Mathematics Education – An enquiry based approach to learning Mathematics…

(This is the third blog post in a series of three blog posts examining where we are with respect to Mathematics education and what we can do additionally. In this blog post, I present an enquiry based approach to learning Mathematics.)

In the first blog post of this three post series, I presented the current advancements in the field of Mathematics education, specifically the proliferation of resources for aiding Mathematics education and associated challenges. I ended the blog post by presenting a need to create a fundamental shift in our approach towards Mathematics education, by asking the question – “What needs to change fundamentally in our approach towards teaching/learning Mathematics?”.

I followed that up with a second blog post where I started exploring this question by examining our current approach towards Mathematics education. I presented my views on our current approach to Mathematics education (which is heavily practice focused) and the issues with this approach. I ended my second blog post with the question – “What can we add to our current approach to enhance Mathematics learning?”

Adding a touch of enquiry can help find some purpose in the practice!

As I mentioned towards the end of my second blog post, I suggest adding an element of enquiry to the teaching/learning process. By encouraging students to enquire into various aspects of a concept and by facilitating such enquiries, we can help students in understanding the need for the concept and how it is related to other concepts. Students build a better understanding of the concept and subsequently, are more open to and better at problem solving.

In my workshops, I have noticed (somewhat to my surprise) that students who were not really good problem solvers to begin with, were not really interested in learning Mathematics. However, when I presented/facilated enquiries regarding the need for a concept, they became interested in the concept. As time went by, they became more open towards solving problems and better at it as well. I guess they were looking for a sense of purpose all the while.

Some of the enquiries that can be presented/explored are:

Why is a particular concept important? What will happen if this concept doesn’t exist? Can we make do with something else instead?
In what way is a concept different from another concept that we have learnt earlier? If the concepts are inter-changeable in some way, what is the need for having both of them?
Is there a relationship between a particular concept and something we have learnt earlier? If so, what is it?
Why does something look different suddenly? Is there any way it can be related to what we have already studied?
Why do we do things a particular way?

To give a concrete feel of how these enquiries can look, I have included below a sample list of the enquiries that I conduct in my workshops/classes:

Why do we need Integers? What would happen if we didn’t have Integers?
How do we intuitively understand the various operations on Integers?
Is the way fractions are multiplied/divided different from the way natural numbers are multiplied/divided? If not, why does it look so?
How do we intuitively understand what are LCM & HCF and the procedure for computing LCM & HCF?
How do we understand the laws of exponents when exponents are rational numbers intuitively from our understanding of the laws of exponents when exponents are integers?

I have also included below a couple of videos where I attempt to look at answers to some of these questions:

But doesn’t the enquiry process get abstract!

A point to be noted is that while such enquiries help students grasp concepts (and the need for learning them) better, the challenge with the enquiry process is that it can tend to be abstract at times and consequently, may need students to think abstract at times. But as we know, students are exposed to formal reasoning techniques only at a later stage. In fact, at early stages of learning, students tend to be more playful and abstract thinking is not the forte for most students.

Concretising the abstract nature of enquiries!

There are a couple of ways in which we can tackle the above challenge.

One way is to figure out a way in which we can make these enquiries more concrete and intuitive. This can be accomplished by performing activities and experiments to accomplish the enquiry process. This keeps students involved and they are more open to abstract out the learnings afterwards. This also brings in an element of fun to the learning process which I feel is very much needed.

As an example of performing an enquiry through an activity, in my workshops I use colored cards with a value relationship (that mimics the place value system) between the different colors, in a game setting to look at questions like:

Can we perform addition/subtraction operations on two natural numbers starting from the highest place value and proceeding towards the right (instead of starting at the units place and proceeding towards the left)? Can we start at any arbitrary place value instead? Will we get the same result? How will the process get impacted?
Why do we perform addition/subtraction of two numbers starting from the units place and proceeding towards the left?
How does the place value system work?

Another way enquiries can be made less abstract is to time the enquiry in such a way that students first gain a basic level of mastery in a concept through basic enquiry and practise before being exposed to advanced enquiries. Unless, the student has a natural flair for Mathematics, I would very much recommend doing this.

But where are the resources for this approach?

As discussed in my first blog post in this series, it is but natural for resource creators to focus their energies on creation of resources in tune with the current approach, as then there is more of a chance of finding a market for the resources that they create. While I am by no means suggesting that resources which facilitate enquiry based learning don’t exist, I would assume that our approach towards teaching/learning Mathematics will need to start changing before resource creators start investing more time and effort in the creation of such resources. As you can see, it is a chicken and egg situation to some extent.

I feel that teachers can take the lead (and I am sure that this must be happening in some schools already) in implementing some elements of enquiry in their classrooms either using available resources or by formulating their own enquiries and creating their own resources to support these enquiries. They can then share results in open forums such that resource creators are motivated to invest more time and effort in creation of resources to facilitate the enquiry based learning approach.

Conclusion

Mathematics is a very beautiful subject and the beauty lies in how it has evolved and how the various concepts are inter-related to each other, in addition to how it serves as a tool for solving problems. Irrespective of whether students take on professions which involve knowledge of certain Mathematical concepts, it is very important that they learn a basic level of Mathematics and have fun while doing so. While there are plenty of resources available for facilitating Mathematics education, they are aligned with our current approach towards learning Mathematics. However, I feel that our current approach towards learning/teaching Mathematics may be making the learning process less enjoyable for students and may be limiting their perspective of the subjective. I suggest adding an element of enquiry to our current approach may help alleviate the issue to some extent. Needless to say, teachers and resource creators would need to take a lead to make this change happen.