As teachers, we don’t typically like to admit when we don’t know something in front of our peers and especially in front of our students. Luckily for us, if we can stall long enough to get to our phone, Google has made it quite handy in making those moments extremely short-lived. The unique opportunity of being a teacher however, is using those moments to reflect on how or why you never learned that particular idea, and in this instance, what the answer really is!
After working through this choral count: https://mathmindsblog.wordpress.com/2015/04/20/choral-counting-decimals/ and https://mathmindsblog.wordpress.com/2015/04/22/investigating-patterns/ my students have come to some really interesting noticings and looked deeply into some proofs of why those patterns are happening each time. Most of this has been focused on properties of multiplication and division and thinking a lot about relationships between factors and multiples. One group of students, however have begun to really play around with the “switching” of the digits in the multiples of .3 (and 3’s since they noticed their similarities) and will rest until they understand why.
I see them working so hard because they WANT to understand. I will completely admit, the closest thing I could come up with is the divisibility rule I “was taught” for 3’s. I wasn’t going to tell them this “rule” because I realized, in that moment, they uncovered something I could not explain to them at all because I never truly learned it. So instead, I sat with them, and we thought through it together. We played around with partial quotients and noticed we could always make dividends that were divisible by 3 any way that we moved the digits around. But, why? I had one student finally just ask…
“Mrs. Gray, do some things just work in math because they just do?“
I quickly said no, but that was exactly what my problem was, I never truly learned why numbers were divisible by 3. I thought it worked because it just did, why would my teacher tell me otherwise? I completely remember copying down all of the divisibility rules, memorizing them and acing the test I took on it. It seemed like a really cool trick that just worked because it did. Today, I know I could easily Google it, find a video with an explanation, but I want to think about it more. I want to play around with the numbers and understand why this works with 3’s, so I can really learn it this time around. I want to be like my students…struggle, persevere and learn.
It is moments like this that make me feel so amazing about the thinking and learning that happens in my classroom and the classrooms of so many of the wonderful colleagues I have in person and on Twitter. We want our students to truly understand the math, not simply just be able to do the math. This is especially true for me in this moment. I could easily have told the class that they can switch the order because the sum of the digits will still be divisible by 3 and that is the rule for determining a multiple of 3, it just works. But I don’t want my students ever thinking math is a series of things that “just work because they do” or something we learn in school and never revisit to think deeper about it. I want them to see us all as learners, which is why I continue to play around with this 3 thing…I will get it:)
-Kristin
MathMinds 
This post is so beautiful for so many reasons. I love how you said that you could look up the explanation and that would be it, but that you’re excited to work on in too as a learner with your students. I feel like there are so many times when I encounter something I know but don’t really know. I have been curious too about divisibility rules and I still remember when I had moments of insight about invert and multiply or moving the decimal point that I never had when I was a student. Choral counting, in particular, has generated some great questions whose answers are not immediately obvious. Like the 9s trick that a lot of students learn about their 9 times table — choral counting has actually helped us develop some good explanations for that in a way that makes sense to grown ups and adults. I am so excited that you shared your thinking. I couldn’t help but to think about Michael Perhan’s shadowcon…wonder about whether this is a situation where we might work on it for a while, need help but want a well constructed hint that still lets us do the thinking! You’re amazing Kristin.
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Thank you so much! I was thinking of Michael also because we never really talk about hints when we, as teachers, don’t know the answer or don’t know to how to even get the students thinking about it! Like when one of mine wondered what happened if there was a fraction in the denominator, I had no idea what to say to get them thinking about it. I lucked out that “Why don’t you play around with what you know about fractions to think about it” worked! Sometimes I luck out!!
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Hi Kristin, thanks for sharing your journey. It’s always inspiring for all of us when you model the learning. Something I still struggle with myself. I’ve just started following your blog since meeting you @ the NCTM conference and am so glad that I did. You now have me intrigued with why the rule works … Hope to continue connecting with you.
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Kristen- I love this post, both for the choral counting aspect and the admission of personal struggle. I recently blogged about a similar experience:
http://exit10a.blogspot.com/2015/04/between-6-and-8.html
In my case, I eventually called on my supervisor for help. Many teachers are afraid to admit that they do not fully understand the math concepts they teach, and see that admission as weakness. I think it’s important that we are very public about our own struggles to understand. Maybe if teachers see that we need help sometimes too, they will be more willing to reach out when they have issues with the content they teach. Rather than weakness, I think it shows courage to make those admissions and reach out to colleagues and administrators for help.
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Thank you Joe, I cannot wait to read your post. Your comment could not be more true and I have to admit, it was a bit scary (and intimidating) to hit Publish on that post. It is so hard to admit we dont know something we think others perceive we should, whether it is students or colleagues. I think courage is a perect word for it Just imagine if everyone could be ok openly saying “i dont know” or “i am confused about..” how much more powerful our PLCs and classrooms could be! Thanks again, it means a lot!
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