QUICK TRIPS OR CONCEPTUAL UNDERSTANDING?

I have always been big on conceptual understanding since I've been teaching. In the earlier years, I referred to it as "Higher-Order Thinking". However, I wasn't for the long drawn out processes to help students understand Math. I had proven results teaching students various tips and tricks to learn Math.

I"m happy that all of that has changed. Sometimes, I feel that I am too much about conceptual understanding. The last couple of years, I have really been proud of myself ensuring that I'm staying away from quick trips to mastery and truly build conceptual understanding of the skills. I repeatedly ask my students to make connections and to explain why certain things work.

HOWEVER...

There are many things that I do want my students to notice. It's like that quick check to ensure their answer is correct. Or that quick way to figure out an answer mentally. There is nothing wrong with me telling my students that, as long as they know why.

FOR EXAMPLE...

A whole number x 10 is that number with a zero in the ones place. I want students to notice that the digits are the exact same, there is just now a zero in the lowest place. I told my kids to imagine the "Rude Zero". When a number is x10, the rude zero comes and pushes the other digits so it can get in the lowest place. 

Prior to telling my students this, I taught them x 10 on the place value chart, representing numbers with disks and bundling. We talked about conceptually what happens when something is times ten through various drawings and diagrams and representations. I taught the overall concept of things that are ten times as many. . However, I want my students to practice like a mathematician and understand that any number x 10 is that number with a zero at the end. That's a shortcut, by exploring patterns when solving. Or if they did use a procedure to solve, that's a check for them to see if their answer is correct. I personally view that as MP.8 of the Standards for Mathematical Practice where students look for and express regularity in repeated reasoning, and look for general methods and shortcuts. As long as I have first taught the conceptual understanding, I do want students to notice patterns to make their work easier and to be able to find shortcuts. I find nothing wrong with that as long as my students understand WHY it works the way that it does.

SUMMARY:

I personally find nothing wrong with students finding patterns after they have been taught the reason WHY the pattern exists.

SUGGESTIONS:

• Make it a habit to ask students WHY? Create what I call a "But Why?" board, where when we notice shortcuts or patterns to help us check our math or find shortcuts, we explain why it works. We notice every whole number multiplied by 5, ends in 0 or 5. We need to use this information to eliminate division and multiplication mistakes. However, we need to understand why any number multiplied by 5 ends in 0 because 5 is half of ten. So when repeatedly adding 5, you are going to either be halfway to the next ten or "at" the next ten.

• Make it habit on quizzes and assessments to show students the value of them knowing why. An example of a question could be:

Tiffany does not understand why when multiplying by ten, the number always end in a zero. What can you say to Tiffany to help her understand why there will always be a 0 in the ones place when a whole number is multiplied by 10?

Hopefully, these two suggestions help! Conceptual Understanding is an absolute must, but knowing shortcuts is a smart way for students who have an understanding to practice. It's what I consider working smart.