Mental Math and Problem Solving

Mental math is when you apply known math rules and facts in an undocumented way. People think of mental math as kids doing calculations in their head. How does this relate to problem solving? Well, what is:

1+1=2 

It is a problem that equates two quantities: "2" and "1+1".

In fact it is an equation that expresses a whole number in two different ways.

Because you can count and you possess intuitive number sense and you can understand the + sign you know that one plus one equals two.  Since you know one plus one equals two without using a calculator or a pencil you know it because of ...'mental math'.  Well what about:

6+16+26=48

6x8=48

6+6+6+4+26=48

6+16+10+10+6=48

6+6+6+4+6+4+6+4+6=48

6+6+6+6+6+4+4+4=48

6+6+6+6+6+12=48

6+6+6+6+6+6+6=48

2x3x2x2x2=48

3x2⁴=48

Over time, you're going to know that the prime factors of 48 are 3 and 2. You're going to know that the square root of 48 is not a whole number. Your new knowledge will be 'mental math'. 

You are going to be asked to 'justify'' your problem solving strategies. That usually means spelling out the mental math you used to find answers. Therefore 'mental math' often becomes personal knowledge that must be proven (in writing) every time you use it.  It must be acknowledged that personal knowledge is always subject to rounding errors: you're unique therefore your understanding of any fact may also be unique. Is your personal knowledge 'good unique' or 'bad unique'? That's what your teachers and colleagues want to know.

I'd like you to review these very elementary 'mental math' facts. Do you know them all? I bet you do!

Next, read page 11 in Math Makes Sense 8.