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What Is It About These Questions?

Today, I gave the 4th graders four questions to get a glimpse into how they think about multiplication and division before starting their multiplication and division unit. Michael Pershan had given the array question to his 4th graders last week and shared the work with me. As we chatted about next steps with his students, I became curious if the students think about multiplication differently depending on the type or setup of the problem.

Here were the questions:

After sorting 35 student responses I found the following:

17 students got the area question wrong but the two multiplication problems on the back correct. Not only correct, but with great strategies based on place value.
8 students got all of the problems correct, however the area was found in many ways, some not so efficient with lots of addition.
10 got more than two of them incorrect. Some were small calculation errors on the back.

So, what makes almost half of the students not get the area?

Here is the perfect example of what I saw on the majority of those 17 papers:

Then I did a Number String with them to hear how they shared their mental strategies. I wanted to get more insight into some of their thinking because a few students had used the algorithm on the back two problems.

They did great. They used the 10 and 20 to help them solve the problems and talked about adding and removing groups of one of the factors. I was surprised on the final problem of 7 x 18 that no one used the 7 x 20 but instead broke the 18 apart to find partial products.

This makes me think there is something about that rectangle that makes them not use the 10s to help them decompose for partial products. I would love others thoughts and ideas!

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After reading the comments about area and perimeter, I wanted to throw another typical example of what I saw to see what others think of this (when I asked her she could easily explain partial products on the second and third problem)

Adding & Subtracting: Tools and Representations

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There is always a lot of talk about students using an algorithm, process or set of rules, for addition and subtraction. Whether talking about “any algorithm” or “the algorithm,” I am certain, in most cases, people are talking about a process that is absent of tools such as a 100 charts, number lines or base ten blocks. But, what happens when we see the tool becoming an algorithm in and of itself? Can moving left or right on a number line, making jumps of 10s and 1s, writing problems vertically, or jumping rows and columns become an algorithm where students lose sight of the numbers themselves because the process is one more thing to learn?

This was the exact conversation I had yesterday with two 3rd grade teachers as I was leaving school. The students had been playing a game called Capture 5 and struggled making various jumps on the 100 chart. The teacher, understandably, was concerned students were confused about adding and subtracting tens and ones. After more conversation, we began to wonder if the students saw the 100 chart as a set of rules to follow in order to add and subtract instead of a place to look for patterns and structure as we add and subtract. Were they getting caught up in the left, right, up, down movement and losing sight of what was actually happening to the number?

As I thought more about it last night, I wondered about other tools and representations students learn that could easily turn themselves into an unhelpful set of procedures. I also wondered how often we make connections between these tools and representations explicit. Like, when is one helpful over another? How are they the same? How are they different?

I emailed the teacher my notes (below) and we decided we would try this out this morning.

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If you can’t interpret from my notes, the plan was to have each student in a group using a different tool or representation as I called out a series of operations to carry out. After the series of addition and subtraction, they compared their answers and discussed any differences. They rotated seats after each series so they had a chance to try out each of the roles.

We came back together to discuss their favorite one. The recording is below…what do I have against writing horizontally, really?? I found this entire conversation SO incredibly interesting!

They found the base ten blocks to be “low stress” because they were easy to count, move and trade, but did agree that bigger numbers would be really hard with them because there would be too many.
They really did not like adding and subtracting on the number line with multiple jumps. It got messy.
They liked mental math because there was nothing (tool) to distract them and they could focus but they didn’t like that you couldn’t check your answer.
The 100 chart’s only perk was they didn’t have to write the numbers in, they were already there for them.
I really loved that they mentioned the equations were they only way they could track their work. So if someone in their group messed up, the equation person was the only one that could help them retrace their steps easily.

I am not sure what I learned today, I am still thinking a lot about this. I know that I loved having them compare the tools and representations and that the teacher felt much better about their ability to add 10s and 1s. I feel like there are so many other cools things to do here, but my brain is fried today so that will have to wait!:)

3rd Grade Dot Image

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Since the 3rd grade classes are about to begin their multiplication unit, the teachers and I wanted to hear how they talk about equal groups to get a sense of where they are in their thinking. What better way to do that than a dot image? I chose the first image because of the 3’s and “look” of 5’s and the second image because of the 2’s,3’s, and 6’s, all of which students can count by easily.

Image 1 went relatively the same in both classrooms and much like I anticipated. There were two things that stood out to me as a bit different between the class responses:

There were more incorrect answers shared in the 1st class than the 2nd class.
In the second class, multiplication came out during the discussion. The “4 groups of 7 and 4 x7 = 28” in the 1st class came out after both images were finished and one student said she knew some multiplication already. She asked to go back to the first image and gave me that.

1st Class

2nd Class

After the first image, I anticipated Image 2 would go much the same, however it was quite different.

1st Class

2nd Class

After the 1st image, I was really surprised at the difference in responses and I have to say it even felt really different. My assumption at this point is that in the 2nd Class one of the early responses was multiplication.

I am left wondering:

Does that early multiplication response shut down other students who don’t know anything about multiplication yet? While I asked her to explain what she meant when she said 4 x 12, I wonder if that intimidated others?
How could I have handled that response differently so others felt OK using skip counting or addition to count the dots?
Can we anticipate that type of reaction from other students when someone starts the discussion with something that may be beyond where the majority of the class is in their thinking?
Was this even the issue at all? Did the 2nd class just see so many more dots and wanted to avoid adding and skip counting?

The 2nd Class ended with a journal entry after a student remarked, “If we know it is 8 groups of 6, then it is also 6 groups of 8.”

I asked if 8 groups of 6 is the same as 6 groups of 8 and the class was split on their response, so they set off to their journals.

The yes’s went with multiplication expressions representing the same product and commutative property:

I loved this no because the picture changes:

I am not sure about this argument but I would love to talk to the student a bit more about the bottom part!

After that talk, I am excited to see what these guys do when they actually start their multiplication unit!

Which One Doesn’t Belong? Place Value

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Since the 3rd grade team begins the year with an addition and subtraction unit in Investigations the teachers and I were having a conversation about how students understand place value. While I don’t see teachers using the HTO (hundreds/tens/ones) chart in their classrooms, students still seem to talk about numbers in that sense. For example, when given a 3-digit number such as 148, students are quick to say the number has 4 tens instead of thinking about the tens that are in the 100. I think a lot of this is because of how we as teachers say these things in our classrooms. I know I am guilty of quickly saying something like, “Oh, you looked at the 4 tens and subtracted…” when doing computation number talks, which could lead students to solely see the value of a number by what digit is sitting in a particular place.

We thought it would be interesting to get a vibe of how this new group of 2nd graders talked about numbers since their first unit deals with place in terms of stickers. A sheet of stickers is 100, a strip of stickers is 10 and then there are the single stickers equal to 1.

I designed a Which One Doesn’t Belong? activity with four numbers: 45, 148, 76, 40

I posted the numbers, asked students to share which number they thought didn’t belong, and asked them to work in groups to come up with a reason that each could not belong. Below is the final recording of their ideas:

I loved the random equation for 148 that emerged and the unsureness of what numbers they would hit if they counted by 3’s or 4’s. One student was sure she would say 45 when she counted by 3’s and was sure she would not say 76 or 40, but unsure about the 148. I wrote those at the bottom for them to check out later.

Since the teacher said she was good on time, I kept going. I pulled the 148 and asked how many tens were in that number. I was not surprised to see the majority say 4, but I did have 3 or 4 students say 14. As you can see below a student did mention the HTO chart, with tallies, interesting.

As students shared, I thought about something Marilyn Burns tweeted a week or so ago…

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So, I asked the students to do their first math journal of the school year (YEAH!):

“For the students who answered 14, what question did you answer?

“For the students who answered 4, what question did you answer?

After the students shared, I revisited the Hundreds, Tens, Ones chart. I put a 14 in the tens column, 8 in the ones column, and asked if that was right. The light bulbs and confusion was great! It was as if I had broken all rules of the HTO chart! Then I put a 1 in the hundreds, 3 in the tens, and they worked out the 18. I look forward to seeing them play around with this some more and wonder if when they go to subtract something 148-92, they can think 14 tens -9 tens is 5 tens.

I had to run out because I was running out of time, but snagged three open journals as I left! (I especially love the “I Heart Math” on the second one!
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Kicking off the 2016/2017 School Year!

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Ah, I am so excited to dust off the blog and get started with the new school year! There are so many exciting changes in our structures for RTI that I just cannot wait to get going! After some really honest conversations at the end of last year on how we were grouping and pulling students for interventions with minimal to no impact, we decided to completely change the way we do RTI, starting with the name. From this point on, we call it WIN, What I Need, time…because, you know, we needed another acronym!;) However, changing the name means nothing without blowing up the structure itself and that is exactly what Erin, the reading specialist, and I intend to do. We spent A LOT of time this summer brainstorming what the new WIN time could look like and imagining how we can use our time with teachers during Learning Labs to support this work. There are so many details and logistics that would make this post way too long, so let’s just start with the first month.

Typically, during the first month, students would be given content screeners used to group them into tiers. These screeners set a terrible tone for how students view the learning they will be doing during the school year (as Tracy blogged about here). The content on the screeners was across so many areas and there were no conversations about the math that, instructionally you got little to no information about what the students actually knew. Not to mention, for the students who had previously been in a tier, they knew this screener would put them right back where they were last year. So, this year, we are starting with getting to know our students in a way that can truly guide our planning and instruction and set a tone of how learning will happen this year!

Since our new WIN time will involve a lot of small group work, Erin and I designed a “First Month of School Plan” for everyone to help guide our future planning conversations. This plan could definitely be adapted by the teachers, but we wanted to give examples, or plan for those that wanted it, of ways to get to know your students and provide opportunities to develop a community of learners.

Without all of the details, it basically looks like this:

Week of September 6th: Talking Points.

Week of September 12th: Ask students what they need as learners. Do Norm-setting.

Week of September 19th: Read Last Stop on Market Street and do activities.

Week of September 26th: Read What Do You Do With A Problem? and do activities.

(If you want more info, the break down of all of this is in our first Learning Letter of the school year.)

For our very first Learning Lab this year, we have asked teachers to come with a formative assessment of what students know about their upcoming content work in October. Using this information we are going to work together to dig into the content and design small groups to help support students where they are. This design could be an extension of the content or address misunderstandings, all with a focus on what the students KNOW, not solely what they don’t.

I am sure it will be a bumpy ride for teachers, Erin and I, but when something needs to change, a bumpy ride is better than the same old crappy way of doing things. I will keep you posted and look forward to sharing our work!

My One Hundred Hungry Ants Obsession

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Lately, I have been obsessed with children’s literature across K-5. My most recent obsession is the book One Hundred Hungry Ants. I did this in Kindergarten and this in 4th grade and today I invaded a 3rd grade classroom with it!

I followed the same pattern I usually do, I read the story aloud and did a notice/wonder. These are all of the things they noticed:

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The last one led perfectly into asking about the ways the ants rearranged themselves. I wrote the combinations they recalled from the book and asked them to chat with a neighbor about patterns they see.

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The discussion started with the 50+50=100, 25+25=25 and 10+10=20. Another student said they had the same things but it sounded different because she saw 50 was half of 100. They moved away from that and went to divisibility by the numbers that did not show up like 3,6,7,8, and 9 and pointed out that all of the second factors were multiples of 5. At this point they were focusing primarily on the second factor until someone pointed out the increasing and decreasing pattern happening. Then we got into the doubling and halving, quadrupling and dividing by 4 and multiplying and dividing by 10 of the factors.

I asked them if that would work with any number I gave them. They were quiet so I threw a number out there for them to think about, 24. They had to move into another activity so I left them with that thought. Before I left, however, one student said yes for 24 because 2×12, 4×6,8×3. Another student said it could be sixteen 1 1/2s and then thirty-two 3/4s! Wow!

Tomorrow they are going to investigate this further to see if they can come up with a conjecture about this work! So excited!

Literature & Algebraic Reasoning

Being THAT Teacher

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Before reading on, please know the point of my post is not testing but instead the poor administrative decisions being made in its administration. In our school and district, testing is fairly uneventful. Since Smarter Balanced, our Department of Education has done a good job lessening assessment requirements across the state and I feel the environment around testing has gotten relatively better. We now have only one testing period, instead of three, during which students spend approximately a week (of just mornings) taking the tests, with their teacher, in their homeroom.

Unfortunately, a school in another district was given the autonomy to treat testing much differently.

Current Situation

This school year, one elementary school spent 1-2 weeks of precious instructional time to give every 8, 9, and 10 year old in the school a standardized interim assessment midyear. This assessment was not mandated by state. It was completely optional, however the administration at that school was given the autonomy to make the students take it. It was the only school in that district to give this assessment.

Based on that assessment each 8, 9 and 10 year old in that school was given a score of 1, 2, 3 or 4. Little did the students know, that score defined how they were grouped from that point forward. The scores were used to group or re-group students for building mandated test preparation for one to two months before they took the spring test. As if each student associating their academic ability with a number was not bad enough, when spring testing began, every other student in the class knew that score as well because they were pulled to test by that score.

This is where the standardized test becomes quite un-standard.

Students in that school were tested in groups based on those interim scores First the 4’s, then the 3’s, then the 2’s, and I am assuming the 1’s were left for last to allow for more test prep time.

Even more interesting is how the proctors rotated mid-test. Let’s say it was the 5th grade “3’s” turn to test. Those students left their homeroom to go test with one of the other 5th grade teachers. The rest of the students in the proctoring teacher’s homeroom, who did not get a 3, were shuffled out into the other 5th grade teachers’ classrooms. Not to worry, that teacher doesn’t have to do that everyday because another teacher from the 5th grade administered the second or third part of the test to the 3’s, and also with the 4’s, 2’s and 1’s. So, technically speaking, a teacher could administer the test three times before they administered it to the 1’s? Interesting. [This changed immediately after I brought it to the district office’s attention. Funny the grouping couldn’t change immediately, but this could.]

As with any great testing situation, it also came with the go-get’m pep talks from the administration. Sending the over-achieving 4’s into panic attacks and the test-prepping 2’s seeing their work as drills before the “big game.”

I can only assume the administration believes there is no harm being done to children administering a test in this manner or why else would they do it, right? There is no way they can think this is bad for students’ mindset and image of themselves as learners or they wouldn’t keep doing it, right? And, worse, what if the superintendent found out, investigated, and did not see enough wrong with the situation to bother changing it immediately?

The most heartbreaking piece was hearing what students were saying and doing….

“I am going with the other dumb kids that got a 2 to do test prep.”

“I am not a 3, so I am not testing today.”

“I don’t need to have any more novels to read because we are test prepping in reading.”

“I didn’t test with my friend ___ because she is a 3 and I am 4.”

“I got called in from recess after 5 minutes because I had to finish my test.”

“Don’t opt me out because then I am bored. At least the test is on the computer.”

An 8 yr old gets sent to the nurse with what seems to be a panic attack after being given the 4’s pep talk before taking the test. Not to mention, she has a pre-existing heart condition after having heart surgery at 2 weeks old. Upon picking her up from the nurse’s office, the parent was told by the nurse to take the child immediately to her pediatrician because she was stark white with an incredibly high heart rate.

And these are only from three students, can you imagine if you collected them all?

After hearing this, I was thinking to myself, there is no way the superintendent would ignore my concerns of such blatant disregard for students’ well-being and emotional safety in a school when brought it to his attention.

Evidently, he can ignore these concerns, for weeks. After three weeks, he responded with a message that basically said, “We have looked into your concerns and your opinion has been considered. I will not meet with you because you do not have children in the school.”

Wait…I cannot be an advocate for children if they are not mine? Isn’t that all part of being an educator? Standing up for what is best for them? Any and All of them? Maybe that is the problem here. They are not the administrations’ children, so why care what is happening to them? I have been teaching for 20 years, not one student has been my child, but I cared about them as if they were.

After three weeks of ignoring my requests, a family member who has children in that district requested a meeting for us. She had previously met with the building administration about this same testing situation so she was invested in this situation. Upon her request, we finally got a meeting date with the superintendent.

Before the meeting, I was still hopeful for change. That was, until we arrived for our scheduled meeting and was told by the curriculum director that the superintendent was meeting with someone else at the moment but we could go “get him if we have any questions.” Really? We waited 3-4 weeks for a meeting and he isn’t there when we explicitly requested he be there? We said we would wait and after the 10-15 minute power trip wait, he finally showed.

The parent started with the firsthand student accounts described above, accompanied by quotes from her children in regards to the teacher testing “pep talks.” It was gut-wrenching as she held back the tears that come when talking about your children. It was so disheartening and equally as infuriating to see the smirk and hear the chuckle that came from the superintendent as if to convey the “Oh, the silly things kids say” message as the parent spoke. I felt me heart actually beating out of my chest at that moment.

How could he not be infuriated by this happening in one of the elementary schools in his district? How did he not feel responsible? How could he give autonomy to a building principal who is so obviously not doing what is best for students?

After establishing the principal had not communicated any of these testing procedures to parents and told blatant untruths about other related items, the meeting continued to be about all of the great things the principal says he is doing to ensure the students do their best on the test, you know, for the student’s sake. Are we really trying to convince ourselves that the test score is not more about the school looking good than the child’s testing pride?

Then the blame game began. I was also told that the teachers in that school requested to give the interim assessment so the principal agreed. So, what, not his fault? Am I to believe he accommodates all of his teachers’ requests as willingly?

After hearing that the building administration “didn’t mean to make the students feel bad,” I couldn’t listen to one more thing. There is too much information out there to use that excuse…you didn’t mean to? Are you kidding me?

It unfortunately ended with the superintendent not being about to assure us this would not happen again next year. He said “we could expect change” but could give no specifics. That is not good enough for me since also finding out that this same issue was brought to his attention last year with no changes.

Fortunately, there are so many wonderfully caring people surrounding this principal and superintendent that I have talked to, that I DO believe these testing/grouping situations will not happen again next year.

The saddest part for me was the dismissive, “my school/district, my way” nature of this principal and superintendent. I am embarrassed for them as educators in this situation.

As teachers, we want to empower our students. We want them to have a voice, share their thoughts and opinions, feel as if they can take control of their given circumstances and make change. However, as adults the harsh reality of what is really like to make change in education leaves us, at times, feeling powerless, like our hands are tied. Oftentimes it is because we don’t want to be THAT parent or THAT teacher, afraid of the repercussions for our children or ourselves in the workplace.

Sadly, I find myself in that place. Knowing this is wrong for students, wanting to make a change, fighting to do so, yet feeling like I am the one in the wrong because I am calling out bad practice.

This entire situation reminds me of a quote from a post by this author to school superintendents about data walls in schools, “This madness in our education has to stop. All of you run schools or districts and you have the power to put an end to this absolute insanity happening in our schools.”

It truly is insanity and this superintendent had the opportunity to make it right and he didn’t. Sad.

I truly always believed if people in education could not answer, “Why are we doing this?” with “Because its what is best for children.” then it would indicate a needed change.

100 Hungry Ants: Math and Literature

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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!

Measuring Tools in 2nd Grade

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Last week, the 2nd grade team and I planned for a measurement lesson. Their measurement unit falls at the end of the year, so this was actually the first lesson of their unit.

We focused on work on the first of these two standards, anticipating the other would be a natural part of the work as well:

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We put out the following measuring tools: square tiles, inch bricks (unlabeled ruler from Investigations), a ruler with inches and cm, and a tape measure in cm.

The teacher launched the lesson by introducing the “Land of Inch,” a context that Investigations uses in the measurement unit. The introduction involved showing a picture of the 4 places in the Land of Inch: the castle, a cottage, apple orchard, and stable. The students discussed why they thought each one was in the Land of Inch.

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On a piece of paper, partners were asked to put the places of the Land where they thought they belonged and measure the distance from the castle to each, choosing whichever tool they thought was appropriate. The only stipulations were that there must be a path from the castle to each and each must be a different distance from the castle.

There were some really cool things that came up as we watched them working:

Every group took only the straightedge ruler and tape measure.
All of the straight lines were measured with the straightedge.
They all noticed the unit difference. We did not state what the unit of each tool was beforehand to see if they noticed.
They labeled 12 inches as 1 foot.
Students measured the curved paths using both the straightedge and tape measure.
Some students wanted to change centimeters to inches because it was the Land of Inch so they lined up the tape measure with the straightedge.

One group recorded their measurements in ranges. They had no interest in starting at the end of the ruler. They just put the ruler down and wrote the two measurements it fell between.

We wrapped up the lesson asking students to talk about why they chose their measuring tools. We had planned for them to share these ideas before they did a different journal prompt we designed last week. However, as they were sharing, there were one or two students doing a lot of talking (great stuff, but a lot) so we decided to have them reflect on their own before having this conversation.

This student did a great job of explaining when they used one tool over another:

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This student discussed why they chose to use the ruler but not the square inch tiles at all because it would take too long. So while both tools were the same unit, one tool has connected units versus individual units that need to be put together.

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This group noticed that the centimeters (on the tape measure) would take them longer than the straightedge because there were more centimeters than there would be inches.

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There were a couple instructional prompts we are revising for the next time this lesson gets taught by one of the other 2nd grade teachers that were there:

We didn’t let them know the paths didn’t have to be straight until after we saw them get started that way. Need to launch with that.
We didn’t have out meter or yard sticks, oops, need those next time. Talked about it during our planning, but we completely forgot.
We didn’t do a poster share which I think we want to incorporate next time because they all wanted to share. So maybe just two groups explaining their choices.
Wondering about the writing connection as they all had interesting reasoning behind where their places were located. Could they write a description about the placement and reasoning for their poster and then have other partners try to match them up?

Next up, reading Inch by Inch and the lesson inspired by the TCM article Inch by Inch in the most recent publication.

MathMinds

by Kristin Gray.