As a teacher, curiosity around students’ mathematical thinking was the driving force behind the teaching and learning in my classroom. To better understand what they were thinking, I needed to not only have great, accessible problems but also create opportunities for students to openly share their ideas with others. It only makes sense that when I learned about routines that encouraged students to share the many ways they were thinking about math such as Number Talks, Notice and Wonder, and Which One Doesn’t Belong?, I was quick to go back to the classroom and try them with my students. It didn’t matter which unit we were in or lesson I had planned for that day, I plopped them in whenever and wherever I could because I was so curious to hear what students would say. Continue reading
A few weeks ago, a conversation about 3rd grade fractions sent me back to the Standards with a #pairedtexts type of lens. Unlike the hashtag’s typical MO of pairing contrasting texts, I was looking for standards that connected in a meaningful, but maybe unexpected way. By unexpected, I don’t mean unintentional, I mean the two standards are not necessarily near-grade or in the same strand, so the connection (to me) is not as obvious as one standard building directly toward another.
The conversation focused on this standard:
With that standard in mind, imagine a 3rd grade student is asked to locate 3/4 on a number line on which only 0 is marked.
I expect a student would mark off the 1/4’s starting at 0 and write 3/4 above the point after the third 1/4 segment. What exactly is the student doing in that process?
Is the student adding?
Is the student counting?
Is the student doing both?
How does adding and counting look or sound the same in this scenario? different?
This is where I find pairing two standards fun and interesting to think about because it demonstrates how important seemingly unrelated ideas work together to build mathematical understandings. It is also really fun to think about how a standard in Kindergarten is so important for work in grades 3-5 and beyond.
In this scenario, I think we instinctively believe students are adding unit fractions when asked to place 3/4 on the number line because the standard is in the fraction strand and therefore we consider all of the work to be solely about fractions. We also sometimes impose our thinking on what students are actually doing in this task. For example, you could imagine the student marking off the fourths, stopping after the third one, writing 3/4 and say the student was adding 1/4+1/4+1/4 to get to the 3/4 because they moved along the number line. If this is the case, then the standard would pair with this 4th grade standard:
Don’t get me wrong, those standards definitely pair as students move from 3rd to 4th grade, however, since the scenario is about a 3rd grade student, pairing it with a higher grade level standard doesn’t seem to make sense in terms of what students are building on. Right here, it is really interesting to pause and think about how building fractions from unit fractions, locating a fraction on a number line, and adding unit fractions are slightly different things a progression.
When I think about the student locating 3/4 in 3rd grade, I hear counting (with a change in units) and would pair that 3rd grade standard with this Kindergarten counting and cardinality standard:
However, because the 3rd grade work is on a number line and the arrangement and order does matter, I would have to add this 2nd grade measurement standard into the mix, but take off the sum and differences part:
So, instead of a #pairedtext, I now think of it more as a #CCSSMashup to create this standard:
With that mashup in mind, I went back to the progressions documents to look for evidence and examples of this.
In the 3rd Grade NF Progression these parts jumped out at me as being representative of this standard mashup:
The goal is for students to see unit fractions as the basic building blocks of fractions, in the same sense that the number 1 is the basic building block of the whole numbers; just as every whole number is obtained by combining a sufficient number of 1s, every fraction is obtained by combining a sufficient number of unit fractions.
The number line reinforces the analogy between fractions and whole numbers. Just as 5 is the point on the number line reached by marking off 5 times the length of the unit interval from 0, so 5/3 is the point obtained in the same way using a different interval as the basic unit of length, namely the interval from 0 to 1 /3 .
There is also a great “Meaning of Fractions” video on the Illustrative Mathematics site that explains this idea with visuals.
There are so many of these great mashups in the standards, especially in the fraction strand, that I find incredibly helpful in thinking about how students coherently learn mathematics.
I look forward to hearing your favorite #CCSSMashup!
The Investigations curriculum and Jessica Shumway’s book, Number Sense Routines contain so many wonderful math routines. Routines designed to give students access to the mathematics and elicit many ways of thinking about the same problem. One of the more open routines, is Today’s Number. In Today’s Number, a number is posed to the class and the teacher can ask students for questions about that number, expressions that equal that number, or anything they know about that number. I love this routine, and while it is more commonly used in the primary grades, I used it often in my 5th grade classroom. While I would capture so much amazing student thinking, I always felt like all of that great thinking was left hanging out there. I could see some students were using what they knew about operations and properties to generate new expressions for the given number, however I wondered how many saw each expression as individual, unconnected ideas.
After I read Connecting Arithmetic to Algebra, I had a different ending to Today’s Number, an ending that pushed students to look explicitly at relationships between expressions. I tried it out the other day in a 3rd grade classroom.
I asked students for expressions that equaled Today’s Number, 48. I was getting a lot of addition, subtraction and multiplication expressions with two numbers, so I asked students if they could think of some that involved division or more than two numbers. I ran out of room so I moved to a new page and recorded their ideas.
After their thinking was recorded, I asked the students which expressions they saw a connection between. This is where my recording could improve tremendously, but I drew arrows between the two expressions as students explained the connection.
In case the mess is hard to see, these are some connecting ideas that arose:
Commutative Property: 3 x 16 and 16 x 3, 6 x 8 and 8 x 6, 12 x 4 and 4 x 12
Fraction and Fraction addition: 48/1 and 24/1 + 24/1 and 24 + 24
Subtracting from 100 and 1000: 100-52 and 1000-952
Multiplication and Repeated Addition: 4 x 12 and 12+12+12+12
Adjusting Addends in similar ways: 38+10 and 18 + 30, 40 + 8 and 48+0
Other ideas that I don’t particularly know how to categorize:
10 x 4 + 8 = 10 x 4 + 4 x 2
58 x 1 – 10 = 58 – 10
The second page got even more interesting:
“Groups of” and Decomposition: 7 x 4 + 2 + 18 = 14 + 14 + 2 + 10 + 8 . This student saw the two 14s as two groups of 7 and then the 18 decomposed into 10 + 8.
Halving and Halving the Dividend and Divisor: 192÷4 = 96÷2. This student actually used the 192 to get the expression with 96.
Another variation of the one above was 200 ÷ 4 – 2 = 100 ÷ 2 – 2.
Other cool connection:
96 ÷2 = (48 + 48) ÷ 2; This student saw the 96 in both expressions since they were both dividing by 2.
I think asking students to look for these connections pushes them to think about mathematical relationships so expressions don’t feel like such individual ideas. I can imagine the more this is done routinely with students, the more creative they get with their expressions and connections. I saw a difference in the ways students were using one expression to get another after just pushing them try to think of some with more than 2 numbers and some division.
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).
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!
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.
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.
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.
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!
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.
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.
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.
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.
Today, I was able to pop into a 3rd grade classroom and have some fun with a true or false equation routine! This routine has become one of my favorites, not only for the discussion during the activity, but more for the journals after the talk. I haven’t figured out quite how to use them with the students, but it gives me such great insight into their understandings that I would love to think about a way to have students reflect on them in a meaningful way. I keep asking myself, what conjectures or generalizations could stem from this work?
I started with 4 x 3 = 3 + 3 + 3 + 3 to get students thinking about the meaning of multiplication and how we can solve for a product using repeated addition. I followed 6 x 4 = 8 + 8 + 4 to see how students talked about the 8’s on the right side. They could explain why it was false by either solving both sides or reasoning about the 8’s as two 4’s in some way.
My final problem was the one below, 8 x5 = 2 x 5 + 2 x 5 + 20. I chose this one because I wanted students to see an equation with multiplication on both sides. Up to this point, I structured them to be multiplication on one side and addition on the other. There was a lot of solving both sides – I think because of the ease of using 5’s – but, as the discussion continued the students made some really interesting connections about why the numbers were changing in a particular way. I really focused on asking them, “Where do you see the 8 and 5 in your response?” to encourage them to think relationally about the two sides.
I ended the talk with 8 x 6 = and asked the students to go back to their journals and finish that equation to make it true.
Some students knew it was equal to 48 right away and started writing equations that were equal to 48. For this student I probably would ask about the relationship between each of the new equations and 8 x 6.
There are so many interesting things in the rest of them, that I am not sure what exactly to ask student to look at more deeply.
In all of them, I see…
- Commutative property
- Multiplication as groups of a certain number
- Distributive property
- Doubling and halving & Tripling and thirding
The student below shared this one with the class during the whole class discussion:
8 x 6 = 7 x 10 – 3 x 10 + 2 x 4
From her explanation, she could explain how both sides were 48, but when I asked her how it related to 8 x 6, her wheels started spinning. You can see she played all around her paper trying to make connections between the two. That is the type of thinking I want to engage all of the students in, but based on their own personal journal writing – but what is the right prompt? “Where is one side in the other?” or “How are they related?” <—that one feels like it will lead to a lot of “They are both 48” so I need a follow up.
I actually left the room thinking about how I would explain how they two sides were related – in particular looking for either 8 groups of 6 or 6 groups of 8 on the right side. I found it was easier for me to find six 8’s, but now want to go back and find eight 6’s for fun. I can see how this could be so fun for students as well, but there is a lot of things going on here so I wonder how to structure that activity for them? Would love thoughts/feedback in the comments!
I just love when students are so excited to extend an activity! During the Notice and Wonder portion of this lesson, a lot of students wondered why those four letters were the ones given. Was it because they are at the beginning of the alphabet? Is it because they have the same area? What would happen with other letters?
Today, Mrs. Sharp gave them the chance to play around with others letters. She asked them to design their own letter and find its area. It is so interesting to see their choice of letters, the way they chose to decompose the shape and their math work all around it. Many of them made multiple shapes because they just wanted to keep going…that is always so AMAZING to hear!
I also see all of this being SO helpful when they find volume of figures composed of two non-overlapping rectangular prisms in 5th grade!
Here are just a few of the creations they came up with in class today:
Since the 3rd graders just wrapped up their unit on area, I thought it was the perfect time to do a task that hit on some really important ideas about area, while also encouraging them to move beyond counting squares. I wanted to see how (or if) students broke apart shapes to find the area, how they used addition and/or multiplication to more efficiently count squares, and if they used any other strategies such as subtracting out blank spaces or decomposing and rearranging pieces to find the area.
I chose this task from Illustrative Mathematics.
We started with a notice/wonder activity:
They had so many great wonders that inspired me to think about a follow up activity about other letters, but I will chat about that later.
Since they wondered if the letters all had the same square units and if they were all the same, I used that as the lead into the activity. Even as I was giving directions, however, I saw a majority of them start to count squares by 1. I paused them, told them I was so excited to see they knew counting the squares would get them the area and knew they all could find the correct area that way so we were going to try something different. I asked them to find the area without counting all of the square by 1.
In looking at the work, I saw them as a bit of a progression of thinking. I put them in order here of how I see students moving through these ideas about area.
As I expected, some counted each row and added them. It was great they know area is additive, but I would love to ask this student if there is a way they could have used multiplication to make it a bit easier.
Some added in chunks, to which I would love to ask the same question. I was excited to see them cutting the shapes up into rectangles in places that made sense.
From there, some used a mix of addition and multiplication. I would love to ask these students how they decided where to make their cuts.
Some students made some larger cuts and I would love to have them meet with the student above and discuss how they decided on their cuts.
Some used some of the strategies above but also relied a bit on symmetry.
Finally, I saw 2 students moving squares from the “bumps” to the empty spaces.
It was always interesting to me that in 5th grade I would still see students find the area of shape on a grid by counting the individual squares even though I know they had better strategies. I think it is the fact that students jump into doing things without thinking about the things first. This is why I think journal writing is so important. It allows students to be more reflective about their decisions.
I asked them to write about which shape was the easiest to find the area and which was the most difficult. It was interesting to see some focus on the size of the number they were working with while others focused on the shape and how it could be partitioned.
As a follow-up activity, I am going to ask them to choose a letter where they would use the same strategy they used with C and a letter where they would use the same strategy they used with B.
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 1: CCSS.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 2: CCSS.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 3: CCSS.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 5: CCSS.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 6: CCSS.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:
It happens every year, in what seems like every grade level…students continually call a rhombus a diamond. Last year, when we heard 3rd graders saying just this, Christopher helped the 3rd grade teachers and me put the students’ thinking to the test with a Which One Doesn’t Belong he created.
This year, at the beginning of the geometry unit, we heard the diamond-naming again along with some conversation about a rectangle having to have 2 long sides and 2 short sides. What better way to draw out these ideas for students to talk more about them than another Which One Doesn’t Belong? We changed the kite to a rectangle this time, hoping we could hear how they talked about it’s properties a bit more.
Overwhelmingly, the class agreed D did not belong because it had “5 sides and 5 corners” and eventually got around to calling it a diamond, which in their words was “not a real shape.”
While we knew a lot of things could arise, our purpose was diamond versus rhombus conversation, so of course the students had other plans and went straight to the square versus rhombus.We wouldn’t expect anything different!:) For every statement someone had about why the square or rhombus did not belong, there was a counter-statement (hence the question marks in the thought bubbles).
Jenn, the teacher, and I were really surprised at how much orientation of A and B mattered to the name they gave the square and rhombus but did not matter for the rectangle. That was just a rectangle, although one student did wonder if a square was also a rectangle (he heard that from his older sister). The students had so many interesting thoughts that we actually had to start a page with things they were wondering to revisit later! That distributive property one blew me away a bit!:)
We then sent them back to journal because we wanted to hear how they were categorizing a square and rhombus. It ended up being really interesting just seeing them try to explain why they were different and change their mind because they just started turning their journals around!
Some stuck with them being different..
Some thought they were different, but one could become the other…
Some were wavering but the square was obviously the “right way.”
Some argued they were the same…
So much great stuff for them to talk about from here! I left wondering where to go from here? In thinking about the math, is it an orientation of shapes conversation? or Is it a properties conversation? In thinking about the activity structure, would you pair them up and have them continue the conversation? Would you throw the rectangle into this conversation? Would you have some playing with some pattern blocks to manipulate? Would you pull out the geoboards? I am still thinking on this and cannot wait to meet and plan with the 3rd grade team!
However, before I left school today, I went back to the 3rd grade standards to read them more closely:
and read the Geometry Learning Progressions, only to find this in 1st grade:
Would love to hear any thoughts and ideas in the comments!
I have never been more intrigued with using Cuisenaire rods in the classroom until I started reading Simon’s blog! I admit, I have read and watched his work from afar…not knowing really where to start with them and was afraid to just jump into another teacher’s classroom and say, “Hey let me try out something!” when I really didn’t know what that something may be. However, after Kassia reached out to Simon on Twitter asking how to get started with Cuisenaire rods and Simon wrote a great blog response, I was inspired to just jump right in!
I am a bit of an over-planner, so not having a really focused goal for a math lesson makes me a bit anxious. I am fairly certain I could anticipate what 4th and 5th graders would notice and wonder about the Cuisenaire rods because of my experience in that grade band, however I wanted to see what the younger students would do, so I ventured into a Kindergarten and 3rd grade classroom with a really loose plan.
Kindergarten (45 minutes)
Warm-up: Let’s notice and wonder!
- Dump out the bags of Cuisenaire rods in the middle of each table of 4 students.
- Tell them not to touch them for the first round.
- Ask what they notice and wonder and collect responses.
Things they noticed:
- White ones looked like ice cubes.
- Orange ones are rectangles.
- End of blue one is a diamond (another student said rhombus)
- Different colors (green, white, orange..)
- They can build things (which is why we did no touching the first round:)
- Orange is the longest.
- They are different sizes.
- We can sort them by colors.
- We can sort them by size.
Things they wondered:
- What do they feel like?
- What can we make with them?
Activity 1: Let’s Sort!
- Tell the students to sort them by size or color. (they quickly realized it was the same thing)
- Discuss their sort/organization and check out how other tables sorted.
I was surprised to see not many sorted them into piles because that is normally how they sort things. I am wondering if the incremental size difference between each rods made them do more of a progression of size than sort into piles? Some groups worked together while others like making their own set with one of each color (and size) and keep making more of those!
Activity 2: Let’s Make an Orange!
Since a lot of students kept mentioning that the orange was the longest, I decided to see if they could build some trains (as Simon calls them) that made an orange.
My time was running out, but it left my mind reeling of where I wanted to go next! My inclination is to ask them if they could assign numbers to some of the rods or if they could build some trains the same length as the different colors? I would love to hear which piece is their favorite piece because a lot of them found the smallest cube really helpful when building the orange.
3rd Grade (60 minutes)
Warm-up: Let’s notice and wonder!
Things they noticed:
- Groups were the same color and length.
- Blue and white is the same length as the orange rod.
- Kind of like adding.
- White is 1 cm.
- Go up by one white cube every time.
- Odd + even numbers
- 2 yellows + anything will be bigger than 0.
- 1 white + 1 green = 1 magenta
Things they wondered:
- Is red 1 inch?
- How long are the rods altogether? (Prediction of 26 or 27 in wide)
- Is orange 4 1/2 or 5 inches?
- Why doesn’t it keep going to bigger than orange?
Activity 1: Let’s build some equivalents!
I found 3rd graders love to stand them up more than Kindergarteners:)
Activity 2: Let’s assign some values!
After they built a bunch, I asked them to assign a value to each color that made sense to them…this was by far my favorite part – probably because it was getting more into my comfort zone!
Again, time was running out, but next steps I am thinking…
- What patterns and relationships do you see in the table?
- What columns have something in common? Which ones don’t have anything in common? Why?
- What if I told you orange was 1? What are the others?
- What if orange was 2? What happens then?
Thank you so much Simon for all of inspiration and Kassia for the push into the classroom with these! Reflecting, I was much more structured than Simon and Kassia, but I look forward to a bit more play with these as the year goes on! I look forward to so much more play with the Cuisenaire rods and continuing Cuisenaire Around Ahe World!