# Fraction Talking Points: 3rd Grade

The 3rd grade is starting fractions this week and I could not be more excited. Fraction work 3-5 is some of my favorite stuff. Last year we tried launching with an Always, Sometimes, Never activity and quickly learned, as we listened to the students, it was not such a great idea. We did not give enough thought about what students were building on from K-2 which resulted in the majority of the cards landing in the “Sometimes” pile without much conversation. And now after hearing Kate Nowak talk about why All, Some, None makes more sense in that activity, it is definitely not something we wanted to relive this year!

We thought starting with a set of Talking Points would open the conversation up a bit more than the A/S/N, so we reworked last year’s statements. I would love any feedback on them as we try to anticipate what we will learn about students’ thinking and the ideas we can revisit as we progress through the unit. I thought it may be interesting to revisit these points after specific lessons that address these ideas.

We were thinking each statement would elicit conversation around each of the following CCSS:

Talking Point 1CCSS.MATH.CONTENT.3.NF.A.3.C
Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram.

Talking Point 2CCSS.MATH.CONTENT.3.NF.A.2
Understand a fraction as a number on the number line; represent fractions on a number line diagram.

Talking Point 3CCSS.MATH.CONTENT.3.NF.A.2.B
Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.

Talking Point 4: CCSS.MATH.CONTENT.3.NF.A.1
Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.

Talking Point 5CCSS.MATH.CONTENT.3.NF.A.3.D
Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

Talking Point 6CCSS.MATH.CONTENT.3.NF.A.3.C Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram.

After the activity, we have a couple of ideas for the journal prompt:

• Which talking point did your whole group agree with and why?
• Which talking point did your whole group disagree with and why?
• Which talking point were you most unsure about and why?
• Which talking point do you know you are right about and why?
• Could any of the talking points be true and false?

Would love your feedback! Wording was really hard and I am really still struggling with #4.

If you want to read more about Talking Points for different areas, you can check out these posts:

# Mathematizing Learning Lab

Each month, teachers choose their Learning Lab content focus for our work together. Most months, 1/2 of the grade level teachers choose to have a Math Learning Lab while the other 1/2 work with Erin, the reading specialist in an ELA Learning Lab. This month, however, we decided to mesh our ELA and Math Labs to do some mathematizing around children’s literature in Kindergarten and 1st grade! This idea was inspired by a session at NCTM last year, led by Allison Hintz, that left me thinking more about how we use read-alouds in our classrooms and the lenses by which students listen as we read.

In The Reading Teacher, Hintz and Smith describe mathematizing as, “…a process of inquiring about, organizing, and constructing meaning with a mathematical lens (Fosnot & Dolk, 2001). By mathematizing books commonly available in classroom collections and reading them aloud, teachers provide students with opportunities to explore ideas, discuss mathematical concepts, and make connections to their own lives.” Hintz, A. & Smith, T. (2013). Mathematizing Read Alouds in Three Easy Steps. The Reading Teacher, 67(2), 103-108.

Erin and I have literally been talking about this idea all year long based on Allison’s work. We discussed the ways we typically see read-alouds used, such as having a focus on a particular text structure or as a counting book in math.

As Erin was reading Kylene Beers & Robert Probst’s book, Reading Nonfiction she pointed me to a piece of the book on disciplinary literacy which automatically had me thinking about mathematizing.

Beers refers to McConachie’s book Content Matters (2010), in which she defines disciplinary literacy as, “the use of reading, reasoning, investigating, speaking, and writing required to learn and form complex content knowledge appropriate to a particular discipline.” (p.15) She continues to say, “…disciplinary literacy “emphasizes the unique tools that experts in a discipline use to engage in that discipline” (Shanahan and Shanahan 2012, p.8).

As I read this section of the book, my question became this…(almost rhetorical for me at this point)

Does a student’s lens by which they listen and/or read differ based on the content area class they are sitting in?

For example, when reading or listening to a story in Language Arts class, do students hear or look for the mathematical ideas that may emerge based on the storyline of the book or illustrations on the page? or Do students think about a storyline of a problem in math class or are they simply reading through the lens of “how am I solving this?” because they are sitting in math class?

Mathematizing gets at just this. To think about this more together, Erin and I decided to jump right into the children’s book  The Doorbell Rang by Pat Hutchins. Erin talked about the ideas she had for using this in an ELA class, I talking through the mathematical ideas that could emerge in math class, and then we began planning for our K/1 Learning Lab where we wanted teachers to think more about this idea with us! We were so fortunate to have the opportunity to chat through some of our thoughts and questions with Allison the day before we were meeting with the teachers. (She is just so wonderful;)

The first part of our Learning Lab rolled out like this…

We opened with this talking point on the board:

“When you change the way you look at things, the things you look at change.”

Everyone had a couple of minutes to think about whether they agreed, disagreed, or were unsure about the statement. As with all Talking Points activities, each teacher shared as the rest of us simply listened without commenting. The range of thoughts on this was so interesting. Some teachers based it on a particular content focus, some on personal connections, while I thought there is a slight difference between the words “look” and “see.”

After the Talking Point, Erin read The Doorbell Rang to the teachers and we asked them to discuss what the story was about with a partner. This was something Allison brought up that Erin and I had not thought about in our planning. I don’t remember her exact wording here, but the loose translation was, “Read for enjoyment. We want students to read for the simple joy of reading.” While Erin and I were so focused on the activity of exploring the text through a Math or ELA lens, we realized that the teachers first just needed to enjoy the story without a purpose.

For the second reading of the book, we gave each partner a specific lens. This time, one person was listening with an ELA lens while, the other, a Math lens. We asked them to jot down notes about what ideas could emerge through these lenses with their classes. You may want to go back and watch the video again to try this out for yourself before reading ahead!

Here are some of their responses:

Together we shared these ideas and discussed how the ELA and Math lenses impacted one another. A question we asked, inspired by Allison, was “Could a student attend to the math ideas without having a deep understanding of the story?”

Many questions came up:

• Could we focus on text structures and the math in the same lesson?
• Could we start with an activity before reading the book, like a probable passage?
• Would an open notice/wonder after the first reading allow the lens to emerge from the students? Do they then choose their own focus or do we focus on one?
• How could focusing on the problem and solution get at both the ELA and Math in the book?
• How could we use the pictures to think about other problems that arise in the book?
• How do we work the materials part of it? Do manipulatives and white boards work for K/1 while a story is being read or is it too much distraction?
• What follow-up activities, maybe writing, could we think about after the book is read?

Unfortunately, our time together ended there. On Tuesday, we meet again and the teachers are going to bring some new books for us to plan a lesson around! So excited!

# Formative Assessment

Assessment always seems to be such a broad, hot topic  There are rubrics to help create assessments, rubrics for reviewing assessments, and tons of reading about the benefit of assessments. While I agree assessment is an important topic of conversation and all of these things can be helpful, I just lose a bit of interest when it becomes so cumbersome. I feel the longer the rubric and steps to create an assessment, the more detached the assessment becomes from student thinking.  This could be completely be my short attention span speaking, however the way assessment is discussed feels either like data (a grade or number-type of data) or a huge process with tons of text in rubrics that I really, quite honestly, don’t feel like reading. Not to mention, I just love looking at student writing and listening to student thinking when planning my immediate next steps (formative) or checking in to see what students have learned over a longer period (summative). This is why I find the work we are doing each month in our Learning Labs such a wonderful way to think about formative assessment in an actual classroom context, in real time.

This passage from NCTM’s Principles to Action really captures how I feel about the work we are doing in our Learning Labs:

In this most recent Learning Lab in 3rd grade, we planned the activity together using the 5 Practices model and reflected after the lesson. Since this blog is always my thoughts about student work, I thought it would be great to hear what the teachers took away from the activities we are doing in terms of the students’ understandings and impact on their future planning, formative assessment.

The teacher mentioned in the blog said, I was surprised by how quick many of the students defended their responses that 1/2 will always be greater than 1/3, and then proving this response using visual representation of the same whole ( which is an idea that we have made explicit). I was impressed with “skeptics” in the crowd that were looking to deepen their understanding around the concept by asking those “What if” questions.  Going forward, I want to create opportunities that push and challenge my student’s thinking. I want them to continue to question and explore math – especially when it uses the word “always.”

Another teacher who taught the same activity after watching it in action in another classroom said, “I learned that almost half of my students assumed they were comparing the same size wholes.  They agreed with the statement, and each student gave at least two different ways to prove their thinking (area and number line model were most common).  The students that disagreed almost all provided their own context to the problem, such as an example with small vs large pizzas, or a 2 different-length races being run.  I found it so interesting that almost all students confidently chose one side or the other, and were able to defend their thinking with examples (and more than one-yeah!)  I was excited to see that they could be so flexible in their arguments as to why they felt as they did.  Three students responded that they were unsure, and gave reasons to support both sides of the argument. This impacted my instruction by giving me such valuable formative assessment information with a simple, non-threatening prompt.  It took about 5 minutes, and gave me tons of information.  It was accessible and appropriate for all.  Students were comfortable agreeing or disagreeing, and in some cases, saying “unsure-and here is why.”  I was most excited about that!”

She also said, From this activity, I learned that I really needed to revisit the third grade standard to see what is actually expected.  It says they should recognize that comparisons are valid only when the two fractions refer to the same whole.  My statement didn’t have a context, so how cool that some were at least questioning this!   This impacted my planning and instruction by reminding me how thinking/wondering about adding a context to the statement would influence their responses.  I am also reminded that I need to stress that students must consider the whole in order to make comparisons accurately.”

Earlier in their fraction unit, the third grade teachers used the talking point below to hear how her students were talking about fractions. (This work is actually from another teacher’s class, but you get the idea;)

A teacher who did this activity reflected, “From this activity, I learned my students had only ever been exposed to a fraction as a part of a whole (and wanted to strictly refer to fractions in terms of pizza). This impacted my instruction by being sure to have the discussion that fractions can represent parts of a whole, but we can also represent whole numbers with fractions.”

To me, these reflections are what assessment should be….the teachers learn about student thinking, the students think about their own thinking, and what we learn helps us plan future lessons with our students’ understandings in mind!

More examples from different grade levels where the teachers and I learned so much about student thinking that impacted future instruction:

Kindergarten: Counting

2nd Grade: Counting and Leftovers

5th Grade: Fraction Number Line

# 3rd Grade Multiplication Talking Points

This week I had the chance to work with a third grade teacher, Andrea! Her class is just about to begin their unit on multiplication and division so she wanted to do Talking Points to see what they knew, and were thinking about, in relation to these operations. During our planning we discussed the ways in which this Investigations unit engages students in these ideas, misconceptions students typically have, bounced around ideas, and played with the wording of the points. Being my first talking points activity with third graders, I was so excited to see how students engaged in the activity. I have found that even during Number Talks, the younger students are very eager to share their own ideas, but listening to others is difficult.

In looking for how students “saw” multiplication and thought about operation relationships, we designed these Talking Points..

Andrea introduced the activity and we did the first talking point as a practice round in which we stopped the groups after each of the rounds to point out the important aspects. We pointed out things like,”I liked how Bobby was unsure and explained why,” and “I liked how everyone was listening to Becky when she was talking,” and “I thought it was great proof when Lily drew something really quick to support her thoughts.” Then we let them go and walked around to listen as well!

During the Talking Point round, some things we found really interesting were:

• How difficult is for them to sit and listen to others without commenting. Not like it is not hard for use as adults though, right? 😉
• How much students struggled to say why they were unsure. Sometimes it was not knowing what the word division meant, yet they struggled to articulate what it was within that talking point that was confusing them. What a great thing for them to be able to think about!?
• How they related the dot images they had been doing in class to multiplication and division.
• How they thought about inverse operations. They said things like, “I don’t know what division is but if I can use subtraction with addition, I probably can use division with multiplication.

We had pulled two of the points that we wanted to discuss, whole group, afterwards, “I can show multiplication as a picture.” and “We can use multiplication problems to solve division problems.” We put them up and just asked them what their table had talking about. The conversation was amazing. Hearing how they thought about multiplication as groups of but 50 x 2 means “fifty two times” while 2 x 50 means “2 fifty times.” We also heard how someone at their table had changed the way they thought about something. And the division conversation was so great and for the students who were unsure because they did not know what division meant, it felt really organic to come out that way… from them, not us.

Of course, we followed with a journal write:) We gave them three choices to write about…

I was so impressed by the way they wrote about their thinking, by 5th grade, they will be amazing!!

The student above, during the Talking Points, said that he could show multiplication as a picture because “an equation IS a picture.” It was lovely to see him make the connection to a visual for an equation in his journal.

I wish the quality of this picture was so much better but her pencil was so light it was hard to see! She does a beautiful writing about how exactly someone at her table changed her mind with such an articulate way of talking about multiplication and division!

This student above explains perfectly why teaching is so difficult…”…sometimes we have facts about math, we all have a different schema. We were taught differently than other kids.” I am curious to hear more about her feeling about the end piece, “some kids know more then other kids.” Is that ok with her and she understands we all will get there just at different times?

These last two were two different ways in which students reflected on the dot images they have been doing in relation to multiplication!

What a great class period! I cannot wait to be back in this class to see how students are working with and talking about multiplication and division!

~Kristin

# Talking Points – Decimals

This week, we are beginning our decimal unit. I decided to start with Talking Points today to hear how the students are thinking about decimals before we dig in. In developing the Talking Points, I asked the #mtbos for some ideas on decimal misconceptions/misunderstandings they see students have each year. Thanks to @MikeFlynn55, @AM_MathCoach, and @MsJWiright2 for  your thoughts! Of course as the students were talking today, I wanted to tweak my wording of the statements!  Here are the points I used, my intended purpose of the statement, what I noticed and possible rewording for future use…

TP1 – My goal in this statement was to hear if students were thinking about multiplying by a number (fraction or decimal) less than one. The first thing I realized is that I have a class obsessed with negative numbers! Then the next thing that other groups talked about a lot was multiplying by zero or 1. I did have some students think about fractions, like in the example below. In future wording, I would probably adjust it to, ” When multiplying by a number other than 0 or 1, the product is greater than the factors.”

TP2: My goal with this statement was to hear if students thought about taking any number and make equivalents. My initial thought was just fraction/decimal equivalents like .3 = 3/10, but I did run into some great conversations about 3=6/2=3.0. The problem was that some were agreeing because they were just taking any number and writing it a fraction, not equivalent, just plopping it into a random fraction, like 7 can be 1/7 can be .7 can by 7/8, as long as there was a 7!  Possible rewording would be “All numbers can be written as equivalent fraction and decimals.”

TP3: This one was to elicit conversation about a comparison problem in which I see students often having a misconception. In comparing, students will think that 0.35 is less than 0.1245 because it has more digits. This one blew up in my face, of course:) The negative numbers arose again, which was interesting however, the some very clever students wrote 00000001 and said it is smaller than 12, but it has more digits. They were right, my statement was bad. Possible rewording, “The more digits a number has after the decimal point, the larger the number is.”

TP4: This one was really to plant a seed for the unit. I wanted students thinking about the place of the digit and its relationship to the digits surrounding it. So often when we decompose numbers, we deal with place values independently but I want to really focus on how the value changes as we move within the number. This one elicited great points made by the students and I plan on revisiting this one often throughout the course of our work together. This is one student’s reflection afterwards about this point:

TP5: My goal was to see if students thought about equivalency between decimals such as .3 and .30. The talk at each table was interesting and it was definitely one that was a split decision at many tables. I would leave this prompt the same based on the student responses, it was a nice mix and the mention of decimals came out vaguely. Student reflection on how another student changed his thinking:

TP6: I used this point to check for equivalency understanding of fractions and percents (because we have worked with them) and then to see any connections to decimals. A lot times, student will take the fraction denominator and put that right behind the decimal point to make equivalents. This one was eh. I like that students knew 1/4 = 25% so the conversation focused on the meaning of the decimal in relation to the other two. I may reword this to, “.4 = 4%” and leave it at that.

After the points, I had the students reflect on two prompts…

“I am still having doubts about Talking Point __- because….” and “When (insert person’s name) said ___________ it changed my mind about Talking Point ___ because…” Some samples of these are above and here are a few more….

# Pre/Post Assessment Reflection

We started our 2D Geometry unit with Talking Points: https://mathmindsblog.wordpress.com/2014/11/13/talking-points-2d-geometry/.  This was the ultimate pre-assessment in which I could hear what the students were thinking around mathematical concepts while at the same time, they had a chance to also hear the thinking of their peers. After the talking points activity, I had the students reflect on a point they were still unsure in their thinking.

We are now wrapping up our Polygon unit, and I thought it would be interesting for them to reflect back on what they were unsure about in the beginning, and get their thoughts now. I have a class full of amazing writings, but here are just two of the great reflections (the top notebook in each picture is the pre-unit and the bottom is post-unit)….

Looking at the class as a whole, it was so interesting to see their math language develop and see them laughing at things they had written before. I loved that the student above wrote, “I am smarter!!!” How amazing they can see their own learning!  During their reflection time, it was so fun to also hear students exclaiming, “See, I KNEW I was right!”

This is the first pre/post assessment I have ever done where I think the students enjoyed it as much as I did! They were as proud of themselves as I was of them!

-Kristin

# Talking Points – 2D Geometry

We are about to start out work with Polygons, so I decided to kick it off with Talking Points. If you have never read about them before you can check out my post or Elizabeth’s post to learn more.

Here were the points my students were discussing:

I had gotten these points by looking back at their fourth grade geometry unit work and thinking about what misconceptions or partial understandings students have each year when we start this unit.

This time, I made a few changes from past experience. In each group I had a facilitator to be sure that everyone got a chance to speak without interruption during Round 1, and a recorder to keep the tally for the group. Also, after the first talking point, I had advise from a math coach in the room filming with me to add individual think time after the reading of the point. LOVED IT! During think time, they were jotting in the journal and getting their thoughts together. I got things like this from just the think time:

It was nice to see them take ownership with their journal without being told to write anything down. They were working on proofs before they started. After the six talking points, I posed three questions on the board for them to reflect upon individually:

1 – What talking point are you sure you were right in your answer? Explain your reasoning.

2 – Which talking point are you unsure about your answer? Why?

3 – Which talking point did your group agree upon easily? Why do you think it was easy for your group to agree on that one?

Here are their reflections:

Never anticipated so much “left angle” talk in my life 🙂 I learned SO MUCH about their understandings and wish I had time right now to add my comments to each journal, but I will very soon!

So, I have a moment here at lunch to reflect on what I learned from these talking points:

TP1 – As I anticipated, this one is always a source of confusion. Every year it seems as if the students know the sentence goes one way or the other but can’t remember it because there is little understanding of the WHY piece. Later on in the unit after we have done more classifications, I do more of these statements with Always, Sometimes, Never, so this is a something I wanted to see how students were thinking about it. Most tables said something to the effect of “I remember last year we said a square is a rectangle or a rectangle is a square, but I can’t remember which one.” Another conversation I heard was that a rectangle has to have two short sides and two long sides.

TP2: I loved this question and was really pleasantly surprised to see some trying to draw it and ending up with unconnected sides. One thing I was so surprised about was the “left angle.” They were not thinking the degrees changed so much from the left to the right angle, but more the orientation of the angle (left side, right side). Interesting.

TP3: I got a great sense that most students knew what area and perimeter were and the best part was that if they didn’t remember, someone at their table did and gave an example. Regardless if they knew they could be the same, I was excited to see a great understanding from most here.

TP4: This one was great. I saw some students drawing a square on their paper, showing the group, rotating the paper and saying, “See, now it is a rhombus.” They all seemed to be in the mindset that a rhombus is a diamond shape, but really not reasoning about the attributes that make it a rhombus.

TP5: They did a very nice job with this one. A lot drew examples of combining two shapes, while I heard others asking their group if the “inside connected side counted” when they were trying to name it.  Also realized that the term polygon was not familiar to most students. I am wondering what they called them in earlier grades? Pattern Blocks? Shapes?

TP6: Interesting one here and it is where we start our 5th grade work with polygons, classifying triangles. Again, the left angle reappeared:) I did hear a few struggling with the name of the angles, obtuse, acute, right but then I had some that said there are other 3-sided shapes that aren’t called triangles. Hmmm, can’t wait to find out what they are! Of course, you always have your comedians who say agree because it could be Bob or Fred.

Can’t wait to start planning this weekend!

-Kristin