##### What is Conceptual Understanding?

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. Something which was not apparent before just clicked into place. It’s the dawning of an “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’ve all had “aha moments”. We all know what the *experience* feels like when a jewel of understanding suddenly emerges in our brain. So, for the purpose of this module, let’s stay with your memory of that *experience*, of that buzz, of that “aha moment,” rather than with a definition of it. And by the way, generating that sort of *experience* in our students is what we are all striving for as teachers of mathematics. That’s the game. It’s the end game. It is the whole game.

##### Why Do We Need Conceptual Understanding?

Ever tried to do something difficult which involved several related steps but you had no understanding of the process… no idea how those steps related to each other? Here’s an example:

You buy a bunk-bed-cupboard ensemble 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.

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!

To be specific, when we ask students to:

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

…and *they do not understand the concepts involved*, so we are setting them up for failure.

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

##### How Do We Get It?

**Part One: the Traditional Approach**

Now here’s something to think about… You can’t teach conceptual understanding!!!!

(What do you mean ‘You can’t teach conceptual understanding’? You *must* be able to teach it!!)

What I mean by ‘you can’t teach conceptual understanding’ is that it is not possible to directly impart our understanding of a concept onto a student.

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’ theorem or trigonometry. We can drill students so they are able to replicate any mathematical procedure. For example, we can’t teach via direct instruction, the concept which underpins decimal rounding or the formula for gradient. We can use direct instruction to teach the rules associated with 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 realize.

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’.

**Part 1.2: The Traditional Approach**

Mathematical instruction has traditionally involved the compartmentalization of mathematics. This has occurred for logical reasons; it just seems obvious to break a skill into smaller parts: teach the parts, then build the whole.

*Is this part-part-part-whole approach a truly effective way to teach?*

(Side Note: ‘Traditional Approach’ means ‘Compartmentalized Approach’)

- 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.
- 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 compartmentalization process applies for almost every topic.

**A Specific Example of a Traditional, Compartmentalized Approach through Trigonometry**

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

- 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 method for trigonometry was that the work output of students was typically poor by the time they reached Step 4 because, students lacked genuine confidence in what they were doing, which leads to a slower work rate. This was highly frustrating for me because despite the fact that I loved trigonometry I had managed, once again, to ‘bore my kids stupid’. Well, 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).

**Does Compartmentalization Work?**

The critical question to ask is *“does this traditional, compartmentalized approach foster genuine understanding in students?”*

To pose the critical question in plainer English, *“is the compartmentalized 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 compartmentalized approach works, then great. Let’s proceed as usual. But if it doesn’t then do we really want to throw our students into more of those confusing ‘flat-pack-bunk-bed-ensemble’ scenarios featured at the start of the article?

I’m convinced that the vast majority of students I taught using the traditional, compartmentalized approach, were answering questions purely from their memory of the learnt routines. Most students did not genuinely understand what was really ‘going on’ with the maths. Furthermore, I’d be very, very surprised if I was in the minority. I suspect most of the students taught by the compartmentalized approach are operating on their memory of routines and little else. I suspect that for hundreds of years we have all, silently, agreed that this is OK.

##### How Do We Gain a Conceptual Understanding?

**Part Two: An example of a Conceptual Approach**

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!

**Conceptually-based Trigonometry**

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 memorize 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.

The video below shows an example of a conceptually-based introduction to trigonometry.

**Following the Introduction**

**Step 1:** The all-important introduction to the trigonometry unit. It should take about 10-20 minutes and be delivered interactively. After the introduction the following steps are recommended.

**Step 2:** Teach the ‘writing of the trig sentence’ (i.e. how to determine whether a question is sine, cosine or tangent situation and to write the initial trig sentence, e.g. sin 36 = x/12)

**Step 3:** Provide students with sets of worksheets (students answer on the sheets) with mixed questions. It is important the students write their answers on the sheets as this saves significant time and keeps their focus on trigonometry rather than on copying triangles. The teacher demonstrates (using direct instruction) the requirements for setting out via a couple of examples from the worksheets.

**Step 4:** Student practice.

Students work through the Trigonometry worksheet series.

**Disclaimer!**

Note that there is nothing complicated about the conceptually-based Trigonometry unit described above. The unit even lacks a practical aspect! This is deliberate so that ‘anyone’ can implement it and the approach does not require highly engaged students in order to work. Of course, other components can be added to the unit. The example provided here is simply a basis for a conceptual approach.

**Other Examples of a Conceptual Approach**

Trigonometry is one unit where the conceptual approach is staggeringly more successful than the compartmentalized approach. However, the approach can be applied to all units in mathematics. Keep in mind that we have shown you just one aspect of the conceptual approach. Throughout the four-month course ‘Engagement – Winning over your mathematics class’ the approach is covered in much greater detail, including an expanded explanation of the Trigonometry unit complete with extensive worksheet series.

**A Conceptual Approach Saves Time**

In the trigonometry example referenced above, a well executed, conceptual approach should bring students to the mixed ‘real-world’ exercises within 2-4 lessons, compared to the 6-7 lessons it takes with the compartmentalized 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.

**Summary**

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

A conceptual approach is a clear alternative to compartmentalized 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 memorizing routines and ‘tricks’.

What do you think of the compartmentalization v conceptual method? Have you experience of either? Let us know in the comments. We would love to learn more and hear your feedback about it.

Watch the video again here.

*Feature image courtesy of Flickr, Tim Geers.*

Source: Fractus Learning