Author Archives: mathmindsblog

Today’s Number: Making Connections

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.

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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:

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“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.

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. 

The Equal Sign

True or False?

5 = 5

5 = 4 + 2

2 + 3 = 1 + 4

After reading so much about the meaning of the equal sign and equality in books such as Thinking Mathematically and About Teaching Mathematics , I anticipated students may think each was false for different reasons…..

5 = 5: There is no operation on the left side.

5 = 4 + 2: The sum comes first or 4+2 is not 5.

2 + 3 = 1 + 4: there is an operation on both sides or because 2+3 =5 (and ignore the 4) or because 2+3+1 ≠ 4.

While I anticipated how students may respond, I was so intrigued by the number of students (probably about 75%) that said false for 5=5. They were about split on the second one, but for many reasons – not many of them being that 5 ≠ 6. The final one left many confused, in fact one student said, “Well now you are just trying to confuse people by putting two plus signs.” So cute.

As they explained their reasoning, my mind was reeling….

  • What questions do I ask to get them to:
    • Think about what the symbols mean?
    • Talk about what is the same?
    • Realize the equal sign in the first one is not a plus sign, so there is no answer of 10?
    • See the equal sign to not mean “the answer is next”?
    • What wording do I use for the equal sign?
      • “The same as” felt wrong because the sides do not look the same in both cases….so, is “Is the same amount” a helpful way for them to think about it?

I got back to my room and starting thinking about what learning experiences would be helpful for students in building their understanding of the equal sign? I talked through it with some colleagues at school and reached out to those outside of school, I needed some serious help!

I started playing around with some cubes and realized how interestingly my thinking changed with each one. I didn’t take a pic of those cubes so I recreated them virtually to talk thru my thinking here.

The first set represents 5 = 5. I can see here where “the same as” works for the equal sign because there are 5 and they are all yellow. But what if I put 5 yellows on the left and 5 red on the right? Then they are the same quantity, but do not look the same.

The second set represents 2+3=5 and is definitely the one students are most comfortable seeing and representing as an equation. It looks and feels like composition to me so I can definitely see why student think the equal sign means “makes” or “the total is.” It looks like 2 and 3 more combine to make 5.

Something interesting happened with the green set. I made two sets of 5 and then broke one set to make the right side – felt like decomposition. I can see why it would feel differently to students. I also realized that when I look at them, I look left to right and much of that lends itself to the way I was thinking about what was happening.

The last set I made by taking my 2 sets of 5 connected cubes and breaking each set differently. Again, “the same as” doesn’t work for me here really well either because they don’t look the same.

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Still thinking of next steps because I always like to put context into play with these types of things, but I am finding that very difficult without forcing the way students represent their thinking which I don’t want to do.

Right now, things I am left thinking about before planning forward:

  • What do students attend to when we ask if things are the same?
  • Our language and recording is SO incredibly important.
  • How can these ideas build in K-1 to be helpful in later grades?
  • If I am thinking of moving students from a concrete to more abstract understanding, how does that happen? Is it already a bit abstract in the way the numbers are represented?
  • Do we take enough time with teachers digging into these ideas? [rhetorical]

I look forward to any thoughts! So much learning to do!

 

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?

 

 

True or False Multiplication Equations

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.

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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.

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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

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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.

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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!

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Following up with the Coordinate Grid

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:

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5th Grade: Decimal Place Value

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:

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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.

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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.

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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.

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So polite in her disagreement, I love it! 

Then, one student explained her reasoning below:

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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.

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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:

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Some students recognized those movements as multiplying or dividing by 10’s:

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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:

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#MTBoS & Tch Collaboration

As a part of my work with Teaching Channel, I am collecting video of students doing various math routines in the classroom. Some of which, I am facilitating, while others are led by teachers in my building! It is such a wonderfully exciting and scary thing…putting our practice out there for everyone.

For this first set of 3rd grade videos, I sporadically tweeted out ideas and asked for feedback on some of the routines we did.

I got some wonderful ideas and started to think it would be great if this project could be even more of a collaborative one. I would love to brainstorm and refine ideas with all of the great minds in and outside of the #MTBoS.

Here is the deal…

Each month I focus on filming a particular grade level. 3rd grade is done so that leaves K-2 and 4-5. There are many routines to choose from however each must be accompanied by the supporting materials (planning page, resources page, and student work) like the ones at the bottom of the righthand column of this page.

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I created this doc for us to collaborate. If you are more comfortable leaving comments here, I can move them to the Google Doc so it is all in one place. I put a column for your name or Twitter handle in case you wanted people to chat with you further on Twitter or if I have more questions it would be really helpful to have a quick way to chat!

The routines we have been using are the following:

  • Number Talk
  • Number String
  • Notice/Wonder
  • Which One Doesn’t Belong
  • Quick Images
  • Dot Images
  • True/False Equations

It is probably easiest to stick with these routines since the students are familiar with them.

They each must also be attached to a CCSS that would appropriate for that grade level. It may be helpful to give you an idea of where our students are, so the routine is not out of left field:

  • I think anything that addresses NBT and OA for K-2 would be great for where students are now. They have just started their 2D geometry units so that would work too.
  • Grade 4 has finished their fraction unit and is now working in the decimal and measurement unit.
  • Grade 5 just started their decimal unit.

Let the lesson planning begin! I put a first image in for K, 1, 2, 4, and 5. I think it would interesting to see how different grade levels think differently about the image! (Simon you will love that one:).

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!