The message of many popular mathematics and statistics videos is harming people’s perceptions of the nature of these disciplines.

I acknowledge the potential for conflict of interest in this post – critically examining the role of video in learning and teaching mathematics and statistics – when StatsLC has a YouTube channel, and also provides videos through teaching and learning systems.

But I do wonder what message it sends when people like Sal Khan of Khan Academy and Mister Woo are applauded for their well-intentioned, and successful attempts to take a procedural view of mathematics to the masses. Video by its very nature tends towards procedures, and encourages the philosophy that there is one way to do something. Both Khan and Woo, and my personal favourite, Rob Tarrou, all show enthusiasm, inclusion and compassion. And I am sure that many people have been helped by these teachers. In New Zealand various classroom teachers ‘flip” their classrooms, and allow others to benefit from their videos on YouTube. One of the strengths, according to Khan, is that individual students can proceed at their own pace. However Jo Boaler states in her book, Mathematical Mindsets, that “Sadly I have yet to encounter a product that gives individualised opportunities and also teaches mathematics well.”

So what is the problem then? Millions of students love Khan, Woo, ProfRobBob and even Dr Nic. Millions of people also love fast food, and that isn’t good as a total diet.

In my work exploring people’s attitudes to mathematics, I find that many, including maths educators, have a procedural view of mathematics, which fails to unlock the amazing potential of our disciplines.

## Procedural maths

Many people have the conception that to do mathematics is to work out the correct procedure to use in a specific instance and use it correctly in order to get the correct answer. This leads to a nice red tick. (Check mark) That was my view of maths for a very long time. I remember being most upset in my first year of university when the calculus exam was in a different format from the ones I had practised on. I was indignant and feared a C at best, and possibly even a failing grade. I liked the procedural approach. I felt secure using a procedural approach, and when I became a maths teacher, I was pretty much wedded to it. And the thing is, the procedural approach has worked very well for most of the people who are currently high school maths teachers.

I recently read the inspiring “Hidden Figures”, about African American women who had pivotal roles in the development of space travel. For many of them, their introduction into life as a mathematician was as a computer. They did mathematical computations, and speed and accuracy were essential. I wonder how much of today’s curriculum is still aiming to produce computers, when we have electronic devices that can do all of that faster and more accurately.

# Open-ended, lively maths

In parallel to the mass-maths-educators, we have the likes of Jo Boaler and Youcubed, Dan Meyer and Desmos, Bobbie Hunter and Mathematics Inquiry Communities, Marian Small, Tracy Zager, Fawn Nguyen and pretty much the entire Math-Twitter-Blogosphere spreading the message that mathematics is open-ended, exciting and far from procedural. Students work in groups to construct and communicate their ideas. Wrong answers are valued as evidence of thinking and the willingness to take risks. Productive struggle is valued and lessons are designed to get students outside of their comfort zones, but still within their zone of proximal development. Work is collective, rather than individualised, and ability grouping is strongly discouraged.

I find this approach enormously exciting, and believe that it could change the perception of the world towards mathematics.

# The problem of the social contract

Thus I and many teachers are keen to develop a more social constructivist approach to learning mathematics at all levels. However, teachers – especially at high school – run into the problem of the implicit social contract that places the teacher as the owner of the knowledge, who is then required to distribute said knowledge to the students in the class. Students want to get the knowledge, to master the procedure and to find the right answers with as little effort or pain as possible. They are not used to working in groups, and find it threatening to their comfortably boring, procedural vision of maths class.

Some years ago I filled in for a maths teacher for a week at a school for girls from privileged backgrounds. I upset one class of Year 12 students by refusing to use up class time getting them to copy notes from the whiteboard. I figured they had perfectly good textbooks, and were better to spend their time working on examples when I was there to help them learn. Silly me! But I was breaking with what they felt was the correct way for them (and me) to behave in maths class. In fact their indignation at my failure to behave in the way they felt I should, actually did get in the way of their learning.

So who is right?

I guess my working theory is that there is a place for many types of learning and teaching in mathematics. Videos can be helpful to introduce ideas, or to provide another way of explaining things. They can help teachers to expand their own understanding, and develop confidence. Videos can provide well-thought-out images and animations to help students understand and remember concepts. They can do something the teacher cannot. I like to think that our StatsLC videos fit in this category. Talking head or blackboard videos can act as “the kid next door” tutor, who helps a student piece something together.

Just as candy cereal can be only “part of a healthy breakfast”, videos should never be anything more than part of a learning experience.

We also want to think about what kinds of learning we want students to experience. We need our students to be able to communicate, to be creative, to think critically and problem solve and to work collaboratively. These are known as the 4 Cs of 21^{st} Century learning. We don’t actually need people to be able to follow procedures any more. What we need is for people to be able to ask good questions, build models and answer them. I don’t think a procedural approach is going to do that.

The following table summarises some ideas I have about ways of teaching mathematics and statistics.

Procedural approach | Social constructivist approach | |

Main ideas | Maths is about choosing and using procedures correctly | Maths is about exploring ideas and finding patterns |

Strengths | Orderly, structured, safe, cover the material, calm | Exciting, fun, annoying |

Skills valued | Computation, memorisation, speed, accuracy | Creativity, collaboration, communication, critical thinking |

Teaching methods | Demonstration, notes, practice | Open-ended tasks, discussion, exploration |

Grouping | Students work alone or in ability grouping | Students discuss as a whole class or in mixed-ability groups |

Role of teacher | Fount of wisdom, guide, enthusiast, coach. | Another learner, source of help, sometimes annoyingly oblique |

Attitude to mistakes | Mistakes are a sign of failure | Mistakes happen when we learn. |

Challenges | Boredom, regimentation, may not develop resilience. | Can be difficult to tell if learning is taking place, difficult if the teacher is not confident. |

Who succeeds? | People like our current maths teachers | Not sure – hopefully everyone! |

Use of worksheets and textbooks | Important – guide the learning | Occasional use to supplement activities |

Role of videos | Can be central | Support materials |

If we are to have a world of mathematicians, as is our goal as a social enterprise, then we need to move away from a narrow procedural view of mathematics.

I would love to hear your thoughts on this as mathematicians, statisticians, teachers and learners. Do we need to be more careful about the messages our resources such as textbooks and videos give about mathematics and statistics?

Hi! Thank you for your lovely and insightful post (to go along with your other insightful posts). I am of course a member of this choir, so I agree throughout. However, I want to emphasize what you say that procedures are only part — and maybe even a decreasing part — of what we want students to be able to do. But if anyone thinks there is no place at all for procedures, I think that would be a mistake. Two main reasons:

(1) As a veteran of the Math Wars in California, I have happily and smugly advocated reduced emphasis on procedures, and had my head handed me on a platter. Although that’s really only an inconvenience, we should all be really really really careful about anything approaching hubris. Not talking about you, Dr. Nic! But the rest of us best be careful. Vigilate ergo…

(2) So one way to approach the issue is to reframe the role of procedures in math education. Try this on: In the future, today’s students will not have to execute procedures for speed and accuracy, but they may have to _create_ procedures and algorithms for others (e.g., machines) to follow. To that end, perhaps we should lead students to _reflect_ on procedure — and for that they have to learn one, or, ideally two or three so they can compare. The key is that students don’t have to master them: they need only know them briefly, and they never have to get particularly proficient. Ideally, and eventually, we’d love it if they understood why the (multiple) procedures worked!

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Hi Tim

Thank you for your kind words and endorsement. Blogs are often more strongly worded than one might really believe. 😉

I really like your way of thinking about the procedures and algorithms. Concepts and procedures go together, definitely, but my concern is that people see and value only the procedures. And when a person wins a national award for producing procedural maths videos, we are only endorsing that view. My aim is to help people embrace the fullness of mathematics, and one thing I would love to do is make a documentary about maths trauma. I believe much maths trauma is caused by over-emphasis on procedures and speed.

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Are you aware of the PISA data which links maths enjoyment to competency? Procedures do this. Cognitive load theory also predicts that procedural mastery would free our working memory up for more analysis. It is clear that you favour the progressive approach in general but there are large amounts of research showing that explicit instruction works better (I have no doubt you do this) while many other seemingly good ideas like problem solving, group work and inquiry perform poorly. This doesn’t mean they are bad I am simply arguing for Hattie’s prescription of explicit active teaching as the bedrock for most of us to start with and if necessary stay with. (Active here means regular q and a, practice questions etc:).

Teaching procedures without understanding is bad teaching, though under pressure and with little time or limited teacher expertise is largely inevitable. These videos do threaten some of your ideas of good teaching. This is because they are simple, clear and scalable in a way many progressive ideas are not. I have little doubt that you use clear explicit teaching with rich examples, group work and analysis but theses are not what trickel down into most classrooms. Instead poorly structured group work (good group work is tricky) poorly supported investigations (before key ideas are understood or even explained) and a Chinese whispers approach to ideas that creates some monstrous lessons. For most teachers learning how to break down ideas into procedures while still explaining and challenging students is for more achievable considering their limited rescources in terms of time, curriculum design support and CPD.

The progressive Vs traditional argument goes back over a 100years. These ideas are not new on either side.

By the way the calculator argument is a red herring. We don’t need to match the old human computers. We do need enough procedural familiarity to free up our working memory and enough accuracy to check our answers are accurate. Without this calculators become error prone and frustrating.

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Hi. Thank you for your comments. There is much I agree with, especially about poorly structured group work.

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Be mindful that students leave school (hopefully by graduating, though not all will) and enter adulthood where they need to be functional responsible individuals. Collaborative groups will not always be available; sometimes we must simply figure out the things we need all on our own. There is no substite for independent problem solving as a life skill.

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Hi Gerald. I hear what you are saying. However, as our school system tends towards individual activities, I suspect there is never going to be a shortage of opportunities for students to work on their own. My experience in the workplace has been that there is just about always someone else to work with, and good collaborative skills are REALLY necessary.

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Couldn’t get your chart out of my head. Modified the procedural side with my interpretation of an explicit but active (often called traditional teacher). I hope you will see that it is a far more rigorous opponent. Annoyingly it would not let me paste a nice chart, sorry.

Explicit but active approach

(Modern traditional teaching

-yes that’s an oxymoron)

Main ideas Maths is about understanding ideas and recognising patterns

(the difference is the lack of emphasis on exploring and finding until we understand and recognise the ideas we already have)

Strengths Orderly, structured, safe, cover the material, calm + satisfying (and engaging after initial competency has been developed)

Skills valued Computation, memorisation, (not speed), accuracy + the ability to evaluate and analyse (your creativity and critical thinking) but this flows from underling knowledge and is difficult to accurately assess.

Teaching methods Demonstration, notes, practice

+ guided discussion and exploration via modelling.

work primarily alone or in ability grouping + have access to constant peer and tutor support (Ability grouping is independent of approach)

Role of teacher Fount of wisdom, guide, enthusiast, coach. (absolutely)

Attitude to mistakes Mistakes happen when we learn. (No theory advocates using them as a sign of failure. Cognitive Load Theory encourages a high percentage of success)

Challenges Boredom, regimentation, could be taught purely to the test

Who succeeds? Anyone with suitable subject knowledge and appropriate training and support.

Use of worksheets and textbooks + online exercises develops mastery and provide assessment for learning. Limits gaps in understanding.

(often replaced with class calibrated examples on the board to provide more targeted work)

Role of videos Reinforces ideas and provides support out of class. (so support materials)

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