Educating the heart with maths and statistics

What has love got to do with maths?

This morning at the Twitter chat for teachers, (#bfc630nz) the discussion question was, How and what will you teach your students about life this year? As I lurked I was impressed at the ideas and ideals expressed by a mixed bunch of teachers from throughout New Zealand. I tweeted:  “I wonder how often maths teachers think about educating the heart. Yet maths affects how people feel so much.”

My teaching philosophy is summed up as “head, heart and hands”. I find the philosophy of constructivism appealing, that people create their own understanding and knowledge through experiences and reflection. I believe that learning is a social activity, and I am discovering that mathematics is a social endeavour. But underpinning it all I am convinced that people need to feel safe. That is where the heart comes in. “People do not care how much you know until they know how much you care.” Relationships are vital. I wrote previously about the nature of teaching statistics and mathematics.

Teachers are people

In the culture of NZ Maori, when someone begins to address a group of people, they give a mihi, which is an introductory speech following a given structure. The mihi has the role of placing the person with respect to their mountain, their river, their ancestors. It enables the listeners to know who the person is before they begin to speak about anything else. I am not fluent in te reo, so do not give a mihi in Maori (yet), but I do introduce myself so that listeners know who I am. Learners need to know why I am teaching, and how I feel about the subject and about them. It can feel self-indulgent, thinking surely it is about the subject, not about me. But for many learners the teacher is the subject. Just look at subject choices in high school students and that becomes apparent.

Recently I began studying art at an evening class. I am never a passive learner (and for that reason do feel sympathy for anyone teaching me). Anytime I have the privilege of being a learner, I find myself stepping back and evaluating my responses and thinking of what the teacher has done to evoke these responses. Last week, in the first lesson, the teacher gave no introduction other than her name, and I felt the loss. Art, like maths, is emotionally embedded, and I would have liked to have developed more of a relationship with my teacher, before exposing my vulnerability in my drawing attempts. She did a fine job of reassuring us that all of our attempts were beautiful, but I still would like to know who she is.

Don’t sweeten the broccoli

I suspect that some people believe that maths is a dry, sterile subject, where things are right or wrong. Many worksheets give that impression, with columns of similar problems in black and white, with similarly black and white answers. Some attempt to sweeten the broccoli by adding cartoon characters and using bright colours, but the task remains devoid of adventure and creativity. Now, as a child, I actually liked worksheets, but that is probably because they were easy for me, and I always got them right. I liked the column of little red ticks, and the 100% at the end. They did not challenge me intellectually, but I did not know any better. For many students such worksheets are offputting at best. Worksheets also give a limited view of the nature of mathematics.

I am currently discovering how narrow my perception of mathematics was. We are currently developing mathematical activities for young learners, and I have been reading books about mathematical discoveries. Mathematics is full of creativity and fun and adventure, opinion, multiple approaches, discussion and joy. The mathematics I loved was a poor two-dimensional faded version of the mathematics I am currently discovering.I fear most primary school teachers (and possibly many secondary school maths teachers) have little idea of the full potential of mathematics.

Some high school maths teachers struggle with the New Zealand school statistics curriculum. It is embedded in real-life data and investigations. It is not about calculating a mean or standard deviation, or some horrible algebraic manipulation of formulae. Statistics is about observing and wondering, about asking questions, collecting data, using graphs and summary statistics to make meaning out of the data and reflecting the results back to the original question before heading off on another question. Communication and critical thinking are vital. There are moral, ethical and political aspects to statistics.

Teaching mathematics and statistics is an act of social justice

I cannot express strongly enough that the teaching of mathematics and statistics is a political act. It is a question of social justice. In my PhD thesis work, I found that social deprivation correlated with opportunities to learn mathematics. My thoughts are that there are families where people struggle with literacy, but mostly parents from all walks of life can help their children with reading. However, there are many parents who have negative experiences around mathematics, who feel unable to engage their children in mathematical discussions, let alone help them with mathematics homework. And sadly they often entrench mathematical fatalism. “I was no good at maths, so it isn’t surprising that you are no good at maths.”

Our students need to know that we love them. When you have a class of 800 first year university students it is clearly not possible to build a personal relationship with each student in 24 contact hours. However the key to the ninety and nine is the one. If we show love and respect in our dealings with individuals in the class, if we treat each person as valued, if we take the time to listen and answer questions, the other students will see who we are. They will know that they can ask and be treated well, and they will know that we care. When we put time into working out good ways to explain things, when we experiment with different ways of teaching and assessing, when we smile and look happy to be there – all these things help students to know who we are, and that we care.

As teachers of mathematics and statistics we have daunting influence over the futures of our students. We need to make sure we are empowering out students, and having them feel safe is a good start.

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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

 

 

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. Sadly, our Kickstarter campaign was unsuccessful, but we hope to work with an established game manufacturer to bring them to the market by the end of 2017.

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

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?

There’s more to reading graphs than meets the eye

There’s more to reading graphs than meets the eye

For those of us who know how to read a graph, it can be difficult to imagine what another person could find difficult. But then when I am presented with an unusual style of graph, or one where the data has been presented badly, I suddenly feel empathy for those who are less graph-literate.

Graphs are more common now as we have Excel to make them for us – for better or worse. An important skill for the citizens of tomorrow and today is to be able to read a graph or table and to be critical of how well it accomplishes its goals.

Here are some stages of reading a graph, much of which also applies to reading a table.

Reading about the graph

When one is familiar with graphs, and the graph is well made, we can become oblivious to the conventions. Just as readers know that English is written from left to right, graph readers understand that the height of a bar chart corresponds to the quantity of something. When people familiar with graphs look at a graph, they take in information unconsciously. This would include what type of graph it is – bar chart, line graph, scatterplot…and what it is about – the title, axis labels and legend tell us this. And they are also able to ignore unimportant aspects. For example if someone has made a 3-D bar chart, experienced graph-readers know that the thickness of the bar does not express information. Colours are generally used to distinguish different elements, but the choice of which colour is used is seldom part of the message. Other aspects about graphs, which may or may not be apparent, include the purpose of the graph and the source of the data.

Beginner graph readers need to learn how to use the various conventions to read ABOUT the data or graph. Any exploration of a graph needs to start with the question, “What is this graph about?”

Identifying one piece of data

When children start making and reading graphs, it is good for them to start with data about themselves, often represented in a picture graph, where each individual observation is shown.  A picture graph is concrete. Each child may point out their particular piece of data – the one that says that they like Wheaties, or prefer mushrooms on their pizza. This is an early stage in  the process of abstraction, that leads eventually to understanding less intuitive graphs such as the box and whisker or a time series chart. It is also important for all graph readers to be aware what each piece of data, or observation, represents and how it is represented.

Identifying one piece of data may help avoid the confusion of graphs which show raw data rather than summary data. For an example, a class may have data about the number of people in households. If this data is entered raw into a spreadsheet, and a graph created, we can end up with something like the graph immediately below (Graph 1).

This is not a good graph, but is what a naive user may well get out of Excel

Graph 1: This is not a good graph, but is what a naive user may well get out of Excel

In this we can identify that each member is represented by a bar, and the height gives the number of people in their family. I usually call this a value graph, as it shows only the individual values, with no aggregation.

A more useful representation of this same data is a summary bar chart, as shown below. (Graph 2) There are two dimensions operating. Horizontally we have the number of people in a household, and vertically we have the number of class members that have the corresponding number of people in their household. Note that it is less intuitive seeing where each class member is. Dividing the bar up into individual blocks can help with that.

Household size

Graph 2: A summary of the size of household for a group of people

Reading off the graph

In order to make sense of a graph, we often need to look at two dimensions simultaneously. If we wish to know how many people in the class come from a household of 5, we need to select along the horizontal axis, the value 5. Then we follow the bar up to the top and take our eye back to the vertical axis to see how high this value is. A ruler can help with this process.  When we read off a graph, our statements tend to be summaries of a single attribute, such as “There are 2 people who come from households of 6.”  “There are 17 dragons that breathe fire.”

Reading within the graph (comparisons, relationships)

Reading within the graph is a more complex task, even with simple graphs. When we read within a graph we are interested in comparisons and relationships. For example we may wish to see which breath type is most common among our herd of dragons. In order to answer this using the graph below, we first need to find the highest bar, by drawing our eye along the top, or drawing a ruler down the page. Then we look down that bar, and read of the name of the breath type. There are many more complex relationships, such as whether green dragons tend to be taller or shorter than red dragons, and which are more likely to be friendly. By introducing another attribute, we are in fact adding a dimension to our analysis.

This is a column chart (or bar chart) summarising the breath types.

This is a column chart (or bar chart) summarising the breath types.

Reading beyond the graph, beyond the data

This idea of reading beyond the data has been suggested as a step towards informal and then formal inference. We can perceive that our data does not represent all existing instances, and can make predictions or suppositions about what might happen in the other instances. For example, for our sample of dragons, we have seen that the green dragons tend to be more likely to be friendly than the red dragons. We could surmise that this holds over the other dragons as well. We can introduce this idea by asking the students, “I wish to have a new dragon join the herd and would prefer it to be friendly. Would I be better to get a green dragon or a red dragon?”

Judging the graph

The advantage of programs like Excel is that many people can make graphs without too much trouble. This is also a problem, as often the graph Excel produces is not really suitable for the task, and can have all sorts of visual clutter which obscures the information displayed. Learners need to think about the graph, either their own, or one they are reading and ask whether it is successful in communicating correctly the information that needs to be communicated. Does the graph serve the purpose it was created for?

I suggest that the steps listed here are a worthwhile structure to use in reading graphs, particularly for beginners. This then leads into another process, summarised as OSEM. You can read about this here in this post, A helpful structure for analysing graphs.

Learning to teach statistics, in a MOOC

I am participating in a MOOC, Teaching statistics through data investigations. A MOOC is a fancy name for an online, free, correspondence course.  The letters stand for Massive Open Online Course. I decided to enrol for several reasons. First I am always keen to learn new things. Second, I wanted to experience what it is like to be a student in a MOOC. And third I wanted to see what materials we could produce that might help teachers or learners of statistics in the US. We are doing well in the NZ market, but it isn’t really big enough to earn us enough money to do some of the really cool things we want to do in teaching statistics to the masses.

I am now up to Unit 4, and here is what I have learned so far:

Motivation and persistence

It is really difficult to stay motivated even in the best possible MOOC. Life gets in the way and there is always something more pressing than reading the materials, taking part in discussions and watching the videos. I looked up the rate of completion for MOOCs, and this article from IEEE gives the completion rate at 5%. Obviously it will differ between MOOCs, depending on the content, the style, the reward. I have found I am best to schedule time to apply to the MOOC each week, or it just doesn’t happen.

I know more than I thought I did

It is reassuring to find out that I really do have some expertise. (This may be a bit of a worry to those of you who regularly read my blog and think I am an expert in teaching statistics.) My efforts to read and ponder, to discuss and to experiment have meant that I do know more than teachers who are just beginning to teach statistics. Phew!

The investigative process matters

I finally get the importance of the Statistical Enquiry Cycle (PPDAC in New Zealand) or Statistical Investigation Cycle (Pose Collect, Analyse, Interpret in the US). I sort of got it before, but now it is falling into place. In the old-fashioned approach to teaching statistics, almost all the emphasis was on the calculations. There would be questions asking students to find the mean of a set of numbers, with no context. This is not statistics, but an arithmetic exercise. Unless a question is embedded in the statistical process, it is not statistics. There needs to be a reason, a question to answer, real data and a conclusion to draw. Every time we develop a teaching exercise for students, we need to think about where it sits in the process, and provide the context.

Brilliant questions

I was happy to participate in the LOCUS quiz to evaluate my own statistical understanding. I was relieved to get 100%. But I was SO impressed with the questions, which reflected the work and thinking that have produced them. I understand how difficult it is to write questions to teach and assess statistical understanding, as I have written hundreds of them myself. The FOCUS questions are great questions. I will be writing some of my own following their style. I loved the ones that asked what would be the best way to improve an experimental design. Inspired!

It’s easier to teach the number stuff

I’m sure I knew this, but to see so many teachers say it, cemented it in. Teacher after teacher commented that teaching procedure is so much easier than teaching concepts. Testing knowledge of procedure is so much easier than assessing conceptual understanding. Maths teachers are really good at procedure. That fluffy, hand-waving meaning stuff is just…difficult. And it all depends. Every answer depends! The implication of this is that we need to help teachers become more confident in helping students to learn the concepts of statistics. We need to develop materials that focus on the concepts. I’m pretty happy that most of my videos do just that – my “Understanding Confidence Intervals” is possibly the only video on confidence intervals that does not include a calculation or procedure.

You learn from other participants

I’ve never been keen on group work. I suspect this is true of most over-achievers. We don’t like to work with other people on assignments as they might freeload, or worse – drag our grade down. Over the years I’ve forced students to do group assignments, as they learn so much more in the process. And I hate to admit that I have also learned more when forced to do group assignments. It isn’t just about reducing the marking load. In this MOOC we are encouraged to engage with other participants through the discussion forums. This is an important part of on-line learning, particularly in a solely on-line platform (as opposed to blended learning). I just love reading what other people say. I get ideas, and I understand better where other people are coming from.

I have something to offer

It was pretty exciting to see my own video used as a resource in the course, and to hear from the instructor how she loves our Statistics Learning Centre videos.

What now?

I still have a few weeks to run on the MOOC and I will report back on what else I learn. And then in late May I am going to USCOTS (US Conference on Teaching Statistics). It’s going to cost me a bit to get there, living as I do in the middle of nowhere in Middle Earth. But I am thrilled to be able to meet with the movers and shakers in US teaching of statistics. I’ll keep you posted!

A Statistics-centric curriculum

Calculus is the wrong summit of the pyramid.

“The mathematics curriculum that we have is based on a foundation of arithmetic and algebra. And everything we learn after that is building up towards one subject. And at top of that pyramid, it’s calculus. And I’m here to say that I think that that is the wrong summit of the pyramid … that the correct summit — that all of our students, every high school graduate should know — should be statistics: probability and statistics.”

Ted talk by Arthur Benjamin in February 2009. Watch it – it’s only 3 minutes long.

He’s right, you know.

And New Zealand would be the place to start. In New Zealand, the subject of statistics is the second most popular subject in our final year of schooling, with a cohort of 12,606. By comparison, the cohort for  English is 16,445, and calculus has a final year cohort of 8392, similar in size to Biology (9038), Chemistry (8183) and Physics (7533).

Some might argue that statistics is already the summit of our curriculum pyramid, but I would see it more as an overly large branch that threatens to unbalance the mathematics tree. I suspect many maths teachers would see it more as a parasite that threatens to suck the life out of their beloved calculus tree. The pyramid needs some reconstruction if we are really to have a statistics-centric curriculum. (Or the tree needs pruning and reshaping – I think I have too many metaphors!)

Statistics-centric curriculum

So, to use a popular phrase, what would a statistics-centric curriculum look like? And what would be the advantages and disadvantages of such a curriculum? I will deal with implementation issues later.

To start with, the base of the pyramid would look little different from the calculus-pinnacled pyramid. In the early years of schooling the emphasis would be on number skills (arithmetic), measurement and other practical and concrete aspects. There would also be a small but increased emphasis on data collection and uncertainty. This is in fact present in the NZ curriculum. Algebra would be introduced, but as a part of the curriculum, rather than the central idea. There would be much more data collection, and probability-based experimentation. Uncertainty would be embraced, rather than ignored.

In the early years of high school, probability and statistics would take a more central place in the curriculum, so that students develop important skills ready for their pinnacle course in the final two years. They would know about the statistical enquiry cycle, how to plan and collect data and write questionnaires.  They would perform their own experiments, preferably in tandem with other curriculum areas such as biology, food-tech or economics. They would understand randomness and modelling. They would be able to make critical comments about reports in the media . They would use computers to create graphs and perform analyses.

As they approach the summit, most students would focus on statistics, while those who were planning to pursue a career in engineering would also take calculus. In the final two years students would be ready to build their own probabilistic models to simulate real-world situations and solve problems. They would analyse real data and write coherent reports. They would truly understand the concept of inference, and why confidence intervals are needed, rather than calculating them by hand or deriving formulas.

There is always a trade-off. Here is my take on the skills developed in each of the curricula.

Calculus-centric curriculum

Statistics-centric curriculum

Logical thinking Communication
Abstract thinking Dealing with uncertainty and ambiguity
Problem-solving Probabilistic models
Modelling (mainly deterministic) Argumentation, deduction
Proof, induction Critical thinking
Plotting deterministic graphs from formulas Reading and creating tables and graphs from data

I actually think you also learn many of the calc-centric skills in the stats-centric curriculum, but I wanted to look even-handed.

Implementation issues

Benjamin suggests, with charming optimism, that the new focus would be “easy to implement and inexpensive.”  I have been a very interested observer in the implementation of the new statistics curriculum in New Zealand. It has not happened easily, being inexpensive has been costly, and there has been fallout. Teachers from other countries (of which there are many in mathematics teaching in NZ) have expressed amazement at how much the NZ teachers accept with only murmurs of complaint. We are a nation with a “can do” attitude, who, by virtue of small population and a one-tier government, can be very flexible. So long as we refrain from following the follies of our big siblings, the UK, US and Australia, NZ has managed to have a world-class education system. And when a new curriculum is implemented, though there is unrest and stress, there is seldom outright rebellion.

In my business, I get the joy of visiting many schools and talking with teachers of mathematics and statistics. I am fascinated by the difference between schools, which is very much a function of the head of mathematics and principal. Some have embraced the changes in focus, and are proactively developing pathways to help all students and teachers to succeed. Others are struggling to accept that statistics has a place in the mathematics curriculum, and put the teachers of statistics into a ghetto where they are punished with excessive marking demands.

The problem is that the curriculum change has been done “on the cheap”. As well as being small and nimble, NZ is not exactly rich. The curriculum change needed more advisors, more release time for teachers to develop and more computer power. These all cost. And then you have the problem of “me too” from other subjects who have had what they feel are similar changes.

And this is not really embracing a full stats-centric curriculum. Primary school teachers need training in probability and statistics if we are really to implement Benjamin’s idea fully. The cost here is much greater as there are so many more primary school teachers. It may well take a generation of students to go through the curriculum and enter back as teachers with an improved understanding.

Computers make it possible

Without computers the only statistical analysis that was possible in the classroom was trivial. Statistics was reduced to mechanistic and boring hand calculation of light-weight statistics and time-filling graph construction. With computers, graphs and analysis can be performed at the click of a mouse, making graphs a tool, rather than an endpoint. With computing power available real data can be used, and real problems can be addressed. High level thinking is needed to make sense and judgements and to avoid wrong conclusions.

Conversely, the computer has made much of calculus superfluous. With programs that can bash their way happily through millions of iterations of a heuristic algorithm, the need for analytic methods is seriously reduced. When even simple apps on an iPad can solve an algebraic equation, and Excel can use “What if” to find solutions, the need for algebra is also questionable.

Efficient citizens

In H.G. Wells’ popular but misquoted words, efficient citizenry calls for the ability to make sense of data. As the science fiction-writer that he was, he foresaw the masses of data that would be collected and available to the great unwashed. The levelling nature of the web has made everyone a potential statistician.

According to the engaging new site from the ASA, “This is statistics”, statisticians make a difference, have fun, satisfy curiosity and make money. And these days they don’t all need to be good at calculus.

Let’s start redesigning our pyramid.

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.

A helpful structure for analysing graphs

Mathematicians teaching English

“I became a maths teacher so I wouldn’t have to mark essays”
“I’m having trouble getting the students to write down their own ideas”
“When I give them templates I feel as if it’s spoon-feeding them”

These are comments I hear as I visit mathematics teachers who are teaching the new statistics curriculum in New Zealand. They have a point. It is difficult for a mathematics teacher to teach in a different style. But – it can also be rewarding and interesting, and you never get asked, “Where is this useful?”

The statistical enquiry cycle provides a structure for all statistical investigations and learning.

We start with a problem or question, and undergo an investigation, either using extant data, an experiment or observational study to answer the question. Writing skills are key in several stages of the cycle. We need to be able to write an investigative question (or hypotheses). We need to write down a plan, and sometimes an entire questionnaire. We need to write down what we find in the analysis and we need to write a conclusion to answer the original question. That’s a whole heap of writing!

And for teachers who may not be all that happy about writing themselves, and students who chose mathematical subjects to avoid writing, it can be a bridge too far.
In previous posts on teaching report writing I promote the use of templates, and give some teaching suggestions.

In this post I am concentrating on analysing graphs, using a handy acronym, OSEM. OSEM was developed by Jeremy Brocklehurst from Lincoln High School near Christchurch NZ. There are other acronyms that would work just as well, but we like this one, not the least for its link with kiwi culture. We think it is awesome (OSEM). You could Google “o for awesome”, to get the background. OSEM stands for Obvious, Specific, Evidence and Meaning. It is a process to follow, rather than a checklist.

I like the use of O for obvious. I think students can be scared to say what they think might be too obvious, and look for tricky things. By including “obvious” in the process, it allows them to write about the important, and usually obvious features of a graph. I also like the emphasis on meaning, Unless the analysis of the data links back to the context and purpose of the investigation, it is merely a mathematical exercise.

Is this spoon-feeding? Far from it. We are giving students a structure that will help them to analyse any graph, including timeseries, scatter plots, and histograms, as well as boxplots and dotplots. It emphasises the use of quantitative information, linked with context. There is nothing revolutionary about it, but I think many statistics teachers may find it helpful as a way to breakdown and demystify the commenting process.

Class use of OSEM

In a class setting, OSEM is a helpful framework for students to work in groups. Students individually (perhaps on personal whiteboards) write down something obvious about the graph. Then they share answers in pairs, and decide which one to carry on with. In the pair they specify and give evidence for their “obvious” statement. Then the pairs form groups of four, and they come up with statements of meaning, that are then shared with the class as a whole.

Spoon feeding has its place

On a side-note – spoon-feeding is a really good way to make sure children get necessary nutrition until they learn to feed themselves. It is preferable to letting them starve before they get the chance to develop sufficient skills and co-ordination to get the food to their mouths independently.