"Every fraction is a division problem!" -- 4th grader, Rowan

This week in 4th grade Room 118, we experienced the true value of math teachers collaborating across the country and around the globe! It led us to discover that every fraction is a division problem.

How?

After I posted a fraction equal sharing lesson from Room 118 (12/5/15) I received a post from one of my twitter colleagues, Kristin Gray (@MathMinds) with a lesson idea to extend our fraction understanding.  Kristin had posed similar equal sharing problems to her students and then asked them to look for patterns in the problems and solutions...what did they notice?  What did they wonder? What did they discover? Fractions as division! (See Kristin's blog: mathmindsblog.wordpress.com/2105/03/03/fractions-as-division-say-what/) Seeing the richness of Kristin's students' noticing, wondering and proofs, I was inspired to roll-out a similar investigation in Room 118.  What's follows is the account of our discoveries and connections, as well as some questions for you all.

Building upon the equal sharing work we had done the prior week, the students of Room 118 pursued the following 3 problems adapted from/inspired by Empson and Levi's book, Extending Children's Mathematics, Fractions and Decimals:

1) There are 4 burritos for 5 children to share.  If they each want to have the same amount, what fraction of the burrito would each child get?

2) Mrs. Cornn accidentally only bought 7 candy bars for her 8 nieces and nephews.  She wants them all to have the same share so how much will they each get?

3) We have 13 liters of juice for 10 children to share equally.  How much juice should each child get?

Figure 1A

Figure 1C

Figure 1B

During Math-by-Myself, many of the students recalled their earlier work with equal sharing problems and immediately began to recognize that there was a pattern...4 burritos for 5 children, 4/5; 7 candy bars for 8 children, 7/8; 13 liters for 10 children, 13/10.  (See Figures 1A, 1B, 1C)  One of these students walked right over to the Conjecture Wall and posted a conjecture about the pattern he was seeing. (See Figure 2)

Figure 2 Kaden's equal sharing conjecture

Another group of students started their work with the guess-and-check strategy...start by splitting the burritos in half, then thirds, then fourths and finally fifths.

While the students worked during Math-by-Myself, and also during pair-share, I collected evidence of students' levels of understanding so that I could scaffold various solution paths during our whole group discussion.  Today in whole group time I chose a student, who is more introverted, to share because she had definitely started to see patterns so I wanted to feature her work.  She described her work with Problem #1 as dividing up each of the four burritos into fifths because their are 5 children as seen in Figure 3.  

Figure 3 Dividing up burritos

We discussed why she would jump right to fifths and all of the students connected with her reasoning.  Before we moved onto Problem #2 I created a chart on the whiteboard and asked the group if they could fill in the chart based on the pattern of problem #1 and the work we had done previously.  The students could see the pattern unfolding as we created the chart in Figure 4.

Figure 3 Summary of Patterns in Equal Sharing Problems

When we got to the problem with 13 liters and 10 children we had a discussion about how this problem was different than the first two.  In this case there were two possible solutions, 1 whole and 3/10 and 13/10.  This was an a-ha moment for many students who had painstakingly divided all 13 liters into tenths.  "Of course!", they exclaimed, "Each person could have 1 whole liter and then the last 3 liters would be divided up among the 10 children."  (See Figure 4)  This realization is a true testament to the value of these type of discussions because I could have easily assumed they already understood that 13/10 was the same as one and three-tenths when only a handful truly did.

We extended our chart to include other possible problems so that we could test the pattern we were seeing.  Watching their faces as their synapses fire and their brains make these connections is always a sight to behold!

I felt is was now time to share Kaden's conjecture and to determine whether or not is was always/sometime/never true.  We have had dedicated conjecture-proof days in Room 118 on three occasions this year so the students are somewhat familiar with the process.  (I wish I had been blogging on those days!)  I was encouraged to pursue this process by my colleague, Mark Pettyjohn.  I was a little reluctant at first because of my own negative experiences with formal proofs in high school and college.  Mark said I should just take it to its fundamental form...I notice a pattern, I wonder if it is always, sometimes or never true...let's find out.  As always with Mark's advice, I am glad I pursued it.  So, today, with Kaden's conjecture, the students were comfortable with the process.  Some asked Kaden to clarify he conjecture, especially the last sentence, "Without doing it backwards."  Here is some background information on Kaden's conjecture to provide context for the sentence.  He originally believed that the numerator would always be the smallest number from the equal sharing problems.  I asked him about his solution to problem #3 where he had 13/10 as his solution.  He then saw the gap in his conjecture and added the sentence "Without doing it backwards," as a way to close the gap.  So, returning to the discussion, with that clarification students began to understand Kaden's thinking. Based upon our class discussion, another student, Andy, said he had created a modified conjecture based on Kaden's and he'd like the class to consider it. Here is his conjecture: "[The] Numerator is always the unit number (burritos) and the denominator is the people (children)."  A moment of silence settled over the class as students reflected on Andy's conjecture.  I could tell that some of the students were starting to grow weary, maybe they were confused, maybe their brains were full.  Consequently, I gave them 2 options: (1) Continue to work on refining Andy's or Kaden's conjectures in their Math Journals at their desks or (2) remain on the carpet with me to further understand the pattern we had noticed in the chart and how these conjectures evolved.  The students whose minds had started to drift came back to life and stayed with me on the carpet while the rest of the class found their next clean page in their journals and wrote their thoughts about the conjectures for the next ten minutes.

I was grateful for the time to share with the group on the carpet.  They had a strong understanding of the chart patterns but were less clear about Andy's conjecture.  It believe they had a clearer understanding as they talked and summarized what they knew and what they wondered.

We reconvened our whole class discussion and Andy was all excited.  He had an idea on how to make his conjecture more precise, in alignment with SMP6, Attend to Precision.  He said, "You know how in a multiplication problem you have factors that are multiplied together to make the product...what are the 'factors' called in a division problem?"  I asked if anyone knew and no one was quite sure.  (This was not surprising to me because the students have not spent much time with formal division up until 4th grade.)  I wrote a simple division problem on the whiteboard 12 divided by 4 equals 3 and defined the 12 as the dividend, the 4 as the divisor and the 3 as the quotient.  Another student, Sullivan, exclaimed, "Hang on a second, we need to write this down in our journals!"  (Don't you just love the enthusiasm of 4th graders!)  I then asked Andy what he thought would be the division equation for the burrito problem.  He said, "Four divided by five is 4/5, so the 4 is the dividend, the 5 is the divisor and the 4/5 is the quotient."  I will confess, at this very moment I had one of those moments when I can hardly contain my math teacher joy!!!  I asked Andy what made him connect this to division and he thought it was because we are dividing up the burrito.  He then asked to share his revised conjecture which is shown in Figure 5.  We then tested his conjecture with the numbers from our table and for that set this conjecture seems to always be true.

Figure 5 Andy's Revised Equal Sharing Conjecture

While Andy was discussing his revised conjecture, another student, Rowan had eagerly expressed an interest to add on to what Andy was saying but he patiently waited while Andy explained.  When Andy wrapped Rowan jumped up and said, "This confirms something I have been thinking about...every fraction is division problem!!!"  Inside my mind I, too, was jumping for joy but my external response to the class was, "Hmmm, that is interesting, Rowan...what do the rests of you think?"  The response was silent, reflecting minds and since the clock on the wall said it was almost time for recess we stopped to reflect in our Math Journals with the promise that we would return to this idea of Rowan's another day...soon!

I don't believe that most of the students have made the fraction as division connection yet but I am so pleased that it is out there as part of our ongoing conversation.

I am wondering where you, my colleagues near and far, would go next to strengthen this connection?  Also, what other connections did we we miss?  What are your thoughts on future Problems of the Day?

Also, what is your advice on creating a more rigorous conjecture-proof process, especially in a 4th grade setting?

I wanted to follow-up with you, Tracy Johnston Zager and Marily Burns, that this series of problems included more rectangular shapes that were not pre-divided and it did not seem to alter the students strategies much from the approached they used with the spherical, subdivided orange.  What connection might I have missed or could have expounded upon?

What's up next from Room 118?  Ms. Burns gave me an idea for a problem from Math Solutions's Math Reasoning Inventory that focuses on student understanding of equivalence.  (See: mathreasoninginventory.com/Home/VideoLibrary )  In an effort to explore all fraction models: area, set and linear, we are going to look into the linear model problem about who lives further from school, the student who lives 3/4 of a mile away or the one who lives 6/8 miles away?  Stay tuned!

Characteristics of a successful math lesson:Our day...your thoughts?

In Room 118 (one-eighteen) today, it was one of those days when everything came together into a beautiful mathematics symphony!

Figure 1: Solutions portrayed akin to Japanese bansho

Students and teachers alike were moved to comment on the power of the fraction lesson including the student that commented, "I love it when we talk about a math problem this way because it helps me really understand math and answers the questions I have."

The questions I am asking now are why, how and what? Why was it successful for all? How did it all come together? What did we all do that made it work? The reason...we want every day to be like today for the shear joy of learning mathematics!

In Room 118 we have a math structure we call the Problem of the Day. There are 4 parts, currently, to the Problem of the Day: Math by Myself, where the students work alone for at least 10 minutes, as the teachers observe student work around the room; Plus Power, where we pair-share; Knowledge Multiplier, where we share our ideas and build on these ideas about the problem as a whole group; and finally, Reflection, where we individually respond in our math journals to the question, "What did this problem make you think about in math today? Why or how?" We have been using this structure since the beginning of the school year including today during our math symphony. So what made today special?

We began our study of fractions a few weeks ago. I decided I wanted to seek out some expert advice on the teaching of fractions so I went to Empson & Levi's book, Extending Children's Mathematics, Fraction and Decimals, and read/refreshed my understanding of equal sharing and multiple group problems. I wanted to pursue a multiple group problem but felt that we needed to scaffold to that type of problem via the route of equal sharing. The authors suggested using a problem that would produce equivalent fraction solutions so I adapted one of their problems:

 "There are 6 candy bars [I used oranges,] for 8 children to share. If each child got the same amount, how much would each child get?" (p. 35)

I knew this problem would be challenging for my students because it produced an answer that would be less than one whole. The 4th graders never cease to amaze me with their thinking and today was no exception.  The many paths to their solutions of 3/4 and 6/8 were intriguing, as well as the rich discussion we would have about each of the solutions presented.  I would say that the majority of the students had an entry point into the problem, during Math by Myself, which contributed greatly to the successful symphony.  I also believe it met most of the students' "intellectual need," something they could really dig into versus just doing a math problem.  Many started splitting the oranges into halves and found that they had too many pieces (12) so some tried thirds while other move quickly to fourths which led to the 3/4 soltuion.  Some combined the halves and fourths and found each child had 1/2 + 1/4 pieces.  A handful of students focused on the fact that there were 8 children in the problem so they divided one orange into eighths by using fraction circle manipulatives.  They moved onto a solution of 6/8s because they knew they had 6 oranges.  I should mention that we have not formally talked about adding or subtracting fractions but they naturally moved to adding the 6 one-eighths to find 6/8.  (See Figure 2.)

Figure 2: Student work that produces a solution of 6/8.

As happens most days, the children completed their Math by Myself work at various paces.  After about 13 minutes I announced that anyone who wished to compare ideas with a partner could move to a designated spot in the room but those who wished to continue their own thinking could do so at their desk.  About half moved and half stayed, which is typical.  Excitement filled the air as they connected with their partner's ideas and wondered why some found 3/4 pieces and others 6/8.  We continued sharing in pairs for 7-8 minutes.

With everyone buzzing, we shifted to the carpet area where we had space to share our ideas on our giant white board in the tradition of the Japanese bansho.  (See Figure 1.)  The first student that shared is someone who is more inclined to follow than lead but this is the student who had found the 6/8 solution and was very proud of her work so she wanted to share.  Many of the students were curious about her solution since the majority had found a solution of 3/4.  At this juncture I tried out a more intentional process of sharing...something known in the math education world as "intentional talk."  I will pause my description of the whole class conversation for a moment to discuss my pursuit of intentional talk.

I have felt that the math conversations in Room 118 were more about sharing-out than truly listening-to and learning from each other.  I went on a mission to find out how to improve our conversations so that all students could learn by contributing, or in the words of SMP 3, to be able to "construct viable arguments and critique the reasoning of others."  Through Twitter, I have discovered many like-minded colleagues including one in Toulouse, France, Simon Gregg (@Simon_Gregg and his blog, followinglearning.blogspot.com) whose rich classroom work directed me towards the idea of intentional talk and the illuminating books by Kazemi & Hintz, Intentional Talk, and Russell, Schifter and Bastable, Connecting Arithmetic to Algebra. By reading postings on Simon's blog and these books I found ideas and methods to make our conversations more intentional.  I am just beginning my informal research and implementation of this work but I decided to take a few ideas and try them out during our conversation today.

Now, returning to our discussion.  I told the students I wanted us to have a conversation today that included all the voices in our class.  Today, if you shared, you would conclude your sharing by asking the other students if they had any "questions and comments" (terminology we use in our Responsive Classroom Morning Meetings when we share) related to their idea.  My role was to make sure we stayed true to that process.  Fourth graders are by nature very eager to share their own ideas but sometimes are challenged to listen to each other.  By making this a part of our conversation structure we did indeed hear from many voices, who don't often contribute, through their questions and comments.  The roof really came off the school building when the conversations zoomed in on how we could have three solutions: 6/8, 3/4 and 1/2 + 1/4!  (Quick background note:  We have created our own fraction kits, a la Marilyn Burns and Math Solutions, and have played both the Cover-up and Uncover games, where equivalent fraction exchanges in the latter game help children quickly learn than 1/2 is the same as 2/4 or 1/4 is the same as 2/8, etc.)  Most of the students quickly saw that 1/2 + 1/4 was the same as 3/4 since 1/2 is equivalent to 2/4 and 2/4 + 1/4 is 3/4.  They were little unsure about 3/4 and 6/8, though.  That surprised me at first but as I thought about it I realized the game, as we play it, doesn't reveal many equivalent fractions where at least one numerator is not 1.  During our discussion we had placed the 8/8 fraction circle manipulatives on the carpet, to see why 6/8s might be a solution, so as we thought about whether 3/4 = 6/8 one student jumped across the carpet with great enthusiasm and said, "Hey!  Look! When I put 3 one-fourth pieces on top of the 6 one-eighth pieces they match exactly so they must be equivalent!"

The student who made this connection is not often totally engaged in the conversation but today, through our structure, it appears he felt more engaged, included and excited about contributing...adding deep percussion to the symphony!

Our final step in the Problem of the Day structure is Reflection time.  I have been inspired by the works of Marily Burns and Magdalene Lampert on journaling in math class and one of my 3rd grade colleagues, April Louthain.  Lately, I have been wondering if the reflections we do in our class are contributing enough towards our learning.  They do give me a small glimpse into student understanding but they are not usually written in enough detail to truly understand what the student is thinking.  (I find our 1-on-1 conversations provide the most insight into their thinking.)  My education partner, Mark Pettyjohn, and I have spent many productive hours thinking about how to create a structure for the most beneficial reflections.  In a recent conversation we decided that we should design the student reflection along the lines of our own general reflection questions so we created, "What did this problem make you think about in math today?"  I eagerly tried the question earlier this week and found some of the student responses were still not specific enough, as when they explain their thinking verbally.  So, for this lesson I added "How or why?" as a follow-up question.  Before I gave the question to the students today we discussed the value of reflection.  Many students said it was important for their learning...it helped them to discuss what they learned and questions they still had about the lesson.    We then contrasted two earlier version of student responses and I asked them students to pair-up and decide which response provided a clear understanding of student thinking:

Reflection 1:  This problem made me think about equal pieces and fractions because we split up oranges.  It also made me think about circles and stars [math game] because...[didn't have time to finish.] 

Reflection 2:  This problem made me think about equality.

The discussion that ensued was enlightening for us all.  Part of the class thought #2 was more specific because the student used a precise mathematical term, "equality."  Others felt response #1 was more specific because the student said a reason for her understanding.  We decided that both reflections had qualities we could incorporate into our own written reflection in the future but including the "because" helped us pinpoint why we felt we had learned or how we had learned it.  When it came time to reflect today I asked them to write for 5 minutes.  They eagerly put pen to paper and they are very eager for me to read their reflections.  I am too!

One final note on Reflections, we built upon one reflection idea from our Calkins' reading and writing lessons, "Think Jots."  I reminded them how they use these jots when they are reading to capture their thoughts in the moment.  This is usually done on post-it notes and placed in their books but we decided we could simply write the notes on our math work papers.  We talked about the importance of writing them just as we are thinking our thoughts so we don't forget our thoughts.  One student commented that on that very day he had brought his notebook to our discussion and taken some notes!  He decided he could use those notes in his daily reflection.  And the symphony conductor smiled!

So, after reading my inaugural blog, why do you think today was a successful day, a mathematical symphony?  What ideas do you have to help us improve our Problem of the Day structure and our intentional talk?

My own reflection is that the success was due to:

> Intentionality in our work and our talk;

> A problem that met our intellectual need and had multiple entry points;

> Connection to prior understanding about equal sharing and equivalent fractions;

> Use of manipulatives and drawings to visualize the action in the problem;

> Modeling and discussing the role of reflections and examples of student reflections;

> Connecting to the work we do in reading and writing;

> Valuable connections I have made through Twitter where colleagues around the world share powerful student-centered, rich ideas.  (Thanks @m_pettyjohn (Mark,) @Simon_Gregg, @TracyZager, @MathMinds (Kristin Gray,) and @mburnsmath, (Marilyn Burns!))

> Taking action to implement the ideas I have learned through colleagues and the user-friendly, inspiring books I have read of late;

> MOST IMPORTANTLY:  The inspiring, enthusiastic, risk-taking 4th grade mathematicians who work hard each and every day!

I eagerly await your thoughts and reflections!  Thank to all of you who have encouraged me to blog!  It has been a enriching day of reflection for me, just as you promised it would be!