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
It is posts like Lisa’s most recent one that make me long for more collaboration K-12. I have to admit, when I saw her Twitter post with the words pre-calculus and simplifying complex fractions, my inclination was to skim right by because I would not understand the post anyway. Literally, my only recollection of simplifying complex fractions like the one at the beginning of her post is through a set of procedures I was explicitly taught step by step. However, when I looked at the accompanying image that showed fraction division, I was curious how my understandings of fraction division connected to her pre-calculus work.
I loved reading Lisa’s process of making the math accessible for her students because I am sure many would have felt like I did if shown the CPM opener from the very beginning. It is that same process of thinking about what students know and how we can build on it that made me get out my journal and start sketching out connections I was making as I read. In no time, my journal was full of problems, diagrams, concepts, questions and every tab on my computer referenced the progressions, standards, references linked in Lisa’s post, and a blank email to Kate and Ashli to jot down my questions for them about the math. Talk about a wonderful rabbit hole to be going down.
The more I read and reread this post, the more I think it could lead to many more posts connecting how students are introduced to ideas in elementary school, the impact it has on later work, and the questions I have as I go. My questions revolve around not only the math, but also how these mathematical ideas build, how our representations impact student understandings, and how there are times when a problems lends itself to one way of thinking versus another.
During my first read, two things I wondered were:
- How does the way the fractions are written impact the way I think about them?
- What happens when I have two ways of thinking about fractions and two ways of thinking about division?
How does the recording of the fractions impact the way I think about them?
As the post progressed from an image of a complex fraction to one of fraction division, I felt like Lisa must have felt, wondering what students may know about the complex fraction and why they may struggle. My initial thought was they may not understand that a complex fraction is even division. This may not be the case for most, however based on what I remember from high school, I saw complex fractions as one thing I did operations on. As an elementary school teacher, it seems similar to the difference between seeing a fraction as a number (introduced in 3rd grade) versus seeing fraction as division (introduced in 5th grade). As I looked at CPM’s complex fraction and how it was written, I only thought about it as multiplying the numerator by the reciprocal of the denominator because of how I was taught. However, when I looked at the fraction division problem written horizontally, I found myself attending more to each fraction as a number, using what I know about division to find the quotient. Less intimidating to me solely because of the way it was written on the paper. I wonder if this compares a bit to how we record computation problems horizontally versus stacked during number talks to encourage thinking about a problem versus always relying on the algorithm?
I know the fraction division problem means the same thing written either way, but how they are written impacts my thinking a lot. From an elementary perspective where we spend so much time attending to developing understanding of fraction as a number, I am not inclined to really think about what it means to divide the two terms when written as a complex fraction. To that end, I wonder if the opening problem written one way versus another evokes a different meaning for some students?
Knowing that there are things to be learned in between the problems listed below, but in terms of seeing the complex fraction as division where I think about the individual pieces as things in their own right, is one possibly a small transition to the other for me or students like me?
Lisa – I would love to hear more about the transition prompt between the fraction division problems the students were solving and the CPM problem. I think that is a really important piece of what you did so beautifully in this lesson.
What happens when we have two ways of thinking about fractions and two ways of thinking about division?
I think about fractions in the two ways I mentioned above: as a number and as division.
I think about division in two ways: how many groups? and how many in each group?
First, fractions: In 3rd grade, students learn a fraction is a number in which the numerator indicates the number of pieces and the denominator (as the denominator of a unit fraction) represents the size of the piece. For example, we say 3/4 is 3 pieces the size of 1/4. This understanding and associated language are so beautiful when students use it to compare fractions and create equivalent fractions. In my 5th grade class, my students were comfortable using complex fractions such as1/2 / 3 when talking about 1/6 because they were thinking ½ a piece the size of ⅓ is ⅙. No division, just reasoning about the pieces and their size. When comparing 4/9 to 5/7, students would use the reasoning that four and a half ninths and three and a half sevenths are equivalent to a half so 5/7 is more than a half and 4/9 is less than a half. I saw a glimpse into how that thinking was not helpful when they asked what happened when there is a fraction in the denominator. This is where understanding fraction as division would have been more helpful.
In 5th grade students also learn about fractions as division. In terms of sharing situations, they learn that 5 things shared by 3 people results in each person getting 5/3 of the things or 5 divided by 3. In these situations, thinking about 5 pieces the size of 1/3 is not particularly helpful in solving, but division is. However, when it comes back to interpreting the solution, 5 pieces the size of 1/3 is needed.
Questions I am thinking about at this point:
- How does the complex fraction in the post relate to either or both of these ways to think about fractions?
- How does the way we represent fraction division relate to one or both of these ways to think about fractions?
Now, division: In 3rd grade, students learn division in two contexts: how many in groups and how many in each group. In 5th grade, students use those understandings to divide whole numbers by unit fractions and unit fractions by whole numbers. Those two meanings of division carry into middle school to divide fractions by fractions and conceptually understand the reason we multiply by the reciprocal.
After reading Kristin and Bill’s series of posts on fraction division, I am now constantly thinking about how the context (interpretation) for division impacts the way students represent and solve a problem. I know changing the way I think about the division context changes how I represent the problem as well as how I operate with the reciprocal.
Questions I am thinking about at this point:
- Does one context of division connect more closely with the CPM complex fraction problem?
- Does the visual fraction model of the the division problem impact the way students approach the complex fraction problem?
- Is an array representing both fractions being divided helpful in this complex fraction?
- Is one bar model representing both fractions on one helpful in this complex fraction?
- Is one way of representing it more helpful than the other?
Obviously, I have a lot to read about how a problem such as the one Lisa posed progresses after middle school but after seeing the division of fraction problem, I am even more intrigued to see how these ideas progress from the time they are introduced. I am so curious when certain ways of thinking are more helpful than others and how we can construct learning experiences that help all students have access to the mathematics in a lesson in the way Lisa did.
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.
I have been meaning to follow up on this post for over a month now!
Over the first 6 slides, we saw so much great confusion when they were trying to figure out how to name the location of the point. We saw some students use measurements from the top and bottom of the screen, while others tried using some fractions. To get a picture of how students’ thinking changed from one grid to the next, I copied their responses from Desmos to this table in Google. In the table, each row is the same student’s response to describing the point location on each of the three different grids.
During the activity, we paused the students after the 6th slide and asked some volunteers to read their directions. After a few misses due to the fact they didn’t name a starting place or they counted squares instead of lines, we asked them to try re-writing some directions on the back of their paper.
We could tell they were finding the axes pretty necessary so after plotting the point successfully a few times as a class, we gave them some of the terminology and conventions for plotting and naming points in a coordinate plane.
This is the place where we thought we may run out of time but we wanted to give them a chance to plot some points that formed rectangles and look for patterns in the coordinates. I copied the things they noticed at the bottom of this table.
If they finished the work with their partner in Desmos, we asked them to complete two final questions about points that would connect to form a rectangle. We wanted to get a picture of which students were graphing them to find out and which were using what they noticed about the x- and y-coordinates. We knew this may be a stretch with some, given the amount of time we had in the class period, but I am always way to curious not to ask! Leigh was revisiting this work over the course of the next few days, so we were in no hurry to expect mastery at this point. It was great to see the mix of thinking about this:
There are some standards I think we do such a great job developing in early elementary, but never revisit explicitly when students learn about different numbers such as fractions and decimals. I blogged about this in reference to even and odd numbers last year, but this past week I have found another:
Early elementary spends SO much time building understanding of the relationships between 1’s, 10’s and 100’s, but I don’t think we do this standard justice as students build their understandings of fractions and decimals.
Leigh’s 5th grade math class just started their work with decimals. To help students make connections to what they learned last year, she and I went back and brushed up on where the students should be in terms of the 4th grade CCSS. It is always so interesting to me how the CCSS authors chose to put decimals in the NF strand in 4th grade because students are learning decimals are just another way to write a fraction with a denominator of 10 or 100. Building on that understanding, the decimal work then moves to the NBT strand in 5th grade as students begin operating with them. Those are the little, thoughtful details in the standards that I really appreciate.
The first lesson or two of the unit, Leigh picked up where the students left off in the curriculum last year – shading 10 by 10 grids in a game called Fill Two. In this game, students only work with tenths and hundredths so in subsequent lessons she introduced thousandths on the same grid, with each small square now partitioned into 10 smaller pieces.
To be confident of the path we were on with decimals, Leigh and I met to revisit the CCSS. We both felt the students were doing 5.NBT.A.1 in a conceptual way, but we were never really making the understanding explicit with students. Since there wasn’t a great place in the curriculum for this, we went to Illustrative Mathematics, found this task and built a lesson around it.
We opened the lesson with this Which One Doesn’t Belong? to see how students related the representations, in particular, how they talked about the picture since the task had something similar.
They came up with some really interesting comparisons and everyone saw the picture of base 10 blocks as 0.12 and “not even close to 1,” which the others were. This is where I think the lack of attention we give the standard above is really apparent. I think students learn each small cube is 1/100, so each skinny tower is 1/10, but are never pushed to think about what other values they could represent…how quickly they forget K-2:).
We then did a Notice/Wonder with the image from the task before they jumped in to work on the task prompt.
They did a great job trying to get into Jossie’s head, so we let those ideas sit there as we gave them 3 minutes of individual work time to begin the full task. After time to work with a partner, we came back together to discuss Jossie’s reasoning and the different values the picture could represent.
As we anticipated, many of the students said the Jossie got the tenths and hundredths pieces confused.
Then, one student explained her reasoning below:
After her explanation, I asked the rest of the class what they thought about how tens and hundreds are related versus tenths and hundredths. It was a great discussion of how tens build to hundreds but hundredths build to tenths. My summary here doesn’t do a bit of justice to how the students were talking about the math.
I collected their possible values for the picture and recorded them on the board.
Students took these values and put them into a place value chart to look for patterns:
Some noticed the 4 and 2 were constant in each number and moved around:
Some students recognized those movements as multiplying or dividing by 10’s:
These last two examples are the ones that really seem to get at the standard and something we want to connect the previous examples to build toward:
This standard in 5th grade always seemed like so much of a “telling lesson” for me.
I never thought it was really addressed in the spirit of this standard in our curriculum, so I was typically like, “Here is what we call a coordinate grid. These are axes, x and y. We name the points like this…” and so on. It is not my usual approach so it always felt blah for me, for lack of a better word. I told them, they practiced plotting some points, and we played a little bit of Battleship (which was really fun).
Last week, I was planning with Leigh, a 5th grade teacher, and we spent a lot of time just talking about what we appreciate about the grid and how we can develop a sense of need in the students for it. Since they are in the middle of their 2D Geometry unit, we thought this could be the perfect place to plot points that connect to form polygons and look at patterns in the ordered pairs.
The questions we wanted to students to reason about through our intro lesson was:
- Why a coordinate grid?
- Why name a point with an ordered pair?
- Why is this helpful?
- What structure do we see?
So, we created this Desmos activity. This was our thinking on the slides and the pausing points we have planned for discussion:
Slide 1: It is really hard to describe a location without guides or landmarks.
Slide 2: Note how difficult it is. Pause and show class the results.
Slide 3: It gets easier. Still need some measurement tool. Notice the intersection of axes.
Slide 4: Note it is a bit easier this time. Pause and show class results.
Slide 5: Much easier because of the grid. Still need a starting point. See it is the distance from axes.
Slide 6: Now it is much easier. Pause show class results. Would love to show all three choices side-by-side (don’t know if this is possible in Desmos).
~Pause~ Ask, “What names of things on the grid would make it easier to talk about the point’s location?” Give students vocabulary and ask them to revisit Slide 6 to describe the location to a partner.
Slide 7: Practice writing some ordered pairs.
Slide 8: Practice writing some ordered pairs.
Slide 9: Start to see some structure in the four ordered pairs of a rectangle.
We are ending with this exit ticket (with grid paper if they choose to use it):
While we are not sure this is the best way to intro the grid, we thought it would generate some interesting conversation. Since we are teaching it tomorrow, there isn’t much time for feedback for change, but we would love your thoughts.