Why We Need Two Teachers in Every Classroom…

This job takes two brains to handle the thoughts of these students.

In class on Friday, one student made the comment that he didn’t really like adding fractions on the clock because it could only be used for certain fractions. When I asked him to expand on that, he explained he could only do halves, 3rd, 4ths, 6ths, 12ths and 60ths easily and what if he wanted to do other fractions like 1/8 or 1/24? He said he couldn’t do that without breaking the minutes up. I am excited at this comment especially because this student is one whose parents have taken him to Kumon math for years for “extra help” and he is most comfortable memorizing procedures over thinking about the math. He thinks changing to “common denominators by multiplying the numerator and denominator by the same number” is faster and easier than this clock.

Upon reflection, I think it is interesting that he stayed with fractions of the fractions we were working…why not pull out 1/9 or 1/11? But my first train of thought in the moment was changing the whole. I wanted to see if he could put the clock in terms of a whole day, 24 hours, 2 rotations around the clock being the whole instead of one. That way 8ths and 24ths would be more apparent.

So I asked him if he could think of a way we could change the clock to do 1/8 or 1/24 without breaking up minutes? His first reaction was no, so I said “That is interesting because there are 24 hours in a day, so I feel like this should work.” Possibly leading him too much but at that point I could see the glazed look in some of the students eyes and I felt like I was losing the class’ attention. I told him that during math workshop that day he could chat with me about it or he could take that thought and work with some more for Monday. He said he wanted to think about it over the weekend…I think mainly because he didn’t want to miss the Math Workshop activities, so we will see what he has for me tomorrow.

After school, I am recapping this lesson for Nancy and saying how difficult I thought it would be for them to grasp two rotations of the clock as the whole for the 24 hours that would allow for 8ths and 24ths more easily. After listening to me ramble for about 5 minutes about this idea, she casually says, “What about military time?” UMMMmmm…DUH. Where was she during that class period?? This job really does take two brains.

So needless to say, I have amended my lesson for tomorrow. I am handing them this military clock and letting them talk about what fractions we can work with easily that are the same as our first clock and which one’s are different. Design addition equations we can solve with this clock that we couldn’t do on the other clock without breaking minutes.

Don’t get me wrong, I still want to get to changing the whole on our original clock, but I think after working with this clock, it may be more accessible for more of the students. I will post later to update on this lesson to show how it went…but good or bad, the questions and thinking that led to this lesson are so worth it!

-Kristin

A Fraction of our Time in Math Class…

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I absolutely love fraction work with my students because there is always something interesting that leaves me pondering the whys and hows of my practice….

Being a K-5 Math Specialist for a couple years offered me the opportunity to really see the trajectory of our fraction work. Now being back in the classroom, I feel I have a much better grasp as to the work the students have previously done within our math program. In third grade, they work tremendously with halves, thirds, and sixths using polygons to represent fractions of a hexagon whole for comparison and addition/subtraction. In fourth grade, students use arrays and known equivalencies to compare and add/subtract fractions with unlike denominators by choosing an appropriate array that works for both fractions (common denominator). In addition, at each grade level, students in need of RTI enrichment, work in Marilyn Burns’ Do The Math Program which utilizes fraction strips to compare and add/subtract fractions. All of this work focuses heavily on the students’ understandings of equivalencies.

Knowing all of this still never prepares you for the power of a new model….time! I have to admit, I am a huge fan of fraction strips and array work, however today I felt the power of clocks in developing equivalencies. I have taught this lesson in previous years and to be completely honest, never really liked it. It felt contrived, like a pizza divided into slices in another form. This year I have realized it was not the context that was lending itself to the “pizza feel,” it was me.

The class began with a discussion of a blank clock face. I asked the class if the minute hand stayed at 12 and the hour hand moved to the 1, what fraction of the clock did it turn? They said 1/12 and we chatted about how we can prove that, divided it up and went from there. Next I asked if the hands were reversed, would that give us a different fraction? Some said no, some said yes and we talked about the equivalency of 5/60.

The student questions that followed took my appreciation of the clock to another level:

“Is this the same as degrees since it is a circle?”

“Could we do the fraction for a whole day (24 hours)?”

“Can we split the minutes in half to do eighths?”

“What fraction does the clock go at the time we go to lunch?”

Holy cow, how many directions could I take this lesson?? I moved forward with having the students work with partners to find all of the fractions they could represent on the clock. Then I asked them to use that model to add 1/3 and 1/4 on the clock. It was interesting to see the students who know how to “find common denominators” by multiplying the numerator and denominator by the same number were challenged to make a proof of their equivalencies on the clock face, while the students who needed the clock as a tool had it as their disposal to see that 1/4 is 3/12 and 1/3 is 4/12. That clock face immediately went from something I saw as just one more pizza, to both a tool and model at the same time in my classroom.

The follow up activity is called Roll Around The Clock (http://tinyurl.com/p8sm7wa). It has fantastic variations to the game and I have student work on the positive/negative scoring system that I will post soon, it was the perfect extension for the students who needed it!

So today, in just a fraction of time, I found a new appreciation for the analog clock and hopefully improved my practice by a fraction!

-Kristin

Reversing the Number Talk

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I am a huge fan of number talks and use Sherry Parrish’s book at least two to three times a week to conduct a number talk with my students. Sometimes I pose just one problem for students to solve mentally and discuss strategies while making connections between them or I do a string of problems targeting a specific strategy. Recently, I have been focusing on partial products and using friendly numbers as strategies to multiply. I noticed that as the string went along, they wanted to try and predict what the final problem (or “the hard problem” as my students would say) in the string would be. I started taking a few predictions each time and the conversation was really intriguing to me.

For example, the other day, the string was:

5 x 10

5 x 50

10 x 50

15 x 50

15 x 49

As they predicted the final problem, they actually made a more difficult prediction than the ending problem, 15 x 49. They predicted problems such as 15 x 47, 30 x 51 and 15 x 52. Their reasonings were targeting the strategy of using friendly numbers without me having to outwardly say it.

So I thought it would be interesting (and fun) to go in the opposite direction and give them the last problem of a string to see if they could develop the string of three problems that would come before it. I gave them “36 x 19” and they ran with it. Here are some ideas i captured from the journals:

This is a great formative assessment for me to see their thought process through our multiplication problems. Definitely adding it to my list of favorite activities!

~Kristin

Number Talks by Sherry Parrish: http://store.mathsolutions.com/product-info.php?Number-Talks-pid270.html

Meaningful Math Conversations…

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I am a true believer that content coaching is a necessity in the improvement and sustainability of math instruction, however we all know that finding time to even use the restroom during the course of the school day is close to impossible! So how do we find time for these important conversations to happen and more importantly, we need to be fortunate enough to have a position in our school that does just that, coach.

Last week, my class was working on finding fraction/percent equivalents using a 10 x 10 grid. They did great with the fourths and eighths, but then we hit 1/3! As I walked around and talked to the students, I saw a range of strategies: shading one out of every 3 squares, shading one out of every 3 rows, then squares, and some just knew that three 33’s was as close as they could get with whole numbers and had just shaded 33. No matter which strategy they chose, the “leftover box” was leaving many perplexed.

After quite a bit of struggling with what to do with this leftover box and some happy to just settle at 1/3 = 33%, Nancy (our math specialist, former 3rd grade teacher, and partner in crime with all things math) came into the room. She helped me by chatting with a group about their thoughts on what do with this 100th box. Class, unfortunately, had to wrap up to go to lunch, and I wasn’t comfortable that some students had had sufficient time to think about it, so I left the class with that leftover box as food for thought that night.

Over lunch, Nancy and I were talking about what she had heard from the students and she made the statement, “It is amazing how they don’t make connections to all of the sharing brownie work we did in 3rd grade when trying to count off by 3’s in the grid..” For those who use Investigations, you will know the exact lessons to which she is referencing, for those who don’t you can probably infer the context 🙂 We discussed the difference of the contexts for students, the array work they do in 4th grade and then tried to figure how to make that connection for my afternoon class. Tall job for the 15 minutes left of lunch, AKA speed eating.

I typically start my class with some type of number talk, so we sketched out a number talk that focused on the brownie problems of years past. Lunch ended and when the class came in the classroom, they headed to the carpet for a number talk.

I did the following sequence of problems, sharing strategies as we went:

How can four people share one brownie?

How can four people share 6 brownies?

How can four people share a pan of 21 brownies?

They did an amazing job and were very confident in their strategies and I definitely put them into a “fraction state of mind.” We then went into finding our percentages and even the strategies for finding the percents equivalent to fourths and eighths seemed smoother and then when we hit 1/3 and that leftover box was much less mysterious. There were still a few who struggled but I definitely could see more perseverance and entry points at problem solving. They seemed to make a connection to the brownie problems at the beginning of the lesson.

This entire rambling of my thoughts really boils down to one thought….Improving instruction is about finding time to have those meaningful math conversations. Had I not had that conversation with Nancy and changed my number talk for the second group, the lesson was going to have the same fate as the first. That conversation helped me make math connections that I could then make my students. Would I have loved to have more time to think out this lesson and retry it the next day, of course, but did Nancy and I improve it…absolutely!

~Kristin

Fraction Talk

3 Replies

It has been forever since I have blogged, and although I have been so inspired from many things I read this summer, nothing inspires me like talking to my 5th graders!

As we begin our venture into fractions, I have to first give some props to my 4th grade teachers. I have never heard so many “Yeah, fractions” and “I love fractions!” ever. I attribute this to a lot of hard work and dedication by Nancy (math specialist), the fourth grade teachers, and the Marilyn Burns’ Do the Math fraction units.

Yesterday in class, to get a feel for what my students know about fractions, we did a “Show What You Know” with problems involving writing, comparing, and adding fractions. They seemed very comfortable with writing fractions, comparing fractions using benchmarks, and finding fraction of a group.

Then we get to the problem asking students if the expression 2/3 > 2/6 is True or False. As they shared their reasonings, I heard many anticipated strategies such as “2/6 is equivalent to 1/3 so 2/3 is bigger than 1/3” and “The pieces are bigger in 2/3 and you have the same amount of each so it has to be more.”

As the conversation was coming to an end, one student raises her hand and sets my wheels spinning. She said “I know that if I just subtract the numerator from the denominator, whichever fraction has the the smallest difference is the larger fraction. But it only works when the numerators are the same.” Huh. I asked her why she thought that worked and she said she didn’t know but proclaimed it would work every time. I told her we would think through that one and revisit it soon because I needed time to think it through. Being the thoughtful student she is, I had this work from her by the end of the day:

I was proud she gave examples and tested even and odd numbers to be sure that didn’t effect the outcome.

So my next question for myself (and anyone else who is reading and feels like offering some advice) was what to do with this…

Nancy and I sat and talked about why this works…here are some points to our discussion:

– When you subtract the numerator from denominator you could finding the fractional piece the fraction is from a whole, assuming you put it back over the denominator.

– But since the denominators are different this would not give you a piece of information that would make this “trick” valuable.

– As the denominator gets larger and the numerator stays the same the fraction gets smaller.

– So the bigger the difference between the numerator and denominator, the smaller the fraction.

– Does it work with improper fractions? Yes.

– Is it worth revisiting in class yet because some students may pick up the “trick” and not be ready for the reasoning behind why it works.

– But isn’t it really simple? 3/4, 3/5, 3/6, 3/7…and so on…the difference of the numerator and denominator is getting greater, so the fraction is getting smaller.

So in closing I have no answer of what to do with this information. I am thinking I will revisit it with the student alone because she is anxious for why this works. I may save it for the rest until I have a better grasp on where they are with their understanding of numerator/denominator relationships, but am I being too cautious? I just don’t want “tricks” to be used because they are easier for some students than the reasoning piece.

Would love any thoughts!

-Kristin

Math in a Movie Trailer

1 Reply

Last Wednesday at a PLC meeting, our district instructional technology specialist did a presentation on Blended Learning. She did a beautiful job of demonstrating apps and web-based activities at various entry levels, so each teacher could participate. One of the fourth grade teachers expressed an interest, and a bit of fear, in trying to use ipads as part of her classroom routines. Since I had been in her room doing some math coaching the previous week, I offered to help her design an activity and give her a hand in the classroom with the ipad piece if she was not comfortable.

We met the next day to start our planning! She was just ending her 3D math unit in which students had been identifying 3D shapes by their silhouettes and attributes and finding volume of a rectangular prism. As a culminating activity, we decided to have the students create a movie trailer in iMovie that “told a story” about the unit. I sent the teacher home with one of the ipads to “play around” with iMovie, since she was not very familiar (or comfortable) with it. I was so excited to come in the next day to see a trailer she had created at home that night! I LOVE when people jump right in!

This is how our lesson played out over the next two days…

– We created a room in “Todays Meet” on their ipads and had students go in and do a test post.

– We posted the question, “What is the purpose of a movie trailer?” in the TM room and let them type as we showed two movie trailers (Percy Jackson 2 and Despicable Me) on the SMARTBoard. When the trailers were over, we switched back to TodaysMeet on the SMARTBoard to go through their comments and have them expand on them. Here is a clip of the conversation:

– Next we asked them to continue chatting about things they learned during this math unit. We noticed they were just writing one or two word things so we asked them to expand a bit and use more of their 140 characters. Sample clip:

– As a class we scrolled back through and had them stop and ask questions of each other if they didn’t understand what someone had posted. They were so engaged and they all kept asking if they could do this at home?!? YES! Next time I will leave the room open for a longer time frame so students can post as they think of things at home! What a great way to open class the following day!

– We took them through a brief “tour” of iMovie and let them move to a place in the room to look through the themes and storyboards and start brainstorming ideas for their trailer.

– To help them organize their thoughts, I had put a template of the storyboards: http://tinyurl.com/c3g5r2e in the Dropbox that was on each ipad. The students exported the PDF to UPad Lite:

and let them play around with how to write on the document with pen width and different colors.

– The following day, students got in their groups (of 2-3 students) to plan out their storyboard and decide on pictures they need for their trailer.

When we meet on Monday, we are taking them around the school and outside to take pictures they need for their trailer. They are working this week finishing up the project, so this story will have To Be Continued…

Mathematically Yours,

Kristin

Who is Coaching Who Here?

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I am so fortunate to be involved in a wonderful state-wide cohort, MiST (Math Instructional Specialist Team), organized by MSERC and made up of specialists from the University of Delaware, the Delaware Department of Education and districts across the state. This year, one of our foci has been content coaching, looking deeply into the structures that need to be in place and how it can be used to develop lead teachers in our schools. Our latest “homework” for the group was to try content (math) coaching with a teacher in our building.

At my school, we do not have a coaching model in place as of this current school year. However, with our implementation of the CCSS, we are writing a district plan that involves structuring a content coaching model into each of our elementary schools. We are fortunate to have both a math and reading specialist in each elementary school, so we are starting with a foundational structure in place. I thought it would be great to “try out” a coaching situation with one of the teachers in my building to bring back to MiST and get a feel for how it would work.

I have a wonderfully open 4th grade teacher in my building who is always excited to learn and willing to have me in her classroom and go through this process. I find one of the more difficult things of coaching is finding that teacher who is open to having someone else in their classroom and sees the value in the learning experience. I really lucked out with her! We met a week prior to the the day I would be in her classroom to choose the lesson and chat about which area(s) she would like to focus. The lesson we chose was on linear equations and in our pre-conference she wanted to focus on the timing of her launch and how to meet the needs of her “done early” and struggling students during the lesson. We planned a day to meet within the next few days to discuss the details of the lesson more thoroughly.

Having never taught this particular lesson before, I read and re-read the lesson, taking notes on the math involved, the launch of the lesson, how to extend and intervene for the students. I really was feeling the pressure to know EVERYTHING about the lesson and be overly prepared for any questions asked of me. I think I was more nervous about this meeting than she was.

The day we were meeting about the lesson (pre-conference), I got to her room only to find out the plan had completely changed. She had just found out that in another RTI group (where @ 6-8 of her students go) they had already done our planned lesson. Oh No! We had to refocus quickly (45 minutes of planning isn’t much time), so we looked ahead to where she would be in her core math class and chose another lesson.

This was probably the BEST thing that could have happened to me, although at that moment, I found myself starting to get nervous. I am a planner. I feel as the “math specialist,” I need to have all of the answers to any questions the teacher may ask. Which is so ironic bc with students, I am completely OK with saying “I don’t know that answer, let’s check it out” but with adults, I put pressure on myself. With this unpredicted switch in lessons, I instantly went from coach to co-learner and it was awesome. We read through the new lesson, asking questions as we went, learning from each other. I was offering ideas, she was offering ideas and we collaboratively “lived” the lesson for her 45 minutes of planning time. We talked through questions (inspired by Lucy West) such as: What is the math in the lesson? What previous experience have the students had? Who will struggle? Who will be done early and what will they do? What will the share out look like? Would you like me to chime in during the lesson?

She taught the lesson the following day, I filmed it, and it went beautifully! (I will be blogging more about the actual lesson soon). Her classroom culture and routines were evident from the way the students respectfully disagreed with one another and moved around the room. I was so impressed with her! I cannot even express to everyone the excitement I felt when I left the room. I felt the success of the lesson as if I had taught it myself. Things we had talked about in the pre-conference came out from the students during the class. Things we had not thought about came out from the students. I feel like we had a mutual investment in the lesson, both feeling equal responsibility. Our post conference is set up for next week, so I will blog about both of our reflections on the lesson and the process….so stay tuned!

It led me to ask myself, what does this coaching structure really look like? Who really coaches who? I would argue that this is really a multidimensional coaching model. I offered insight into “the math” of the lesson, the classroom teacher offered questions and insight into her students’ minds, the students offered comments for us to think about in upcoming lessons. When our district revisits our CCSS implementation plan and structure for coaching, this type of experience is critical in setting up those structures.

So thank you to my MiST peeps for the knowledge, motivation and safe environment to learn and share experiences. Thank you to my amazing 4th grade teacher for being so open to having me in the classroom and willing to learn through this with me. Thank you to the students who teach me something new every day I walk in the building.

Mathematically Yours,

Kristin

Negative Talk Is Not Always a Bad Thing.

Connecting the Dots in 1st Grade Math Centers

Isn’t Math Really Just How You Look At It?

1 Reply

Over the past year and a half, I have attended numerous CCSS trainings, read the standards and examined the CCSS learning trajectories. It is evident there is an emphasis placed on understanding of the properties of operations in the elementary grades. I don’t know about anyone else, but I remember it being taught to me as a lesson: Commutative Property is a+b=b+a… and such. No meaning behind it, simply some symbols, that if you could memorize and recite each, you were considered successful (as far as grades were concerned) in math class.

Fast forward to my second year as a K-5 math specialist. Having taught nothing below 5th grade in my previous 15 years in education, I am slowly wrapping my head around the depth of conceptual knowledge in grades K-1. I always knew K-1 was very “hands-on” but I have to admit, I really did not understand the complexity and beauty in the way kindergarteners “see” math until this year.

The other day I did a number talk with a class of kindergarten students. I displayed various dot images with anywhere from 5-10 dots arranged in different patterns. My goal was to have students subitizing the dot patterns and writing addition equations to match the groupings.

I flashed the first dot image on the smartboard for @ 2 seconds and the students wrote the number of dots they saw on their dry erase board. Students shared their answer with a partner and showed me their boards. I put the image back up and asked how they saw (visualized) the dots. We talked about different groupings, circled the dots for each, and practiced writing a couple equations together.

Feeling confident about the goals i had set for the number talk, i began to rethink them a bit after the following image:

Students quickly shared the answer of seven and then I asked, “How did you see the dots?”

The first student said,”I saw 2, 1 ,1,1, 2.” I had him circle the dots the way he saw them on the SMARTBoard and asked the students to write an equation for that grouping. Many successfully wrote a version (with some backwards 2s) of 2 + 1 +1+ 2+1=7. As I was looking around, I noticed one little girl had written all of the possible ways to arrange the 2s and 1s in the equation on her dry erase board. I realized at that moment, THIS is the commutative property in action! We shared all of the equations and I wrote them on the Smartboard. I posed the wondering to the class: How can these equations look different but still have the same answer? They talked to their neighbor and the common response was because no dots left the picture…not exactly what I was looking for, but good answer. I thought maybe it was too many numbers in the equation to see the commutative property or i just asked the question wrong, so i continued.

I asked for another way they saw it. Tons of thumbs went up (this is our sign for having a strategy) and the next student came to the board and circled 5 and 2. She knew it was a five, she explained because of a dice and she just knew two (there was the subitizing i wanted, but at this point we were going deeper). I asked students to write an equation for that grouping. They shared with their partner and we recorded 2+5=7 and 5+2=7. I was excited because two students had already written both equations on their boards before the share out. Now I posed the same type of question, worded differently, “What do you notice about the two equations we just wrote?”

I got responses like:
“The have the same numbers”
“Seven is at the end”
“Seven is the answer”
“He took my eraser” (all a part of the kindergarten learning curve)
“5,2,7 are there, mixed up”

I went with that comment and pressed further… “So how can the 5 and 2 be mixed up and still have the same answer?”

After a minute or two, one little girl said, “It’s just how you look at it. From that way (she pointed left) it is 2 then 5. If you look that way (she pointed right) it is 5 then 2.”

So there you have it teachers…the commutative property is “just the way you look at it.” Simple and beautiful.

MathMinds

by Kristin Gray.

Why We Need Two Teachers in Every Classroom…

A Fraction of our Time in Math Class…

Reversing the Number Talk

Meaningful Math Conversations…

Fraction Talk

Math in a Movie Trailer

Who is Coaching Who Here?

Negative Talk Is Not Always a Bad Thing.

Connecting the Dots in 1st Grade Math Centers

Isn’t Math Really Just How You Look At It?