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

And 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. I’m not suggesting the ideas are new. However, an attempt has been made to present them in a fresh way. Despite the lack of data used, 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', '

I found the ‘most people are capable of K-12 mathematics’ statement somewhat surprising. In the past, I always assumed that the ‘Most people are capable of’ statement applied only to junior high mathematics. Here’s why …

*the vast majority of people are fully capable of learning K-12 mathematics*' (Willingham, 2010, p. 1).I found the ‘most people are capable of K-12 mathematics’ statement somewhat surprising. In the past, I always assumed that the ‘Most people are capable of’ statement applied only to junior high mathematics. Here’s why …

## The ongoing, unintentional maths survey

For four decades, upon discovering I’m a mathematics educator, people have shared with me, unasked, their school maths experience. 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% of them have been of the ‘I never got maths at school’ type. Granted, this is not hard research. Nevertheless, when I share this anecdote with other maths teachers, they all relate to the experience and almost all agree 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?

To explain why I am surprised by Willingham’s ‘Most people can cope with Yr 12 Mathematics’ statement, we’ll address the question - 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? I don’t believe so. I have reason to be certain that those who claim to have 'bombed' with school maths never made it to the higher levels of Yr 12 maths. Rather, they failed to cope with 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 it is not 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. Hence my surprise at Willingham’s statement.

## Houston, we have a problem!

Allow me to make another experienced-based proposition; that the ‘results’ of the above unintentional maths survey also apply to current Australian high school students. In other words, a majority of Australian high school students struggle with junior high mathematics despite possessing the capacity to succeed. (This likely applies to students from other countries but given my experience is Australian we will keep the proposition local.)

My 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 not hard research, but is the proposition outrageous? I don’t think so. I’m confident most maths teachers with a wide range of experience would agree. 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!”

My 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 not hard research, but is the proposition outrageous? I don’t think so. I’m confident most maths teachers with a wide range of experience would agree. 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 part of the reason behind the reality that many students are not coping with maths is a lack of understanding. However, the phrase ‘lack of understanding’ is too vague for my liking. To be more specific, let us 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 for students to possess in order 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 for 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 for 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 by students of procedures. Of course, however, 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

In case you are wondering, I am not criticising teachers. 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, we cannot help but have students spending time working with those procedures in the dark, so to speak. 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. Having an understanding of how a procedure works is clearly important. However, we need to 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 concepts underpinning any given procedure.

Furthermore, I’m suggesting that the understanding that follows the correct practice of procedures is mostly an understanding of how the procedures work. Having an understanding of how a procedure works is clearly important. However, we need to 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 concepts underpinning any given procedure.

## 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, learning and understanding are closely linked.

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

Of course, when we first tackle something new, there is usually a period of ‘not getting it’. Working through these periods of confusion builds resilience, provided our efforts, fairly quickly, lead us 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 default flaw of the Procedures-first mindset. As we have already established, Procedures-first, by default, has (at least some) 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?’

Herein lies the default flaw of the Procedures-first mindset. As we have already established, Procedures-first, by default, has (at least some) 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 firstly striving to have students understand the concepts upon which those procedures are based?

Teachers who walk into their maths classroom with an Understanding-first, (procedures-second) mindset realise the importance of procedures, realise the importance of 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, realise the importance of 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 make a point. 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 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.

- Strive to have students understand the activities they are working through for the majority of lesson time.
- Use activities that are mostly open-ended and student-centred.
- Ensure the activities enable students to work with the concepts via simple situations that allow them to use their own thinking rather than learned procedures.
- Have students determine the associated procedure (if appropriate).

## 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 why not 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.

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.

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 will provide you with three resources for you to use with your students. You will need to sign up first.

The same principles are employed through the Conceptual Approach to Trig - 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 engagement is way higher. Vastly more students understand what they are doing for the majority of lesson time and, in my experience, three to four lessons-worth of time can be saved over a ten-twelve lesson unit. There is way too much detail to share here so again, check out the free tutorial.

The same principles are employed through the Conceptual Approach to Trig - 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 engagement is way higher. Vastly more students understand what they are doing for the majority of lesson time and, in my experience, three to four lessons-worth of time can be saved over 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 a conventionally taught Coordinate Geometry 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 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 they are finally exposed to procedures based on these formulas, the response is “Oh, but we’ve already been doing this!” Moreover, and in my experience, 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. In addition, 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. More importantly, students work through the conceptually-based activities in a far more engaged and efficient manner 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.

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 they are finally exposed to procedures based on these formulas, the response is “Oh, but we’ve already been doing this!” Moreover, and in my experience, 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. In addition, 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. More importantly, students work through the conceptually-based activities in a far more engaged and efficient manner 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.

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## 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 they should. The teacher then concludes “I tried that, but it didn’t really work.” 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 are still 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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