Hello again! Welcome back to my blog :)
It's been a while since my last blog post but there is lots to catch you guys up on. We've been discussing a lot about the three phases of work that help us in achieving a resolution to a mathematical problem. These three very important phases of work are The Entry Phase, The Attack Phase, and The Review Phase.
Basically, these three phases are the framework of reaching a resolution to any mathematical problem. Let me break each one down for you...
The Entry Phase
This phase begins as soon as you are faced with a problem! In this phase, it is really important that you really read the problem to absorb the information and find out what the question is asking. After establishing a genuine understanding of the problem, it is helpful to answer these three questions:
- What do I KNOW?
- What do I WANT?
- What can I INTRODUCE?
These three questions will help you break down the question in order to start off the process in reaching a resolution. What you KNOW allows you to digest the context of the problem, what you WANT directs your attention to what you have to do, and what you can INTRODUCE aids you in finding what you want. It is important to break to what you WANT into manageable pieces of information for clearer objectives for yourself. If you do find yourself a little STUCK or confused, a diagram is often a big help to understanding the problem. This phase prepares us for the next phase of work....
The Attack Phase
The thinking that you did in The Entry Phase has prepared you for The Attack Phase! You now have a complete understanding of the question and feel like you can tackle the problem. This phase is full of STUCK! and AHA! moments that you will engage in as you specialize and generalize. This phase is also full of conjectures. If you don't know what a conjecture is, it is a statement that appears reasonable, but whose truth has not yet been established. It is up to you to analyze the problem and develop conjectures that you believe may be the resolution to your problem. From here, you can check the conjecture with examples, try to distrust it with counter examples, and verify its validity or lack thereof. This process will lead you to your resolution!
This is the final phase that you will engage in once you have reached a reasonably satisfactory resolution or when you are about to give up. This is where you look back at what you have done so far to:
- CHECK the resolution
- REFLECT on the key ideas and key moments
- EXTEND to a wider context
These three processes will allow you to self-validate your resolution and make sure it makes sense to you. Despite checking our resolution as we do it, it is more reliable to CHECK once the resolution has been reached and we are more relaxed and dispassionate. REFLECTING on key ideas and is the way to build on your mathematical experience. Having written notes of your thinking along the way is very helpful in recalling key moments throughout the resolution. Lastly, mathematical thinking does not really begin until you are engaged by a question. However, the most engaging question is always your own and this is why EXTENDING the problem to a wider context is important in the review phase as it will build on your mathematical thinking abilities.
Well... there you have it! These are the three phases of work when engaging in a mathematical problem to reach a resolution. As you may have noticed, these three phases make use of the two key mathematical processes that I discussed in my last blog post: specializing and generalizing. By specializing, we are able to find out what we KNOW, WANT, and can INTRODUCE. The specializing will enable us to uncover patterns that will lead us to generalizing information. By generalizing, we are able to come up with conjectures that we can CHECK and EXTEND to a wider context.
As you can see, everything we have learned and discussed so far is all connecting as we continue to move through the course!
I hope you guys learned something new today or gained a deeper understanding of what you already knew! Until next time....
Signing off,
Ms. Blackwell
