Four Examples of the Understanding-first,
Procedures-second Approach in Action
Updated 23.10.21
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If this is your first exposure to the Understanding-first, Procedures-second approach (U-1, P-2) then the articles Why Students Need To Understand The Concept Before We Teach Them The Procedure and Why We Need An Understanding-first, Procedures-second Approach To Teaching Mathematics provide some background.
In short, U-1, P-2 is a simple way of presenting mathematics that has more students spending more time in more lessons understanding the mathematics work they are tasked to complete. 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 that we are all too familiar with from too many students. But what does U-1, P-2 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 four examples:
ONE: An Understanding-first, Procedures-second 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.
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Therefore, if you have, for example, several 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 compartmentalising will make the learning easier for students. And logically, this makes sense. When you teach a beginner how to serve in tennis, for example, 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 part-part-whole pedagogy to the learning of mathematics.
However, learning mathematics that has been compartmentalised is problematic. And the problem is this: compartmentalisation makes mathematics MORE DIFFICULT to learn, not less!
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. Ideally, you need to experience the alternative in order to understand the principle.
However, learning mathematics that has been compartmentalised is problematic. And the problem is this: compartmentalisation makes mathematics MORE DIFFICULT to learn, not less!
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. 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 F-D-P 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 zero-decimal-two-five or twenty-five 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 compartmentalising the most.
Some examples are given below …
However, it is within each component of any given unit that we should avoid compartmentalising 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 students to use their own thinking, i.e. open-ended 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; i.e. activities that force them to think.
- Ask students to create and demonstrate questions based on simple percentages of quantities.
- Workshop some related open-ended 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 Understanding-first principles to create activities for working with decimals and for the remaining tasks for the fractions-decimals-percentages unit.
DISCLAIMER - The next three examples refer to courses.
Below are the summaries of two mini courses and a short course comprising a total of five-plus hours of content. These thee courses each unpack the Understanding-first, Procedures-second approach within a specific context - in great detail. In this article the best I can do is give a brief overview of the five-plus hours of course content.
For those who are wary of articles that promote courses, it would be ridiculous of me to give these brief overviews without mentioning the courses.
For those who are wary of articles that promote courses, it would be ridiculous of me to give these brief overviews without mentioning the courses.
TWO: An Understanding-first, Procedures-second approach to Right-angled Trigonometry
The mini course An Understanding-1st Approach To Right-Angled Trigonometry unpacks an approach that gives students an appreciation of the concepts underpinning right-angled trig, then has them work through exercises that draw on those concepts. And this is in preference to having students work through banks of blocked questions for sine, cosine and tangent questions.
On the surface, this approach may appear similar to the Procedures-first equivalent, however, in reality, Understanding-first is fundamentally different. When implemented correctly, the level of understanding and engagement in students tends to be much higher via the Understanding-first approach.
The mini course contains a detailed explanation, two videos and three excellent resources to use with your students.
Below are highlighted some of the Understanding-first, Procedures-second aspects of this approach:
Aspect #1: The Conceptual Introduction.
This features a pre-made GeoGebra file, however, other ways could be used to impart the following principles.
Students are shown how to ascertain whether each trig question is a sine, cosine or tangent situation.
At the same time as showing students how to ascertain the trig ratio type of questions, students are taught the trigonometric sentence.
The interleaved worksheet saves time (no need to draw diagrams). Importantly, because the questions are mixed the students' focus remains on the broader trigonometry picture rather than just sine or cos or tan questions.
and is provided. To understand interleaved questions, you need to understand Blocked Practice.
Aspect #4: Greater self-direction.
The main thing we should be striving for in students is agency. Once students commence the interleaved sheets students experience more autonomy than usual and this helps to breed agency in students.
This approach should save three to four lessons-worth of time over the course of a ten-twelve lesson unit.
The mini course comes with some free preview pages. This allows you to preview the course for free (sign in required).
On the surface, this approach may appear similar to the Procedures-first equivalent, however, in reality, Understanding-first is fundamentally different. When implemented correctly, the level of understanding and engagement in students tends to be much higher via the Understanding-first approach.
The mini course contains a detailed explanation, two videos and three excellent resources to use with your students.
Below are highlighted some of the Understanding-first, Procedures-second aspects of this approach:
Aspect #1: The Conceptual Introduction.
This features a pre-made GeoGebra file, however, other ways could be used to impart the following principles.
- That right-angled trigonometry is essentially based on similar triangles.
- That there are three pairs of sides we can compare.
- That given the length of one side we can calculate the other.
- Students are focused on the WHOLE trigonometric principle, not just ONE of the ratios.
Students are shown how to ascertain whether each trig question is a sine, cosine or tangent situation.
- Note: This is a fundamental key to the U-1, P-2 approach.
- Requiring students to ascertain the trigonometric type places their focus conceptually on the entire (right-angle) trigonometric landscape rather than the minutia of following one procedure for one trigonometric ratio at a time.
- The PowerPoint resource can be instrumental in assisting students to ascertain each question's trigonometric ratio type.
At the same time as showing students how to ascertain the trig ratio type of questions, students are taught the trigonometric sentence.
- This procedure is taught explicitly.
- However, it is taught as a trigonometric principle rather than as a sine procedure OR a cosine or tan procedure.
- Note: The name Trigonometric Sentence fits with the idea of focusing on the whole rather than on just one part at any time.
The interleaved worksheet saves time (no need to draw diagrams). Importantly, because the questions are mixed the students' focus remains on the broader trigonometry picture rather than just sine or cos or tan questions.
and is provided. To understand interleaved questions, you need to understand Blocked Practice.
Aspect #4: Greater self-direction.
The main thing we should be striving for in students is agency. Once students commence the interleaved sheets students experience more autonomy than usual and this helps to breed agency in students.
This approach should save three to four lessons-worth of time over the course of a ten-twelve lesson unit.
The mini course comes with some free preview pages. This allows you to preview the course for free (sign in required).
THREE: An Understanding-first, Procedures-second 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 (Procedures-first) approach, and by the 3-5 lessons mark, all but the high-achieving 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 conceptually-based straight-line 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 Understanding-first 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 middle-of-the-road students! Being able to determine an equation simply by looking at its graph, and being able to sketch a simpler graph from its equation is empowering. It elicits the response in students ‘This feels like difficult maths, yet I’m finding it easy.’ It's because the process immerses students in 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 conceptually-based rather than procedurally-based, 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 30-minute slot during in-person 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 an Understanding-first approach but were using the only approach they were familiar with - the traditional Procedures-first approach.
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 facilitator-of-learning, an approach that many mathematics teachers are initially uncomfortable with yet rapidly adopt when they see it in action.
Through this (Procedures-first) approach, and by the 3-5 lessons mark, all but the high-achieving 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 conceptually-based straight-line 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 Understanding-first 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 middle-of-the-road students! Being able to determine an equation simply by looking at its graph, and being able to sketch a simpler graph from its equation is empowering. It elicits the response in students ‘This feels like difficult maths, yet I’m finding it easy.’ It's because the process immerses students in 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 conceptually-based rather than procedurally-based, 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 30-minute slot during in-person 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 an Understanding-first approach but were using the only approach they were familiar with - the traditional Procedures-first approach.
What is more important than the resource is a roadmap for implementation. The roadmap needs to:
- Guide teachers into becoming comfortable with the student-centred 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 larger-than-normal 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 facilitator-of-learning, an approach that many mathematics teachers are initially uncomfortable with yet rapidly adopt when they see it in action.
FOUR: The Workshopping of Concept-Specific Open Ended Questions
The Workshopping of Concept-Specific Open Ended Questions is one of the the most effective whole-class methods of bringing understanding of specific concepts to students. However, it is a unique approach. I used to have a comprehensive page of information in one of the courses, including a detailed 7-minute video. However, upon reading the implementation reports it was obvious that most teachers had not grasped some of the key principles of the approach. What they implemented was good, but quite different to the approach taught and with less potential for change. This is why I created the short course on workshopping concept-specific questions.
What are concept-specific Open Ended Questions (OEQs)?
First, here is an example of an Open Ended Investigation:
Now, an example of a concept-specific OEQ.
Let's say we are teaching perimeter and area. Examples of concept-specific OEQs we might use are:
The Workshopping of Concept-Specific Open Ended Questions explained
The approach features two equally important aspects:
In a nutshell, the workshopping process looks like this:
Key aspects of the approach and why it can be so effective
The Workshopping of Concept-Specific Open Ended Questions:
The crux of the approach is:
Disclaimers - The Workshopping of Concept-Specific OEQs:
Get a free preview of the course by accessing the free preview pages (sign in required).
What are concept-specific Open Ended Questions (OEQs)?
First, here is an example of an Open Ended Investigation:
- Design a water tank for the town of Wangilendra - population 2500 - if, on average, each person consumes 30 litres of water and the tank needs to hold water for a period of time based on the rainfall charts provided.
Now, an example of a concept-specific OEQ.
Let's say we are teaching perimeter and area. Examples of concept-specific OEQs we might use are:
- Sketch several rectangles with a perimeter of 24 units.
- Now use sides with lengths containing decimals.
- Sketch several isosceles triangles with a perimeter of 14 units.
- Sketch several scalene triangles with a perimeter of 14 units.
- Sketch a composite figure (rectangle and triangle) with a perimeter of 30 units.
- Repeat with similar questions but based on area
The Workshopping of Concept-Specific Open Ended Questions explained
The approach features two equally important aspects:
- The first is the concept-specific OEQ.
- The second is the workshopping of the concept-specific OEQ.
In a nutshell, the workshopping process looks like this:
- Pose the concept-specific OEQ ... have students work for a short time ... take a sample response ... workshop the response ... more student work ... more sample responses ... more workshopping of those responses ... and so on.
Key aspects of the approach and why it can be so effective
The Workshopping of Concept-Specific Open Ended Questions:
- Is engaging.
- Is somewhat student-centred (hence engaging).
- Gives the teacher a high-degree of control over proceedings.
- Is extremely easy to differentiate on the fly (by adding constraints to a question for specific students).
- Involves questions that can only be answered through applying the concept - they cannot be answered via a procedure. This is a vital point.
- Is an approach that can be used specifically to address a gap in conceptual understanding (across many students) as it arises in a lesson.
The crux of the approach is:
- Students need to understand the associated concept in order to create a correct response.
- Students who do not understand the associated concept get to see the concept in action through the workshopping process (which is why the workshopping is so critical).
- Hence the Workshopping of Concept-Specific OEQs is a powerful way to impart concepts to students en masse.
Disclaimers - The Workshopping of Concept-Specific OEQs:
- Does not replace normal teaching. It is simply an approach that can be used occasionally - perhaps 1-2 times per week for 5-10 minutes at a time during many units of work.
- Does not work with all concepts and topics.
Get a free preview of the course by accessing the free preview pages (sign in required).
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 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
Let's stop forcing our mathematics students to play the memory game! - here
Why We Need An Understanding-first, Procedures-second Mindset When Teaching Mathematics - here
Too Many Kids Hate Maths - here
Why Compartmentalising Is A Bad Idea When Teaching Mathematics - here
Could These Four Aspects Of Mathematical Understanding Change The Way We Teach Maths? - here
70% Of Capable Students Are Failing Mathematics. What Can We do? - here
Call to Action
Do you have questions? Do you see merit in the Understanding-first examples given above?
Are you tempted to adopt these principles? Have you already, in part, adopted them?
Your input is welcomed.
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Are you tempted to adopt these principles? Have you already, in part, adopted them?
Your input is welcomed.
NOTE: Create a Hyvor account before commenting (click LOGIN) -that way you'll be notified of replies and you won't be anonymous.
If you don't create an account, please state your name at the start of your comment. Thanks.
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