been teaching for five years or shorter, and 20% had been teaching for more than twenty years. There was no correlation between the perceived ideal class size and the experience of the teacher.

The graph below displays the results, comparing the ideal class-size for the different year levels. Each dot represents the response of one teacher. It is clear that the teachers believe the younger classes require smaller classes. The median value for the ideal class size for a New Entrant, Year 1 and/or Year 2 class is 16. The median value for the ideal class size for Year 3/4 is 20, for Year 5/6 is 22 and for year 7/8 is 24. The ideal class size increases as the year level goes up. It is interesting that even numbers are more popular than odd numbers. In the comments, teachers point out that 24 is a very good number for splitting children into equal-sized groups.

These dotplot/boxplots from iNZight show each of the responses, and the summary values.

These dotplot/boxplots from iNZight show each of the responses, and the summary values.

It is interesting to compare the maximum class size the teachers felt would lead to good learning outcomes. I also asked what class size they will be teaching this year.  The table below gives the median response for the ideal class size, maximum acceptable, and current class size. It is notable that the current class sizes are all at least two students more than the maximum acceptable values, and between six and eight students more than the ideal value.

Median response
Year Level Number of respondents Ideal class size Maximum acceptable Current
New Entrant Year 1/2 56 16 20 22
Year 3/4 40 20 24.5 27.5
Year 5/6 53 22 25 30
Year 7/8 46 24 27 30

Financial considerations

It appears that most teachers will be teaching classes that are considerably larger than desired. This looks like a problem. But it is also important to get the financial context. I asked myself how much money would it take to reduce all primary school classes by four pupils (moving below the maximum, but more than the ideal)? Using figures from the Ministry of Education website, and assuming the current figures from the survey are indicative of class sizes throughout New Zealand, we would need about 3500 more classes. That is 3500 more rooms that would need to be provided, and 3500 more teachers to employ. It is an 18% increase in the number of classes. The increase in salaries alone would be over one hundred million dollars per year. This is not a trivial amount of money. It would certainly help with unemployment, but taxes would need to increase, or money would need to come from elsewhere.

Is this the best way to use the money? Should all classes be reduced or just some? How would we decide? How would it be implemented? If you decrease class sizes suddenly you create a shortage of teachers, and have to fill positions with untrained teachers, which has been shown to decrease the quality of education. Is the improvement worth the money?

My sympathies really are with classroom teachers. (If I were in charge, National Standards would be gone by lunchtime.) I know what a difference a few students in a class makes to all sorts of things. At the same time, this is not a simple problem, and the solution is far from simple. Discussion is good, and informed discussion is even better. Please feel free to comment below. (I will summarise the open-ended responses from the survey in a later post.)

Has the Numeracy Project failed?

The Numeracy Development Project has influenced the teaching of mathematics in New Zealand. It has changed the language people use to talk about mathematical understanding, introducing the terms “multiplicative thinking”, “part-whole” and “proportional reasoning” to the teacher toolkit. It has empowered some teachers to think differently about the teaching of mathematics. It has brought “number” front and centre, often crowding out algebra, geometry, measurement and statistics, which are now commonly called the strands. It has baffled a large number of parents. Has the Numeracy Development Project been a success? If not, how can we fix it?

I have been pondering about the efficacy and side-effects of the Numeracy Project in New Zealand. I have heard criticisms from Primary and Secondary teachers, and defense and explanation from advisors. I have listened to a very illuminating podcast from one of the originators of the Numeracy Project, Ian Stevens, I have had discussions with another educational developer who was there at the beginning. I even downloaded some of the “pink booklets” and began reading them, in order understand the Numeracy Project.

Then I read this article from the US organisation, National Council of Teachers of Mathematics, Strategies are not Algorithms,  and it all started to fall into place.
The authors explain that researchers analysed the way that children learn about mathematics, and the stages they generally go through. It was found that “Students who used invented strategies before they learned standard algorithms demonstrated better knowledge of base-ten number concepts and were more successful in extending their knowledge to new situations than were students who initially learned standard algorithms.” They claim that in the US “(t)he idea of “invented strategies” has been distorted to such a degree that strategies are being treated like algorithms in many textbooks and classrooms across the country.” I suspect this statement also applies in New Zealand.

Strategies taught as algorithms

Whitacre and Wessenberg refer to a paper by Carpenter et al, A Longitudinal Study of Invention and Understanding in Children’s Multidigit Addition and Subtraction. I was able to get access to read it, and found the following:
“Although we have no data regarding explicit instruction on specific invented strategies, we hypothesize that direct instruction could change the quality of children’s understanding and use of invented strategies. If these strategies were the object of direct instruction, there would be a danger that children would learn them as rote procedures in much the way that they learn standard algorithms today.” (Emphasis added)

Were they right? Are the strategies being taught as rote procedures in some New Zealand classrooms? Do we need to change the way we talk about them?

How I see the Numeracy Development Project (NDP)

The NDP started as a way to improve teacher pedagogical content knowledge to improve outcomes for students. It was intended to cover all aspects of the New Zealand Mathematics and Statistics curriculum, not just number. Ian Stevens explained: “Numeracy was never just Number. We decided that in New Zealand numeracy meant mathematics and mathematics meant numeracy.”

The Numeracy Development Project provided a model to understand progression of understanding in learning mathematics. George Box once said “All models are wrong and some models are useful.” A model of progression of understanding is useful for identifying where we are, and how to progress to where we would like to be, rather like a map. But a map is not the landscape, and children differ, circumstances change, and models in education change faster than most. I recently attended a talk by Shelley Dole, who (I think) suggested that by emphasising additive thinking in the early school years, we may undo the multiplicative and proportional thinking the students had already. If all they see is adding and subtracting, any implication towards multiplicative and proportional thinking is stifled. It is an interesting premise.
The Numeracy Project (as it is now commonly called) suggested teaching methods, strongly based around group-work and minimising the use of worksheets. Popular invented strategies for arithmetic operations were described, and the teaching of standard algorithms such as vertical alignment of numbers when adding and subtracting was de-emphasised.
An unintended outcome is that the Numeracy Project has replaced the NZ curriculum in some schools, with “Number” taking centre stage for many years. Teachers are teaching invented strategies as algorithms rather than letting students work them out for themselves. At times students are required to know all the strategies before moving on. Textbooks, worksheets and even videos based around the strategies abound, which seems anathema to the original idea.

Where now?

So where do we go from here?

To me empowerment of teachers is pivotal. Teachers need to understand and embrace the beauty of number theory, the practicality of measurement, the art and challenge of geometry, the detective possibilities in data and the power of algebra to model our world. When mathematics is seen as a way to view the world, and embedded in all our teaching, in the way literacy is, maybe then, we will see the changes we seek.

10 hints to make the most of teaching and academic conferences

Hints for conference benefit maximisation

I am writing this post in a spartan bedroom in Glenn Hall at La Trobe University in Bundoora (Melbourne, Australia.) Some outrageously loud crows are doing what crows do best outside my window, and I am pondering on how to get the most out of conferences. In my previous life as a University academic, I attended a variety of conferences, and discovered some basic hints for enjoying them and feeling that my time was productively used. In the interests of helping conference newcomers I share them here. They are in no particular order.

1. Lower your expectations

Sad, but true, many conference presentations are obvious, obscure or dull. And some are annoying. If you happen to hit an interesting and entertaining presentation – make the most of it. I have talked to several newbies this afternoon whose experience of the MAV conference could be described as underwhelming. This is not the fault of the conference, but rather a characteristic of conferences as a whole. My rule of thumb is that if you get one inspiring or useful presentation per day you are winning. (Added later) You can generally find something positive in any presentation, and it is good to tweet that. (Thanks David Butler for reminding me!)

2. Pace yourself

When I first went to conferences I would make sure that I attended every session, feeling I needed to fulfil my obligations to the University that was kindly funding (or in those days, part-funding) my trip and attendance. Fortunately I was saved from exhaustion by my mentor, who pointed out that you had diminishing returns, if not negative returns on continued attendance beyond a certain point. Consequently I have learned to take a break and not attend every single presentation I can. Some down-time is also good for contemplating what you have heard. Conferences are also a chance to step back from the daily grind, and think about your own teaching practice or research.

3. Go to something out of your usual area of interest.

When I used to teach operations research, many of the research talks went whizzing over my head. But every now and then I would find a gem, which for me would be a wonderful story I could tell in lectures of how operations research had saved money, lives or the world from annihilation. You never know what you might find.

4. Remember “Names” are just people too.

It may be my colonial cringe, but I tend to be a little in awe of the “big names” in any field. These are the people who have been paid to attend the conference, who give keynote addresses, and you have actually heard of before. Next year at the NZAMT conference in October, Dan Meyer is going to be a keynote speaker. I have to say I am a little in awe of him, but at the same time know that that is silly. Dick de Veaux is one of my favourite keynote speakers and you could not ask for a nicer or more generous person. The point is that speakers are people too, and are playing a certain role at a conference, which means that they should give the punters some of their time. – So this is my advice to paid keynote speakers – be nice to people. It can’t hurt, and it can make a real difference in their lives. Because of my YouTube videos I have a small level of celebrity among some teachers and learners of statistics in New Zealand. (I said it was small) I LOVE it when people talk to me, and hope no one would feel reluctant. If it is in your power to do good, do it

5.Talk to people.

This can be daunting and tiring, but is essential to make the most of a conference opportunity. The point of conferences is to bring people together, so if you do not talk to anyone other than the people you came with, you could have stayed home and watched presentations on YouTube. I am learning that some conversation topics are easy starters : “Where are you from?”, “What do you teach/research?”, “Have you been to any good sessions?” “What did you think of the Keynote?” are all reasonably safe. To my surprise, criticising the US President elect was not universally well received, so I have learned to avoid that one. Being positive is a good idea, and one I need to remember at all times. When I do not agree with what a speaker is saying I have a tendency to growl in a Marge Simpsonesque way. This can be disturbing to the people around me and I am attempting to stop it.

At the 2016 MAV conference I had yellow hair, and immediately found kinship with a delightful and insightful young teacher with magenta hair. Now if we could just have found an attendee with cyan hair we could have impersonated a printer cartridge! I went to Sharon’s presentation and she to mine, and I believe we were both the better for it.

We have Yellow and Magenta - but where is Cyan?

We have Yellow and Magenta – but where is Cyan?

6. Be brave and give a presentation

The biennial NZ Association of Maths teachers conference is being held in Christchurch on 3rd to 6th October 2017. I strongly believe we need more input from primary teachers, and more collaboration across primary, secondary and tertiary. It would be SOO wonderful to have many primary teachers giving workshops or presentations of work they are doing in their maths classrooms.

The abstracts are due by the end of May and if any primary teachers would like some help putting one together, I would be really happy to help.

7. Visit the trade displays

The companies that have trade displays pay a considerable amount for the right to do so. I believe that teachers need producers of educational resources, and when you visit producers and give them the opportunity to talk about their product, it makes it worthwhile for them to sponsor, thus keeping the price down. And you never know – you might find something really useful!

8. Split up to maximise benefit.

If two or more of you come from the same school or organisation, it is a good idea to plan your programme together. When there are 40 – or even 10 presentations to choose from in any one slot, it is more sensible to attend different ones.

9. Plan ahead

It is really helpful to know when conferences are approaching, so I have added links below to the maths teaching conferences I know about, in the hope that many of you may think about attending. Do let me know any you know about that I haven’t listed.

10. Wear sensible shoes

This particularly applies to the MAV conference at La Trobe University. It is held on a massive campus, which is particularly confusing to get around, so one tends to cover far more ground than intended. I was pleased I sacrificed style for comfort in this particular instance, after a bad attack of blisters last year.

11.Add your own hints

Any other conference attenders here – what other suggestions could you make?

Mathematics and statistics teaching conferences in New Zealand and Australia

Primary Mathematics Association 25 March 2017, Auckland

AAMT 11 – 13 July 2017, Canberra, Australia

2017 MANSW Annual Conference 15-17 September 2017.

NZAMT 3 – 7 October 2017 Christchurch New Zealand

MAV Early Dec 2017 Melbourne, Australia

 

 

Mathematics activities using Lego bricks

I love Lego. And I love making up mathematics and statistics activities for people of all levels of attainment. So it makes sense that I would make up maths discussion activities using Lego.

Whenever I have posted my ideas on Twitter (hashtag @Rogonic) and Facebook (Statistics Learning Centre) they have proved popular. So I thought it would be good to put them in a less transient location – this blog.

Here is one to start with:

Which of the models, A to H is most like the one in the middle?

Which of the models, A to H is most like the one in the middle?

 

You can ask any question you like. I suggest, “Which of the models, A to H is most like the model in the middle?”

Then listen to what your learners have to say. Feel free to vote here:

I would love to hear what comes of this discussion. Please put your ideas in the comments below. In a follow-up post, I will talk about some of the concepts that might have arisen in discussions. A follow-up activity for your students (or you) is to come up with a new model that is most like the one in the middle, but not exactly the same.

Math/Maths Lego/Legos

Explanation of the placement of the letter ‘s’ with respect to Maths with Lego. The Danish company that makes Lego does not approve of the use of Legos as a word. The plural of one Lego brick is two Lego bricks. In New Zealand we talk about Lego as a collective noun, as in “I am going to play with my Lego” and “Pick up your Lego before I stand on it.” We also follow the UK tradition of talking about the subject of Maths, rather than Math. I am aware that my friends in the US would talk about Math with Legos, but I am not in the US so I reserve the right to talk about Maths with Lego. I am refraining from making any statement about the state of politics in the US at present… With difficulty.

Disclaimer

This website uses Lego (R) bricks to teach mathematical and statistical concepts. There is no official affiliation with the Lego(R) company. Lego is a registered trademark of the Lego company.

Talking in class: improving discussion in maths and stats classes

Maths is right or wrong – end of discussion  – or is it?

In 1984 I was a tutor in Operations Research to second year university students. My own experience of being in tutorials at University had been less than inspiring, with tutors who seemed reserved and keen to give us the answers without too much talking. I wanted to do a good job. My induction included a training session for teaching assistants from throughout the university. Margaret was a very experienced educational developer and was very keen for us to get the students discussing. I tried to explain to her that there really wasn’t a lot to discuss in my subject. You either knew how to solve a set of linear equations using Gauss-Jordan elimination or you didn’t. The answer was either correct or incorrect.

I suspect many people have this view of mathematics and its close relations, statistics and operations research.  Our classes have traditionally followed a set pattern. The teacher shows the class how to do something. The class copies down notes and some examples into their books, and then they individually work through exercises in the textbook – generally in silence. The teacher walks around the room and helps students as needed.

Prizes can help motivate students to give answers in unfamiliar settings

Prizes can help motivate students to give answers in unfamiliar settings

So when we talk about discussion in maths classes, this is not something that mathematics and statistics teachers are all familiar with. I recently gave a workshop for about 100 Scholarship students in Statistics in the Waikato. What a wonderful time we had together! The students were from all different schools and needed to be warmed up a little with prizes, but we had some good discussion in groups and as a whole. One of the teachers  commented later on the level of discussion in the session. Though she was an experienced maths teacher she found it difficult to lead discussion in the class. I am sure there are many like her.

It is important to talk in maths and stats classes

It is difficult for many students to learn in solitary silence. As we talk about a topic we develop our understanding, practice the language of the discipline and experience what it means to be a mathematician or statistician. Explaining ideas to others helps us to make sense of them ourselves. As we listen to other people’s thinking we can see how it relates to what we think, and can clear up misconceptions. Some people just like to talk, (who me?) and find learning more fun in a cooperative or collaborative environment. This recognition of the need for language and interaction underpins the development of “rich tasks” that are being used in mathematics classrooms throughout the world.

I have previously stated that “Maths learning should be communal and loud and exciting, not solitary, quiet and routine.”

Classroom atmosphere

One thing that was difficult at the Scholarship day was that the students did not know each other, and came from various schools. In a regular classroom the teacher has the opportunity of and responsibility for setting the tone of the class. Students need to feel safe. They need to feel that giving a wrong answer is not going to lead to ridicule. Several sessions at the start of the year may be needed to encourage discussion. Ideally this will become less necessary over time as students become used to interactive, inquiry-based learning in mathematics and statistics through their whole school careers.

Number talks” are a tool to help students improve their understanding of number, and recognise that there are many ways to see things. For example, the class might be shown a picture of dots and asked to explain how many dots they see, and how they worked it out. Several different ways of thinking will be discussed.

Children are encouraged to think up multiple ways of thinking about numbers  and to develop discussion by following prompts, sometimes called “talk moves”. Talk moves include revoicing, where the teacher restates what she thinks the student has said, asking students to restate another students reasoning, asking students to apply their own reasoning to someone else’s reasoning (Do you agree or disagree and why?), prompting for further participation (Would someone like to add on?), and using wait time (teachers should allow students to think for at least 10 seconds before calling on someone to answer. These are explained more fully in The Tools of Classroom talk.

Google Image is awash with classroom posters outlining “Talk moves”. I have been unable to trace back the source of the term or the list, and would be very pleased if someone can tell me the source,  to be able to attribute this structure.

Good questions

The essence of good discussion is good questions. Question ping pong is not classroom discussion. We have all experienced a teacher working through examples on the board, while asking students the answers to numerical questions. This is a control technique for keeping students attentive, but it can fall to a small group of students who are quick to answer. I remember doing just this in my tutorial on solving matrices, when I didn’t know any better.

Teachers should avoid asking questions that they already know the answers to.

It is not a hard-and-fast rule, but definitely a thing to think about. I like to use True/False quizzes to help uncover misconceptions, and develop use of statistical language. I just about always know the answer to the question, but what I don’t know is how many students know the answer. So  I ask the question not to know the answer, but to know if the students do, and to provoke discussion. Perhaps a more interesting question would be, how many students do you think will say “True” to this statement. It would then be interesting to find out their reasoning, so long as it does not get personal!

Multiple answers and open-ended questions

Where possible we need to ask questions that can have a number of acceptable answers. A discussion about what to do with outliers will seldom have a definitive answer, unless the answer is that it depends! Asking students to make a pictorial representation of an algebra problem can lead to interesting discussions.

The MathTwitterBlogosphere has many attractive ideas to use in teaching maths.

I rather like “Which one doesn’t belong”, which has echoes of “One of these things is not like the other, one of these things doesn’t belong, can you guess…” from Sesame Street. However, in Sesame Street the answer was usually unambiguous, whereas  with WODB there are lots of ways to have alternative answers. There is a website dedicated to sets of four objects, and the discussion is about which one does not belong. In each case all four can “not belong” for some reason, which I find a bit contrived, but it can lead to discussion about which is the strongest case of not belonging.

Whole class and group discussion

Some discussions work well for a whole class, while others are better in small groups or pairs. Matching or ordering paper slips with expressions can lead to great discussion. For example we could have a set of graphs of the same data, and order them according to how effective they are at communicating the aspects of the data. Or there could be statements of possible events and students can place them in order of likelihood. The discussion involved in ordering them helps students to clarify the nature of probability. Desmos has a facility for teachers to set up card matching or grouping exercises, which reduces the work and waste of paper.

Our own Dragonistics data cards are great for discussion. Students can be given a number of dragons (more than two) and decide which one is the best, or which one doesn’t belong, or how to divide the dragons fairly into two or more groups.

It can seem to be wasting time to have discussion. However the evidence from research is that good discussion is an effective way for students to learn mathematics and statistics. I challenge all maths and stats teachers to increase and improve the discussion in their class.

Play and learning mathematics and statistics

The role of play in learning

I have been reading further about teaching mathematics and came across this interesting assertion:

Play, understood as something frivolous, opposed to work, off-task behaviour, is not welcomed into most mathematics classrooms. But play is exactly what is needed. It is only play that can entice us to the type of repetition that is needed to learn how to inhabit the mathematical landscape and how to create new mathematics.
Friesen(2000) – unpublished thesis, cited in Stordy, Children Count, (2015)

Play and practice

It is an appealing idea that as children play, they have opportunities to engage in repetition that is needed in mastering some mathematical skills. The other morning I decided to do some exploration of prime numbers and factorising even before I got out of bed. (Don’t judge me!). It was fun, and I discovered some interesting properties, and came up with a way of labelling numbers as having two, three and more dimensions. 12 is a three dimensional number, as is 20, whereas 35 and 77 are good examples of two dimensional numbers. As I was thus playing on my own, I was aware that it was practising my tables and honing my ability to think multiplicatively. In this instance the statement from Friesen made sense. I admit I’m not sure what it means to “create new mathematics”. Perhaps that is what I was doing with my 2 and 3 dimensional numbers.

You may be wondering what this has to do with teaching statistics to adults. Bear with…

Traditional vs recent teaching methods for mathematics

Today on Twitter, someone asked what to do when a student says that they like being shown what to do, and then practising on textbook examples. This is the traditional method for teaching mathematics, and is currently not seen as ideal among many maths teachers (particularly those who inhabit the MathTwitterBlogosphere or MTBoS, as it is called). There is strong support for a more investigative, socially constructed approach to learning and teaching mathematics.  I realise that as a learner, I was happy enough learning maths by being shown what to do and then practising. I suspect a large proportion of maths teachers also liked doing that. Khan Academy videos are wildly popular with many learners and far too many teachers because they perpetuate this procedural view of mathematics. So is the procedural approach wrong? I think what it comes down to is what we are trying to teach. Were I to teach mathematics again I would not use “show then practise” as my modus operandi. I would like to teach children to become mathematicians rather than mathematical technicians. For this reason, the philosophies and methods of Youcubed, Dan Meyer and other MTBoS bloggers have appeal.

Play and statistics

Now I want to turn my thoughts to statistics. Is there a need for more play in statistics? Can statistics be playful in the way that mathematics can be playful? Operations Research is just one game after another! Simulation, critical path, network analysis, travelling salesperson, knapsack problem? They are all big games. Probability is immensely playful, but what about statistical analysis? Can and should statistics be playful?

My first response is that there is no play in statistics. Statistics is serious and important, and deals with reality, not joyous abstract ideas like prime numbers and the Fibonacci series – and two and three dimensional numbers.

The excitement of a fresh set of data

But there is that frisson of excitement as you finally finish cleaning your database and a freshly minted set of variables and observations beckons to you, with SPSS, SAS or even Excel at your fingertips. A new set of data is a new journey of discovery. Of course a serious researcher has already worked out a methodical route through her hypotheses… maybe. Or do we mostly all fossick about looking for patterns and insights, growing more and more familiar with the feel of the data, as if we were squeezing it through our fingers? So yes – my experience of data exploration is playful. It is an adventure, with wrong turns, forgetting the path, starting again, finding something only to lose it again and finally saying “enough” and taking a break, not because the data has been exhausted, but because I am.

Writing the report is like cleaning up

Writing up statistical analysis is less exciting. It feels like picking up the gardening tools and putting them away after weeding the garden. Or cleaning the paintbrushes after creating a masterpiece. That was not one of my strengths – finishing and tidying up afterwards. The problem was that I felt I had finished when the original task had been completed – when the weeds had been pulled or the painting completed. In my view, cleaning and putting away the tools was an afterthought that dragged on after the completion of the task, and too often got ignored. Happily I have managed to change my behaviour by rethinking the nature of the weeding task. The weeding task is complete when the weeds are pulled and in the compost and the implements are resting clean and safe where they belong. Similarly a statistical analysis is not what comes before the report-writing, but is rather the whole process, ending when the report is complete, and the data is carefully stored away for another day. I wonder if that is the message we give our students – a thought for another post.

Can statistics be playful?

For I have not yet answered the question. Can statistics be playful in the way that mathematics can be playful? We want to embed play in order to make our task of repetition be more enjoyable, and learning statistics requires repetition, in order to develop skills and learn to differentiate the universal from the individual. One problem is that statistics can seem so serious. When we use databases about global warming, species extinction, cancer screening, crime detection, income discrepancies and similarly adult topics, it can seem almost blasphemous to be too playful about it.

I suspect that one reason our statistics videos on YouTube are so popular is because they are playful.

helen-has-attitude

Helen has an attitude problem

Helen has a real attitude problem and hurls snarky comments at her brother, Luke. The apples fall in an odd way, and Dr Nic pops up in strange places. This playfulness keeps the audience engaged in a way that serious, grown up themes may not. This is why we invented Ear Pox in our video about Risk and screening, because being playful about cancer is inappropriate.

Ear Pox is imaginary disease for which we are studying the screening risk.

Ear Pox is imaginary disease for which we are studying the screening risk.

Dragonistics data cards provide light-hearted data which yields worth-while results.

A set of 240 Dragonistics data cards provides light-hearted data which yields satisfying results.

When I began this post I did not intend to bring it around to the videos and the Dragonistics data cards, but I have ended up there anyway. Maybe that is the appeal of the Dragonistics data cards –  that they avoid the gravitas of true and real grown-up data, and maintain a playfulness that is more engaging than reality. There is a truthiness about them – the two species – green and red dragons are different enough to present as different animal species, and the rules of danger and breath-type make sense. But students may happily play with the dragon cards without fear of ignorance or even irreverence of a real-life context.

What started me thinking about play with regards to learning maths and statistics is our Cat Maths cards. There are just so many ways to play with them that I can see Cat Maths cards playing an integral part in a junior primary classroom. This is why we created them and want them to make their way into classrooms. You can help by supporting our Kickstarter crowdfunding campaign. Click the picture to pledge and get a box, provide a box for a school, or make a corporate donation.

We'd love your help.

We’d love your help.

Your thoughts about play and statistics

And maybe we need to be thinking a little more about the role of play in learning statistics – even for adults! What do you think? Can and should statistics be playful? And for what age group? Do you find statistical analysis fun?

 

Mathematics teaching Rockstar – Jo Boaler

Moving around the education sector

My life in education has included being a High School maths teacher, then teaching at university for 20 years. I then made resources and gave professional development workshops for secondary school teachers. It was exciting to see the new statistics curriculum being implemented into the New Zealand schools. And now we are making resources and participating in the primary school sector. It is wonderful to learn from each level of teaching. We would all benefit from more discussion across the levels.

Educational theory and idea-promoters

My father used to say (and the sexism has not escaped me) “Never run after a woman, a bus or an educational theory, as there will be another one along soon.” Education theories have lifespans, and some theories are more useful than others. I am not a fan of “learning styles” and fear they have served many students ill. However, there are some current ideas and idea-promoters in the teaching of mathematics that I find very attractive. I will begin with Jo Boaler, and intend to introduce you over the next few weeks to Dan Meyer, Carol Dweck and the person who wrote “Making it stick.”

Jo Boaler – Click here for official information

My first contact with Jo Boaler was reading “The Elephant in the Classroom.” In this Jo points out how society is complicit in the idea of a “maths brain”. Somehow it is socially acceptable to admit or be almost defensively proud of being “no good at maths”. A major problem with this is that her research suggests that later success in life is connected to attainment in mathematics. In order to address this, Jo explores a less procedural approach to teaching mathematics, including greater communication and collaboration.

Mathematical Mindsets

It is interesting to  see the effect Jo Boaler’s recent book, “Mathematical Mindsets “, is having on colleagues in the teaching profession. The maths advisors based in Canterbury NZ are strong proponents of her idea of “rich tasks”. Here are some tweets about the book:

“I am loving Mathematical Mindsets by @joboaler – seriously – everyone needs to read this”

“Even if you don’t teach maths this book will change how you teach for ever.”

“Hands down the most important thing I have ever read in my life”

What I get from Jo Boaler’s work is that we need to rethink how we teach mathematics. The methods that worked for mathematics teachers are not the methods we need to be using for everyone. The defence “The old ways worked for me” is not defensible in terms of inclusion and equity. I will not even try to boil down her approach in this post, but rather suggest readers visit her website and read the book!

At Statistics Learning Centre we are committed to producing materials that fit with sound pedagogical methods. Our Dragonistics data cards are perfect for use in a number of rich tasks. We are constantly thinking of ways to embed mathematics and statistics tasks into the curriculum of other subjects.

Challenges of implementation

I am aware that many of you readers are not primary or secondary teachers. There are so many barriers to getting mathematics taught in a more exciting, integrated and effective way. Primary teachers are not mathematics specialists, and may well feel less confident in their maths ability. Secondary mathematics teachers may feel constrained by the curriculum and the constant assessment in the last three years of schooling in New Zealand. And tertiary teachers have little incentive to improve their teaching, as it takes time from the more valued work of research.

Though it would be exciting if Jo Boaler’s ideas and methods were espoused in their entirety at all levels of mathematics teaching, I am aware that this is unlikely – as in a probability of zero. However, I believe that all teachers at all levels can all improve, even a little at a time. We at Statistics Learning Centre are committed to this vision. Through our blog, our resources, our games, our videos, our lessons and our professional development we aim to empower all teacher to teach statistics – better! We espouse the theories and teachings explained in Mathematical Mindsets, and hope that you also will learn about them, and endeavour to put them into place, whatever level you teach at.

Do tell us if Jo Boalers work has had an impact on what you do. How can the ideas apply at all levels of teaching? Do teachers need to have a growth mindset about their own ability to improve their teaching?

Here are some quotes to leave you with:

Mathematical Mindsets Quotes

“Many parents have asked me: What is the point of my child explaining their work if they can get the answer right? My answer is always the same: Explaining your work is what, in mathematics, we call reasoning, and reasoning is central to the discipline of mathematics.”
“Numerous research studies (Silver, 1994) have shown that when students are given opportunities to pose mathematics problems, to consider a situation and think of a mathematics question to ask of it—which is the essence of real mathematics—they become more deeply engaged and perform at higher levels.”
“The researchers found that when students were given problems to solve, and they did not know methods to solve them, but they were given opportunity to explore the problems, they became curious, and their brains were primed to learn new methods, so that when teachers taught the methods, students paid greater attention to them and were more motivated to learn them. The researchers published their results with the title “A Time for Telling,” and they argued that the question is not “Should we tell or explain methods?” but “When is the best time do this?”
“five suggestions that can work to open mathematics tasks and increase their potential for learning: Open up the task so that there are multiple methods, pathways, and representations. Include inquiry opportunities. Ask the problem before teaching the method. Add a visual component and ask students how they see the mathematics. Extend the task to make it lower floor and higher ceiling. Ask students to convince and reason; be skeptical.”

All quotes from

Jo Boaler, Mathematical Mindsets: Unleashing Students’ Potential through Creative Math, Inspiring Messages and Innovative Teaching

Teachers and resource providers – uneasy bedfellows

Trade stands and cautious teachers

It is interesting to provide a trade stand at a teachers’ conference. Some teachers are keen to find out about new things, and come to see how we can help them. Others studiously avoid eye-contact in the fear that we might try to sell them something. Trade stand holders regularly put sweets and chocolate out as “bait” so that teachers will approach close enough to engage. Maybe it gives the teachers an excuse to come closer? Either way it is representative of the uneasy relationship that “trade” has with salaried educators.

Money and education

Money and education have an uneasy relationship. For schools to function, they need considerable funding – always more than what they get. In New Zealand, and in many countries, education is predominantly funded by the state. Schools are built and equipped, teachers are paid and resources are purchased with money provided by the taxpayer. Extras are raised through donations from parents and fund-raising efforts. However, because it is not apparent that money is changing hands, schools are perceived as virtuous establishments, existing only because of the goodness of the teachers. This contrasts with the attitude to resource providers, who are sometimes treated as parasitic with their motives being all about the money. It is possible that some resource providers are in it just for the money, but it seems to me that there are richer seams to mine in health, sport, retail etc.

Statistics Learning Centre is a social enterprise

Statistics Learning Centre is a social enterprise. We fit in the fuzzy area between “not-for-profit” and commercial enterprise. We measure our success by the impact we are having in empowering teachers to teach statistics and all people to understand statistics. We need money in order to continue to make an impact. Statistics Learning Centre has made considerable contributions to the teaching and learning of statistics in New Zealand and beyond for several years. This post lists just some of the impact we have had.  We believe in what we are doing, and work hard so that our social enterprise is on a solid financial footing.

StatsLC empowers teachers

Soon after the change to the NCEA Statistics standards, there was a shortage of good quality practice external exams. Even the ones provided as official exemplars did not really fit the curriculum. Teachers approached us, requesting that we create practice exams that they could trust were correct and aligned to the curriculum. We did so in 2015 and 2016, at considerable personal effort and only marginal financial recompense. We see that as helping statistics to be better understood in schools and the wider community.

We, at Statistics Learning Centre, grasp at opportunities to teach teachers how to teach statistics better, to empower all teachers to teach statistics. Our workshops are well received, and we have regular attenders who know they will get value for their time. We use an inclusive, engaging approach, and participants have a good time. I believe in our resources – the videos, the quizzes, the data cards, the activities, the professional development. I believe that they are among the best you can get. So when I give workshops, I do talk about the resources. It would seem counter-productive for all concerned, not to mention contrived, to do otherwise. They are part of a full professional development session. Many mathematical associations have no trouble with this, and I love to go to conferences, and contribute.

I am aware that there are some commercial enterprises who wish to give commercial presentations at conferences. If their materials are not of a high standard, this can put the organisers in a difficult position. Consequently some organisations have a blanket ban on any presentations that reference any paid product. I feel this is a little unfortunate, as teachers miss out on worthwhile contributions. But I understand the problem.

The Open Market model – supply and demand

I believe that there is value in a market model for resources.  People have suggested that we should get the Government to fund access to Statistics Learning Centre resources for all schools. That would be delightful, and give us the freedom and time to create even better resources. But that would make it almost impossible for any other new provider, who may have an even better product, to get a look in. When such a monopoly occurs, it reduces the incentives for providers to keep improving.

Saving work for the teachers, and building on a product

Teachers want the best for their students, and have limited budgets. They may spend considerable amounts of time printing, cutting and laminating in order to provide teaching resources at a low cost. This was one of the drivers for producing our Dragonistics data cards – to provide at a reasonable cost, some ready-made, robust resources, so that teachers did not have to make their own. As it turned out we were able to provide interesting data with clear relationships, and engaging graphics so that we provide something more than just data turned into datacards.

Free resources

There are free resources available on the internet. Other resources are provided by teachers who are sharing what they have done while teaching their own students. Resources provided for free can be of a high pedagogical standard. Having a high production standard, however, can be prohibitively expensive for individual producers who are working in their spare time.  It can also be tricky for another teacher to know what is suitable, and a lot of time can be spent trying to find high quality, reliable resources.

Teachers and resource providers – a symbiotic relationship

Teachers need good resource providers. It makes sense for experts to create high quality resources, drawing on current thinking with regard to content specific pedagogy. These can support teachers, particularly in areas in which they are less confident, such as statistics. And they do need to be paid for their work.

It helps when people recognise that our materials are sound and innovative, when they give us opportunities to contribute and when they include us at the decision-making table. Let us know how we can help you, and in partnership we can become better bed-fellows.

What do you think?

 

(Note that this post is also being published on our blog: Building a Statistics Learning  Community, as I felt it was important,)

 

Enriching mathematics with statistics

Statistics enriches everything!

In many school systems in the world, subjects are taught separately. In primary school, children  learn reading and writing, maths and social studies at different times of the day. But more than that, many topics within subjects are also taught separately. In mathematics we often teach computational skills, geometry, measurement and statistics in separate topics throughout the school year. Textbooks tend to encourage this segmentation of the curriculum. This causes problems as students compartmentalise their learning.  They think that something learned in mathematics can’t possibly be used in Physics. They complain in mathematics if they are asked to write a sentence or a report, saying that it belongs in English.

I participated in an interesting discussion on Twitter recently about Stretch and Challenge. (Thanks #mathschat) My interpretation of “Stretch and challenge” is ways of getting students to extend their thinking beyond the original task so that they are learning more and feeling challenged. This reminds me a lot of the idea of “Low floor High Ceiling” that Jo Boaler talks about. We need tasks that are easy for students to get started on, but that do not limit students, particularly ones who have really caught onto the task and wish to keep going.

Fractions

As a statistics educator, I see applications of statistics and probability everywhere. At a workshop on proportional thinking we were each asked to represent three-quarters, having been told that our A5 piece of paper was “one”. When I saw the different representations used by the participants, I could see a graph as a great way to represent it. You could make a quick set of axes on a whiteboard, and get people to put crosses on which representation they used. The task of categorising all the representations reinforces the idea that there are many ways to show the same thing. It also gets students more aware of the different representations. Then the barchart/dotplot provides a reminder of the outcome of the task. Students who are excited about this idea could make up a little questionnaire to take home and get other family members to draw different fractions, and look at the representations, adding them to the graph back at school.

Measurement

Measurement is an area of the mathematics curriculum that is just begging to be combined with statistics. Just physically measuring an object leads to a variation in responses, which can be graphed. Getting each child to measure each object three times and take the middle value, should lead to a distribution of values with less spread. And then there is estimation. I love the example Dan Meyer uses in his Ted talk in 2010 of filling a tank with water. Students could be asked their estimate of the filling time, simply by guessing, and then use mathematical modelling to refine their estimate. Both values can be graphed and compared.

Area and Probability

Area calculations can be used nicely with probability. Children can invent games that involve tossing a coin onto a shape or shapes. The score depends on whether the coin lands within the shape, outside the shape or on a line. They can estimate what the score will be from 10 throws, simply by looking at the shape, then try it out with one lot of ten throws. Now do some area calculations. Students may have different ways of dealing with the overlap issue. Use the area calculations to improve their theoretical estimates of the probability of each outcome, and from there work out the expected value. Then do multiple trials of ten throws and see how you need to modify the model.  So much learning in one task!

Statistics obviously fits well in much topic work as well. The Olympics are looming, with all the interest and the flood of statistics they provide. Students can be given the fascinating question of which country does the best? There are so many ways to measure and to account for population. Drawing graphs gives an idea of spread and distribution.

There is so much you can do with statistics and other strands and other curriculum areas!  Statistics requires a context, and it is economical use of time if the context is something else you are teaching.

Can you tell me some ways you have incorporated statistics into other strands of mathematics or other subject areas?

Introducing Probability

I have a guilty secret. I really love probability problems. I am so happy to be making videos about probability just now, and conditional probability and distributions and all that fun stuff. I am a little disappointed that we won’t be doing decision trees with Bayesian review, calculating EVPI. That is such fun, but I gave up teaching that some years ago.

The reason probability is fun is because it is really mathematics, and puzzles and logic. I love permutations and combinations too – there is something cool about working out how many ways something can happen.

So why should I feel guilty? Well, in all honesty I have to admit that there is very little need for most of that in a course about statistics at high-school or entry level university. When I taught statistical methods for management, we did some probability, but only from an applied viewpoint, and we never touched intersection and union signs or anything like that. We applied some distributions, but without much theoretical underpinning.

The GAISE (Guidelines for Assessment and Instruction in Statistics Education) Report says, “Teachers and students must understand that statistics and probability are not the same. Statistics uses probability, much as physics uses calculus.”

The question is, why do we teach probability – apart from the fact that it’s fun and makes a nice change from writing reports on time series and bivariate analysis, inference and experiments. The GAISE report also says, “Probability is an important part of any mathematical education. It is a part of mathematics that enriches the subject as a whole by its interactions with other uses of mathematics. Probability is an essential tool in applied mathematics and mathematical modeling. It is also an essential tool in statistics.”

The concept of probability is as important as it is misunderstood. It is vital to have an understanding of the nature of chance and variation in life, in order to be a well-informed, (or “efficient”) citizen. One area in which this is extremely important is in understanding risk and relative risk. When a person is told that their chances of dying of some rare disease have just doubled, it is important that they know that it may be because they have gone from one chance in a million to two chances in a million. Sure it has doubled, but it still is pretty trivial. An understanding of probability is also important in terms of gambling and resistance to the allures of games of chance. And more socially acceptable gambling, such as stockmarket trading, also requires an understanding of chance and variation.

The concept of probability is important, and a few rules of probability may help with understanding, but I suspect the mathematicians get carried away and create problems that are unlikely (probability close to zero) to ever occur in reality. Anything requiring a three-way Venn Diagram has moved from applied problem to logic puzzle.This is in stark contrast to the very applied data-driven approach used in teaching statistics in New Zealand.

Teaching Probability

The traditional approach to teaching probability is to start with the coin and the dice and the balls in the urns. As well as being mind-bogglingly boring and pointless, this also projects an artificial certainty about the probabilities, which is confusing when we start discussing models. If you look at the Khan Academy videos (but don’t) you will find trivial examples about coloured balls or sweets or strangely complex problems involving hitting a circular target. The traditional approach is also to teach probability as truth. “The probability of getting a boy is one-half”. What does that even mean?

I am currently reading the new Springer volume, Probabilistic Thinking, and intend to write a review and post it on this blog, if I can get through enough before my review copy expires. It is inspiring and surprisingly gripping (but I don’t think that is enough of a review to earn me a hard copy to keep.). There are many great ideas for teaching in it, that I hope to pass on in due time.

The New Zealand approach to teaching probability comes from a modelling perspective, right from the start. At level 1, the first two years of schooling, children are exploring chance situations, playing games with a chance element and describing possible outcomes. By years 5 and 6 they are assigning numeric values to the likelihood of an occurrence. They (in the curriculum) are being introduced to model estimates and experimental estimates of probability. Bearing in mind how difficult high school maths teachers are finding the new approach, I don’t have a lot of confidence that the primary teachers are equipped yet to make the philosophical changes, let alone enact them in the classroom.