Conceptual vs Procedural mathematics Teaching (10-Minute Survey)

A conceptual vs procedural teaching survey (for teachers of mathematics)

If you're a mathematics teacher then this checklist-survey is designed for you. Please read the bullets below before commencing the survey.

The survey will take you at least ten minutes with some careful reading.

Your aim as you work through the survey should not be to gain the highest score. Instead, your goal should be to gain some insight into your teaching practice.

The survey contains many pointers to key strategies used in quality mathematics teaching. It was written with the intention to give teachers ‘food for thought’.

One of the benefits of the survey is to point out to you some strategies you already use as 'default' but which you may not have realised were considered 'strategies' as such. In other words, you may receive some reassurance of your current teaching practice.

Understand that the survey was written from a pro-conceptual approach perspective. Therefore, it will be easy for you to ascertain the option worth ranked highest on the ‘conceptual scale’. Avoid being put off by this. Again, your aim should be to gain insight into your current teaching practice and to ponder whether some of the strategies mentioned might be worth exploring.

Scoring

The survey comprises ten items.

Each item offers either 2 or 3 options and you are to choose one option for each item.

Each option is allocated a score somewhere between 1 and 5.

You will gain an automatically collated score at the end of the survey. Your score is in no way definitive - it is simply a guide.

The value of each option is listed at the bottom of this form.

Click Next to start the survey

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Item 1. Conceptual instruction vs procedural instruction when teaching and planning lessons

Option A My focus is on teaching procedures so that students can successfully answer questions in assessment tasks. I am interested in my students gaining a conceptual understanding of the mathematics we are working through, and I do what I can to explain my understanding of the concepts as well as possible.

Option B My main focus is on teaching routines well. I believe that if students gain sufficient practice with the routines, they’ll eventually understand the mathematics.

Option C I aim to have my students understand the concepts underpinning the mathematics being taught. A lack of conceptual understanding makes the task of learning routines unnecessarily difficult. I strive to teach in a way that maximises the number of 'aha moments' that my students experience. I also teach procedures; however, concepts come first.

Item 1. - Choose your response
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Option A Option B Option C

Item 2. Your relationship to ‘student spread’ within a unit of work

Option A When I’m teaching my aim is to allow students to progress at their own pace throughout a unit. However, I occasionally have teacher-directed lessons where the entire class is working on the same task. I find teaching in this student-centred way to be much more engaging for students. This affords me more opportunities to give specific help to students who most require it and helps me to facilitate my students’ learning more effectively.

Option B When I am teaching my aim is to teach the unit step by step in an orderly fashion to reduce the possibility of students getting way ahead or of other students falling behind.

Option C I aim to teach the unit step by step in an orderly fashion to reduce the possibility of students getting way ahead of others or falling behind. However, I extend my more-able students by encouraging them to work ahead, and I scaffold the work for ’less able’ students so that they can succeed.

Item 2. - Choose your response
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Option A Option B Option C

Item 3. Student-centredness vs teacher-centredness vs explicit instruction

Option A I prefer to plan and direct the way my class runs as it is my role to explicitly teach my students what they need to know within any given mathematics unit.

Option B I am the teacher in the room, and it is primarily my role to guide student learning. However, occasionally I instruct students to work on open-ended tasks and investigations. I use these different tasks because they offer a change of routine rather than because I think students necessarily learn more from these activities than from ’normal’ lessons.

Option C My role in the classroom is to facilitate 'aha moments' for students. I realise that I cannot directly ‘teach’ understanding to students, but I can ask leading questions, use analogies, draw diagrams and use other means that lead students towards moments of understanding. To maximise learning, I run most of the activities in an organised, student-centred manner but give explicit procedural instruction as students require them. As much as possible I structure activities so that students work with the mathematics ‘from their own understanding’ before they get to see and use the related routine or procedure.

Item 3. - Choose your response
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Option A Option B Option C

Item 4. The importance of student engagement

Option A My role is to orchestrate a controlled, orderly classroom where my students complete as much work as possible in the shortest possible time. My role is to teach. I’m not there to entertain. If the students are engaged, that is a bonus.

Option B It is very important to me that my students are genuinely engaged in lessons. My view is that if my students are not genuinely engaged in the work, I set them, then their ability to learn is greatly diminished. The three main principles that I use to maximise student engagement are: 1) To engineer a sense of ownership over learning in students through having a robust student-centred aspect to the unit, 2) To allow for student collaboration in (most) lessons, and 3) To cultivate an engaged learning environment in the classroom.

Option C Student engagement is important, however, it is not a major factor when I’m planning activities for my students. Having said that my personality lends itself to engaging my students, and I (mostly) enjoy teaching mathematics, and I (mostly) enjoy teaching my students.

Item 4. - Choose your response
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Option A Option B Option C

Item 5. Questioning techniques

Option A I ask questions to the whole class, to groups of students and individuals as part of the natural role of a mathematics teacher. However, I usually call upon the same 4-5 more confident students who, to be honest, answer the bulk of the questions within a group setting (because it allows me to progress through the lesson effectively.

Option B I ask questions to the whole class, to groups of students and individuals as part of the natural role of a mathematics teacher. As I review work during whole-class ‘mini-lectures’, I ask questions to ascertain where the students are at as a whole. I am aware of the trap of always asking the same few students (who I know are likely to have the correct answer) because it is quicker to do so. However, I also make a point of encouraging those who don’t usually volunteer answers to ‘have their say’. I admit I am not particularly good at engaging the whole class when questioning and would benefit from some extra ideas and guidance regarding how to include more students in the questioning process.

Option C I make a point of regularly asking tactical questions, especially of those students who rarely volunteer answers. I use strategies such as ‘The nine-second rule’ (waiting 9 seconds before asking for an answer), which gives less confident students time to formulate an answer. I also make a point of validating all students who volunteer honest answers regardless of whether their answers are correct or not.

Item 5. - Choose your response
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Option A Option B Option C

Item 6. Open-Ended Questions (OEQs)

Option A As often as is reasonably possible, I utilise Open-Ended Questions (OEQs) by workshopping them with the students. Importantly, I use the workshopping of OEQs as a primary means to impart conceptual understanding to students. For example, I often use OEQs at the start of a unit to lay a conceptual foundation before teaching the required routines. Examples of this are the teaching of rounding, fractions, measurement, algebra and the principles of range, mode, median and mean when introducing or revising statistics. The workshopping of OEQs is very much a teacher-controlled activity, one which is engaging and collaborative and in which students experience many 'aha moments'.

Option B I rarely or never use OEQs. The work I set for students through any given unit contains mostly closed questions - questions with one correct answer. Closed questions allow me to provide students with familiar routines to follow.

Option C Occasionally, I include OEQs in lessons in worksheets that students complete alone or in pairs. I incorporate at least one OEQ into most assessment tasks.

Item 6. - Choose your response
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Option A Option B Option C

Item 7. Peer teaching within your classroom

Option A My mission is to get my students through the syllabus as efficiently as possible. Therefore, I am the one who does the teaching and I rarely, if ever, use peer teaching. The one exception is when a student has been absent and returns to class behind schedule.

Option B I strongly encourage peer teaching in my classes, some of which is structured and at other times students work together naturally in a collaborative manner.

Option C Although I do most of the teaching in my lessons I do acknowledge the value of peer teaching and use it occasionally. Mostly the peer teaching is structured and with a specific purpose.

Item 7. - Choose your response
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Option A Option B Option C

Item 8. The journey or destination?

Option A When working with students, I am much more interested in the mathematical thinking that students are using than I am their answer. I want my students to be using their own, mathematically-sound thinking so that they have a genuine understanding of the mathematics they are undertaking and of the errors they make.

Option B When I am working with students, I am focused on getting through the work in the most efficient manner and giving students as much practice in the correct use of procedures as possible

Option C For me, the journey and the destination are of equal importance. I acknowledge the importance of developing mathematical thinking and allow some time for students to explore and investigate. However, my main aim is for students to learn the correct procedures.

Item 8. - Choose your response
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Option A Option B Option C

Item 9. Compartmentalisation vs holistic approach

Option A Learning mathematics is a bit like playing with (old school) lego - by learning individual facts and routines (mathematical building blocks), they can be assembled to solve increasingly complex problems. This is why I make a point of explicitly teaching each routine/procedure and ensuring students gain sufficient practice so that they can solve complex problems. I do not see how mathematics can be taught more efficiently and effectively using a non-compartmentalised approach.

Option B A compartmentalised approach limits students’ understanding of mathematics. When I teach holistically, students become immersed in the over-arching concept, students better understand what is going on, and as a result, students tend to use their own logic to tackle questions rather than relying on their memory of a routine or procedure.

Item 9. - Choose your response
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Option A Option B

Item 10. Creating a need to learn

Option A One way to engage students in mathematics is to create in them ‘a need to learn’ as often as possible. This is a strategy I use regularly. For example, rather than teaching students how to calculate an angle using right-triangle trigonometry BEFORE they face such a question (after they know how to calculate side lengths) I allow them to meet such a question unprepared. This creates interest and the need to learn. It sounds insignificant, yet I find this to be a powerful strategy, one that can be applied through every mathematics topic.

Option B I know the order in which the unit is best learnt and I know the common mistakes which students make. Therefore, I aim to help students avoid making these mistakes by showing them tips and tricks to avoid the traps before they encounter them.

Item 10. - Choose your response
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Option A Option B

Your total

Interpreting your score

A score of 10 - 19

You are most likely a very dedicated teacher!

You probably use a procedural approach to mathematics teaching almost exclusively.

You are more likely to believe that the key to successful mathematics teaching is teacher-direction, explicit teaching of routines and procedures.

You may have explored conceptually-based activities in the past and found that they didn’t work. or you may not be convinced of the merit of conceptually-based activities.

You may believe that adopting a conceptual approach is synonymous with ‘not teaching procedures’.

A score of 20 - 37

You are most likely a very dedicated teacher!

You are more likely to be a procedurally-based mathematics teacher who sees some value in running some conceptually-based activities.

You are probably open to considering exploring conceptually-based activities, but only if you can be assured that the activities were advantageous and time-effective. You would welcome some implementation-based easy-to-follow guidance re such activities.

A score of 38 - 45

You are most likely a very dedicated teacher!

It would appear you are a conceptually-oriented mathematics teacher.

Your teaching practice encourages students to experience aha moments as frequently as possible.

You most likely see the importance of the explicit teaching of procedures but believe that understanding needs to come first.

Please remember that these interpretations serve only as a guide!

Your feedback please (4 multiple-choice questions)

Interesting or not?
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I don't want to give feedback I found the survey very interesting I found it somewhat interesting

Benefits?
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I’m definitely a bit more curious now about this whole conceptual approach idea. I always thought a conceptual approach meant 'not teaching procedures', but this information infers the two go together. The survey opened my eyes to some new-to-me possibilities that I could use in the classroom. I already teach using a highly-structured, student-centred, conceptually-based approach which allows for the explicit instruction of procedures. Therefore there was nothing new for me. I saw some new-to-me ideas, but I'd need to see how these ideas work before adopting some/any of them. To me, most/all of the ideas aligned to 'conceptual' are a waste of time. I am not open to the possibility that I could benefit from these conceptual ideas. I'm currently learning about these principles.

Are there any other aspects you’d like to see covered?

No, it’s fine how it is. Yes, see below in 'Other'.

Next steps?
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I assume there must be a course. I'm interested. Please send information. (I am keen for a 2-3 email conversation - I will supply my email address below) I'd like to see in more detail how to implement the conceptual strategies mentioned in the survey. I realise this would require a course of some kind of course or PL program because a series of articles would be insufficient to guide teachers through all the strategies mentioned. However, I am sceptical because many/most courses don’t actually impact the classroom. (I am keen for a 2-3 email conversation - I will supply my email address below). There's nothing here that I wish to pursue. I'm currently undertaking the course featuring the principles mentioned in this survey.

Your name - optional.

First Name Last Name

Email (Required if you have requested a 2-3 email conversation)

example@example.com

Your state/region
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Any other comments?

Click 'Next' to submit and to see the score assigned to each option

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The values assigned to each option

Item 1 Option A = 2 Option B = 1 Option C = 5

Item 2 Option A = 4 Option B = 1 Option C = 2

Item 3 Option A = 1 Option B = 2 Option C = 5

Item 4 Option A = 1 Option B = 5 Option C = 2

Item 5 Option A = 1 Option B = 2 Option C = 5

Item 6 Option A = 5 Option B = 1 Option C = 2

Item 7 Option A = 1 Option B = 4 Option C = 2

Item 8 Option A = 4 Option B = 1 Option C = 2

Item 9 Option A = 1 Option B = 4

Item 10 Option A = 4 Option B = 1

The 'Submit' button (A note from Richard, owner, Learn Implement Share)

Clicking 'Submit' simply saves your responses. It will be interesting to see the results after hundreds of teachers have responded. Submitting also sends your feedback to me.