This is a longer article than I usually write. So hang onto your hats. My suggestion: Read the first half as though it's a short article - then you’ll probably want to read the rest!

A quick search of the net will bring up numerous long-winded, tedious definitions of conceptual understanding within a mathematics context. This one from Dreambox is more palatable than most:

Conceptual understanding is knowing more than isolated facts and methods. The successful student (with conceptual understanding) understands mathematical ideas and can transfer their knowledge into new situations and apply it to new contexts.

The problem is, all definitions of conceptual understanding are poor representations of the actual experience of understanding a concept conceptually, especially the experience at the time the concept first 'clicks'. The moment of gaining an understanding is, after all, an EXPERIENCE. It’s a tangible ‘buzz’. In fact, at the risk of sounding dramatic, it is a ‘revelation experience'. The lights come on and something which was not apparent before clicks into place. It’s the dawning of a new ‘aha moment’.

Of course, many commentators are compelled to repackage the EXPERIENCE into a definition comprising several measurable quantities. This might be necessary to meet the demands of the age we live in, but the fact is that as soon as we slap a definition onto an EXPERIENCE, we lose our grip on that EXPERIENCE. To put this another way, conceptual understanding is more significant than any definition would suggest.

We all know what the EXPERIENCE feels like when a jewel of understanding suddenly emerges in our brain. So, for now, let’s stay with your memory of that EXPERIENCE, of that buzz, of that aha moment, rather than with any definition of it. I want you to stay with your memory of the aha EXPERIENCE because generating that sort of EXPERIENCE in our students is what we are all striving for as teachers of mathematics.

Striving for aha moments - it’s the game, it’s the end game, it’s the whole game.

Of course, many commentators are compelled to repackage the EXPERIENCE into a definition comprising several measurable quantities. This might be necessary to meet the demands of the age we live in, but the fact is that as soon as we slap a definition onto an EXPERIENCE, we lose our grip on that EXPERIENCE. To put this another way, conceptual understanding is more significant than any definition would suggest.

We all know what the EXPERIENCE feels like when a jewel of understanding suddenly emerges in our brain. So, for now, let’s stay with your memory of that EXPERIENCE, of that buzz, of that aha moment, rather than with any definition of it. I want you to stay with your memory of the aha EXPERIENCE because generating that sort of EXPERIENCE in our students is what we are all striving for as teachers of mathematics.

Striving for aha moments - it’s the game, it’s the end game, it’s the whole game.

Ever tried to do something difficult which involved several related steps but you had no understanding of the process, had no idea how those steps related to each other? How’s this for an analogy:

You buy a bunk-bed-cupboard as a flat pack. The instructions were written by a Spanish computer nerd, and before reaching you they were translated into Chinese, then Russian, then Hindi and finally into broken English. There are 317 pieces, not including 12 types of fasteners, and you swear a heap of them are missing. You are 3 hours into the maze and have no idea how to get out. You have done your best to follow the instructions but there is no conceptual connection between the steps - they simply do not make sense!

When applied to the maths classroom, asking students to ‘do maths’ without them having the related conceptual understanding is a lot like asking them to build that flat pack piece of furniture with instructions written by that Spanish computer nerd, and translated to Chinese to Russian to ... you get the picture! It’s damn near impossible!

**To be specific, when we ask students to:**

Asking them to blindly follow steps is akin to following those flat-pack instructions that were impossible to understand.

So, friends, THAT is why we need our students to have conceptual understanding.

You buy a bunk-bed-cupboard as a flat pack. The instructions were written by a Spanish computer nerd, and before reaching you they were translated into Chinese, then Russian, then Hindi and finally into broken English. There are 317 pieces, not including 12 types of fasteners, and you swear a heap of them are missing. You are 3 hours into the maze and have no idea how to get out. You have done your best to follow the instructions but there is no conceptual connection between the steps - they simply do not make sense!

When applied to the maths classroom, asking students to ‘do maths’ without them having the related conceptual understanding is a lot like asking them to build that flat pack piece of furniture with instructions written by that Spanish computer nerd, and translated to Chinese to Russian to ... you get the picture! It’s damn near impossible!

- add fractions, or
- round off numbers, or
- find equations of lines, or
- find intersection points, or
- find the discriminant, or
- discuss the standard deviation of a data set, or
- calculate a bearing, or
- calculate the surface are or volume of a composite solid, or
- construct the ultimate package for a box of chocolates, or
- any one of an unlimited number of other mathematical tasks

Asking them to blindly follow steps is akin to following those flat-pack instructions that were impossible to understand.

So, friends, THAT is why we need our students to have conceptual understanding.

Now here’s something to think about …
You can’t teach conceptual understanding !!WHAT?? What I mean by ‘You can’t teach conceptual understanding’ is that it is not possible to directly impart our understanding of a concept into a student. To explain: Using direct instruction we can teach a child to calculate the mean of a data set, or the length of a side of a right-triangle side using ‘Pythagoras' or trigonometry. We can drill students so they are able to replicate any mathematical procedure. But, we can’t teach, via direct instruction, the concept which underpins, for example, place value … or decimal rounding … or the formula for gradient. We can use direct instruction to teach the rules associated with place value, decimal rounding and gradient but we cannot directly impart the understanding of those concepts to students. |

This is an important point for any teacher of mathematics to realise. To impart the understanding of a mathematical concept to students - i.e. to generate aha moments in students - we need an approach which is different to direct instruction. We need a conceptual approach. We’ll get to the conceptual approach soon. First however, we need to unpack what I call the ‘traditional approach to mathematics education’.

Mathematical instruction has traditionally beed procedural, the teaching of procedures the teacher decides are ideal for solving the questions contained in the unit of work. This process tends to compartmentalise the mathematics.

Compartmentalising has occurred for logical reasons, it just seems obvious to break a skill into smaller parts, teach the parts, then build the whole. But is this part-part-part-whole approach a truly effective way to teach?

Compartmentalising has occurred for logical reasons, it just seems obvious to break a skill into smaller parts, teach the parts, then build the whole. But is this part-part-part-whole approach a truly effective way to teach?

- In basic trigonometry, we traditionally teach the routines to find the lengths of sides based on sine, then based on cosine, then based on tangent, each in isolation from the other. Then we teach the process to find angles.
- Traditionally, we teach fractions as distinct from decimals as distinct from percentages. We have traditionally taught these as three separate topics despite the fact that they are each founded on common concepts.
- In Coordinate Geometry we traditionally teach, in isolation, the distance formula, then the midpoint formula, then the gradient formula, and then a method to determine equations of lines.
- Traditionally, the same compartmentalisation process applies for almost every topic.

Below is one example of how Trigonometry has been traditionally tackled.

One problem I always found with the above-compartmentalised method for trigonometry was that the work output of students was typically poor by the time they reached Step 4 because, by this stage, students lacked genuine confidence in what they were doing, and this lead to a slower work rate. This was frustratingly painful for all concerned and especially for me because, despite the fact that I loved trigonometry, I had managed, once again, to ‘bore my kids stupid’. Most of them at least. Their lack of confidence continued through the unit, meaning it generally took about 6-7 lessons to reach the mixed exercises (Step 9).

- Step 1: Introduction to trigonometry.
- Step 2: One of the trig ratios is presented (e.g. sine). Students copy notes, write down the rule.
- Step 3: An example using sine is presented. Students copy.
- Step 4: Students work through several similar examples.
- Step 5: Repeat for second trig ratio.
- Step 6: Repeat for third trig ratio.
- Step 7: Teach 'how to find an angle'. Students take notes, copy the example, practice multiple questions.
- Step 8: Move onto similar examples using pictures ('real life' examples).
- Step 9: Move onto mixed exercises.
- Step 10: Teach bearings.
- Step 11: Assessment (mostly fluency questions).

One problem I always found with the above-compartmentalised method for trigonometry was that the work output of students was typically poor by the time they reached Step 4 because, by this stage, students lacked genuine confidence in what they were doing, and this lead to a slower work rate. This was frustratingly painful for all concerned and especially for me because, despite the fact that I loved trigonometry, I had managed, once again, to ‘bore my kids stupid’. Most of them at least. Their lack of confidence continued through the unit, meaning it generally took about 6-7 lessons to reach the mixed exercises (Step 9).

The critical question to ask is 'Does this traditional, compartmentalised approach foster genuine understanding in students'?

To pose the critical question in somewhat plainer English - is the compartmentalised approach effective in enabling students to actually understand what they are doing? Or does it result in students answering questions purely from their memory of the learnt routines?

If the compartmentalised approach works, then great. Let’s proceed as usual. But if it doesn’t lead to the vast majority of students understanding the maths - and being engaged by it - then should we continue to keep feeding our students those confusing 'flat-pack-bunk-bed-ensemble' tasks referenced at the start of the article?

No, I think not.

I’ll bet England to a pound that the vast majority of students I taught using the traditional, compartmentalised approach were answering questions purely from their memory of the learnt routines and that most of them did not genuinely understand what was really ‘going on’ with the maths. Furthermore, I don’t think I’m a unique case! I suspect most of the students taught by the compartmentalised approach are operating primarily on their memory of routines. I suspect that for hundreds of years we have all, silently, agreed that this is OK.

To pose the critical question in somewhat plainer English - is the compartmentalised approach effective in enabling students to actually understand what they are doing? Or does it result in students answering questions purely from their memory of the learnt routines?

If the compartmentalised approach works, then great. Let’s proceed as usual. But if it doesn’t lead to the vast majority of students understanding the maths - and being engaged by it - then should we continue to keep feeding our students those confusing 'flat-pack-bunk-bed-ensemble' tasks referenced at the start of the article?

No, I think not.

I’ll bet England to a pound that the vast majority of students I taught using the traditional, compartmentalised approach were answering questions purely from their memory of the learnt routines and that most of them did not genuinely understand what was really ‘going on’ with the maths. Furthermore, I don’t think I’m a unique case! I suspect most of the students taught by the compartmentalised approach are operating primarily on their memory of routines. I suspect that for hundreds of years we have all, silently, agreed that this is OK.

As was stated above, to impart understanding of mathematical concepts to students - to generate aha moments - we need an approach which is different to direct instruction. I call that approach ‘conceptual teaching’ or ‘conceptually-based instruction’. However, these are only labels.

Hang onto your hats ... an example is coming.

Hang onto your hats ... an example is coming.

We have already seen through using a traditional approach to trigonometry, each component is taught separately and in isolation. Students often experience this as having to memorise numerous rules which, to them, seem unrelated.

A conceptual approach, on the other hand, first presents the trigonometric principle - a principle based on similar triangles. Following this conceptually-based introduction, students tackle a series of questions mixed with sine, cosine and tangent situations. This sounds counter-intuitive and more difficult than the traditional, compartmentalised approach. But it isn’t. On the contrary, it is far easier for students to understand and, far more engaging. What’s more, it saves time! As students tackle the mixed questions, finding side lengths and angles using all three ratios in the same block of questions, the students are drawing upon the principles of trigonometry, rather than their memory of routines.

A conceptual approach, on the other hand, first presents the trigonometric principle - a principle based on similar triangles. Following this conceptually-based introduction, students tackle a series of questions mixed with sine, cosine and tangent situations. This sounds counter-intuitive and more difficult than the traditional, compartmentalised approach. But it isn’t. On the contrary, it is far easier for students to understand and, far more engaging. What’s more, it saves time! As students tackle the mixed questions, finding side lengths and angles using all three ratios in the same block of questions, the students are drawing upon the principles of trigonometry, rather than their memory of routines.

The video above has been lifted from the online course ‘Engagement - Winning over you mathematics class’. The course expands beyond the introduction - time doesn’t permit further elaboration here - as well as numerous other examples of a conceptual approach, most not requiring dynamic geometry software.

It is important students are able to write their answers on the sheets and are not required to draw the diagrams as this saves significant time and keeps their focus on trigonometry rather than on copying triangles.

In the trigonometry example referenced above, a well executed, conceptual approach should bring students to the mixed ‘real-world’ exercises (Step 9 of the compartmentalised approach) within 2-4 lessons, compared to the 6-7 lessons it generally takes with the compartmentalised approach. That is a saving of between two and five lessons! Similar sorts of time saving can be expected from teaching other topics using a conceptual approach.

Note that there is nothing complicated about the conceptually-based Trigonometry unit described above. The unit even lacks a practical exercise! This is deliberate so that 'anyone' can implement it and the approach can be applied to students who are not especially engaged. Of course, other components can be added to the unit. The example provided here is simply a basis for a conceptual approach to basic trigonometry.

Note that there is nothing complicated about the conceptually-based Trigonometry unit described above. The unit even lacks a practical exercise! This is deliberate so that 'anyone' can implement it and the approach can be applied to students who are not especially engaged. Of course, other components can be added to the unit. The example provided here is simply a basis for a conceptual approach to basic trigonometry.

Trigonometry is one unit where the conceptual approach is staggeringly more successful than the compartmentalised approach. However, the approach can be applied to all units in mathematics. Keep in mind that the above is just one aspect of the conceptual approach. When I work with teachers the approach is covered in much greater detail, including an expanded explanation of the Trigonometry unit complete with extensive worksheet series, the ‘Brick Wall Strategy’ and the intricacies of creating a need to learn in students. Nevertheless, what has been shared here should prove to be a valuable start.

Although compartmentalisation makes intuitive instructional sense because it is a genuine attempt to ‘keep it simple’, this article proposes that compartmentalising information, in most cases, creates an unnecessary level of confusion for students.

A conceptual approach is a clear alternative to compartmentalised, procedural instruction. It presents the mathematics holistically and draws upon the students' ability to understand rather demanding them to remember seemingly unrelated routines and procedures.

A well executed, conceptual approach, fosters understanding. It requires students to understand a concept in order to proceed. Progress through a conceptually-based unit requires students to understand the concepts because the work is less dependent on memorising routines and ‘tricks’.

Note that a conceptual approach, as outlined in this article, is not devoid of procedural instruction. Procedures are taught but are secondary to attempts to provide students with 'aha' moments in the first instance.

A conceptual approach is a clear alternative to compartmentalised, procedural instruction. It presents the mathematics holistically and draws upon the students' ability to understand rather demanding them to remember seemingly unrelated routines and procedures.

A well executed, conceptual approach, fosters understanding. It requires students to understand a concept in order to proceed. Progress through a conceptually-based unit requires students to understand the concepts because the work is less dependent on memorising routines and ‘tricks’.

Note that a conceptual approach, as outlined in this article, is not devoid of procedural instruction. Procedures are taught but are secondary to attempts to provide students with 'aha' moments in the first instance.

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