I have always been a proponent of conceptual understanding throughout my teaching career. In the earlier years, I referred to it as "Higher-Order Thinking." However, I wasn't in favor of the long, drawn-out processes used to help students understand math. I had proven results teaching students various tips and tricks to learn math.
I'm happy that things have changed. At times, I feel that I focus too much on conceptual understanding. In the last couple of years, I have taken pride in ensuring that I stay away from quick trips to mastery and instead focus on building a solid conceptual understanding of the skills. I repeatedly ask my students to make connections and explain why certain things work.
HOWEVER...
There are many things that I want my students to notice – that quick check to ensure their answer is correct or that rapid way to figure out an answer mentally. I find nothing wrong with me bringing these shortcuts to my students' attention, as long as they understand why.
FOR EXAMPLE...
A whole number multiplied by 10 is that number with a zero in the ones place. I want students to notice that the digits remain the exact same; there is just now a zero in the lowest place. I told my kids to imagine the "Rude Zero." When a whole number is multiplied by 10, the rude zero comes and pushes the other digits so it can get in the lowest place.
Prior to conveying this to my students, I taught them multiplication by 10 using the place value chart, representing numbers with disks and bundling. We discussed conceptually what happens when something is multiplied by ten through various drawings, diagrams, and representations. I taught the overall concept of things that are ten times as many. However, I want my students to practice like mathematicians and understand that any whole number multiplied by 10 is that number with a zero at the end. That's a shortcut by exploring patterns when solving. If they used 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 it does.
SUMMARY:
I think it's perfectly fine for students to notice patterns once they understand why those patterns exist. It's like having a helpful tool in math that makes things easier. So, after they understand the concept, I encourage my students to look for shortcuts or patterns that can make their work simpler. This helps them practice math in a smart way and can help improve efficiency.
SUGGESTIONS:
Make it a habit to ask students why shortcuts or patterns exist. You can create what I call a "But Why?" board. Our "But Why?" board is a space we use to explain why shortcuts or patterns we notice work. Students are allowed to put patterns or shortcuts they think they've noticed on the board, and we routinely check the board to discuss them. For instance, one student may notice that every whole number multiplied by 5 has a 5 or 0 in the ones place. They may use what they've noticed to eliminate division and multiplication mistakes. However, if they do not understand how this is possible, they could put it on the "But Why?" board. After that, students have the opportunity to discover, or we may just discuss as a class that any whole number multiplied by 5 has this pattern. 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 a routine in quizzes and assessments to highlight the importance of students knowing why. An example of a question could be:
I hope these two suggestions prove helpful! Conceptual understanding is absolutely essential, but incorporating shortcuts is a clever approach for students with a solid understanding to apply their knowledge effectively. It's what I consider working smart.
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