The Understandingfirst, Proceduressecond Approach in Action  Three Examples.
If this is your first exposure to the Understandingfirst, Proceduressecond approach (U1, P2) then the articles Why Students Need To Understand The Concept Before We Teach Them The Procedure and Why We Need An Understandingfirst, Proceduressecond Approach To Teaching Mathematics provide some background.
In short, U1, P2 is a simple way of presenting mathematics that has more students spending more time in more lessons understanding the work that is in front of them. Having students understand the maths they are working through is critical  because the opposite  having students NOT understand what they are working through for extended periods of time prevents learning, breeds disengagement and fosters the ‘I hate maths’ mantra of which we are all too familiar. But what does U1, P2 look like in action?
It’s all well and good to talk up an approach to teaching mathematics. What is more important is to gain a sense of what it looks like in action, Below are three examples:
ONE: An Understandingfirst, Proceduressecond approach to Fractions, Decimals and Percentages
Note I chose the three related topics there, not either fractions or decimals or percentages. That is because fractions, decimals and percentages are, in reality, ONE topic based on ONE common concept (part vs whole) and, logically, should be presented as one unit, not three.

Therefore, if you have, for example, four units throughout Years 7, 8 and 9 in which fractions OR decimals OR percentages are presented, you could:
 Keep the unit names unchanged
 In the decimals unit, place a stronger emphasis on decimals but weave in fractions and percentages
 In the percentages unit, place a stronger emphasis on percentages but weave in fractions and decimals
 … and similarly with fractions
Compartmentalising mathematics is a bad idea!
The irony is that the reason we compartmentalise fractions, decimals and percentages into three distinct topics is because we believe that compartmentalisation will make the learning easier for students. And logically, this makes sense. When you teach a beginner how to serve in tennis, you compartmentalise the skill  you teach the toss first and gradually work up to the full action. The same applies to all physical skills. So it makes logical sense to apply this idea to the learning of mathematics.
However, there is a problem when learning mathematics that has been compartmentalised. And it is this: compartmentalisation makes mathematics HARDER to learn, not easier!
Explaining why this is the case is near impossible within the confines of an article. I did my best to explain it in the article Compartmentalisation When Teaching Mathematics Is A Bad Idea. However, Ideally, you need to experience the alternative in order to understand the principle.
However, there is a problem when learning mathematics that has been compartmentalised. And it is this: compartmentalisation makes mathematics HARDER to learn, not easier!
Explaining why this is the case is near impossible within the confines of an article. I did my best to explain it in the article Compartmentalisation When Teaching Mathematics Is A Bad Idea. However, Ideally, you need to experience the alternative in order to understand the principle.
Reducing the compartmentalisation …
We can begin to reduce compartmentalisation within a unified FDP unit by interchanging the references of fractions and decimals and percentages. For example, when referencing 0.5, say ‘a half’ or ‘fifty percent’. And when referencing ‘one quarter’, say zerodecimaltwofive or twentyfive percent. In this way, we reinforce the idea that fractions, decimals and percentages are simply three ways of expressing the same mathematical idea.
More ways to reduce compartmentalisation …
I’ve already stated that we should unify, rather than compartmentalise the units Fractions, Decimals and Percentages; teach all three as one topic but via several units throughout junior high school.
However, it is within each component of any given unit that we should avoid compartmentalisation the most.
Some examples are given below …
However, it is within each component of any given unit that we should avoid compartmentalisation the most.
Some examples are given below …
Simplifying fractions:
To lead students towards the procedure of simplifying fractions, first establish their understanding of the related concepts through activities that require them to use their own thinking, i.e. openended activities based on fractions that simplify easily.
 Ask students to demonstrate and explain, with equipment and/or diagrams, why a given set of equivalent fractions are equivalent.
 Ask students to create fractions equivalent to ½, ⅓, ¾, and so on.
 Ask students to create and answer their own ‘simplify these fractions’ questions.
 Challenge students to establish a procedure for simplifying fractions.
 After  and only after  students have been immersed in the concept of simplifying fractions, teach them (or consolidate their findings of) the procedures for simplifying fractions.
Operating with percentages:
To lead students towards operating with percentages, immerse them in activities that require their own thinking based on situations that they already understand. In other words, design activities based on operating with percentages that don’t allow them to use learned procedures; activities that force them to think.
 Ask students to create and demonstrate questions based on simple percentages of quantities.
 Workshop some related openended questions.
 Challenge students to determine the procedure for more difficult situations.
 After  and only after  students have been immersed in the concept of percentages, teach the procedures based on percentages.
Operating with decimals
Use the above Understandingfirst principles to create activities for working with decimals and for the remaining tasks for the fractionsdecimalspercentages unit.
TWO: An Understandingfirst, Proceduressecond approach to Rightangled Trigonometry
To see the Understandingfirst approach applied to rightangled trigonometry, you can 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 optin.)
The Understandingfirst principles are employed through the Conceptual Approach to Trigonometry, namely, to give students an appreciation of the concepts underpinning rightangled trig, then have them work through exercises that draw on those concepts (rather than working through banks of blocked questions for sine, cosine and tangent questions).
On the surface, this approach may appear similar to the Proceduresfirst equivalent. But in reality, Understandingfirst is fundamentally different. When implemented correctly, the level of understanding and engagement in students tends to be much higher via an Understandingfirst approach.
As a bonus, the approach should save you three to four lessonsworth of time over the course of a tentwelve lesson unit.
There is way too much detail to share here, so again, check out the free tutorial.
The Understandingfirst principles are employed through the Conceptual Approach to Trigonometry, namely, to give students an appreciation of the concepts underpinning rightangled trig, then have them work through exercises that draw on those concepts (rather than working through banks of blocked questions for sine, cosine and tangent questions).
On the surface, this approach may appear similar to the Proceduresfirst equivalent. But in reality, Understandingfirst is fundamentally different. When implemented correctly, the level of understanding and engagement in students tends to be much higher via an Understandingfirst approach.
As a bonus, the approach should save you three to four lessonsworth of time over the course of a tentwelve lesson unit.
There is way too much detail to share here, so again, check out the free tutorial.
THREE: An Understandingfirst, Proceduressecond approach to Coordinate Geometry (straight line graphing)
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) and then learn a procedure to algebraically calculate the equation of a line.
And so on.
Through this (Proceduresfirst) approach, and by the 35 lessons mark, all but the highachieving students tend to have that ‘get me out of here’ look in their eyes. At best, the class is working compliantly, gaining correct answers but with minimal conceptual understanding (and only by following the teacher’s procedural instructions). At worst, students become antagonistic, unhappy, and have little clue of the relationships between ordered pairs, gradients and related equations nor between tables of values and the related equations of the graphs.
However, in a conceptuallybased straightline graphs unit, students become immersed with the concepts of gradient, midpoint and distance in ways they can understand BEFORE they see any formulas. Then, when students are finally exposed to the procedures, students typically respond with, “Oh, but sir, we’ve already been doing this!”
Moreover, through this Understandingfirst approach, almost all students are able to, by their second or third lesson of the unit, determine the equation of a simple straight line simply by looking at the graph! And they are able to sketch the graph of a simple straight line equation without calculating any ordered pairs.
This is an extraordinary achievement for your middleoftheroad students! Being able to spot an equation from its graph and being able to sketch a simpler graph from its equation is empowering  eliciting the response in students ‘This feels like difficult maths, yet I’m finding it easy.’ It is because the process forces students to physically and practically employ the principles of gradient and intercept rather than follow an abstract procedure involving ordered pairs.
All of the activities contained in the comprehensive student handout are conceptuallybased rather than procedurallybased, meaning students are using their own thinking and understanding rather than attempting to replicate procedures from the board.
Note that I’m not providing the student handout here. And for a good reason. I used to present the approach and the handout to teachers during a 30minute slot during inperson workshops. But the approach never received the response it deserved. I had teachers inform me later, “I tried it but it didn’t work!” The reason became obvious. Teachers were trying to implement a unit of work that required a studentcentred approach but using the only approach they were familiar with  the traditional teachercentric model.
What is more important than the resource is a roadmap for implementation. The roadmap needs to:
It is for these reasons that this Conceptual Coordinate Geometry unit requires a detailed roadmap (online course) to walk teachers through the entire Conceptual Coordinate Geometry unit and related pedagogies. The course enables teachers to deal with a wider spread of students and requires the teacher to be a facilitatoroflearning, an approach that many mathematics teachers are initially uncomfortable with yet rapidly adopt when they see it in action.
And so on.
Through this (Proceduresfirst) approach, and by the 35 lessons mark, all but the highachieving students tend to have that ‘get me out of here’ look in their eyes. At best, the class is working compliantly, gaining correct answers but with minimal conceptual understanding (and only by following the teacher’s procedural instructions). At worst, students become antagonistic, unhappy, and have little clue of the relationships between ordered pairs, gradients and related equations nor between tables of values and the related equations of the graphs.
However, in a conceptuallybased straightline graphs unit, students become immersed with the concepts of gradient, midpoint and distance in ways they can understand BEFORE they see any formulas. Then, when students are finally exposed to the procedures, students typically respond with, “Oh, but sir, we’ve already been doing this!”
Moreover, through this Understandingfirst approach, almost all students are able to, by their second or third lesson of the unit, determine the equation of a simple straight line simply by looking at the graph! And they are able to sketch the graph of a simple straight line equation without calculating any ordered pairs.
This is an extraordinary achievement for your middleoftheroad students! Being able to spot an equation from its graph and being able to sketch a simpler graph from its equation is empowering  eliciting the response in students ‘This feels like difficult maths, yet I’m finding it easy.’ It is because the process forces students to physically and practically employ the principles of gradient and intercept rather than follow an abstract procedure involving ordered pairs.
All of the activities contained in the comprehensive student handout are conceptuallybased rather than procedurallybased, meaning students are using their own thinking and understanding rather than attempting to replicate procedures from the board.
Note that I’m not providing the student handout here. And for a good reason. I used to present the approach and the handout to teachers during a 30minute slot during inperson workshops. But the approach never received the response it deserved. I had teachers inform me later, “I tried it but it didn’t work!” The reason became obvious. Teachers were trying to implement a unit of work that required a studentcentred approach but using the only approach they were familiar with  the traditional teachercentric model.
What is more important than the resource is a roadmap for implementation. The roadmap needs to:
 Guide teachers into becoming comfortable with the studentcentred approach that this resource requires.
 Support them in differentiating and catering to the various needs of students working through the resource.
 Give pedagogies for various parts of the unit.
 (And importantly) guide teachers into dealing with the largerthannormal amount of student spread within the unit of work
It is for these reasons that this Conceptual Coordinate Geometry unit requires a detailed roadmap (online course) to walk teachers through the entire Conceptual Coordinate Geometry unit and related pedagogies. The course enables teachers to deal with a wider spread of students and requires the teacher to be a facilitatoroflearning, an approach that many mathematics teachers are initially uncomfortable with yet rapidly adopt when they see it in action.
Learn Implement Share and the Understandingfirst Approach
If you or your department are looking for some quality guidance through the transition to an Understandingfirst, Proceduressecond approach, then you’ll find either of the two online PD options here and here well worth your time.
Related Articles
Why Students Need To Understand The Concept Before We Teach Them The Procedure  here
Why We Need An Understandingfirst, Proceduressecond Mindset When Teaching Mathematics  here
Why Compartmentalising Is A Bad Idea When Teaching Mathematics  here
70% Of Capable Students Are Failing Mathematics. What Can We do?  here
The Case For NOT Teaching Procedures  here
Call to Action
Do you see merit in the Understandingfirst examples given above?
Are you tempted to adopt these principles?
Have you already, in part, adopted them?
Do you have questions?
Your input is welcomed.
Are you tempted to adopt these principles?
Have you already, in part, adopted them?
Do you have questions?
Your input is welcomed.