## There’s an Elephant in the (Maths Class)Room!

Why we need to have students understanding the work we give them for most of each lesson

## Introduction

This article sets out to illuminate ‘The Elephant’, a set of related, unseen conditions that I’m proposing has been lurking in maths classrooms for, it would seem, a very long time. Our Elephant can be described as follows:

A significant portion of school maths students spend considerable lesson time NOT understanding the work they encounter in lessons. The main reason for the extended periods of lesson time students spend ‘not getting maths’ is that maths is mostly taught via a 'Procedures-first, (understanding-second)' approach which, by default, requires students to spend significant lesson time practising procedures - while not understanding - in order to gain some understanding. This ‘working without understanding’ causes students to disengage, closing the door to learning. Meanwhile, their teachers (the vast majority), through no fault of their own, are unable to relate to what it’s like for students who ‘don’t get maths’. This is because the vast majority of us have always understood maths - we have never had the experience of being a student who ‘doesn’t get maths’. This helps to prevent the status quo from changing, and so The Elephant remains.

A significant portion of school maths students spend considerable lesson time NOT understanding the work they encounter in lessons. The main reason for the extended periods of lesson time students spend ‘not getting maths’ is that maths is mostly taught via a 'Procedures-first, (understanding-second)' approach which, by default, requires students to spend significant lesson time practising procedures - while not understanding - in order to gain some understanding. This ‘working without understanding’ causes students to disengage, closing the door to learning. Meanwhile, their teachers (the vast majority), through no fault of their own, are unable to relate to what it’s like for students who ‘don’t get maths’. This is because the vast majority of us have always understood maths - we have never had the experience of being a student who ‘doesn’t get maths’. This helps to prevent the status quo from changing, and so The Elephant remains.

## The Unintentional Survey

We maths teachers have been conducting a survey - unintentionally - since we first started our teacher training. The unintentional survey could be called ‘Ascertaining the extent to which adults coped with maths when they were at school’. Here’s how it works. We find ourselves in conversation with someone new who, upon discovering we are a mathematics teacher, without invitation, tells us how they fared with school mathematics. Poignantly, the most common response is something along the lines of “I was hopeless at maths”.

I’ve been unintentionally conducting this survey for around four decades. Over this time, my intel suggests that around 70% of the adult population speak negatively of their school maths experience. Statements such as “I bombed out”, “I never got maths”, “I hated maths lessons” as well as variations on this theme feature prominently.

When I tell this story to mathematics teachers, as I often do, many suggest the percentage to be nearer 80%! For the purpose of this article, let’s call it 70% - it might be a bit less, or a bit more.

I’ve been unintentionally conducting this survey for around four decades. Over this time, my intel suggests that around 70% of the adult population speak negatively of their school maths experience. Statements such as “I bombed out”, “I never got maths”, “I hated maths lessons” as well as variations on this theme feature prominently.

When I tell this story to mathematics teachers, as I often do, many suggest the percentage to be nearer 80%! For the purpose of this article, let’s call it 70% - it might be a bit less, or a bit more.

## What types of maths?

What sort of mathematics is The 70% claiming they struggle with? Locus? Matrices? Imaginary numbers? Integral calculus? Not likely! Our 70% never make it to the higher levels of school mathematics! Our 70% struggled with fractions, decimals and percentages. They failed to ‘handle’ basic algebra. They never ‘got’ basic trigonometry. Pythagoras’ Theorem caused headaches. They found measurement formulas confusing. Essentially, The 70% ‘didn’t get’ junior high mathematics.

The next question to ask, then, is this: Is the average human capable of handling junior high school mathematics? Did The 70% of people, when they were at school, possess the ability to succeed with basic algebra, right-angled trigonometry, Pythagoras’ Theorem and the use of simple formulas - and walk away from school believing “I was OK at maths”?

Last time I checked, upon breaking these topics down into concepts, junior high mathematics is not all that difficult to understand!

Cognitive scientist Daniel T Willingham reinforces this in his article 'Is it true that some people can't do math' by stating that in actuality, 'the vast majority of people are fully capable of learning K-12 mathematics' (Willingham, 2010, p. 1).

To me, the surprise here is that Willingham is suggesting most of us are capable of K-12 mathematics; I was suggesting the cut off was junior high mathematics!

The next question to ask, then, is this: Is the average human capable of handling junior high school mathematics? Did The 70% of people, when they were at school, possess the ability to succeed with basic algebra, right-angled trigonometry, Pythagoras’ Theorem and the use of simple formulas - and walk away from school believing “I was OK at maths”?

Last time I checked, upon breaking these topics down into concepts, junior high mathematics is not all that difficult to understand!

Cognitive scientist Daniel T Willingham reinforces this in his article 'Is it true that some people can't do math' by stating that in actuality, 'the vast majority of people are fully capable of learning K-12 mathematics' (Willingham, 2010, p. 1).

To me, the surprise here is that Willingham is suggesting most of us are capable of K-12 mathematics; I was suggesting the cut off was junior high mathematics!

## Houston, we have a problem!

Following on from Willingham’s claim, if the vast majority of humans are fully capable of learning K-12 mathematics, yet most adults are stating that they never 'got Maths' at school then... Houston, we have a problem!

## Mathematical Understanding is Key

It seems obvious to me that mathematical understanding - or rather, the lack of it - is a prominent reason behind the majority of adults insisting they never succeeded in maths despite their capabilities. It doesn't take an Einstein to deduce that the vast majority of those who love maths, who succeed with maths, also understand most of the mathematics that they undertake and that the reverse is also true. In other words, if you didn't 'get' maths at school - if you didn't understand it - then you will have disengage and will now be part of The 70%.

While writing this article, a significant effort was made searching for research looking into the portion of students who struggle with school maths, i.e. perform poorly and consider themselves to cope poorly with the subject. Alas, we didn't find any. So we will resort to using some common-sense logic instead.

Had we discovered surveys of high school students from, say, Australia, the UK and the US to ascertain the percentage of current students who struggle with maths, I’d be very surprised, based on experience and anecdotal evidence, if the findings would have arrived at a number much different to our 70-ish% from our unintentional adult survey.

Regardless of our predictions, we can safely say that the group of students who struggle with school mathematics, whether they be the current cohort or the one who are now adults, is a high portion of the total. For simplicity, let’s assume that the 70% includes our group of school maths strugglers - both the adults and current students.

While writing this article, a significant effort was made searching for research looking into the portion of students who struggle with school maths, i.e. perform poorly and consider themselves to cope poorly with the subject. Alas, we didn't find any. So we will resort to using some common-sense logic instead.

Had we discovered surveys of high school students from, say, Australia, the UK and the US to ascertain the percentage of current students who struggle with maths, I’d be very surprised, based on experience and anecdotal evidence, if the findings would have arrived at a number much different to our 70-ish% from our unintentional adult survey.

Regardless of our predictions, we can safely say that the group of students who struggle with school mathematics, whether they be the current cohort or the one who are now adults, is a high portion of the total. For simplicity, let’s assume that the 70% includes our group of school maths strugglers - both the adults and current students.

## Only people who struggled with school maths are able to appreciate what it’s really like for students who (also) struggle with maths. By default, this discounts the vast majority of maths teachers!

We high school maths teachers, due to no fault of our own, are the least equipped people to stand in the shoes of students who do not understand the work we give them in lessons. This is because the vast majority of us understood maths at school - we never experienced extended periods of time, not understanding the maths we were given.

This point is worth labouring some more - the fact is that most of us cannot truly appreciate what the experience is like for those (many) students who lack understanding of the work we give them. It’s easy for us to agree with the literature in support of the importance of students understanding mathematics. However, many of us do not fully appreciate just how important it is that we teach in a way that has students understanding, for the majority of lesson time, the work we give them to do. And the reason we do not fully appreciate this is because we don’t know what it is like to spend extended periods of lesson time working through tasks that we do not understand.

This point is worth labouring some more - the fact is that most of us cannot truly appreciate what the experience is like for those (many) students who lack understanding of the work we give them. It’s easy for us to agree with the literature in support of the importance of students understanding mathematics. However, many of us do not fully appreciate just how important it is that we teach in a way that has students understanding, for the majority of lesson time, the work we give them to do. And the reason we do not fully appreciate this is because we don’t know what it is like to spend extended periods of lesson time working through tasks that we do not understand.

## The conventional ‘Procedures First (understanding second)' mindset explained

Let’s get one thing clear - the teaching of procedures is important for students to be successful with mathematics. In my early years of teaching, I was a ‘Procedures-first’ teacher. Allow me to explain, in simple terms, what I think it means to be a Procedures-first teacher before explaining why the teaching of procedures needs to be our second priority rather than our first.

As a Procedures-first teacher, I would walk into a classroom having, as my top priority, to teach the next few procedures of the current unit and to have students successfully gain practice with those procedures. Of course, I’d try to make the lesson interesting, and I’d handle students’ questions as best I could. However, by far, my number one priority was to teach those procedures. In fact, I didn’t even consider this a priority because the teaching or procedures was what I thought maths teaching was!

As a Procedures-first teacher, I would walk into a classroom having, as my top priority, to teach the next few procedures of the current unit and to have students successfully gain practice with those procedures. Of course, I’d try to make the lesson interesting, and I’d handle students’ questions as best I could. However, by far, my number one priority was to teach those procedures. In fact, I didn’t even consider this a priority because the teaching or procedures was what I thought maths teaching was!

## What about understanding?

My view was that understanding followed sufficient practice of procedures. Of course, however, I knew that for some students, the understanding never came, and for many, the understanding was little more than a vague sense of how the procedures worked.

I’ll refer to the approach described above as ‘Procedures-first (understanding-second)’. Having been inside some 200 maths classrooms and having observed numerous maths lessons, I’m confident that the Procedures-first, Understanding-second approach is still the dominant one used in the majority of maths classrooms today.

I’ll refer to the approach described above as ‘Procedures-first (understanding-second)’. Having been inside some 200 maths classrooms and having observed numerous maths lessons, I’m confident that the Procedures-first, Understanding-second approach is still the dominant one used in the majority of maths classrooms today.

## The Procedures First (understanding second) approach has, by default, a major flaw

Despite the many exceptional proceduralists operating in classrooms today, there is an issue with the ‘Procedures First, Understanding Second’ approach. By default, if we enter maths lessons holding as our #1 aim to teach procedures then we cannot help but have students NOT understand what they are working on for significant amounts of lesson time because, at best, the Procedures-first approach requires students to practise repeatedly (while they don’t understand) in order to gain understanding.

As has already been mentioned, the understanding that results from the practice of procedures is mostly ‘an understanding of how procedures work’. Yet what we need is for students to gain an understanding of the underlying concepts upon which the procedures are based. And for the record, the understanding of a concept (the experience of an ‘aha moment’) is rarely acquired through the practice of procedures.

However, as we all know through experience, when students do not understand what they are working on, they begin to dislike maths - they turn off. Herein lies the default flaw of the Procedures first (understanding second) approach.

As has already been mentioned, the understanding that results from the practice of procedures is mostly ‘an understanding of how procedures work’. Yet what we need is for students to gain an understanding of the underlying concepts upon which the procedures are based. And for the record, the understanding of a concept (the experience of an ‘aha moment’) is rarely acquired through the practice of procedures.

However, as we all know through experience, when students do not understand what they are working on, they begin to dislike maths - they turn off. Herein lies the default flaw of the Procedures first (understanding second) approach.

## The Elephant in the (maths class)room:

Our Elephant comprises a few related facets. It can be described as follows:

- We have a majority of maths students spending considerable lesson time struggling and NOT understanding the work they encounter in lessons,
- who disengage due to this struggle,
- who are mostly taught via a Procedures-first (understanding-second) approach,
- which, by default, cannot help but have most students NOT understanding the work they encounter for significant chunks of lesson time,
- and taught by teachers, the majority of who, through no fault of their own, cannot relate to what it’s like for students who struggle to understand maths.

## Another way of putting this is as follows:

Students disengage when they don’t understand. Therefore they need to be taught in a way that allows them - for the majority of lesson time - to understand what they are doing. Yet most students are taught via an approach which, by design, has them NOT understanding for the majority of lesson time (because significant practice is required before understanding is gained). If teachers could relate to the experience of students who ‘don’t get maths’ then there would be an impetus for change. However, because most of us never experienced extended difficulties with maths, we are blind to the reality described, and The Elephant remains hidden.

## Addressing The Elephant

If we are to address The Elephant, then we need to address two key questions:

- For those of us who did not struggle with school mathematics (and are therefore inhibited in our ability to relate to students who ‘don’t get maths’), what can we do?
- How can we have students understand what they are working on for the majority of lesson time?

## ONE: What to do if we have never struggled with mathematics?

Call me an extreme optimist, but I am hoping that some teachers, after reading this article, will be inspired to look ‘with new eyes’ at the reality of those students who are not coping with maths.

However, if readers need more motivation, then here are some suggestions:

Welcome to the reality of many school maths students today.

Assuming you have never struggled with maths, you’ll find the above tasks challenging because you are being asked to experience something you have never experienced - a bit like asking a native Central Australian who has never been to the coast to describe the ocean. Nevertheless, you should be able to glimpse a snippet of the struggle, perhaps ‘nightmare’ that is school mathematics for a great many students. Once you begin to appreciate the experience of students‘ not getting maths’ you’ll hopefully want to switch to an approach that aims to have students understand what they are working on for the majority of most lessons. Which leads us into the next point ...

However, if readers need more motivation, then here are some suggestions:

- Spend a minute or so thinking about place value. Unpack your understanding of place value. Then, try to imagine NOT having this understanding. View, for example, 539.71 as simply a string of digits - you roughly have an appreciation of the number 500 but don’t fully recognise it in this string. And you have absolutely no idea what the ‘decimal 71’ means. Now, with this imagined lack of understanding, answer any yr 7-8 number problem and see how it feels.

- Do the same for any fraction-based question. Imagine you have no idea what a numerator is, no idea what role the denominator plays, no idea, for example, what 2/7 of 21 means. You don’t understand cancellation nor how nor why it works, nor why anyone would bother cancelling a fraction in the first place. Nor what the point is of an equivalent fraction. I could go on. Embrace your newfound, imaginary lack of understanding and tackle some fraction-based questions. See what you find. See how it feels.

Welcome to the reality of many school maths students today.

Assuming you have never struggled with maths, you’ll find the above tasks challenging because you are being asked to experience something you have never experienced - a bit like asking a native Central Australian who has never been to the coast to describe the ocean. Nevertheless, you should be able to glimpse a snippet of the struggle, perhaps ‘nightmare’ that is school mathematics for a great many students. Once you begin to appreciate the experience of students‘ not getting maths’ you’ll hopefully want to switch to an approach that aims to have students understand what they are working on for the majority of most lessons. Which leads us into the next point ...

## TWO: Striving to have students understand what they are working on for the majority of lesson time

Teachers who walk into their maths classroom with an Understanding First, (Procedures Second) mindset realise the importance of procedures. However, they also know that having students understand what they are working on for the majority of lesson time is their most important aim. This can be achieved, for example, in a fractions-decimals-percentage unit (and why would we teach these three aspects of the same concept separately?) by giving students activities that are based on the underlying concepts; activities that allow students to use their own thinking. Once they have consolidated their understanding, then teach the procedures for working with more complicated questions.

As another example, in a conceptually-based straight-line graphs unit students can get ‘their hands dirty’ dealing with gradient, midpoint, distance and equations of lines BEFORE they see any formulas. This approach, when implemented well, is ‘ridiculously’ more successful at engaging students and having them - the students who usually struggle with this topic - understand what they are doing.

As another example, in a conceptually-based straight-line graphs unit students can get ‘their hands dirty’ dealing with gradient, midpoint, distance and equations of lines BEFORE they see any formulas. This approach, when implemented well, is ‘ridiculously’ more successful at engaging students and having them - the students who usually struggle with this topic - understand what they are doing.

## A glimpse into an Understanding First (procedures-second) approach

Transitioning from a Procedures-first (understanding-second) approach to an Understanding-first (procedures-second) approach requires a paradigm shift in thinking for most teachers. At risk of sounding like a marketing insert - and, of course, it is (!) - the sole reason for developing and running courses like this one and this one (Conceptual Coordinate Geometry) is to walk teachers through the paradigm shift from a Procedures-first approach to an Understanding-first approach. This transition needs mapping out for the simple reason that implementing conceptually-based activities with a procedures-based mindset simply will not work. This is a mistake many procedurally-based teachers make when first adopting conceptually-based activities. The pedagogies required for a successful Understanding-first approach are fundamentally different from those required by the familiar Procedures-first approach.

An Understanding-first (procedures-second) approach requires the teacher to become a facilitator of learning. Activities need to be highly structured yet somewhat student-centred. Therefore, teachers need support in creating and implementing units that are designed to generate initial understanding. Teachers also benefit from guidance with fostering collaborative learning, from strategies that help create a ‘need to learn’ in students and from support in dealing with a wider spread of students (resulting from a student-centred approach).

To explore in detail what an Understanding-first (procedures-second) approach looks like would take another ten+ articles like this one and a series of quality videos (otherwise known as an online course). However, to gain a glimpse of what an Understanding-first approach looks like check out this article which includes an understanding first introduction to a trigonometry unit.

Also, this article will shed some light on an Understanding-first (procedures-second) approach.

An Understanding-first (procedures-second) approach requires the teacher to become a facilitator of learning. Activities need to be highly structured yet somewhat student-centred. Therefore, teachers need support in creating and implementing units that are designed to generate initial understanding. Teachers also benefit from guidance with fostering collaborative learning, from strategies that help create a ‘need to learn’ in students and from support in dealing with a wider spread of students (resulting from a student-centred approach).

To explore in detail what an Understanding-first (procedures-second) approach looks like would take another ten+ articles like this one and a series of quality videos (otherwise known as an online course). However, to gain a glimpse of what an Understanding-first approach looks like check out this article which includes an understanding first introduction to a trigonometry unit.

Also, this article will shed some light on an Understanding-first (procedures-second) approach.

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## Call to Action

Has this article helped to shine a light on The Elephant for you? Or, have you always known it was there and actively use some strategies to combat the syndrome? Please feel free to share and comment below.

Otherwise, if you’d like to read more about this Understanding-first (procedures second) approach, check out this article.

Finally, you’ll find some useful strategies that fit into an Understanding-first approach on the newly revamped blog.

Otherwise, if you’d like to read more about this Understanding-first (procedures second) approach, check out this article.

Finally, you’ll find some useful strategies that fit into an Understanding-first approach on the newly revamped blog.