Expand Students' Horizon of Thinking

I have heard some very wise saying, it says "What we know isn’t what we need to know. Familiarity, process, and our comfort zones are only holding us back."

Too much familiarity is really bad for creativity. Too much experiences with the common problem may narrow down the choices that you have to deal with the problem at hand, you stick to the old ways of solving problems and you are unable to produce new ideas.

But creativity is incredibly important in problem-solving – if you're less creative, you'll struggle to understand the issues surrounding a problem, and you're unlikely to produce the best solutions. You're good if you could find solutions that work, but what happens when you could not solve the problem at all? That will definitely put you in very precarious position and you would lose your status or even your job when someone else appears and proved to be more creative than you.

Therefore, to compete effectively, one needs to have the quality standards in place. As mathematics educators, we need to start now to bring in interactive and engaging activities that promote productive thinking, exploration and experimentation.

Take for example, knowing the following that are so common and familiar are no longer enough:

$x^2+2xy+y^2=(x+y)^2$

$x^2-2xy+y^2=(x-y)^2$

$x^2-5x+6=(x-2)(x-3)$

$x^2-6x+9=(x-3)^2$, etc.

You have to toy around with the expressions:

[MATH]\color{yellow}\bbox[5px,purple]{(x+1)(y+1)=xy+x+y+1}[/MATH]

[MATH]\color{black}\bbox[5px,pink]{(x-1)(y-1)=xy-x-y+1}[/MATH]

[MATH]\color{yellow}\bbox[5px,green]{\begin{align*}(x+1)(y+1)(z+1)&=xy(z+1)+x(z+1)+y(z+1)+1(z+1)\\&=xyz+xy+xz+yz+x+y+z+1\end{align*}}[/MATH]

[MATH]\color{black}\bbox[5px,yellow]{\begin{align*}(x-1)(y-1)(z-1)&=xy(z-1)-x(z-1)-y(z-1)+1(z-1)\\&=xyz-xy-xz-yz+x+y+z-1\end{align*}}[/MATH]

[MATH]\color{yellow}\bbox[5px,indigo]{(x+2)(y+2)=xy+2x+2y+4}[/MATH]

[MATH]\color{black}\bbox[5px,orange]{\begin{align*}(x+2)(y+2)(z+2)&=xy(z+2)+2x(z+2)+2y(z+2)+4(z+2)\\&=xyz+2xy+2xz+2yz+4x+4y+4z+8\end{align*}}[/MATH]

Use your imagination and experience to think of other list of expressions for your students to improve and expand their horizons,and as usual, if you've something to say, you're welcome to post it in the comment section below this post.