Category Archives: 1st Grade

Number Talk: Which Numbers Are Helpful?

I think Number Talks are such a powerful routine in developing students’ fluency and flexibility with operations, but maybe not for the reason most think. One of the most highlighted purposes of a Number Talk is the ability to elicit multiple strategies for the same problem, however, an even more important goal for me during a Number Talk is for students to think about the numbers they are working with before they begin solving. And then, as they go through their solution path, think about what numbers are helpful in that process and why.

The struggle with trying to dig deeper into that thinking is simply, time. If the opportunity arises, I ask students about their number choices during the Talk but often students just end up re-explaining their entire strategy without really touching on number choices. Not to mention the other 20ish students start losing interest if they take too long. I do think it is a particularly tough question if students are not used to thinking about it and when the thinking happens so quickly in their head, they don’t realize why they made particular choices.

Last week in 2nd grade I did a Number Talk with two problems, one addition and one subtraction. During the addition talk, I noticed students using a lot of great decomposition to make friendly numbers (the term they use to describe 10’s and 100’s).

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During the subtraction problem, I saw the same use of friendly numbers, however in this one I actually got 100 as an answer. My assumption was because the student knew he was using 100 instead of 98, but got stuck there so went with 100 as the answer. I was really impressed to see so many strategies for this problem since subtraction is usually the operation teachers and I talk endlessly about in terms of where students struggle. I find myself blogging on and on about subtraction all of the time!

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When the Number Talk ended, I looked at the board and thought if my goal was to elicit a lot of strategies, then I was done – goal met. However, I chose the numbers in each problem for a particular reason  and wanted students to dig more into their number choices.

This is where I find math journals to be so amazing. They allow me to continue the conversation with students even after the Number Talk is finished.

I went back to the 100, circled it and told the class that I noticed this number came up a lot in both of our problems today. I asked them to think about why and then go back to their journal to write some other problems where 100 would be helpful.

Some used 100 as a number they were trying to get to, like in this example below. I really liked the number line and the equations that both show getting to the 100, but in two different ways.

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This student got to 100 in two different ways also. I thought this was such a clear explanation of how he decomposed the numbers to also use 10’s toward the end of their process as well.

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This student used the 100 in so many ways it was awesome! She got to 100, subtracted by 100 and adjusted the answer, and then added up to get to 100.

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While the majority of the students chose to subtract a number in the 90’s, this student did not which I find so incredibly interesting. I would love to talk to him more about his number choices!

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I didn’t give a clear direction on which operation I wanted them to use, so while most students chose subtraction because that was the problem we ended on, this one played around with both, with the same numbers. I would love to ask this student if 100 was helpful in the same or different way for the two problems.

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As I said earlier, this is a really tough thing for students to think about because it is looking deeper into their choices and in this case apply it to a new set of numbers. This group was definitely up for the challenge and while I love all of the work above, these two samples are so amazing in showing the perseverance of this group.

In this one, you can see the student started solving the problem and got stuck so she drew lines around it and went on to subtract 10’s until she ran out of time. I love this so much.

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This student has so much interesting work. It looks as if he started with an addition problem involving 84, started adding, then changed it to subtraction and got stuck.

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This is what I call continuing the conversation. They wrote me notes to let me know Hey, I am not done here yet and I am trying super hard even though there are mistakes here. That is so powerful for our learners. So while there was no “right” answer to my prompt, I got a glimpse into what each student was thinking after the Number Talk which is often hard to do during the whole-group discussion.

If you want to check out how I use journals with other Number Routines, they are in the side panel of all of my videos on Teaching Channel. 

1st Gr Number String: Missing Number

Yesterday, I wrote a quick post as I was trying to decide which of two number talks I should do with a 1st grade class. I got some great feedback and went with the first one in the post! It was amazing and completely evident that the teacher, Ms. Williams, does a great job asking students to share their thinking regularly. The students were so clear in explaining their reasoning and asking questions of one another.

The first problem drew out exactly what I was hoping and more. One student shared counting on and a few students shared how they decomposed the 4 and added 2 and then 2 more. I was not expecting the use of a double, but two students used 8+8 in their reasoning. The use of their “double fact” reminded me of the solving equations conversations I have with Michael Pershan but in a much more sense-making way than I personally think about it. The students said they “knew 4 and 4 made 8 so they took 4 away and that changed the answer.” I tried to get out of them that they subtracted the 4 from the 16 as well, but it just made sense to them the 16 changed to 12 because he subtracted 4 from the 8. I am so glad I videoed this talk because I want to talk more about it after I re-watch it!

The second problem was as tricky, as I anticipated, and split the class between the answers 1 and 9. The students seemed very used to having the difference on the lefthand side of the equal sign which is great, but some still wanted to add 1 to the 4 instead of subtract the 4 from the missing number. I moved on to the final question because we were at a bit of a standstill at this point. Hindsight, I wish I did that problem last, but I had them journal about it after the talk.

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The final problem, which I wish was my first problem – what was I thinking in this order? – was great! They decomposed the 5, made 10 and talked their way through the two incorrect responses.

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I asked them to journal about the second problem when we finished. The prompt was to explain which answer, 9 or 1, they thought it was and why. Here are few examples:

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I think I would love to post the following string (all at once) on the board to start tomorrow’s lesson:

? – 4 = 5

5 = ? – 4

? + 4 = 5

5 = ? + 4

Ask what the question mark is in each one and which equations seem most similar.

Such a great day in 1st grade!

 

1st Grade Number Talk

I am planning for a number talk tomorrow with a 1st grade class. I have been playing around with two different problem strings that I would love feedback on, because I can’t make a decision!

I would particularly like feedback on:

  • What could we learn about student thinking?
  • What would you be curious to find our about their thinking at the end?
  • Do you think one would be better before the other or doesn’t it matter?

Here are the two I am playing around with (sorry, I had them written on Post-its):

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My thoughts:

  • 1st problem – Do they add to 10 and then add on? For example, 8+2=10 and since 12 is 2 more the answer is 4 or do they subtract 8 from 12?
  • 2nd problem – How do they do with the missing number on the right side of the equation? Do they visualize a 10 frame, taking 4 off of the bottom row to leave 5? Do they add 5 and 4?
  • 3rd problem – Do they decompose the 5 into 4+1 to use the 1 with the 9? Do they count on from 9?

Prompt at the end – How are these problems different? Which was your favorite to do? Why?

Second option:

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My thoughts:

  • 1st problem: Set the stage for expressions on both sides of equal sign. Notice you can’t add more to the same number and stay equal. Did we need to solve both sides to know that?
  • 2nd problem: Both equal 10 but did we need to solve both sides to know it is equal? Take one from one addend and add it to the other and still remain equal.
  • 3rd problem: Commutative property.
  • 4th problem: Now that I just wrote the commutative for problem 3, I want to switch the 8+5 on this one to 5+8 so that they might also think about taking 2 from one addend and adding it to the other.

Prompt at the end: Write two of your own equations that would fit something you noticed in our problems today. (wording is rough on that one).

Chances are I will have the opportunity to do both of them and I think they both hit on different, interesting things. I would love feedback on both and know if you think one is better before the other or if it doesn’t matter?

 

 

Asking Better Questions

I am sure we have all seen it at one time or another – those math questions that make us cringe, furrow our brow, or just plain confuse us because we can’t figure out what is even being asked. Sadly, these questions are in math programs more often than they should be and even though they may completely suck, they do give us, as educators, the opportunity to have conversations about ways we could adapt them to better learn what students truly know. These conversations happen all of the time on Twitter and I really appreciate talking through why the questions are so bad because it pushes me to have a more critical lens of the questions I ask students. Through all of these conversations, I try to lead my thinking with three questions:

  • What is the purpose of the question?
  • What does the question tell students about the math?
  • What would I learn about student thinking if they answered correctly? Incorrectly?

Andrew posted this question from a math program the other day on Twitter….

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I tried to answer my three questions…

  • What is the purpose of the question? I am not sure. Are they defining “name” as an expression? Are they defining “name” as the word? What is considered a correct answer here?
  • What does the question tell students about the math? Math is about trying to interpret what a question is asking and/or trick me because “name” could mean many things and depending on what it means, some of these answers look right. 
  • What would I learn about student thinking if they answered correctly? Incorrectly? Correctly? I am not sure I even know what that is because I don’t know what “name” means in this case. Is it a particular way the program has defined it?

On Twitter, this is the conversation that ensued, including this picture from, what I assume to be, the same math program:
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When a program gives problems like this, we not only miss out on learning what students know because they get lost trying to navigate the wording, but we also miss out on all of the great things we may not learn about their thinking. For example, even if they got the problem correct, what else might they know that we never heard?

The great thing is, when problems like this are in our math program, we don’t have to give them to students as is. We have control of the problems we put in front of students and can adapt them in ways that can be SO much better. These adaptations can open up what we learn about student thinking and change the way students view mathematics.

For example, if I want to know what students know about 12, I would just ask them. I would have them write in their journal for a few minutes individually so I had a picture of what each student knew and then would share as a class to give them the opportunity to ask one another questions.

After I saw those the problem posted on Twitter, I emailed the 2nd grade teachers in my building and asked them to give their students the following prompt:

Tell me everything you know about 12. 

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Ms. Thompson’s Class

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Mrs. Leach’s Class

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Mrs. Levin’s Class

Look at all of the things we miss out on when we give worksheets from math programs like the one Andrew posted. I do believe having a program helps with coherence, but also believe it is up to us to use good professional judgement when we give worksheets like that to students. While it doesn’t help us learn much about their thinking it also sends a sad message of what learning mathematics is.

I encourage and appreciate conversations around problems like the one Andrew posted. I think, wonder, and reflect a lot about these problems. To me, adapting them is fun…I mean who doesn’t want to make learning experiences better for students?

Looking for more like this? I did this similar lesson with a Kindergarten teacher a few years ago. Every time I learn so much and they are so excited to share what they know!

100 Hungry Ants: Math and Literature

This week the Kindergarten and 1st grade teachers planned with Erin, the reading specialist, and I for an activity around a children’s book. This planning was a continuation of our previous meeting about mathematizing. We jumped right into our planning by sharing books everyone brought, discussing the mathematical and language arts ideas that could arise in each. I made a list of the books the teachers shared here.

We chose  the book One Hundred Hungry Ants and planned the activity for a Kindergarten class. We decided the teacher would read the story and do a notice/wonder the day before the activity. We thought doing two consecutive readings may cause some students to lose focus and we would lose their attention. Based on Allison Hintz’s advice, we wanted the students to listen and enjoy the story for the first read-through. Here is an example from one classroom:

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So many great problem and solutions, cause and effects, illustration and mathematical ideas were noticed by the students.

The following day, the teacher revisited the things students noticed and focused the students’ attention on all of the noticings about the ants. She told the students she was going to read the story one more time but this time she wanted them to focus on what was happening with the ants throughout the story. We had decided to give each student a clipboard and blank sheet of paper to record their thoughts.

We noticed a few great things during this time..

  • Some students like to write a lot!
  • After trying to draw the first 100 ants, students came up with other clear ways to show their thinking. I love the relative size of each of the lines in these!
  • A lot of students had unique ways of recording with numbers. Here is one that especially jumped out at me because of the blanks:

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Students shared their recordings at the end of the reading and it was great to hear so many students say they started the story by trying to draw all of the ants, but changed to something faster because 10o was a lot!

After sharing, we asked students, “What could have happened if they had 12 or 24 ants?” We put out manipulatives and let them go! So much great stuff!

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Next time I do this activity, I would like to see them choose their own number of ants.

Just as I was telling Erin that I could see this book being used in upper elementary grades when looking at generalizations about multiplication, I found some great posts by Marilyn Burns on this book for upper elementary and middle school:

Excited to do this in a 1st grade classroom today!

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:

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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: Adding

Kindergarten: Counting

1st Grade: Fractions and Adding

2nd Grade: Counting and Leftovers

4th Grade: Division

5th Grade: Fraction Number Line