# Dragon Trainer rich mathematical task

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

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.

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?

# 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?

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%.

## 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.

## 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.

# Maths trauma and earthquakes

Trauma is a deeply distressing or disturbing experience. Many people in my home town of Christchurch still suffer from post traumatic stress disorder (PTSD) as a result of our earthquakes five or so years ago. I know I will never be the same again. The trauma began with the original terrifying experience of having the ground move in a way you never thought was even possible. It was perpetuated by over eighteen months of never knowing when the next earthquake (deceptively called aftershock) would hit. And the trauma still continues for many as they struggle to sort out their homes, and jobs, and their families. (Even now the thought of earthquakes can bring me to tears, and heavy machinery undertaking drainage work happening in my street is not helping.)

People might question if the impact of bad maths experiences can really be likened to the trauma people experience as the result of a series of earthquakes. I listened recently to a webinar about maths trauma, hosted by Global Math Department, and presented by Dr Kasi Allen. Math Trauma: Healing Our Classrooms, Our Students, and Our Discipline The webinar occurred in April 2016, but thanks to the amazing global maths community, it is still available and has had over 1000 views. Dr Allen calls herself a “math activist who studies math trauma and promotes teaching mathematics for social justice”. I see myself and the work we do at Statistics Learning Centre in that vein also.

Three students engrossed in a maths event

I have reproduced a few of the ideas in the webinar, but would recommend visiting it yourself to get the full value.

Dr Allen’s proposition is that what is commonly called math anxiety is probably better described as math trauma. She teaches preservice elementary school teachers. A watershed experience has been seeing people bolt from the room in tears, simply looking at the syllabus at the start of a maths course.

## Too many people think they are not maths people

I am frequently told by people that they do not have a maths brain, could never learn maths, that they are not a maths person. I have had middle-aged women tell me of formative experiences that happened over fifty years previously that have shaped their relationship with mathematics. Recently I asked my Facebook friends both mathematically inclined and not so mathematically inclined about how they picture numbers. Time and again their responses included the statement that they are not good at maths.

## Maths anxiety and trauma

The term “math anxiety” dates back to the 1950s and is still used today. There are decades of research into how math anxiety disproportionately affects students who are female, low income and non-white. What Dr Allen (and I)  found disturbing was that among college students, undergraduate education majors are the most maths anxious, both in terms of number and severity. These are the people who are entrusted with teaching mathematics to the next generation. Primary school teachers too often have an unhealthy relationship with maths – that is NOT their fault. They were taught in a way that did not work for them and they carry the burden with them.

Dr Allen suggests that maths trauma is a more fitting description than maths anxiety. Jo Boaler talks about people as having been maths traumatised. The negative experiences people have with mathematics, are described as painful and damaging. Traumatic events can be grounded in everyday life, and do not need to come from one catastrophic event. It is the subjective response that matters. Dr Allen gives the following definition:

“Math trauma stems from an event, a series of events, or a set of circumstances experienced by an individual as harmful or threatened such that there are lasting adverse effects on the individual’s functioning and well-being in the perceived presence of mathematics.”

## Healing teachers and students from maths trauma

Dr Allen has suggestions to help heal maths trauma. One suggestion is to acknowledge past negative experiences and their effects. We can listen and express sympathy and even apologise for the harm people have felt. We can provide opportunities for students to tell their maths stories. We can help them nurture their mathematics identities.

We also need to work on prevention of maths trauma. Classroom culture is important. Students need to feel safe and brave and they need to move. And we need to end traumatizing traditions. I have reproduced a screen shot of the slide about ending traumatizing traditions. Timed tests in mathematics have to stop. Now. Forever.

The question is, how do we (Statistics Learning Centre) help teachers to recover from maths trauma, so they can feel the fun and excitement that can be had in maths? Teachers matter for themselves, as well as for the good they can do their students. Maths educators need to be part of the solution and part of the prevention – to be maths activists. People are not born with maths trauma and it does not exist in all cultures. We need to do better.

## Call for comment

So here is my question. Do you or someone you know suffer from maths trauma? Let me tell you now – it is not your fault. It is not their fault.
What needs to happen for you to feel better about maths? What needs to happen so that maths trauma can be eliminated from our schooling?

# Graphs – beauty and truth (with apologies to Keats)

## A good graph is elegant

I really like graphs. I like the way graphs turn numbers into pictures. A good graph is elegant. It uses a few well-placed lines to communicate what would take a paragraph of text. And like a good piece of literature or art, a good graph continues to give, beyond the first reading. I love looking at my YouTube and WordPress graphs. These graphs tell me stories. The WordPress analytics tell me that when I put up a new post, I get more hits, but that everyday more than 1000 people read one of my posts. The YouTube analytics tell me stories about when people want to know about different aspects of statistics. It is currently the end of the North American school year, and the demand is for my video on Choosing which statistical test to use. Earlier in the year, the video about levels of measurement is the most popular. And not many people view videos about statistics on the 25th of December. I’m happy to report that the YouTube and WordPress graphs are good graphs.

Spreadsheets have made it possible for anyone and everyone to create graphs. I like that graphs are easier to make. Drawing graphs by hand is a laborious task and fraught with error. But sometimes my heart aches when I see a graph used badly. I suspect that this is when a graphic artist has taken control, and the search for beauty has over-ridden the need for truth.

Three graphs spurred me to write this post.

## Graph One: Bad-tasting Donut on house occupation

The first was on a website to find out about property values. I must have clicked onto something to find out about the property values in my area, and was taken to the qv website. And this is the graph that disturbed me.

Graphs named after food are seldom a good idea

Sure it is pretty – uses pretty colours and shading, and you can find out what it is saying by looking at the key – with the numbers beside it. But a pie or donut chart should not be used for data which has inherent order. The result here is that the segments are not in order. Or rather they are ordered from most frequent to least frequent, which is not intuitive. Ordinal data is best represented in a bar or column chart. To be honest, most data is best represented in a bar or column chart. My significant other suggested that bar charts aren’t as attractive as pie charts. Circles are prettier than rectangles. Circles are curvy and seem friendlier than straight lines and rectangles. So prettiness has triumphed over truth.

## Graph Two: Misleading pictogram (a tautology?)

It may be a little strong to call bad communication lack of truth. Let’s look at another example. In a way it is cheating to cite a pictogram in a post like this. Pictograms are the lowest form of graph and are so often incorrect, that finding a bad one is easier than finding a good one. In the graph below of fatalities it is difficult to work out what one little person represents.

What does one little person represent?

A quick glance, ignoring the numbers, suggests that the road toll in 2014 is just over half what it was in 2012. However, the truth, calculated from the numbers, is that the relative size is 80%. 2012 has 12 people icons, representing 280 fatalities. One icon is removed for 2013, representing a drop of 9 fatalities. 2011 has one icon fewer again, representing a drop of 2 fatalities. There is so much wrong in the reporting of road fatalities, that I will stop here. Perhaps another day…

## Graph Three: Mysterious display on Household income

And here is the other graph that perplexed me for some time. It came in the Saturday morning magazine from our newspaper, as part of an article about inequality in New Zealand. Anyone who reads my blog will be aware that my politics place me well left of centre, and I find inequality one of the great ills of the modern day. So I was keen to see what this graph would tell me. And the answer is…

See how long it takes for you to find where you appear on the graph. (Pretending you live in NZ)

I have no idea. Now, I have expertise in the promulgation of statistics, and this graph stumped me for some time. Take a good look now, before I carry on.

Graphs are the main way that statistical analysts communicate with the outside world. Graphs like these ones do us no favours, even if they are not our fault. We need to do better, and make sure that all students learn about graphs.

## Teaching suggestion – a graph a day

Here is a suggestion for teachers at all levels. Have a “graph a day” display – maybe for a month? Students can contribute graphs from the news media. Each day discuss what the graph is saying, and critique the way the graph is communicating. I have a helpful structure for reading graphs in my post: There’s more to reading graphs than meets the eye;

Here is a summary of what I’ve said and what else I could say on the topic.

• The choice of graph depends on the purpose
• The text should state the purpose of the graph
• There is not a graph for everything you wish to communicate
• Sometimes a table communicates better than a graph
• Graphs are part of the analysis as well as part of the reporting. But some graphs are better to stay hidden.
• If it takes more than a few seconds to work out what a graph is communicating it should either be dumped or have an explanation in the text
• Truth (or communication) is more important than beauty
• There is beauty in simplicity
• Be aware than many people are colour-blind, or cannot easily differentiate between different shades.

## Feedback from previous post on which graph to use

Late last year I posted four graphs of the same data and asked for people’s opinions. You can link back to the post here and see the responses: Which Graph to Use.

The interesting thing is not which graph was selected as the most popular, but rather that each graph had a considerable number of votes. My response is that it depends.  It depends on the question you are answering or the message you are sending. But yes – I agree with the crowd that Graph A is the one that best communicates the various pieces of information. I think it would be improved by ordering the categories differently. It is not very pretty, but it communicates.

I recently posted a new video on YouTube about graphs. It is a quick once-over of important types of graphs, and can help to clarify what they are about. There are examples of good graphs in there.

I have written about graphs previously and you can find them here on the Collected Works page.

I’m interested in your thoughts. And I’d love to see some beautiful and truthful graphs in the comments.

# 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.

# STEM, STEM-Ed, STEAM and Statistics

STEM is a popular acronym in educational circles and is used to refer to careers and educational tasks. Though most know that the four letters stand for Science, Technology, Engineering and Mathematics, it can be difficult to pin down what exactly it means. In this post I suggest that there are two related uses of STEM as a description.

# STEM

The term, STEM, originated in the USA in the late 1990s to describe specific careers and education for these careers. There seems to be no universally agreed-upon definition of STEM. From a careers perspective, the focus is on making sure that there are enough skilled workers in the STEM areas for future development. A common (engineering?) analogy is that of a pipeline. Industrialised nations need people with STEM skills, so need to ensure there are enough people entering and staying in the pipeline in order to fill future demand. There are also identified equity issues, as STEM jobs tend to be higher-paying, and also tend to be dominated by white males. A consequence of the higher demand and pay for people with STEM skills and qualifications is that there is often a shortage of teachers in STEM subjects.

This graphic illustrates the analogy of a STEM pipeline. Are people the drips?

# STEM Education

There are multiple ways of viewing STEM Education. One category is specific STEM Projects which I refer to as STEM-Ed, and another is education in STEM subjects, as they currently exist in the school curriculum.

## STEM-Ed

It is believed that one way to encourage children and young people to continue in STEM subjects, is to embed STEM into the curriculum. There has been a move towards specific STEM-based lessons or projects, particularly at middle-school or older primary level. Pinterest is full of attractive STEM-Ed lessons based around engineering and the design process. These include tower and bridge building, making boats to carry certain weights, creating a mechanism that will protect an egg from a fall or launching projectiles a maximum or specified distance. STEM-Ed lessons use a wide range of materials, including Lego bricks, spaghetti, marshmallows, masking tape, newspaper, recycled materials – just about anything you can think of. The makerspace movement ties in with STEM-Ed.

A good STEM-Ed project is described by Anne Jolly in her post Perfect STEM lessons. Anne Jolly suggests that a “perfect” STEM lesson uses an engineering approach as a framework, applies maths and science content through authentic experiences, deals with real world issues, involves hands-on and open-ended exploration with multiple right answers for students working in teams with the teacher in a facilitator role. A STEM project should also engage students in communicating, remove the fear of failure, appeal equally to boys and girls and promote authentic assessment.

It seems that when primary/elementary school teachers talk about STEM, it is usually STEM-Ed they are referring to. Certainly material under the STEM label on Pinterest, a popular source of inspiration for teachers, tends to be STEM-Ed.

A screenshot of some STEM tasks found on Pinterest

## Education in STEM subjects

In order to encourage and enable students to continue on to STEM careers, they must study the individual subjects that make up STEM. At school level, maths, physics and chemistry are often the areas where students make decisions that limit their later opportunities in STEM areas. (Where they leak out of the pipeline?) This is where teachers of STEM subjects have a part to play. Tying their subjects to authentic, real world contexts and teaching using STEM-Ed projects can help engagement.

However, there is also a need to learn the mathematics that does not appear in a “good STEM lesson”. Current mathematics education thinking aims to enable children to become mathematicians, not just engineers. To quote Tracey Zager’s excellent book, “Becoming the math teacher you wish you’d had“, mathematicians take risks, make mistakes, are precise, rise to a challenge, ask questions, connect ideas, use intuition, reason, prove, and work together and alone. Mathematics curriculum overlaps well with STEM-Ed in the areas of measurement, geometry and statistics. Number skills are practised in context. However, to provide enough exposure to other areas of mathematics, specific STEM-Ed lessons would need to be carefully designed. I suspect that there are areas of the curriculum that are more effectively learned through other methods than STEM-Ed.

# STEAM

There is a push to add Arts to STEM, making it STEAM. The relevance of this addition depends on viewpoint. It does not seem relevant to include Arts when talking about high-shortage career paths. But at the same time, STEM jobs also require other skills, not the least being communication skills. There is a strong link between fine art and technology, through design. The inclusion of A in STEM also depends on the definition of Art. The term “Arts” can include painting, music, dance, literature, film, design and the even the humanities. Including these into STEM (as a career or subject description) seems a trifle incongruous and begs the question whether there is anything that is not included in STEAM. Physical education and foreign languages?

However, when we look at STEM-Ed, there is a rationale for the inclusion of art. Good design does have an artistic component, as is all too clear when we look at some communist-era architecture and much amateur web-design. And written and oral communication are well-developed in many STEM-Ed projects.

# Statistics

Statistics clearly has a place in both STEM and STEM-Ed. There is a demand for statisticians, and people who can use statistics in what they do. The study of the discipline of statistics gives important insights into the nature of variability in our world. STEM-Ed projects could involve collecting and analysing data in a non-trivial way, though I have not seen evidence of this. The barrier to this is the statistical understanding of the teachers creating the STEM-Ed tasks, and points to an area where statistics educators need to be involved. Another barrier can be the time taken to collect an adequate sample, clean the data and analyse it. This is why specific tasks need to be designed for this.

# Concerns about STEM and STEM-Ed

We do need to think about the focus on STEM and wonder about the philosophical underpinnings. Are we educating our students to provide workers for the industrial machine? Is this the right thing to be doing? I found a very interesting book: Philosophy of STEM Education: A Critical Investigation by Nataly Z. Chesky and Mark R. Wolfmeyer . They ask these important questions.

STEM-Ed also needs to be approached carefully. Dayle Anderson, a lecturer in science education emphasised at a Primary Maths Symposium that teachers need to keep their eye on the learning. When a project is engaging it can be seductive to think that the learning is taken care of. There are so many demands on time in a school, that STEM-Ed lessons need to be well-designed for specific learning.

# STEM, STEM-Ed and 21st Century Skills

I am quite taken with the 4 Cs of 21st Century Skills, which have been defined as Critical thinking and problem solving, Communicating, Collaborating and Creativity and innovation. These correspond well to the five Key Competencies in the New Zealand curriculum – thinking, using language, symbols, and texts, managing self, relating to others and participating and contributing. These skills are needed by people who work in STEM jobs. They need to be able to communicate and work with others.

These 21st Century skills can be developed in STEM-Ed lessons, as students are required to work together, solve problems, think, innovate and communicate their results.

# Closing thoughts

Overall I am excited about STEM and STEM-Ed. A knowledge of how the world around us works is empowering to all, whether or not they join the STEM pipeline. Making mathematics, statistics and other related subjects more relevant and desirable is always going to be a good thing. Statistics educators need to be involved make sure that statistics is a vital part of STEM-Ed.

Any suggestions on how this is best achieved? And if you are interested in STEM-Ed, please Like and Follow our Facebook page to keep up with the discussion and find out about our contribution.

# 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.”

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.

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.)

# Why people hate statistics

This summer/Christmas break it has been my pleasure to help a young woman who is struggling with statistics, and it has prompted me to ask people who teach postgraduate statistical methods – WTF are you doing?

Louise (name changed) is a bright, hard-working young woman, who has finished an undergraduate degree at a prestigious university and is now doing a Masters degree at a different prestigious university, which is a long way from where I live and will remain nameless. I have been working through her lecture slides, past and future and attempting to develop in her some confidence that she will survive the remainder of the course, and that statistics is in fact fathomable.

# Incomprehensible courses alienating research students

After each session with Louise I have come away shaking my head and wondering what this lecturer is up to. I wonder if he/she really understands statistics or is just passing on their own confusion. And the very sad thing is that I KNOW that there are hundreds of lecturers in hundreds of similar courses around the world teaching in much the same way and alienating thousands of students every year.

And they need to stop.

Here is the approach: You have approximately eight weeks, made up of four hour sessions, in which to teach your masters students everything they could possibly need to know about statistics. So you tell them everything! You use technical terms with little explanation, and you give no indication of what is important and what is background. You dive right in with no clear purpose, and you expect them to keep up.

Frequently Louise would ask me to explain something and I would pause to think. I was trying to work out how deep to go. It is like when a child asks where babies come from. They may want the full details, but they may not, and you need to decide what level of answer is most appropriate. Anyone who has seen our popular YouTube videos will be aware that I encourage conceptual understanding at best, and the equivalent of a statistics drivers licence at worst. When you have eight weeks to learn everything there is to know about statistics, up to and including multiple regression, logistic regression, GLM, factor analysis, non-parametric methods and more, I believe the most you can hope for is to be able to get the computer to run the test, and then make intelligent conclusions about the output.

There was nothing in the course about data collection, data cleaning, the concept of inference or the relationship between the model and reality. My experience is that data cleaning is one of the most challenging parts of analysis, especially for novice researchers.

# Use learning objectives

And maybe one of the worst problems with Louise’s course was that there were no specific learning objectives. One of my most popular posts is on the need for learning objectives. Now I am not proposing that we slavishly tell students in each class what it is they are to learn, as that can be tedious and remove the fun from more discovery style learning. What I am saying is that it is only fair to tell the students what they are supposed to be learning. This helps them to know what in the lecture is important, and what is background. They need to know whether they need to have a passing understanding of a test, or if they need to be able to run one, or if they need to know the underlying mathematics.

Take for example, the t-test. There are many ways that the t-statistic can be used, so simply referring to a test as a t-test is misleading before you even start. And starting your teaching with the statistic is not helpful. We need to start with the need! I would call it a test for the difference of two means from two groups. And I would just talk about the t statistic in passing. I would give examples of output from various scenarios, some of which reject the null, some of which don’t and maybe even one that has a p-value of 0.049 so we can talk about that. In each case we would look at how the context affects the implications of the test result. In my learning objectives I would say: Students will be able to interpret the output of a test for the difference of two means, putting the result in context. And possibly, Students will be able to identify ways in which a test for the difference of two means violates the assumptions of a t-test. Now that wasn’t hard was it?

# Like driving a car

Louise likes to understand where things come from, so we did go through an overview of how various distributions have been found to model different aspects of the world well – starting with the normal distribution, and with a quick jaunt into the Central Limit Theorem. I used my Dragonistics data cards, which were invented for teaching primary school, but actually work surprisingly well at all levels! I can’t claim that Louise understands the use of the t distribution, but I hope she now believes in it. I gave her the analogy of learning to drive – that we don’t need to know what is happening under the bonnet to be a safe driver. In fact safe driving depends more on paying attention to the road conditions and human behaviour.

# Assumptions

Louise tells me that her lecturer emphasises assumptions – that the students need to examine them all, every time they look at or perform a statistical test. Now I have no problems with this later on, but students need to have some idea of where they are going and why, before being told what luggage they can and can’t take. And my experience is that assumptions are always violated. Always. As George Box put it – “All models are wrong and some models are useful.”

It did not help that the lecturer seemed a little confused about the assumption of normality. I am not one to point the finger, as this is a tricky assumption, as the Andy Field textbook pointed out. For example, we do not require the independent variables in a multiple regression to be normally distributed as the lecturer specified. This is not even possible if we are including dummy variables. What we do need to watch out for is that the residuals are approximately modelled by a normal distribution, and if not, that we do something about it.

You may have gathered that my approach to statistics is practical rather than idealistic. Why get all hot and bothered about whether you should do a parametric or non-parametric test, when the computer package does both with ease, and you just need to check if there is any difference in the result. (I can hear some purists hyperventilating at this point!) My experience is that the results seldom differ.

# What post-graduate statistical methods courses should focus on

Instructors need to concentrate on the big ideas of statistics – what is inference, why we need data, how a sample is collected matters, and the relationship between a model and the reality it is modelling. I would include the concept of correlation, and its problematic link to causation. I would talk about the difference between statistical significance and usefulness, and evidence and strength of a relationship. And I would teach students how to find the right fishing lessons! If a student is critiquing a paper that uses logistical regression, that is the time they need to read up enough about logistical regression to be able to understand what they are reading.They cannot possibly learn a useful amount about all the tests or methods that they may encounter one day.

If research students are going to be doing their own research, they need more than a one semester fly-by of techniques, and would be best to get advice from a statistician BEFORE they collect the data.

# Final word

So here is my take-home message:

Stop making graduate statistical methods courses so outrageously difficult by cramming them full of advanced techniques and concepts. Instead help students to understand what statistics is about, and how powerful and wonderful it can be to find out more about the world through data.

# 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.

# Which graph to use?

A student asked me on our Facebook page to help with an assignment. It got me thinking again about the nature of answers in statistics, and the challenge of communicating through graphs.

The student gave no explanation, but rather a set of data and a choice of graph-type. I have changed the data a little. As the data was about numbers of men and women in different roles in an organisation, I assume the question regards gender equity.

I have used the ubiquitous Microsoft Excel to create some candidate graphs, and I would like you to vote on which you think is the most useful. (And yes – I know that it depends, but in the absence of further information, I’m interested to see what the crowd comes up with.) Pie chart was not an option. Pie chart is almost never an option.

Here are four possible answers for the student. If you click on the graphic it should get bigger and clearer.

Which is the best graph to use?

I will post later with my opinion, but I would rather get yours first.