# Writing in Math: After a Number String

Many people ask me when and how I use journals in math class. At those moments,  I always seem to have so many reasons that it is hard to pinpoint just one to focus on during the conversation. And even when I seem to find a coherent way of explaining when and how I use journals, I often forget the reasons that seem to happen naturally in the classroom. The other day I had one of those moments that I think Joan Countryman, author of Writing to Learn Mathematics, would classify as continuing the conversation.

During Number Talks or Number Strings it always seems to happen…one student has a way of solving the problem that, as he or she gets midway through the explanation, the rest of the class begins to disengage either because it is a long explanation or they are lost in what is being said mathematically. Journals help me continue that conversation with the student who is sharing. I attempt to clearly capture what is being said, but ask the student to tell me more in their journal because I am so interested to hear all of their thinking.

This particular string was in a 3rd grade class who has been working with multiplication. I wanted to see how they thought about changing one of the factors in a particular way. This was the string:

3 x 4

4 x 4

6 x 4

12 x 4

The majority of students shared strategies that involved either skip counting or using repeated addition of one of the factors. Some used previous problems (which was my goal) to help them with the new one, however there was one student who started using 5’s for the last two problem instead of either of the factors. He had a very clear way of explaining it, but I could tell many students were beginning to get lost in the explanation. I encouraged the students to ask him some clarifying questions, but that conversation began to stretch this number talk a bit too long time-wise. Not to mention, many had stopped listening at this point.

I was so curious to hear more about his strategy because to be honest, I was getting a bit lost in his explanation of 12 x 4 using 5’s. I told him I wanted to hear all of his thinking but we needed to finish up the number talk to get started with class. I asked him to explain to me what he as doing with 12 x 4 in his journal and I would be sure to check it out later! He went right to work and knocked out this beautifully clear explanation, not just for 12 x 4 but EACH of the problems!

The thing that I appreciated most was the opportunity if gave me to continue this conversation with him. I could feel he wasn’t done explaining his strategy during the talk and this also gave him the chance to think about how he could clearly communicate it to me in his writing. What a powerful thing for a student to be able to do! It was amazing to me he had done all of that decomposition, adjusting, and adding in his head!

So, if you asked me this week for a reason I have students write in math class, it is to continue the conversations that are not quite ready to end during our class time together.

A couple of weeks ago, I blogged about my planning with a first grade teacher here.  After teaching the lesson, the students did an amazing job with the dot images we chose to use. Some students moved the dots to make the dice look the same on both sides of the equal sign while others solved both sides. On the last image they easily decomposed the 4 into the two 2’s to prove both sides were equal so that was something we were hoping to see transfer into the dot image activity.

We walked around, recorded the expressions we saw students writing, and asked students questions about their strategies for choosing cards. As I do with many lessons, in thinking about their strategies beforehand, I referred to the Learning Progressions to see how students progress through algebraic reasoning.  If they didn’t know the the addition expression from memory, like 3+3 or 5+5, this clip from the progressions best describes how I was seeing students arrive at the first expression written for each given sum. Because the commutative property was the way most students found the second expression for each sum the day before, this particular day we told the students they had to use different cards than their partner in thinking about writing their expression.

I especially loved this passage in the Progressions about counting on…I had never thought of counting on as seeing the first addend embedded in the total, although it makes complete sense now! I wonder how understanding that could impact the way in which I question students about their thinking when adding?

What we were looking for as we walked around in particular was how students were using either this Level 2 method above or, what the progressions would call it, Level 3:

It is hard to convey all of the conversations we heard, however here are some of the game boards I captured after the finished playing the game. (Some boards were 6,9,10,15 and others were 8,9,12,16)

These partners seemed to think individually about their expressions on the left and right sides of the board. The student on the left appears to use facts they know such as 7+3 to arrive at 4+3+3 (since there were no 7 cards). I love the use of the equal sign between the two columns!

The other two pairs appears to have done the same thing…

The two groups below, I remember talking to because I was so interested in how closely their sides were related. After the student on the left had written their expression, the student on the right either combined or decomposed numbers to write an equivalent expression. I would love to talk to both groups about the sum for 12 because I am curious if they are decomposing and making a “new” number based on what they are “taking from” another number.

After playing the game, we put the equations we saw for each of the sums on the board and asked students what they noticed. Some noticed relationships between the expressions for a given sum while others looked at expressions for various sums. For example, when looking at the expressions for 10 and 15, they noticed that each expression added 5. Then we discussed whether that 5 was always a 5 and students were really comfortable saying that it could be a 2 and 3 or a 4 and 1. They could have shared their noticings for quite a while so we asked them to go back to their journals and describe something they were noticings among any of the equations.

It was at this moment when I started to detach myself from the math for a quick second and began seeing how journaling really begins. I found I take it for granted that when I say write in your journal about something, that they understand how we explain our mathematical thinking. I know that writing at various grade levels differs based on so many things such as vocabulary, writing experience, and just how they write words in general. However, one thing I did not think so much about is how students view writing in math. I did not realize until I saw this student showing all of his compensation in numbers by connecting the numbers that were staying the same with lines and showing the number that was “one less” by writing -1 when going from an expression that totals 10 to a sum of 9. He explained it so beautifully but was having trouble communicating that on paper. When he finished talking a girl next to him, asked me, “Can we use words too?” <—- that is when I had an aha! Do students think about writing in math as only communicating numerically? Do we ever explicitly tell them it is ok to write about math in numbers, words, or we can use both numbers and words? I think I have always assumed they knew.

Then I came back later and the very same girl had written all of this wonderful thinking…

This student showed a wonderful connection to what was happening when he went from 6 to 9 and then from 10 to 15:

After they had finished journaling, the students moved to recess, however this student sat for another 20 minutes explaining to me all of the wonderful thoughts he had in his journal. The arrows were movement of numbers that were changing however being able to clearly communicate that in his writing was not something he was able to capture clearly. THIS is the power of writing in math I think…learning to take all of the amazing thoughts and communicate it clearly because the more he talked it out to me, the more arrows he drew, the more he elaborated on his thoughts.

Moving forward from here there is so much to think about for me….in addition to moving students thinking about addition and relating that to subtraction, how do I begin to think more about journaling in math, how does it really start?

For Dot Addition game I am wondering if we could allow some students the option to use subtraction? Make the range of card choices larger to allow for students to play around with that relationship. It is something that I thought about as I looked at the table in the Learning Progressions..

So much to think about each time I leave a classroom!

~Kristin

# What DO they know?

I love reading and giving feedback on my students’ journals,  I make time for it every day. But the mere thought of grading papers (feedback or not) makes me world’s biggest procrastinator. Unfortunately, my reality is that I need grades for progress reports and report cards, so I make the best of it. I try to make the assignments valuable for both the students and myself in their learning, however  I always wonder why I don’t approach the papers the same, they are both student work right?

I had a realization yesterday while I was grading, as to why I make time to read their journals vs the aversion I have to grading papers. While I was grading, my mind was focused on what the students DON’T know, what they aren’t getting, aggravation at the careless mistakes, aggravation that I didn’t “reach” that child and why they don’t all have 100%. As I graded, I was busy making notes in my own journal of the students who were missing certain items so I could make my plan for next week to help them better understand the material. And while I know this is invaluable in planning to better teach my students, I realized I was completely glancing over what they DO know. I was checking off the problems they were getting correct and focusing solely on the wrong. Don’t be mistaken, I LOVE mistakes in math, I love analyzing what students could have been thinking, misconceptions and/or misunderstandings, but when grading, the feeling is still not the same.

This focus on “wrong” wasn’t the only thing that bothered me though. I also wasn’t “feeling” my students’ voices in the assignments, like I do their journals. Maybe it is because I love hearing them talk about math so much, their journals are the next best thing when they have left class for the day.  Maybe it is the freedom for them to take more chances in their journals or simply say, “I don’t understand it from this point on..” that makes them so much more enjoyable. Or maybe it is the mere fact I don’t have to put a grade to their thinking. As I read their journals, I am looking at everything they DO know and how that led them to where they are instead of the other way around.

Their journals feel more like the way we learn then grades do. We try, we make mistakes, people help us along the way with advice, we try again, we test things out, we look back at what we did to build on it….no number is attached to that, so why grades? I would like to think I try my best to not have grades be a focus in my classroom and instead be a snapshot of where students are right now in their learning, but those assignments still do not hold the same value that their journals do for me.

Maybe someday standards based grading will make its way into our district but until then I will continue to read their journals for things just like this…

– Kristin

# Math Class Through My Students’ Eyes…

Each January, I like to ask the students to do a reflection on the first half of the year…things they liked, didn’t like, things they still want to learn, questions they have, etc…

Some students gave me a list of things they have learned by topic, others suggested that their seat be moved because they think they would work much better with their best friends, while some offered the suggestion of doing a “math project” that they work on over the course of a month or two (like their science fair project). I do like this last idea and looking into some type of ideas for this:)

I could really post all of them, because I just think my students are the coolest, most honest people I know, but for the sake of time, I chose two to reflect on tonight because I think it says a lot about what I hope students leave my class thinking about each year….

This is exactly why I started the Class Claim wall! I SO love that this student enjoys proving why things work, and even better that she started the sentence with the word “Actually,” like it was not expected! I also think it is so awesome that she said multiplied fractions before she even realized she was multiplying fractions! It makes me feel so good about all of the planning and work for the cornbread task which launched this unit.

This one just made me chuckle at the subtrahend and minuend talk. That came out of a number talk one day when they were calling them the “one you’re taking away from” and the “one you are taking away” and wanted an easier, less wordy way to say it (don’t know if those words are, but stuck for this student). It did make me reflect on my work with Virginia Bastable this summer when she said (I am putting quotation marks, but this is not verbatim),  “Vocabulary should be a gift for the students in their explanations, developed out of need.”

The second part was just too funny and completely what I do to these poor kiddos all of the time! He has learned that when he has a finding or “idea,” I don’t just give him an answer, but instead send him back to think about it and see if they can figure out why that is happening. Then with another idea, the same process ensues…but at least, “it is not as hard as it seems.”

This is exactly what I want, curious students who work to explore their ideas and strategies and learn the processes of “doing math” without knowing there are procedures in place to do exactly what they are doing. I want them to see the “hard” math work they do as fun and an invaluable part of their learning.

They would probably be very surprised to find out that they make me do all of these same things before, during and after each lesson….

-Kristin