How Planning Mistakes Can Lead To Great Student Thinking….

The other day I did this fraction clothesline activity with a 5th grade class and today I had the chance to do it again with another 5th grade teacher, Leigh. It is always so nice to get to have a do-over after having time to reflect and think more about what the students thought about both during and after the activity.

I really thought the conversation was great during the clothesline activity, but it took too long the first time. We noticed that some students began to disengage. To try and improve upon that, Leigh and I decided to give only one card to every pair of students instead of each student having one. However, due to us wanting to keep a few important cards we wanted to hear them talk about, some pairs had two.

I also did not like my placement of 0 being at the very end of the left (when looking at it) end of the string. I moved it out some and talked about the set of numbers that falls on either side of the 0. I felt much better about that this time around!

In the planning of the first clothesline activity, we took fractions from the work the students had been doing with percents and decided on putting 100% in there, completely thinking it would be at 4/4. As the student placed it, however, I started realizing that I never thought about the difference of 100% in terms of the area representations the students had been using versus 100% when talking about distance on a number line. But now, having time to reflect on the card, I thought it would make a great journal entry!

As we neared the end of the card placements, I handed the 100% card to a student and told her it was going to probably cause a lot of discussion but just put it where she thought it went. She said she got it, walked up there and placed it on top of the 2 (the highest number on the line). There were some agree signals going on and some other hands that shot right up to disagree. We talked about it a bit and then we asked them to journal their ending thoughts so we could move on with the rest of the lesson about different sized wholes.

Some thought that 100% was at 4/4 on the number line because it equals 1….

Some thought it was at 4/4, but because of the conversation became a bit unclear…

Some thought it goes on the 2 because it is the biggest number on the number line…

Some related it to different contexts with different wholes…

And one student said it can be anywhere with beautiful adjustments as it moves….

What a great day revisiting my planning mistake!

-Kristin

Today, I had the chance to plan and teach with a 5th grade teacher and it was awesome! Last week, this class had just finished a bunch of 100s grid shading in thinking about fraction/percent equivalencies, so we picked up planning the lesson in Investigations with the fraction/percent equivalent strips. Instead of the 10-minute math activity, we thought it would be really interesting to do the clothesline number line to kick off the class period.

We chose fractions (and one percent I will talk about later) based on the fractions the students had been working with on the grids. We chose fractions based on different comparison strategies that could arise such as:

• Partitioning sections of the line
• Distance to benchmarks
• Equivalent Fractions
• Common Denominator
• Greater than, Less than or equal to a whole or 1/2

We settled upon the following cards:

1/4, 3/4, 4/4, 1/3, 4/3, 5/10, 2/5, 100%, 3/8, 1 5/8, 1 7/8, 4/5, 11/6, 1 6/10, 1/10, 9/8, 12/8, 2

Then, we moved into the fraction cards. We gave each pair of student two cards. In hindsight, for times sake, I would probably only do one card per pair. I gave them one minute to talk about everything they knew about the fractions they had and then we started. I asked for volunteers who thought their card would help us get started and called on a boy with the 1 7/8 card. He went up and stood all of the way to the right and said he couldn’t put his on. I asked why and he said that since the cards were all fractions the line could only go to 1 so his is more than one and can’t go on here. I asked if anyone in the class had a card that may help us out and a student with the 2 card raised her hand. She placed her card all of the way to the right, said “maybe it goes to two” and the other student placed it just to the left of it because, “it is only 1/8 from 2.” Awesome!

We went along with the rest of the cards and so many amazing conversations, agreements and disagreements happened along the way. There are a few things that stand out in my mind as some great reflections on the activity:

1. A student had placed 5/10 halfway between 0 and 1. The next student placed 2/5 just to the left of the 5/10 because, “I know 2 and a half fifths is a half so that means that 2/5 has to be less than 5/10. It is a half of a fifth away.” The NEXT student volunteered and placed 3/8 overlapping just the edge of the 2/5 card on the left. I was expecting percentages to come out, since that was their most recent work with those fractions, however the student said they knew 3/8 was an 1/8 from a half and 2/5 was a 1/10 from a half and an 1/8 and 1/10 are close but an 1/8 is just a little bit further away. Awesome and definitely not what I expected!
2. I wish I had not put the zero so far to the left. Looking back I am wondering if that instills misunderstandings when they begin their work with negative numbers on a number line similar to the original misconception that launched the activity with the 1 7/8.
3. Oh, the 100% card….complete mistake on my part, although it may have been a great mistake to have! In the first class, the student with the 100% card came up and said, “I have 100% and that is 100/100 which is 1” and put it in the appropriate place on the line. Just as she did that, I started thinking how I never really thought about the distinct difference between percent in relation to area (like the grids they had been shading) and 100% when dealing with distance on a number line. No one seemed to notice and since I didn’t know exactly what to ask at that point because I was processing my own thoughts, I waited until another student placed 4/4 on top of it and erased it from my immediate view!
• I stayed for the next class and this time I was prepared for that card and now really looking around to see what students’ reactions were when it was placed. As soon as the student placed it at the 1 location, I heard some side whispers at the tables. I paused and asked what the problem was and they said, “100% is the whole thing.” The next student who volunteered had the 2 card, picked up the 100% card on the way to the right side and put the 2 down and the 100% on top. Lovely and just what I was thinking.

I have never had students reflect on the difference of talking about percentages with distance versus area because I had never thought about it! It definitely feels like an interesting convo to have and a great mistake that I am glad I made!!

I will be back in another 5th grade class tomorrow and will see what happens…it could make for a great journal writing!

-Kristin

Fraction/Percent Equivalents

It goes without saying that I miss talking 5th grade math with my students each day. But I am so lucky this year to have a new, wonderful teacher in 5th grade who lets me plan and teach some lessons with her! This lesson was one of her first lessons of Unit 4, Name That Portion.

Since in 4th grade the students do a lot of work with comparing fractions, we designed a Number Talk string in which students were comparing two fractions. We wanted to hear how they talked about the fractions. In the string we used a set with common denominators, common numerators, and one unit from a whole. On each problem we were excited to hear talking about the “size of the piece” being the unit and the numerator telling us how many of those pieces we have. Our 4th grade teachers really do a beautiful job with this work. They also used equivalents to have common denominators to compare and a few used percents, since they had done a some grid work with that they day before.

They then worked in pairs to shade 1/8 and 3/8 and we came back to discuss. We noticed as we walked around that the shading was wonderful on their papers, but when asked to write the fraction and percent, most were blank. I remember this lesson from last year during decimals where the same thing happened. So, we asked them what they thought the fraction was as we got these three answers…

12 r4/100

12 1/2/100

12.5/100

They were not overly comfortable with any of them so we asked them to journal which one “felt right” to them and why…

We loved to see what they knew about decimal fraction relations, but we especially liked the “it sounds more fifth grady to use 12.5.”

-Kristin

#IntentTalk Chapter 1

Since it was a bit too much for me to continually tweet, I decided to do a quick blog at lunch!

Principle 1: Discussions Should Achieve a Mathematical Goal

The first week of school my mathematical goals revolve around discussions about students’ mindset in terms of math, as well as the mathematical practices. I found this year that Talking Points and one of the tasks I found on Fawn’s blog fostered those goals. I blogged about it here. My tables are all arranged in groups and the students know from the very first day that discussions will be a huge part of our work together.

Principle 2: Students Need to Know What and How to Share

To support this in my classroom at the start of the school year, I have the students agree upon our class norms. They originate after doing a Number Talk together and reflecting on what we expect as a group during our discussions. I reference these norms throughout the course of the school year.

Principle 3: Teachers Need to Orient Students to One Another and the Mathematical Ideas

I find a lot this happens during our Number Talks and in then daily in our journal reflections. This is such a focus on my planning of questioning. Asking things such as, “Can you re-explain their thinking in your own word?” or “Did something ____ say change your mind about that way you were thinking?”

Principle 4: Teachers Must Communicate That All Students Are Sense Maker and That Their Ideas are Valued

I think this principle emerges during our very first round of Talking Points of the year when the students my go around their circle with no commenting from others. It allows students the opportunity to speak their ideas without judgement or comment. Being able to change their response on the second round, lets the group know that as we make sense of problems and listen to others, we change our mind, just as we do when we make errors. The freedom I see in their journal entries also reinforces the idea that I value their thinking and know that there is reasoning behind everything they write and do in my class.

“Talk is an important way to build that sense of community and to help children grapple with important mathematical ideas.”

-Kristin

Last Day of Math Class :(

Today was the last official day I had my students for math. It was a bit sad for me and it was nice to hear some of them say it was “kinda sad” for them too. In moving into a K-5 Math Specialist position next year, I know it will not be the same experience watching a group of students grow over the course of the school year.  It will be great in different ways, but I am really appreciating all of the amazing work my students have done this year.

So…what to do on the last math day after they just had field day yesterday followed by our PBS bowling field trip tomorrow? It is a tough planning!

I first had them look through their two math journals, one from the first half of the year and one from the second half. As their last writing piece, I asked them to write things they noticed in their work over the course of the year after looking through their journal. I only had time to grab one journal today because the end of the year craziness is kicking in, but I plan on following up with a more detailed post later. This one was so powerful and truly gave me goosebumps….

After they finished that, I asked them to revisit some of the claims they had written over the course of the year and see if they still thought they were true and could be proven or were not true and needed to be revised. This student had written a claim that when you are multiplying fractions, you could multiply the numerator and denominator to get your answer. As he was proving it just worked for multiplication, he stumbled upon the realization that it worked for division as well. He then worked through a few more division problems and it was such an amazing explanation!

He revised his claim…

I promise to follow up with some really amazing work they did on the last day when summer is here and there is a chance to breathe 🙂

-Kristin

Patterns and Perseverance

Today in math was a test in perseverance. The students were working on the growth pattern of an animal called the Fastwalker. It was fairly easy for them to complete up to the 10th year, graph it and answer the questions regarding the line they graphed. The book did not require them to do any generalizing of a rule, however they had other plans! Here is a completed table of one of my students:

We worked on this up until lunch, discussed the line and how it was different than the constant rate of change graphs we had seen earlier. They kept asking if there was a rule for this one, so I asked them to play around with it and see what they thought. One of the students noticed that if you added all of the terms before with the term number you were trying to find, it gave her the height, so she started adding to see if it worked for the 100th term (nothing like starting small:).

While she was working on adding, another student, who had done a consecutive sums task earlier in the year in RTI w/me, realized there was an easier way to add those numbers, and you can see on the top of the page where he started playing around with pairing up the numbers:

It was so interesting to see the groups working with them and asking questions as they tried different ideas. The two of them finally saw that pairing up the numbers was working and began to generalize based on what they had done with the numbers. It was awesome because they began generalizing based on an even or odd number term because of the pairings and needing to divide the term number by 2. At the bottom of the first paper earlier in this post, you can see she wrote an even and odd rule for the pattern, while this student realized that if should work with even and odd because the decimal didn’t make a difference.

Shew…..they were sooo proud of themselves (and I was so proud of them) at the end of all of this hard work! The student who did the paper above said, “Wow, that one problem took us almost two hours!” And it was SO worth it to see the accomplishment on their faces. THIS is the reason we must make time for students to investigate their own mathematical curiosities and give them the time they need to persevere through these problems!

-Kristin

Growth Patter Number Talk….3rd Times a Charm

Over the past couple of days, with my homeroom, I have tried a few strings of numbers to bring out the different ideas that are important when thinking about growth patterns and finding any term in a sequence, Here and here. Both days brought out many great ideas, conversations, and disagreements, however I couldn’t help but feeling the ideas we talked about in two days, could have been achieved in one and felt a little more connected. I knew it was completely the way I posed the problems, so when my second class came in yesterday, after missing a couple days of math due to testing, I was excited to try and adjust my previous work.

Apologize for the messy board, but I still cannot seem to get a handle on that recording thing…

After that, I asked them to to count by 3’s starting with 6 and stopped them at 15… Asked for the 10th term and got, as expected, 30 and 33. Then the conversation took off with proofs and some really important ideas that was hoping would emerge. I love it when the class is practically divided in half on an answer, we had the 30’s and the 33’s. I asked a 30 to explain how he got the answer and he quickly said 3 x 10=30. I saw a lot of agreement, so I asked for a 33 to share their reasoning. A student said that we “need the beginning number, three, to find out where the tenth one is. 3 x 10 is 30 but then you started three ahead of that so you add 3 to 30.” I wrote that down on the board.

A student then said something that made me have a realization, “It shouldn’t change because you are still doing 10 jumps of 3, so it HAS to be 30. 33 is 3 x 11.” In my last class I had a student who kept insisting that the 10th term remain the same no matter where we started and I could not figure out what they were trying to articulate. NOW, I understand. 30 will always be the distance between wherever we start in the sequence and the 10th term, but not the tenth from the true beginning. AH HA!

So, the beginning number was suddenly becoming very important and articulating “10th term from where” was having students agreeing that the 10th term starting from the 6 was going to be 33 but when thinking about a rule for the pattern we needed the true beginning. We were just about to head back to our desks to continue our work when a student (different than the one who had originally said it) said that we could write this one “3 x n + 3 = A” because you have to “add the three you are missing from the beginning to get the answer.” I had them turn, talk and try a few terms out and see what they thought. It was all wrapping up nicely (I was excited about it) when another student said, “You could also write 6 + (3 x n) since you are starting at 6” ….oh goodness, they just don’t ever let it end and I love it:) A disagreement arose that it would have to be “6 + (3 x n -3) because of that extra jump of 3 to start at 6.”

I always hate to say that time got the best of me, but I had missed this group for 2 days of math and I saw this conversation going lonnnnng so I had them write those ideas down in their journal to kick off our class on Monday!

I love when I have the chance to refine ideas that don’t go exactly as I had hoped they would, especially when I know it was completely how I posed the problem or asked the question. After a couple days of talks not connecting as I hoped they would, third time was a charm!

-Kristin

Growth Patterns – Number Talk Attempt…

After Tuesday’s talk, I wanted to continue having the students look for patterns within sets of numbers. They found it really easy to find any future term in our last talk because our starting term was the change value. For example, they knew the 10th term when counting by 3’s and starting at 3 was as easy as multiplying 3 x 10. I was curious how they would think about predicting future terms if the series did not start with the same number they were by which they were counting. I brainstormed a few possible strings students could begin to think about this and, if time went well, they could look for connections between:

I decided on the following three series:

12, 18, 24, 30….

12, 15, 18, 21…

6, 11, 16, 21….

In each one I was going to have them figure the 10th term and discuss ways they were thinking about it. The idea that I wanted to emerge is the importance of accounting for the number at which they were starting and I also wanted to see how they made their predictions. I was going to end the talk by asking what the graphing story would look like if the first term was a starting height and it continued growing at this rate to connect back to our graphing stories from the previous day. As it sometimes happens, I did not make it as far as I had hoped because some amazing conversations were happening in the very first set of numbers.

I had a student count by 6’s while I recorded, starting at 12 and stopped him after I wrote 30 on the board. I purposefully stopped there because I was curious to see if students would think about the next (5th term) and double to find the 10th as they did the day before. There was an overwhelming agreement for 72 for that exact reason, but since I got a few different answers for the 10th term, I wrote them all on the board and the proving, agreeing and disagreeing began. There was one, lone 66.

There were two proofs for 72:

– Found the 5th term as 36 and doubled it to get the tenth term.

– Did 6 x 10 to get tenth term but then added 12 because he started at 12. I was excited to see he was acknowledging where the series started and the idea of the start being important.

The lone 66, then did a simple continuous count to the 10th term proving that it would be 66. Heads tilted and eyes squinted. I realized at that moment how much I appreciated that the students looked for ways to think about the 10th term without having to count up to it, but also realized that we needed to do a little more work in thinking about what was happening in the sequence.

Since I knew I was not making it past this series of numbers, I decided to connect this set of numbers to a set in which the start was 6. I wrote them on top of each other:

12, 18, 24, 30……

6, 12, 18, 24…..

It then became clear to most that the first set’s 10th term had to be 6 ahead of the bottom one because of the start. The idea of term number and the increase from term to term started to emerge. One student said the bottom series “started one term earlier so it has to end 6 earlier than the top one.” Another student bounced off of that with “A term is 6, right?”

The debate continued and great ideas were coming out about what being the 1st term meant and then one student (the 66) said well it has to be right because (the term number +1) x 6 works for every one of them. That threw most kids for a loop and definitely not a place that I thought most of the class would be ready to engage in conversation around. I let a few students question what he meant, and I let him answer them. The biggest disconnect for students was how the term number factors into finding the number in future terms. To them the term number was just labeling and not really relevant in the values.

It was time to move into our lesson for the day and I was happy with the ideas that were emerging so I had them go back to their journals and do a quick 3 minute writing of either: what they noticed between the two, what someone else said that cleared up something for them, or something they were confused about still. It was interesting to see the word “group” popping up when that really didn’t come up in the talk…

The “R” was my writing on the board….the SMARTBoard jumped and my 2 looked a bit like an R…they would not let it go of course:)

and of course there is always one that I want to find more about because it seems nothing like what the others thought about..

After this talk, we went into some pattern building with rectangular arrays and finding the nth term. So much to write about that too, but will have to save that work for another night!

Tomorrow, I want to go back to second number talk set I had intended to do today and see how the conversation builds on our thoughts from today. Do they think about the starting number now? Do they talk about the numbers as “terms”? I think I will have them journal about what they find is most important when predicting what future terms would be in the series.

-Kristin

Growth Patterns…the Beginning

It is finally here, our last unit of the year and I have to say I am so excited to make it to Growth Patterns! Before we did some reorganization of units due to CCSS shifts, I could never get to this unit, however now having fewer units, I finally make it to the end!

If you have never looked at this Investigations unit before, here is a brief description:

This unit is really a beautiful culmination of all of the pattern noticings my student do all year long.

Being the end of the year, I know I am going to miss constantly having a class to try out my ideas on during the course of the summer, so I am making the most of the time I have left to test some different number routines.  After reading through this unit, two things jumped out at me that could potentially be incorporated into our number routine work before the lesson: predicting nth terms and relationships between two sets of numbers. So, I thought it may be interesting to do some counting and then think about relationships between our sets we make.

I first had a student count by 3’s four times: 3,6,9,12…. I asked the 10th term and everyone looked at me like I was crazy because this seemed too simple and said “30.” I asked how they knew and immediately thumbs went up and they all agreed that 3 x 10 = 30.I asked for another way you could get there? Next I got, “If you add one more 3 and get 15, you can just double it because it is doing the same thing all the way.” I wrote (12 + 3) x 2 = 30. Another student said, “You can double/half and do 5 x 6 = 30.” Me, “How does double/halving look here?” Silence for a minute, so I asked them to chat with a neighbor and then they came to thinking about jumps on a number line and explained that you were doing jumps twice as big so you need half as many.

Next I did the same process with 4’s: 4, 8, 12, 16 and I got much of the same as above. I asked if we could make any connections to the first set. I had an idea this one may be tough (but I had a plan for it in the next set), however I did have two students who noticed it was one more every time you counted by 4’s, so by the time you counted 10 times, you would have to add 10 to your total, 30+10=40.

The third problem, I wanted to give them the chance to make some connections to the 3 and 4 counts, so I did 12’s: 12, 24, 36, 48… Again, I got many of the above strategies, but then they made some connections to doing “3 times as many jumps of 4 in 12” and “4 times as many jumps of 3 in 12” and adjusting their 10th terms.

Because some students were really comfortable moving between the sets of numbers, others were still staying within one set of numbers in describing the 10th term, I gave them two sets to go back to their journal to reflect on: 2’s: 2, 4, 6, 8…. and 2/3’s:  2/3, 4/3, 6/3, 10/3….

I asked them to either talk about how they could arrive at the 10th term or make a connection to one of the previous sets of numbers…

It was interesting to see their connections between the sets, but I think for next time I need to think more about either the predictions of future terms or comparing two sets, this was a bit too much at once. Next time I may have them look at sets with constant change but not starting at a multiple of that number….and then as I was typing that, I think it would be cool to come up with a set with a missing number at the end that could be varying numbers depending on how they see the pattern, then give a term after the missing number and narrow it down to which pattern it actually is….hmmmm…have to think more about this one…

-Kristin

Listening Carefully to Student Thinking

Recently, I have been reviewing a new “CCSS-Aligned” middle school curriculum and find myself completely frustrated with the overabundance of scaffolding and lack of student thinking required on every assignment. Not having the days/weeks it would take for teachers to engage in the mathematics as both learners and teachers, I needed a short, powerful way to show that this is not how students should experience/learn mathematics.

As I looked at the fraction page like this, my thought was “Why just two ways?” quickly followed by “Why those two ways?” quickly followed by “My students are doing this now, flexibly.”

Right then, I realized the perfect proof of why NOT to do this, was the work my students already do when given the freedom to reason about a problem and do more than just procedurally compute an answer. So, I put the proof in their hands.  I simply asked them to solve 2/5 x 7/10 as many ways as they could. Some got creative after a couple of ways, and by no means am I saying some of these are “efficient,” but they show so much flexibility.

This felt perfect. Why would we want to miss out on all of the great conversations that can happen around this work by making them answer in just 2 ways, and more specifically, those 2 ways they show you how to do…step-by-step?

and THEN this happened which validated my thoughts even further and instantly made me reflect on my friend Christopher’s talk at ShadowCon (video coming soon) around listening carefully to student thinking…

The students were working on 2/5 x 7/10 as I was walking around the room observing their work. I glanced over a student’s shoulder and saw “Doubling and Halving” written on her paper with the correct answer. Assuming it was doubling/halving in the sense of doubling one factor and halving another factor, I was excited to see the use of the strategy.

I asked her how she did it, she said, “I double/halved” and I was about to move on to get ready for our sharing. When I glanced down, however, it was not at all like I had imagined. I asked her to explain further…“I halved this numerator and doubled this denominator [points to 2/5] then I doubled this numerator and halved this denominator [points to 7/10].”. Ok, now THIS is much different than I thought!!

I had her share, and others immediately said they had double/halved also but did not get those fractions to multiply and wondered if that worked every time (I love that they ask that now:). I let them play around with it for a bit but since we had some division work to do I told them to keep thinking about that and we will revisit it tomorrow. By the end of the next class period, I had a student come up and say, “She didn’t double/half really, she quadrupled/fourthed.” I asked him to write down his explanation for me because it was lovely.

So glad I listened carefully and didn’t makes assumptions on her understandings because how amazing is this work? I am also so glad that I can appreciate a curriculum that allows for these reasonings and conversations to happen.

-Kristin