In math class right now, we’re learning about Divisibility Rules. The rules themselves are outlined for the students, but the students are, *overall*, really struggling with them. The most common issues seem to be one of these three:

**1) The students confuse the rules.**

**2) The students know the rules, but try to create their own.**

**3) The students ignore the rules and just divide.**

When we started this new topic on Monday, I gave the students a description of the rules, and they had to work together to try and figure out which number was for which rule. I demonstrated how I can pick a number that I know is divisible by a certain number (e.g., I know that 20 is divisible by 10) and start looking at the answer and noticing patterns between answers. I showed how this can help with determining Divisibility Rules. *I thought that this activity would be difficult, but that students would meet with success.* I didn’t see the success that I thought I would though. Working with individual groups, I noticed that many students didn’t know their division facts. Many students also didn’t understand what it means to divide.

I decided to take a step back. I did a review of division. I demonstrated long division again. I showed patterns in division, and I tried to explicitly teach the Divisibility Rules. It looked like more students were understanding them. The next day, I had students use three different Divisibility Rules to sort numbers in a Venn Diagram and explain their answers. I was feeling better about this activity, but a new problem emerged. Students were sorting numbers, but they either weren’t explaining how they got their answers, or they explained how they got them, but they used long division instead of the Divisibility Rules.

Okay, I knew what to do now. I would develop a similar type of activity, but state that the students *had* to use the Divisibility Rules to explain their answers. *This would work! *It did work somewhat, but then I noticed that students confused the rules.

Here’s a student video that I used in my introductory math activity today to re-look at Divisibility Rules:

I’ll admit that when I watched this video last night, I was starting to think that the Divisibility Rule for 6 was wrong. It looked like this student’s new approach could work, but I eventually figured out the problem. Instead of just telling the students the problem, I thought I’d try something different. After playing this student video, I played a YouTube Divisibility Rules video that a fellow teacher shared with me.

Students watched this video, and then the student that demonstrated the math problem in the video, quickly piped in with, “I did it wrong.” Instead of just saying, “yes,” I decided to say, “prove it.” I asked all of the students in the class to work in small groups and try this student’s approach to dividing by 6 and see if it works. *The goal: prove that this is another Divisibility Rule for 6, or prove that it’s not.*

Tonight, I was looking through many different videos from today, and most have similar answers to this screencast here:

*Yes,* these students find out that this is not a Divisibility Rule for 6, but they don’t try out the actual Divisibility Rule for 6. They also don’t come to the conclusion that if the rule doesn’t work every time, then it’s not a rule. Instead, students see this as a strategy that works just sometimes.

Tomorrow I need to clarify this through a Math Congress. I also need to get students working with more Divisibility Rules, and explaining how these rules are helping them solve different math problems. We’ll see what happens tomorrow.

While this week’s math classes have been a lot of trying, changing, and trying again, the one approach that does seem to be working is small group guided instruction. Below is a pencast of one of my guided math groups from today. (Please note that I know I need to work on talking less and giving more open-ended questions, but that’s another blog post in itself. :))

Having the chance to really talk with these students about their strategies and really clarify the Divisibility Rules are allowing the students to be successful. I’ll definitely continue this small group support. **What else would you suggest? How do you get students to understand difficult math concepts?** I’d love to hear your thoughts on this!

Aviva

Hi, Aviva!

I have often found that many/any problems with division / divisibility is that they’re really not seeing the inverse of multiplication. I’ve done similar work as you’ve demonstrated in this blog post… in the end, we’ve gone with posting them in the classroom and referring back to them periodically, through math word prob of the day or that sort of thing. A math mentor one told me that sometimes, you’ve just got to let it go and revisit later. Likely not the solution you were looking for… but a different perspective nonetheless…

Good Luck,

@dawnbheag

@ruralteacher

Thanks for sharing your experiences with this, Dawn. What you said here makes a lot of sense, and I can definitely see how this is the case. Without a doubt, we’re going to have to keep re-exploring divisibility throughout the year. While I hate to move on when the students are still trying to figure this out, I can understand why there might be the need to do so.

Thanks again!

Aviva

Hi Aviva!

I really enjoyed reading your post, as Ann and I are now at the point in the year in which our students are receiving their individual math goals. Your reflection helped me to think more deeply about what is just around the corner for us with our students. Divisibility rules will be a strategy we soon share with some of them.

I tried a few different approaches to this concept last year with some of our 4th and 5th graders. Initially I provided them with just the conceptual term and they conducted a partner research, finding information/samples/charts they connected to. We then met in small groups and I confirmed their findings and then we worked with just a few rules at a time.

What I do believe, thinking about our group this year, is the conceptual understanding can only be developed with our students who have deep number sense. Those who understand basic multiplication and division, factoring, and a variety of strategies for explaining these processes will be ready to truly understand the rules. They MUST have a foundational knowledge to anchor this new thinking to. I watched one of our 5th graders last year struggle to understand the “rules” simply because he didn’t truly know that division was about dividing into equal groups. Another struggled because she did not understand how to explain the purpose of factoring; so how were they to connect to this grander topic?

Many students can learn to memorize step by step processes, algorithms, and facts without having a foundation to stand on, those layers of mathematics that provide us the reasoning. I learned quickly last year that many of our students could understand the HOW but were lost in translation understanding the WHY. For many we ended up taking several steps backwards to strengthen their conceptual “building blocks”, and it was well worth the time.

Not sure if this helps, in regards to where you are with your students, but I would love to hear your thoughts. Are you noticing the sames things in terms of developing their conceptual understanding? Thank you for providing a venue for me to think further on this topic; math has been on my brain a lot lately! 🙂

~ Celina

Thanks for your comment, Celina! I absolutely agree with you. In many cases, I think that this is exactly the problem. I’ve found that when working with small groups and developing these conceptual understandings (especially with regards to what division really means), it’s making a difference on their understanding of the rules themselves. I will definitely be continuing with this small group intervention, even as we move onto some new concepts in math. And yes, I hate to move one when I know that some students still aren’t quite ready, but I’m feeling the pressure of how much I still have to teach in math, so it becomes a bit of a balancing act. We’ll move forward, but continue reviewing these difficult concepts throughout the year.

Aviva