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.

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?

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

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.

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.)

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.

Choosing your level

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.

Your word

Am I right or is my preaching of the devil? Please add your comments below.

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.

Talking in class: improving discussion in maths and stats classes

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

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

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

Prizes can help motivate students to give answers in unfamiliar settings

Prizes can help motivate students to give answers in unfamiliar settings

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

It is important to talk in maths and stats classes

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

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

Classroom atmosphere

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

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

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

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

Good questions

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

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

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

Multiple answers and open-ended questions

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

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

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

Whole class and group discussion

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

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

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

Play and learning mathematics and statistics

The role of play in learning

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

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

Play and practice

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

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

Traditional vs recent teaching methods for mathematics

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

Play and statistics

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

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

The excitement of a fresh set of data

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

Writing the report is like cleaning up

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

Can statistics be playful?

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

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

helen-has-attitude

Helen has an attitude problem

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

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

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

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

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

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

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

We'd love your help.

We’d love your help.

Your thoughts about play and statistics

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

 

The nature of mathematics and statistics and what it means to learn and teach them

I’ve been thinking lately….

Sometimes it pays to stop and think. I have been reading a recent textbook for mathematics teachers, Dianne Siemon et al, Teaching mathematics: foundations to middle years (2011). On page 47 the authors asked me to “Take a few minutes to write down your own views about the nature of mathematics, mathematics learning and mathematics teaching.” And bearing in mind I see statistics as related to, but not enclosed by mathematics, I decided to do the same for statistics as well. So here are my thoughts:

The nature of mathematics

Mathematicians love the elegance of mathematics

Mathematicians love the elegance of mathematics

Mathematics is a way of modelling and making sense of the world. Mathematics underpins scientific and commercial endeavours as well as everyday life. Mathematics is about patterns and proofs and problem structuring and solution finding. I used to think it was all about the answer, but now I think it is more about the process. I used to think that maths was predominantly an individual endeavour, but now I can see how there is a social or community aspect as well. I fear that too often students are getting a parsimonious view of mathematics, thinking it is only about numbers, and something they have to do on their own. I find my understanding of the nature of mathematics is rapidly changing as I participate in mathematics education at different ages and stages. I have also been influenced by the work of Jo Boaler.

To learn mathematics

My original idea of mathematics learning comes from my own successful experience of copying down notes from the board, listening to the teacher and doing the exercises in the textbook. I was not particularly fluent with my times-tables, but loved problem-solving. If I got something wrong, I was happy to try again until I nutted it out. Sometimes I even did recreational maths, like the time I enumerated all possible dice combinations in Risk to find out who had the advantage – attacker or defender. I always knew that it took practice to be good at mathematics. However I never really thought of mathematics as a social endeavour. I feel I missed out, now. From time to time I do have mathematical discussions with my colleague. It was an adventure inventing Rogo and then working out a solution method. Mathematics can be a social activity.

To teach mathematics

When I became a maths teacher I perpetuated the method that had worked for me, as I had not been challenged to think differently. I did like the ideas of mastery learning and personalised system of instruction. This meant that learners progressed to the next step only when they had mastered the previous one. I was a successful enough teacher and enjoyed my work.

Then as a university lecturer I had to work differently, and experimented. I had a popular personalised system of instruction quantitative methods course, relying totally on students working individually, at their own pace. I am happy that many of my students were successful in an area they had previously thought out of their reach. For some students it was the only subject they passed.

What I would do now

If I were to teach mathematics at school level again, I hope I would do things differently. I love the idea of “Number talks” and rich tasks which get students to think about different ways of doing things. I had often felt sad that there did not seem to be much opportunity to have discussions in maths, as things were either right or wrong. Now I see what fun we could have with open-ended tasks. Maths learning should be communal and loud and exciting, not solitary, quiet and routine. I have been largely constructivist in my teaching philosophy, but now I would like to try out social constructivist thinking.

Statistics

And what about statistics? At school in the 1970s I never learned more than the summary statistics and basic probability. At uni level it was bewildering, but I managed to get an A grade in a first year paper without understanding any of the basic principles. It wasn’t until I was doing my honours year in Operations Research and was working as a tutor in Statistical methods that things stared to come together – but even then I was not at home with statistical ideas and was happy to leave them behind when I graduated.

The nature of statistics

Statistics lives in the real world

Statistics lives in the real world

My views now on the nature of statistics are quite different. I believe statistical thinking is related to mathematical thinking, but with less certainty and more mess. Statistics is about models of reality, based on imperfect and incomplete data. Much of statistics is a “best guess” backed up by probability theory. And statistics is SO important to empowered citizenship. There are wonderful opportunities for discussion in statistics classes. I had a fun experience recently with a bunch of Year 13 Scholarship students in the Waikato. We had collected data from the students, having asked them to interpret a bar chart and a pie chart. There were some outliers in the data and I got them to suggest what we should do about them. There were several good suggestions and I let them discuss for a while then moved on. One asked me what the answer was and I said I really couldn’t say – any one of their suggestions was valid. It was a good teaching and learning moment. Statistics is full of multiple good answers, and often no single, clearly correct, answer.

Learning statistics

My popular Quantitative Methods for Business course was developed on the premise that learning statistics requires repeated exposure to similar analyses of multiple contexts. In the final module, students did many, many hypothesis tests, in the hope that it would gradually fall into place. That is what worked for me, and it did seem to work for many of the students. I think that is not a particularly bad way to learn statistics. But there are possibly better ways.

I do like experiential learning, and statistics is perfect for real life experiences. Perhaps the ideal way to learn statistics is by performing an investigation from start to finish, guided by a knowledgeable tutor. I say perhaps, because I have reservations about whether that is effective use of time. I wrote a blog post previously, suggesting that students need exposure to multiple examples in order to know what in the study is universal and what applies only to that particular context. So perhaps that is why students at school should be doing an investigation each year within a different context.

The nature of understanding

This does beg the question of what it means to learn or to understand anything. I hesitate to claim full understanding. Of anything. Understanding is progressive and multi-faceted and functional. As we use a technique we understand it more, such as hypothesis testing or linear programming. Understanding is progressive. My favourite quote about understanding is from Moore and Cobb, that “Mathematical understanding is not the only understanding.” I do not understand the normal distribution because I can read the Gaussian formula. I understand it from using it, and in a different way from a person who can derive it. In this way my understanding is functional. I have no need to be able to derive the Gaussian function for what I do, and the nature and level of my understanding of the normal distribution, or multiple regression, or bootstrapping is sufficient for me, for now.

Teaching statistics

I believe our StatsLC videos do help students to understand and learn statistics. I have put a lot of work into those explanations, and have received overwhelmingly positive feedback about the videos. However, that is no guarantee, as Khan Academy videos get almost sycophantic praise and I know that there are plenty of examples of poor pedagogy and even error in them. I have recently been reading from “Make it Stick”, which summarises theory based on experimental research on how people learn for recall and retention. I was delighted to find that the method we had happened upon in our little online quizzes was promoted as an effective method of reinforcing learning.

Your thoughts

This has been an enlightening exercise, and I recommend it to anyone teaching in mathematics or statistics. Read the first few chapters of a contemporary text on how to teach mathematics. Dianne Siemon et al, Teaching mathematics: foundations to middle years (2011) did it for me. Then “take a few minutes to write down your own views about the nature of mathematics, mathematics learning and mathematics teaching.” To which I add my own suggestion to think about the nature of statistics or operations research. Who knows what you will find out. Maybe you could put a few of your ideas down in the comments.

 

Mathematics teaching Rockstar – Jo Boaler

Moving around the education sector

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

Educational theory and idea-promoters

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

Jo Boaler – Click here for official information

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

Mathematical Mindsets

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

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

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

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

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

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

Challenges of implementation

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

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

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

Here are some quotes to leave you with:

Mathematical Mindsets Quotes

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

All quotes from

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