Kindergarten Dot Image Number Talk

Pre/Post Assessment Reflection

We started our 2D Geometry unit with Talking Points: https://mathmindsblog.wordpress.com/2014/11/13/talking-points-2d-geometry/. This was the ultimate pre-assessment in which I could hear what the students were thinking around mathematical concepts while at the same time, they had a chance to also hear the thinking of their peers. After the talking points activity, I had the students reflect on a point they were still unsure in their thinking.

We are now wrapping up our Polygon unit, and I thought it would be interesting for them to reflect back on what they were unsure about in the beginning, and get their thoughts now. I have a class full of amazing writings, but here are just two of the great reflections (the top notebook in each picture is the pre-unit and the bottom is post-unit)….

Looking at the class as a whole, it was so interesting to see their math language develop and see them laughing at things they had written before. I loved that the student above wrote, “I am smarter!!!” How amazing they can see their own learning! During their reflection time, it was so fun to also hear students exclaiming, “See, I KNEW I was right!”

This is the first pre/post assessment I have ever done where I think the students enjoyed it as much as I did! They were as proud of themselves as I was of them!

-Kristin

Developing Claims – Rectangles With Equal Perimeters

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Yesterday’s math lesson launched based on this student activity book page in Investigations…

Students read the introduction and I first asked them, “If we were to build these in Minecraft, was feet an appropriate unit of measure?” Some thought that feet seemed too small for a garden and instead wanted to use yards, that was, until one student schooled us all on Minecraft. Come to find out, Minecraft uses the metric system with each block representing one cubic meter. We then changed the unit to meters and were on our way.

I gave them 10 minutes, either with a partner or individually, to build as many different rectangular garden designs as they could with 30 meters of fencing. As I walked around, it was interesting to see how students were designing their gardens. Some started building random dimensions and adding/subtracting fence links to eventually hit 30 while others had thought of one rectangle to start and adjusted from there by subtracting/adding from sides. It was fun to see the ones that discovered a pattern in their building and sped through the rest of the rectangles. Some beautiful patterns emerged….

After 10 minutes, I asked them fill in the table on their activity page and chat with their table to see if anyone had any dimensions they didn’t have. As a table, they came up with some noticings based on their table, Minecraft builds, and process of building.

The class ended after this group work. Talking with another colleague about the lesson at the end of the day, we chatted about what claims could come out of this lesson. It was so interesting to me to think about not only the geometric claims, in terms of rectangles and dimensions, but also the numeric claims that can evolve from these conversations as well. And to think that these number claims are grounded in a visual connection is pretty awesome to me!

I was so I started today by pairing up every two tables to share with one another. One table read their noticings while the job of the other table was to ask clarifying questions. That gave me time to circulate, listen, and choose the claims for our class share out. After the table shares, we reconvened as a whole class and chatted about a few of the claims as a group.

I asked them to think about similarities and if we could combine some of them to form one claim based on our work.

Class ended with a journal entry in which they worked independently to begin to form a claim or the beginnings of one. Here is where I will start tomorrow…

How fun!

As a follow up on this post, this is the assessment on their work with perimeter and area. During the class period, only one student started working on finding the square that would give them the largest area with a perimeter of 30. After this idea came out in our class discussion, using fractional dimensions was something that others were thinking about during the assessment. Here is a student’s (different than the initial student) work that I thought was pretty fantastic:

-Kristin

Math Class Through My Students’ Eyes…

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Each January, I like to ask the students to do a reflection on the first half of the year…things they liked, didn’t like, things they still want to learn, questions they have, etc…

Some students gave me a list of things they have learned by topic, others suggested that their seat be moved because they think they would work much better with their best friends, while some offered the suggestion of doing a “math project” that they work on over the course of a month or two (like their science fair project). I do like this last idea and looking into some type of ideas for this:)

I could really post all of them, because I just think my students are the coolest, most honest people I know, but for the sake of time, I chose two to reflect on tonight because I think it says a lot about what I hope students leave my class thinking about each year….

This is exactly why I started the Class Claim wall! I SO love that this student enjoys proving why things work, and even better that she started the sentence with the word “Actually,” like it was not expected! I also think it is so awesome that she said multiplied fractions before she even realized she was multiplying fractions! It makes me feel so good about all of the planning and work for the cornbread task which launched this unit.

This one just made me chuckle at the subtrahend and minuend talk. That came out of a number talk one day when they were calling them the “one you’re taking away from” and the “one you are taking away” and wanted an easier, less wordy way to say it (don’t know if those words are, but stuck for this student). It did make me reflect on my work with Virginia Bastable this summer when she said (I am putting quotation marks, but this is not verbatim), “Vocabulary should be a gift for the students in their explanations, developed out of need.”

The second part was just too funny and completely what I do to these poor kiddos all of the time! He has learned that when he has a finding or “idea,” I don’t just give him an answer, but instead send him back to think about it and see if they can figure out why that is happening. Then with another idea, the same process ensues…but at least, “it is not as hard as it seems.”

This is exactly what I want, curious students who work to explore their ideas and strategies and learn the processes of “doing math” without knowing there are procedures in place to do exactly what they are doing. I want them to see the “hard” math work they do as fun and an invaluable part of their learning.

They would probably be very surprised to find out that they make me do all of these same things before, during and after each lesson….

-Kristin

Number Routine PD: What Do I Know About…

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My colleague Nancy and I facilitated a K-2 afternoon professional development session yesterday afternoon. The 2.5 hour session was with a wonderful group of teachers from across our state who we are fortunate to work with several times over the course of the school year. Our major focus over the course of this school year centers around connecting arithmetic to algebra based on a book by Virginia Bastable, et al, that I blogged about here: https://mathmindsblog.wordpress.com/2014/11/20/articulating-claims-in-math/ I thought blogging about this experience would be helpful for any of our teachers who could not attend and for any others who facilitate PD.

I find planning for professional development is much like planning for the classroom. Many of the same questions arise:

What content will be engaging and relevant? (especially being an afternoon session when everyone is winding down on a Friday)

What is the trajectory of the content?

Where are they? Where are they heading?

What questions or prompts will encourage conversation?

When are points for table conversation? Whole Group conversation?

How will be know where they are in terms of the content when they leave us?

How will we follow up?

After much planning, videoing, and organizing this was the flow of the afternoon:

We opened by getting into grade level groups to discuss the homework from last month, doing a group planned Number Talk with their students. They used this form to plan together and brought back recording sheets of their work to discuss these two questions:

With the number talk being planned by the group, I felt a sense of ownership over the results in the classroom and, really, who doesn’t like talking about all of the wonderful things our students say during a number talk?

We continued with a quick recap of last month’s session on the book, “Connecting Arithmetic to Algebra” to plant the seed for our routine of focus that day, What Do You Know About….?

Now into the really fun stuff! Working with a Kindergarten teacher in my school (@jennleachteach) who is also a part elementary pd group, we planned and videoed a math routine called “What Do You Know About 15?” in Jenn’s class.

We mixed the grade level PD groups up at this point so there was a range of K-2 teachers (and a few math coaches) in each group. They got a blank planning sheet to brainstorm what they think the planning would look like for this routine in a Kindergarten classroom in January. It was great conversation, with the Kindergarten teachers being the experts at each table. I thought this was such an interesting dynamic since we often tend to pose a mathematical idea and ask what previous understandings K-2 need to build to get there, however, with this opportunity, it was starting in the opposite direction and really focusing on what Kindergartners know at this point of the school year.

After they predicted what our planning sheet would look like, Nancy brought 6 teachers up to act as students in a fishbowl enactment of the Number Routine. The other teachers in the room were observers focusing on two particular aspects of the talk, what you notice about the teacher recording and what you notice the “students” noticing. Importance of recording was a previous topic in an earlier pd, so we wanted to be sure that resurfaced. Nancy did the routine with the teachers and we came back as a group to discuss the observations of our focus questions. Our discussion also touched on the use of the talk moves she used to clarify and illustrate student thinking.

We then watched Jenn’s Kindergarten class do the same exact Number Routine, focusing now on the follow up piece of the planning sheet. What did they notice the students noticing? I wish I had permissions from everyone because Jen did a beautiful job in facilitating the talk and her students said some amazing things. We also took a look at the planning sheet that Jenn, Nancy and I had done for this routine. Here is the planning sheet and anchor chart that arose from the talk:

As a group we discussed what they noticed the students noticing that could lead to future “claims” in their classroom. Teachers noticed such things as, “We can count by fives to get to 15” “It is three fives” (of course I am thinking about groups of and multiplication right there!) “A teen number is a group of ten and some more” “Looking at equality with related equations” and “The 1 means one ten”

Jenn then gave students “random” journal entries to see how students were thinking about the numbers after the talk. To differentiate, we decided to give students 12, 19, or 21 depending on where we thought their entry level was into this thinking. After students completed the journals they chatted with someone who had a different number, to talk about their ideas.” Here are the student samples our PD group looked at and discussed:

We ended with Virginia’s conclusion slide about Connecting Arithmetic to Algebra and our homework for the group:

We also gave an Exit sheet to help us in future planning. We got some very useful information as to where the teachers feel they are. I am very excited to hear about everyone’s journey back in their classrooms next month!

-Kristin

My #oneword

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Since I am having my students choose one word that will be their inspiration this year, I have been thinking a lot about which word I will choose. In the chaos of everyday school life, my initial thought was the word “No” simply to remind myself to say no to more things to de-crazy my life. If you are anything like me, which most teachers who I find on Twitter and blogging are, you are juggling a million things at once. Ideas sound so exciting to me, that I jump on board and say yes without really thinking about how much work it will be and what other projects will sacrifice time because of the new project.

Today, I thought more about it and the word “No’ seemed a bit too negative and I wouldn’t really want that to be my motto for the year, however I have come up with a more powerful word to help guide my actions this year….Less.

Less initiatives that I have to be the driving force behind, less projects that I am not passionate about that will take time away from things in which I am, less things that go around in circles and never make much progress, less, less, less…..

-Kristin

Area/Perimeter of Squares…Help.

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Through my work each year with area and perimeter, I learn more and more about how I learned (was taught) math. I can work out a “proof” algebraically, however I struggle in connecting it conceptually to the problem. When this happens, I am so hesitant to reach out bc, truthfully, it is a bit embarrassing. I mean, I teach this stuff, right? But I finally hit a point, after I started blogging, when I learned that I will never learn more if I am not open to where I am. Since I encourage my students to write honestly about their understandings, I thought it only fair to throw my thoughts out there into the wonderful #mtbos. So here it is…

Here was last year’s example that I worked my way through: https://mathmindsblog.wordpress.com/2014/01/27/area-and-perimeter-of-squares-student-noticings/

And now here is this year’s: https://mathmindsblog.wordpress.com/2014/12/20/areaperimeter-my-homework-over-vacation/

I am finding the same thing is happening again…algebraically I have it, but struggling with the conceptual connection. I have a bunch of theories scattered on my papers this morning, but nothing that is satisfying me mathematically and would appreciate any thoughts….

So when thinking about area related to perimeter in squares, I know that n^2 x 4/n = 4n, but I am stuck at that 4/n. I marked off what I thought 4/n looked like in my squares, I messes around with ratios, found some patterns, but still not seeing (or putting together) what I want. … so I went to this drawing bc the side was increasing by 2 every time…. Then I went to this…

And while I would love to play with this for a bit longer, I have so many other things to do to get ready for school tomorrow! I feel like I have it somewhere here, I just cannot make a connection that works for me. Would love any pieces to the puzzle put together for me:)

Thanks!!

-Kristin

Intuition in Learning Math

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Yesterday, Malke tweeted this…

…and it led to such an interesting conversation that I honestly can say, I had never really given much thought. The conversation is here if you would like to read it now, or you may want to save it until after you read my rambling thoughts. 🙂

At first read of the tweet, my initial thought was how hard it was for me to make a distinction between intuition and making sense of problems. What makes them different? The amount of time it is given? The context of the situation? The math experience behind it? My questions could go on and on. I needed time to process these thoughts and let them sit with me for a bit. I tried reading some of the links to works about intuition in learning, but until I could figure out how I was thinking about intuition and put it in the context of my classroom experiences, the readings were not making much sense to me.

Luckily, I had a lot of car-riding time yesterday to think about this and jot some thoughts down. Disclaimer: these ideas are quite scattered, do not form a cohesive string of thoughts, and you will probably be left with more questions than answers by the end, however it is really fun to think about all of the ways “Intuition” takes shape in learning.

I thought it would be good for me to start with a definition and work from there. First, I tried Webster and got this one:

eh. I didn’t like the “without any proof or evidence” piece of this because I feel that our intuitions do come with proof or evidence, they are maybe not explored or articulated yet, however I think they are there. Then I found this one that I felt I could work from a bit better:

This definition by no means encapsulates how I envision “intuition”, however it had some really interesting points that led me to more questions….

– Does our intuition mean we have an “understanding”?

– Where does our “conscience reasoning” come from?

– Is our intuition always what is “likely“?

– Where do I see these hunches, inklings, notions in my students’ learning of math?

– Where do I see these same things in myself as a teacher?

Up to this point, I was gaining more questions than answers, so I began thinking about these questions in terms of my students and myself as a teacher. I am a person always in need of examples, so I needed to form some “example cases” to start to clarify these thoughts.

In this first example, from an Illustrative Task, the student was asked to determine if the answer to the problem could be solved using multiplication. Although the student came to the correct answer of 2/40, his intuition is telling him it still doesn’t make sense. Is this intuition based on previous experiences? Are all of our intuitions based on past experiences or non-experiences? Has he never seen an area less than 1 so it can’t make sense? The array has to be whole numbers? For me, the use of miles intuitively feels weird when I read it. Even as an adult, I hear miles and think of distance, bars, and do not like it so much with area.

In class each day we do Estimation 180. In this example, I specifically look at Day 23. I never really thought much about this, but I think it is intuitive of the student to look at the size of the item being packaged in another when thinking about capacity and volume. The student doesn’t look at the color of the paper or types of lines on it, but instead intuitively looks at the size of each part. It is something that happens so fast, that perhaps these are the quick, small moments in my classroom where students are acting intuitively.

On Day 36 in the example below, a student’s reasoning is that it is “usually a weird number” and on Day 37 says “My family never gets blow pops so I guessed.” Both of these seem to stem from experience/non-experience. The student has noticed that in previous days the packaging numbers have not been “friendly numbers” so the students is intuitively thinking it is a non-friendly number. If he/she was given this problem on the first day of the estimation questions, would their intuition have led them in a different direction? Day 37, shows a glimpse into, what I call, non-experience. This student intuitively goes to guessing because of never having them at home before, however is still only 20 off. This then leads me to question, that if a student doesn’t intuitively think of a reasonable estimate, do they then move into a more concrete strategy? To be within 20, I feel like the student used counting in the picture to some extent.

In this second grader’s work, I see such an interesting intuition in the second part of this question…

She seems to think that she cannot possibly know all of the possibilities because she is not older. She associates getting older as getting closer to knowing “all” of something. Little does she know that as you get older, you find you know less of “all” than you originally thought 🙂 Ha!

Here is where I get muddled between intuition based on experience and making sense based on prior knowledge. I asked this question to my students last year and here is one example of a student who went beyond yes or no and started to give a proof. I would love to hear others thoughts on intuition here….

This example below makes me think about how conjectures are made by students. Do conjectures stem from intuition and proofs that we can’t fully explain? The student said that when comparing two fractions, with the same numerator, she can subtract numerator from the denominator and the smaller difference is the larger fraction. Is this false intuition in dealing with the numerator and denominator as whole numbers? Thinking you can just pull them apart and operate with them as wholes?

This one may not be an example of intuition, but it is how I think about my own intuitions in learning….I try to explain them, prove them, revise them, edit them….. I would love every student paper to look like this…

In writing so far, I am really thinking that a student’s intuitions in learning math come from a “conscience reasoning” based on prior experiences and exposure. I could truly rack my brain over this for a while, but before I do, I wanted to think about myself as a teacher also. How much do we do as teachers that is intuitive? How does that intuition change as we evolve as educators?

There are many things I do during the course of the school day that just feel like routine or habit. The hard part is deciphering where it is not just habit or routine, but instead intuition.

When writing my lesson plans, I would say I use my intuition often in the respect of anticipation. From the minute I read the lesson, I have intuition on how I feel the lesson will “go over” with the students. I have a gut feeling if they will be interested in it, which students will be able to easily enter into the problem and which will struggle, and what strategies will emerge. All of these anticipations are based on my experiences with the students. So are these anticipations, intuitions?

As the lesson is happening, I think so much of my questioning is based on my intuition. I could not possibly have a list of questions to ask students during the course of every lesson, I have to rely on my intuition. As a student is explaining something, I am thinking to myself, “I think it would be interesting to ask _______.” This is something that has definitely evolved based on my experience, however because of the “newness” of every day and every class, I have to rely on intuition of similar case scenarios. Something like, “I asked this question the other day and it got me nowhere, how can I ask it differently to push student thinking?” This inner dialogue during a lesson happens in an instant which makes me believe it is intuitive.

If all of this is true, then I would say that when I first started teaching, my intuitions were not as fine-tuned as they are now. Does that makes sense, can you fine-tune intuition? Is there a point beyond thinking something is a good idea/bad idea or makes sense/doesn’t make sense that is still intuition but a more detailed, specific intuition? Intuitively, I think there is:)

A lot to think about still….Thanks to Malke, Tracy, Simon, Bridget, Kassia for a great (to be continued) conversation!

-Kristin

Area/Perimeter – My homework over vacation

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It is always fun when I can look back at my past blog posts, see what I learned from a lesson, and reflect on student reasonings before I teach that same lesson again. This past week we were working on this lesson from last year: https://mathmindsblog.wordpress.com/2014/01/27/area-and-perimeter-of-squares-student-noticings/.

The lesson unfolded in much the same way, with the class patterns looking like this:

I anticipated all of them, however, like last year, there always has to be one that throws me a curve ball and leaves me math homework over Christmas vacation:)

The 5th statement looked like this in his math journal:

With these other noticings below it…

His explanation focused on the pattern of the fraction the perimeter is of the area. The numerator stayed a two and the denominator went up by one on every even dimension. I honestly didn’t know what to ask him because the question “Why is that happening?” seemed way to broad. He played around with building the squares and was not getting anywhere. I am thinking, after chatting with Christopher (@trianglemancsd) on Twitter, that focusing on the meaning of the fraction first may be the way to go??

Basically, I first have to sit down and reason about his on my own…gotta love math homework on vacation!