# When My Students Uncover Something I Never Learned….

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

# Investigating Patterns

Due to ELA testing, I luck out with an extra 45 minutes of math time twice this week, and today was one!! I wanted my students to revisit the choral count we did on Monday and look deeper into the patterns they noticed. To extend that thinking, I wanted them to make some predictions about decimals that may or may not show up if we continued counting by 0.3 (Thanks so much Elham for the suggestion:)!

We revisited the count and the noticings…

I then wrote some decimals on the board, shown inside the rectangles (kinda) in the first picture above. I asked them to try and use the patterns they discovered to decide if the decimals would show up if we kept counting by 0.3. I was sure to choose a range of options so everyone had an entry into the investigation and focused on the patterns we had discussed. I loved the way they explored their patterns and it completely intrigued me the manner in which they do so.

Some explored by multiples of 3 by looking at wholes and then tenths…

Some used the patterns that involved just one place value but did not look at the decimal as a number…

This group looked at the decimal as a number and chose one pattern they know would work for any number. They broke each decimal into partial quotients to see if each part was divisible by 3…

Other groups used a variety of patterns, noticing that some would work nicely for certain decimals and not others…

The next two especially caught my attention because I had not anticipated the connections being made (I ADORE the way they think:)..

Let’s look at the first one…He saw the “switching the digits around and the other decimal always shows up” pattern working every time and decided to examine the why. His approach was so interesting. He decided to look at the missing addend between the number and its “switch” each time.  He noticed the missing addend was always a multiple of 0.9. He then started to look at the relationship between the original numbers and their missing addend. For example (and I so wish you could hear his thinking on this) the missing addend from 1.2 and its switch was 0.9 and the missing addend from 5.7 and its switch was 1.8, so what is the relationship between 1.2 and 5.7 that explains why the missing addend doubles? My curiosity is..what makes that be the next step for some students while others just notice it the missing addend is a multiple of 0.9 and are content. Loved this moment today because I got such insight into how students look at different pieces of a “puzzle” and choose to explore different relationships.

This one was so funn…

She noticed that any two numbers in her list (table), added together, had a sum that also appeared in the table or would appear, if extended. I asked her how she knew that and she showed me a few examples. “Ok, but why?” She thought for a while and then said, “Okay, it is kind of like the even plus and odd number will always give you an even number.” I could tell she was starting to make sense of the structure of numbers but having such a struggle in explaining it. To her, it seemed to just make sense and I think (hard not to make assumptions) that she was thinking about that 0.3 being a factor of both so duh, it just is.

She came back up, an hour later (she kept working on it when she left me:), and said she had it…”it is like DNA.” Ok, now I am intrigued. She explained it to me and I asked her if she could write that down for me because I thought it was so cool…

It seems like a stretch and I am still thinking about the connections, but I am stuck on the piece in which she says, ” …may look different but act similar…or act different but look similar….”

How many connections to factors and products, addends and sum and such ring true in this statement?? I love when they leave me with something to think about!!!

Another great day in math!

-Kristin