## Why Students Need To Understand What They Are Working On For The Majority Of Lesson Time

Three Ways To Achieve This With Your Mathematics Students

Disclaimer #1: This article is written mostly from a thought perspective. The assumptions are derived from experience and from connecting ideas logically. The ideas are not new. However, an attempt has been made to present them in a fresh way. Despite the lack of supportive data, the arguments are, I believe, valid. All feedback is welcome, for or against.

## All people are capable of K-12 maths?

Cognitive scientist Daniel T Willingham states in his article 'Is it true that some people can't do math Is It True That Some People Can't Do Math, '

I find the statement ‘most people are capable of K-12 mathematics’ somewhat surprising because I've always assumed junior high mathematics to be the standard that most people might be capable of, not K-12 mathematics. Here’s why …

*the vast majority of people are fully capable of learning K-12 mathematics*' (Willingham, 2010, p. 1).I find the statement ‘most people are capable of K-12 mathematics’ somewhat surprising because I've always assumed junior high mathematics to be the standard that most people might be capable of, not K-12 mathematics. Here’s why …

## The ongoing, unintentional maths survey

For four decades, upon discovering I’m a mathematics educator, people have freely shared with me their school maths experience - despite me never having asked for this. Poignantly, the most common response has been something along the lines of “I was hopeless at maths”. At least a hundred people must have summarised for me their school maths experience, and I’d say around 70% have been of the ‘I never got maths at school’ type. Granted, this is far from solid research. Nevertheless, when I share this anecdote with other maths teachers, they all relate to the experience and almost all concur with the 70% figure. At the very least we can assume that a majority of people faired poorly with school maths.

## What types of maths?

What types of mathematics did this majority of responders struggle with? Was it with Yr 12 topics such as locus, matrices, imaginary numbers and integral calculus? No! Clearly, those who claim to have 'bombed out' with school maths never made it to the higher levels. Rather, they failed to cope with simpler topics such as fractions, decimals, percentages, basic algebra, Pythagoras, indices, right-angled trigonometry and most procedures involving formulas. This is junior high mathematics! And when we break junior high maths into concepts, we discover that none of it is very difficult. Therefore, in the past, I had concluded that because a majority of people struggle with junior high mathematics - maths that is not especially difficult - many of these people would therefore not be capable of Yr 12 mathematics. This is why I I'm surprised at Willingham’s statement.

## Houston, we have a problem!

Here's another experienced-based proposition ... I see no reason why a figure similar to our 70% from the unintentional maths survey would not apply to current Australian high school students. In other words, I'm proposing that a majority of current Australian high school students likely struggle with junior high mathematics - in the same ways that the people from our 'unintentional maths survey' tell us they struggle - despite, ackording to Willingham, possessing the capacity to succeed.

Note: This likely applies to students from other countries but given my experience is Australian we will keep the proposition local.

The proposition is derived from twenty-plus years experience - i.e. as a maths teacher and Head of Department plus some consultant work - through which I taught in and/or witnessed over 200 maths classrooms. Again, this is far from solid research. But is the proposition outrageous? I don’t think so. I’m confident most maths teachers with a wide range of experience would be in agreeance. The simple fact is that many - possibly a majority of - high school students struggle with maths.

And when we run this through the lens of ‘most students are capable of learning K-12 mathematics’, then Houston, we have a problem!

Note: This likely applies to students from other countries but given my experience is Australian we will keep the proposition local.

The proposition is derived from twenty-plus years experience - i.e. as a maths teacher and Head of Department plus some consultant work - through which I taught in and/or witnessed over 200 maths classrooms. Again, this is far from solid research. But is the proposition outrageous? I don’t think so. I’m confident most maths teachers with a wide range of experience would be in agreeance. The simple fact is that many - possibly a majority of - high school students struggle with maths.

And when we run this through the lens of ‘most students are capable of learning K-12 mathematics’, then Houston, we have a problem!

## Mathematical Understanding is Key

It doesn’t take an Einstein to conclude that a lack of mathematical understanding by students is, in part, a reason behind the many students who are not coping with maths. However, the phrase ‘lack of mathematical understanding’ is too vague for my liking.

To be more specific, let's put it this way -

And this is despite the extraordinary efforts of hard-working, determined maths teachers trying to make a difference. Sure, there are other factors at play - poor home backgrounds, learned helplessness, low expectations and negative peer environments, to name a few. Nevertheless, if more students understood - for the majority of each lesson - what they were working on, then, surely, fewer students would be falling through the cracks.

To be more specific, let's put it this way -

*too many students are spending too much lesson time not understanding the activities they are working through.*And this is despite the extraordinary efforts of hard-working, determined maths teachers trying to make a difference. Sure, there are other factors at play - poor home backgrounds, learned helplessness, low expectations and negative peer environments, to name a few. Nevertheless, if more students understood - for the majority of each lesson - what they were working on, then, surely, fewer students would be falling through the cracks.

## The conventional Procedures-first mindset explained

Let’s get one thing clear - procedural knowledge is important if students are to be successful with mathematics. Therefore, the teaching of procedures is clearly important.

In my early years of teaching, I was a Procedures-first teacher. Allow me to explain, simply, what I mean by the term ‘Procedures-first teacher’.

As a Procedures-first teacher, I would walk into a classroom having, as my top priority, to teach the next series of procedures of the current unit and to have students successfully gain practice with those procedures. Of course, I tried to make lessons interesting and relevant and I tried to do my best to cater to the varied needs of different students. However, by far, my number one priority was to teach procedures. In fact, I didn’t even consider this to be a priority because the teaching of procedures was what I thought maths teaching was.

Again, from experience, I think it is safe to assume that, although nobody teaches ONLY via a Procedures-first mindset, Procedures-first is likely to be the predominant pedagogy employed in maths classrooms today.

In my early years of teaching, I was a Procedures-first teacher. Allow me to explain, simply, what I mean by the term ‘Procedures-first teacher’.

As a Procedures-first teacher, I would walk into a classroom having, as my top priority, to teach the next series of procedures of the current unit and to have students successfully gain practice with those procedures. Of course, I tried to make lessons interesting and relevant and I tried to do my best to cater to the varied needs of different students. However, by far, my number one priority was to teach procedures. In fact, I didn’t even consider this to be a priority because the teaching of procedures was what I thought maths teaching was.

Again, from experience, I think it is safe to assume that, although nobody teaches ONLY via a Procedures-first mindset, Procedures-first is likely to be the predominant pedagogy employed in maths classrooms today.

## What about understanding?

As a Procedures-first teacher, my view was that understanding followed sufficient practice of the procedures by students. Of course, I knew that for some students, the understanding never came and that for many, their understanding was little more than a vague sense of how the procedures worked.

## The Procedures-first (approach) has, by default, a major flaw

I am not criticising teachers here. What I am suggesting, however, is that despite the many exceptional proceduralists operating in classrooms today, there is, I believe, an issue with the Procedures-first approach.

If we enter maths lessons holding as our #1 aim to teach procedures, then, by default,

This means that a Procedures first approach, by default,

Furthermore, I’m suggesting that

Having an understanding of how a procedure works is clearly important. However, let's not confuse this with an understanding of the concepts upon which the procedure is based. Surely, we want students to acquire conceptual understanding in addition to procedural knowledge. We want students to gain an understanding of the related concepts.

If we enter maths lessons holding as our #1 aim to teach procedures, then, by default,

*we cannot help but have students spending time working with those procedures with a lack of understanding.*This means that a Procedures first approach, by default,

*results in many students NOT understanding what they are working on for significant amounts of lesson time.*Furthermore, I’m suggesting that

*the understanding that follows the correct practice of procedures is mostly an understanding of how the procedures work. Rarely is it a genuine understanding of the concepts underpinning those procedures.*Having an understanding of how a procedure works is clearly important. However, let's not confuse this with an understanding of the concepts upon which the procedure is based. Surely, we want students to acquire conceptual understanding in addition to procedural knowledge. We want students to gain an understanding of the related concepts.

## Understanding as a pre-requisite for learning

The Committee on Developments in the Science of Learning, (2000, p. 8) cited 'learning with understanding (as) one of the hallmarks of the new science of learning'. In other words, understanding needs to be present for learning to occur. This is clearly a blatantly obvious point, but there you have it - it has been researched!

## If students don’t understand what they are learning, they won’t learn it

Of course, when students first tackle something new, there is usually a period of ‘not getting it’. Working through these periods of confusion builds resilience, provided the students' efforts, fairly quickly, lead them to understanding.

However, as we have all observed, when students don’t understand the maths they are working on for extended periods of time, they begin to dislike maths - they switch off, and the door to meaningful learning shuts.

Herein lies the flaw of the Procedures-first mindset. As we have already established, Procedures-first, by default, has (some/many) students spending extended periods of time not understanding the activities they have been given. This leads to disengagement, which, in turn, inhibits learning.

The question to ask then, is How can we have students understand what they are working on for the majority of lesson time?

However, as we have all observed, when students don’t understand the maths they are working on for extended periods of time, they begin to dislike maths - they switch off, and the door to meaningful learning shuts.

Herein lies the flaw of the Procedures-first mindset. As we have already established, Procedures-first, by default, has (some/many) students spending extended periods of time not understanding the activities they have been given. This leads to disengagement, which, in turn, inhibits learning.

The question to ask then, is How can we have students understand what they are working on for the majority of lesson time?

## The Understanding-first (procedures-second) approach

Rather than having, as our first priority, to teach our students procedures, why don’t we prepare students for the coming procedures by presenting students with activities that allow them to understand the very concepts upon which the related procedures are based?

Teachers who walk into their maths classroom with an Understanding-first, (procedures-second) mindset realise the importance of procedures and procedural knowledge. However, they also know that having students understand what they are working on for the majority of lesson time is their most important aim.

Teachers who walk into their maths classroom with an Understanding-first, (procedures-second) mindset realise the importance of procedures and procedural knowledge. However, they also know that having students understand what they are working on for the majority of lesson time is their most important aim.

Disclaimer #2: The terms Understanding-first and Procedures-first are terms I’m deliberately using to help get this message across. I’m not suggesting these are new ideas. In other articles, I refer to the Understanding-first approach as a Hybrid conceptual-procedural approach. In all cases, the terms fit the context. (I’m stating this for anyone reading this through an academic lens.)

## Some principles of an Understanding-first (procedures-second) approach:

The following bullets are common to any Understanding-first approach. More could be added.

Below are three examples of how this works.

- Strive to have students understand the activities they are working through for the majority of lesson time.
- Where possible, use activities that have at least some degree of open-endedness and student-centredness.
- Use activities that allow students to get their hands 'mathematically dirty'. In other words, where possible use activities that enable students to use their own thinking (rather than learned procedures) in ways that have them working deeply with the concepts associated with the soon-to-be-taught procedures.
- Where possible and when appropriate, have students determine the associated procedure.

Below are three examples of how this works.

## ONE: An Understanding-first (procedures-second) approach to Fractions, Decimals and Percentages

Before we start, if we are going to approach a fractions-decimals-percentage unit in a way that will have students understanding what they are working on for the majority of lesson time, then it is best to approach these as one unified topic. After all, the three are based on the same ‘parts-vs-whole’ concept. Compartmentalising and teaching fractions, decimals and percentages as separate units only adds to students’ confusion. But that is the subject of another article.

To lead students towards the procedure of simplifying fractions, first establish their understanding of the related concepts through activities that require students to use their own thinking i.e. open-ended activities based on fractions that simplify easily.

To lead students towards operations with percentages, immerse them in activities that require their own thinking based on situations that they understand.

Use these Understanding-first principles to create activities for the remaining procedures for fractions-decimals-percentages.

**Simplifying fractions:**To lead students towards the procedure of simplifying fractions, first establish their understanding of the related concepts through activities that require students to use their own thinking i.e. open-ended activities based on fractions that simplify easily.

- Ask students to create fractions equivalent to ½, ⅓, ¾, and so on.
- Ask them to demonstrate, with equipment or diagrams, why they are equivalent fractions.
- Ask students to create and answer their own ‘simplify these fractions’ questions.
- Challenge students to establish a procedure for simplifying more complicated fractions.

**Operating with percentages:**To lead students towards operations with percentages, immerse them in activities that require their own thinking based on situations that they understand.

- Ask them to create and demonstrate questions based on simple percentages of quantities.
- Again, challenge students to determine the procedure for more difficult situations.

**More …**Use these Understanding-first principles to create activities for the remaining procedures for fractions-decimals-percentages.

## TWO: An Understanding-first (procedures-second) approach to Right-angled Trigonometry

To see the Understanding-first approach applied to right-angled trigonometry, you are encouraged to work through this free tutorial. It contains a detailed explanation, two videos and three resources for you to use with your students. (You will need to sign up.)

The same principles as above are employed through the Conceptual Approach to Trigonometry, namely, to give students an appreciation of the concepts underpinning right-angled trig, then have them work through exercises that draw on those concepts (rather than 20 questions of the same type).

On the surface, this approach may not seem very different from the Procedures-first equivalent. Yet the level of understanding and engagement is much higher. Vastly more students understand what they are doing for the majority of lesson time and, in my experience, the approach should save you three to four lessons-worth of time over the course of a ten-twelve lesson unit.

There is way too much detail to share here so again, check out the free tutorial.

The same principles as above are employed through the Conceptual Approach to Trigonometry, namely, to give students an appreciation of the concepts underpinning right-angled trig, then have them work through exercises that draw on those concepts (rather than 20 questions of the same type).

On the surface, this approach may not seem very different from the Procedures-first equivalent. Yet the level of understanding and engagement is much higher. Vastly more students understand what they are doing for the majority of lesson time and, in my experience, the approach should save you three to four lessons-worth of time over the course of a ten-twelve lesson unit.

There is way too much detail to share here so again, check out the free tutorial.

## THREE: An Understanding-first (procedures-second) approach to Coordinate Geometry (straight line graphs)

The problem with a conventionally taught Coordinate Geometry unit is that in such a unit students are required to work through a series of seemingly unrelated procedures (usually involving the gradient, midpoint and distance formulas) and then through a procedure to algebraically calculate the equation of a line. And so on.

By this time we are 3-5 lessons into the unit and all but the high-achievers have that ‘get me out of here’ look in their eyes.

At best, the class works compliantly, gaining correct answers but not really understanding the relationship between an ordered pair, a gradient and it’s equation nor between a table of values, the equation of the line and its graph.

At worst, you are looking at your class and staring at the potential for mutiny (!)

Footnote: If it is not already obvious, this was my experience of teaching Coordinate Geometry for many years.

However, in a conceptually-based straight-line graphs unit, students get their hands dirty by dealing with gradient, midpoint, distance-between-two-points formulas and equations of lines - in a way that they can understand - BEFORE they see any formulas.

When students are finally exposed to the procedures based on these formulas, students typically respond with “Oh, but we’ve already been doing this!”

Moreover, through this Understanding-first approach, almost all students are able to, by their second or third lesson, determine the equation of a simple straight line (e.g. y = 2x + 3) simply by looking at the graph.

Also, students are able to sketch the graph (of, for example, y = -3x + 2) without calculating any ordered pairs!

This is made possible because the activities are conceptually-based rather than procedurally-based, meaning students are using their own understanding rather than attempting to replicate a procedure from the board.

Most importantly, students work through the conceptually-based activities in a far more engaged and efficient manner and with greater understanding than when taught using a Procedures-first approach.

If you think the above coordinate geometry example is a bit light on in detail, understand that this one requires an online course to walk teachers through the entire Conceptual Coordinate Geometry unit as well as the related pedagogies.

By this time we are 3-5 lessons into the unit and all but the high-achievers have that ‘get me out of here’ look in their eyes.

At best, the class works compliantly, gaining correct answers but not really understanding the relationship between an ordered pair, a gradient and it’s equation nor between a table of values, the equation of the line and its graph.

At worst, you are looking at your class and staring at the potential for mutiny (!)

Footnote: If it is not already obvious, this was my experience of teaching Coordinate Geometry for many years.

However, in a conceptually-based straight-line graphs unit, students get their hands dirty by dealing with gradient, midpoint, distance-between-two-points formulas and equations of lines - in a way that they can understand - BEFORE they see any formulas.

When students are finally exposed to the procedures based on these formulas, students typically respond with “Oh, but we’ve already been doing this!”

Moreover, through this Understanding-first approach, almost all students are able to, by their second or third lesson, determine the equation of a simple straight line (e.g. y = 2x + 3) simply by looking at the graph.

Also, students are able to sketch the graph (of, for example, y = -3x + 2) without calculating any ordered pairs!

This is made possible because the activities are conceptually-based rather than procedurally-based, meaning students are using their own understanding rather than attempting to replicate a procedure from the board.

Most importantly, students work through the conceptually-based activities in a far more engaged and efficient manner and with greater understanding than when taught using a Procedures-first approach.

If you think the above coordinate geometry example is a bit light on in detail, understand that this one requires an online course to walk teachers through the entire Conceptual Coordinate Geometry unit as well as the related pedagogies.

**Footnote:**I used to share the above unit during a one-hour timeslot of face-to-face PD. However, this was an insufficient walk-through for most teachers to be able to successfully implement the approach with their students. This ‘PD failure’ gave birth to the current online course.Subscribe to receive free tutorials, information and to stay in touch

## A common implementation mistake

The common mistake made by teachers new to an Understanding-first (conceptual) approach when trialling such activities is to fall back on their familiar teacher-directed pedagogies when delivering the activities.

As a result, the implementations often fail, or, unbeknown to the teacher, work less successfully than anticipated. The teacher logically but incorrectly concludes “I tried that, it didn’t really work, this style of activity isn't so great after all.”

However, the reality is, this highly-structured, conceptually-based, student-centred approach requires the adoption of pedagogies based on the teacher being a facilitator-of-learning; an approach that many maths teachers appear to be relatively unfamiliar with.

As a result, the implementations often fail, or, unbeknown to the teacher, work less successfully than anticipated. The teacher logically but incorrectly concludes “I tried that, it didn’t really work, this style of activity isn't so great after all.”

However, the reality is, this highly-structured, conceptually-based, student-centred approach requires the adoption of pedagogies based on the teacher being a facilitator-of-learning; an approach that many maths teachers appear to be relatively unfamiliar with.

## Learn Implement Share and the Understanding-first Approach

If you or your department are looking for some quality guidance through the transition to an Understanding-first (procedures-second) approach, then you’ll find either of the two online options here and here worthy of your time.

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