Why are there teachers out there still teaching multiplication (and division) with decimals where the decimal is moving?

The answer to this question is simple. It’s easy. It doesn’t take as long to teach (though when you look at all of the time spent on remediation, I tend to disagree here). And it takes little preparation.

Let’s take a look at this. First: It’s easy for teachers. It is! I agree. If the teacher has this procedural understanding down, all they need is to find a set of computations (usually in the form of a worksheet with no context) Unfortunately, this procedural understanding breeds more procedural understanding and neglects the sense making necessary in learning mathematics. Teaching any mathematical procedures at the expense of making sense is like teaching only phonics with no connection to literature and comprehension.

By teaching moving the decimal, teachers are undoing any understanding of place value (and this is often surface understanding). Think about it. In first grade, students learn that when you get too many (10) popsicle sticks they need to get grouped together to make one ten. This requires sense making on the students’ part. The students are beginning to think of the group of ten as a unit. The “ten” is a unit and they can work with that unit in much the same way as they work with a “one.” In terms of place value understanding, the physical grouping and the representation on a place value chart help students make the connection between the digits and the values of those digits due the quantities of popsicle sticks (or any other material). The digits are moved to a different place value based on the quantity. Quantities connected to groupings connected to place value. It makes sense to students when they experience it consistently.

- Two popsicle sticks are represented by a digit 2 on a place value chart
- When we get to 20 popsicle sticks (10 times as much as 2), that digit 2 that was in the ones place is now moved to the tens place. The digits are placed based on the quantity they represent.

Flash forward to 5th grade (for example). A student is learning to multiply decimals and the teacher is teaching procedural methods where students are told to move the decimal. What if the student gets the incorrect product? Do they know? Are they aware that their computation is off? Most likely not. They have been taught to follow procedures (often blindly) and if they do, they’ll get the correct product. So, when they do make an error, they are not concerned, because they’re being taught to be robots. Follow these steps and you’ll get the right answer. Here’s how it might sound in a classroom:

**Problem on the board: 10.030 x 0.03**

Teacher to student: You made a mistake.

Student: (answer 0.03009) But I followed the steps.

Teacher: You made a mistake. Please check your work.

Student: Ok. (after a few minutes) I got the same thing. I checked my steps.

Teacher: Did you check your multiplication? Maybe your error is in the facts.

Student: Yes. I checked the multiplication – all of my facts were correct. I don’t know what I did wrong.

Teacher: Let me see. (a few minutes pass) Right here. Your decimal is in the wrong place.

Student: But I counted the places and counted back. Why did I get the wrong answer?

Teacher: The answer is 0.3009.

Student: But if you count the decimal places, the decimal should go 5 places back, not 4.

Teacher: Hmm? Thank you for bringing this to my attention. I’ll take a look at it. . .

This scenario is very informative. First, it’s obvious that no one in this situation “owns the math.” The teacher is trying to be the owner, and in the student’s mind, it may be the case – as soon as the teacher says, “let me see.” The student is trying to make sense (once the teacher corrects him), but can’t and doesn’t even know where to begin, due to the limiting procedural understanding in place. Based on the “rules” the student learned, he is correct. So why is the answer incorrect?

One thing that the teacher did well is admit that he doesn’t know and that he wants to try to make sense of the situation, but that’s really just the beginning. Students should also make sense of why the rule fails here. Instead of blindly following rules, students should be estimating and using what they know to make sense (about 10 x 0.03 = 0.3 so my answer should be really close to 3 tenths). This should be a part of every student’s math day. It can’t just be told to students. They need to experience the value of estimation through problem solving situations on a daily basis. Over time, students adopt this valuable strategy and use it readily in multiple situations.

Learning (and teaching) mathematics is about making sense, not just procedures. There’s no better time to start than the present!

This makes every bit of sense, Michael. I’m glad to see someone bringing this dialogue to the pedagogical forefront.

Thanks, Peter! More to come!