Dragon Trainer rich mathematical task

I love rich mathematical tasks. Here is one for all levels of schooling. What do you think?

Background to rich tasks

A rich task is an open-ended task that students can engage with at multiple levels. I use the following information from the nrich website when I am talking to teachers about rich tasks.

Some important aspects of rich mathematical tasks

Background to Dragonistics data cards

In this task we use our Dragonistics data cards, which are shown here. For a less colourful exercise you could use 24 pieces of card with numbers 1 to 8 on them.

A small sample of Dragonistics data cards

Each dragon has a strength rating of between 1 and 8, shown by the coloured dragon scales on the right-hand side of the card. The distribution of dragon strengths is not uniform, but is clustered around the middle, and depends to a certain extent on other aspects of the dragon, such as their species, gender and behaviour.

The students will already be familiar with the dragon cards, and each group of students has a set of about 24 dragonistics data cards. As there are a total of 288 dragons, each group will have a different set of dragons. Some may or may not have dragons of each strength rating.

The task

A dragon team trainer says that teams of two dragons chosen at random nearly always have a combined strength of between 7 and 11.
Is this true?
Provide evidence to support your conclusion.

Try it yourself

If you do not have any dragons of your own, make up about 20 pieces of card, with the numbers 1 to 8 on them, so you can explore the problem. Like Tracy Zager, we emphasise the necessity of exploring the maths ourselves before the children.

Possible approaches

What is great about this exercise is that you can explore it experimentally or theoretically. It has a low entry point, as encouraged on Youcubed. This is sometimes called “low floor, high ceiling”.For younger children, it is a good start to take pairs of dragons, add their strengths, and write down the answer. Then they need to work out a recording method, possibly a tally table.  You can have discussions about what it means for the dragons to be chosen randomly. You can also discuss what “nearly always” means.

Recently I used this task with a group of ten-year-olds. After they had made an attempt at solving it, I asked what they thought would be the most common team strength, and one said 9 or 10 because it is in the middle. I should have explored this idea further. What I did do, was start working out how many different combinations were possible. It is not possible to have a team of strength 1, and there is only one way to get a team of strength 2. How many ways to get strength 3? By the time we got to strength 6, they could see a pattern, that the number of combinations is one less than the total strength. So then I jumped to the other end of the distribution, asking “What is the strongest team we could possibly get?” As it happened, they did have two dragons of strength 8 in their set of dragons, so they correctly estimated the answer to be 16. So then I asked how many different ways they could get 16, and using their previous rule, they suggested 15 ways.  Then when I asked them to tell me what they were, they realised that there was only one way. From there we started working down the numbers. Unfortunately this was during a holiday programme, so I didn’t have time to pursue this further. However we will be using this exercise in our rural rich maths events.

Lessons to bring out at different levels

There are three main ways to approach this problem. The first is to experiment by randomly taking pairs of dragons, and recording their total strengths. A simple theoretical model involves thinking about all the possible outcomes and seeing what proportion of the outcomes lies between the chosen values. Then a more refined model would take into account the distributions of strengths for the given dragons.  The learners may well come up with some interesting other ways to go about this.

Extension questions

A teacher can encourage further thinking with questions such as:

Would this answer be the same for every group of dragons? Is it possible to find a set of dragons so that the only team strengths are between 5 and 11? What would happen if you had teams of three dragons. Does it make a difference if you select one team at a time, and shuffle, or divide into lots of teams and record, before shuffling? How many different team possibilities are possible? What if you only had green dragons – would this make a difference?

Show them the maths

It is important to point out the mathematical skills they are exercising as they tackle rich tasks. This task improves number skills, encourages persistence and risk-taking, develops communication skills as they are required to justify their conclusion. At higher levels it is helping to develop understanding of probability distributions, and you could also introduce or reinforce the idea of a random variable – in this case the team strength.

It would also be interesting to look at the spread for single dragons, two dragon teams and three dragon teams. With enough repetitions (and at this point a spreadsheet could be handy) the central limit theorem will start to be apparent. As you can see, there is great potential to expand this.

Transferring

We need to look at ways this is also applicable in daily life, and not just for dragon trainers. The same sort of problem would occur if you had people buying different numbers of items, or different weights of suitcases. You might like to think of the combined strengths as similar to total scores in sports events. At higher levels you might discuss the concept of independence.

So rich – so many possibilities! Thoughts?

Mathematics and statistics lessons about elections

Study elections in mathematics because it is important

Too often mathematics is seen as pure and apolitical.  Maths teachers may keep away from concepts that seem messy and without right and wrong answers. However, teachers of mathematics and statistics have much to offer to increase democratic power in the upcoming NZ general elections (and all future elections really). The bizarre outcomes for elections around the world recently (2016/2017 Brexit, Trump) are evidence that we need a compassionate, rational, informed populace, who is engaged in the political process, to choose who will lead our country. Knowledge is power, and when people do not understand the political process, they are less likely to vote. We need to make sure that students understand how voting, the electoral system, and political polls work. Some of our students in Year 13 will be voting this election, and students’ parents can be influenced to vote.

There are some lessons provided on the Electoral Commission site.   Sadly all the teaching resources are positioned in the social studies learning area – with none in statistics and mathematics. Similarly in the Senior Secondary guides, all the results from a search on elections were in the social studies subject area.

Elections are mathematically and statistically interesting and relevant

In New Zealand, our MMP system throws up some very interesting mathematical processes for higher level explorations. Political polls will be constantly in the news, and provide up-to-date material for discussions about polls, sample sizes, sampling methods, sampling error etc.

Feedback

It would be great to hear from anyone who uses these ideas. If you have developed them further, so much the better. Do share with us in the comments.

Suggestions for lessons

These suggestions for lessons are listed more or less in increasing levels of complexity. However I have been amazed at what Year 1 children can do. It seems to me that they are more willing to tackle difficult tasks than many older children. These lessons also embrace other curriculum areas such as technology, English and social studies.

Physical resources

Make a ballot box, make a voting paper. Talk about randomising the names on the paper. How big does the box need to be? How many ballot boxes are being made for the upcoming election? How much cardboard is needed?

Follow the polls

Make a time series graph of poll results. Each time there is a new result, plot it on the graph over the date, and note the sample size. At higher levels you might like to put confidence intervals on either side of the plotted value. A rule of thumb is 1/square root of the sample size. For example if the sample size is 700, the margin of error is 3.7%. So if the poll reported a party gaining 34% of the vote, the confidence interval would be from 33.3% to 37.7%.

You can get a good summary of political polls on Wikipedia.

From NZ maths  – On the Campaign Trail (CL 4)

Figure it Out, Number sense  Book 2 Level 4 – has an exercise about finding fractions, decimals, and percentages of amounts expressed as whole numbers, simple fractions, and decimals.

From NZ maths – Whose News (CL 4)

This is a guide to running an analysis on the level of representation of different geopgraphic areas in the news. The same lesson could be used for representation of different parties or different issues.

Graphical representations

The newspapers and online will be full of graphs and other graphical representations. Keep a collection and evaluate them for clarity and attractiveness.

How many people will be employed on election day?

This inquiry uses a mixture of internet search, mathematical modelling, estimation and calculation.

  • How many electorates are there?
  • How many polling booths per electorate?
  • How many people per booth?
  • How long are they employed for?

Fairness of booth provision

  • Is the location of polling booths fair?
  • What is the furthest distance a person might need to travel to a voting booth?
  • What do people in other countries do?

The mathematics of MMP

This link provides a thorough explanation of the system. A project could be for students to work out what it is saying and make a powerpoint presentation or short video explaining it more simply.

Overhang and scenario modelling

Overhang occurs when a party gets more electoral MPS elected than their proportion allows. Here is a fact sheet about overhang and findings of the electoral review. Students could create scenarios to evaluate the effect of overhang and find out what is the biggest overhang possible.

Small party provisions

How might the previous two election results have been different if there were not the 5% and coat-tailing rules?

Gerrymandering

Different ways of assigning areas to electorates get different results. The Wikipedia article on Gerrymandering has some great examples and diagrams on how it all happens, and the history behind the name.

Statistical analysis of age and other demographics

Statistics should be analysed in response to a problem, rather than just for the sake of it.
Suggested Scenario: A new political party is planning to appeal to young voters, under 30 years of age. They wish to find out which five electorates are the best to target. You may also wish to include turn-out statistics in your analysis.

Resource: Enrolment statistics by electorate – some graphs supplied, percentages for each electorate.

Statistical analysis of turn out

In the interests of better democracy, we wish to have a better voter turnout. Find out the five electorates with the best voter turnout and the worst, and come up with some ideas about why they are the best and the worst. Test out your theory/model by trying to predict the next five best and worst. Use what you find out to suggest how might we improve voter turnout.

Resource: Turn out statistics – by electorate or download the entire file

Happy teaching, and fingers crossed for September.

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.

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.

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.

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?

 

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?