Blog Post #5: Ready, Set, Attack!

Hi Everyone and Welcome Back to my Blog!

In my last blog post, I talked about what we do when we get stuck. Well, now I have a follow-up question: What do we do when we finally become unstuck? The answer is actually quite simple... we ATTACK the problem. The answer may be simple, but realistically not many people know how to go about this phase of attack. But, that's what I am here for! Today we will be breaking down how we should tackle the attack phase when engaging in a mathematical problem. 

The first main component of the ATTACK phase is conjecturing. A conjecture is essentially a statement that appears reasonable, but has not yet proven to be true. When we are solving a mathematical problem, it is important that we conjecture because they form the backbone of mathematical thinking. Half the battle when it comes to mathematical thinking is getting a sense of the question and establishing what might be true and articulating these conjectures. When we form these conjectures, we are focusing our attention on useful components of the question. This allows you to ask what can be done by generalizing and specializing because when conjectures are made, a larger pattern emerges. Thus, we come back to specializing and generalizing being two very important aspects involved in conjecturing. Therefore, when we conjecture, we must become open to new interpretations of the problem and articulate, test, and modify our conjectures to form the backbone for our resolution. Just think... conjectures are kind of like butterflies. Like butterflies, conjectures are not easy to capture, but once one comes by, there are many to follow. 

The second main component of the ATTACK phase is justifying & convincing. This phase is all about seeking and explaining why. It is one thing to conjecture WHAT is happening in a question, but it is much more difficult to see WHY it is happening. Justification and convincing is necessary to frame an argument for your conjectures. Articulating the link between what you KNOW and what you WANT is the essence of one's justification for a conjecture. This can be done following three basic steps:

1. Convince yourself
2. Convince a friend 
3. Convince a skeptic

Justifying every step of your argument will convince both yourself and others behind the WHY of your conjecture. It is also important to develop your own internal skeptic in order to question your own steps allowing you to CHECK your work without the assumption that you completed your resolution without any errors. 

The only way to question and test the correctness of your conjecture is by attempting to refute it - be your own skeptic. This is just another way to sharpen your critical skills in looking at your own and other people's arguments to decide whether or not you are convinced. 

The last main component of the ATTACK phase is dealing with being still STUCK. Sometimes, when all else fails, we realize that we are still stuck with the problem we are facing. When we are working on a problem, we have to accept the fact that there is no way of getting around the work we need to do that is necessary to carrying out the problem. However, if you encounter still being stuck, there are three options you can choose from:

1. abandon the problem altogether;
2. put it aside for a while;
3. keep going. 

Here is what we can do when trying to push through being stuck:

• distilling the problem to a sharp question;
• intentionally mulling;
• more extreme specializing and generalizing. 

To distill the problem to a sharp question we must articulate as clearly and succinctly as possible the essence of the problem. Sometimes it helps to do this by explaining it to someone else. This will help us juggle the components of the question so that new combinations and connections can be formed. When we are able to make these new combinations and questions, we are mulling the problem. Lastly, to engage in more extreme specialization and generalization, we are altering the current problem that we cannot solve until we can solve it. Sometimes the problems we are attempting become clearer when stated more abstractly with the inessential details removed. 

It is important that we try to remove our hidden assumptions from the question because a changed perspective or insight may be exactly what you need to effectively ATTACK the problem. 

Well... now you know. This is what you have to do to ATTACK! Build on your mathematical thinking and keep going! I hope I taught you a thing or two in this blog post.

Until next time...

Signing off, 

    

            Ms. Blackwell