Matt Coaty

Fraction Blocks and Strategies

My second and third graders started a unit on fractions last week. Students are used to identifying typical pie fraction pieces. Generally, I find students are introduced to fractions using this type of visual representation. Students then count the amount of pieces and place that number as the numerator. I find moving towards mixed-numbers has some students changing their strategy as they can’t just count the pieces, but they have to recognize that a certain amount of equal parts are one whole. Based on their pre-assessment results, it seems as though my second grade and some of my third grade students are at this point.

Using a number line has helped. Placing the fractions on the line has brought a better understanding of the placement of fractions in relation to a whole number. Currently, students can identify certain benchmark fractions on a number line. We’re working on bolstering this skill and connecting it to fraction computation in the near future. Before that happens I want to ensure that they have a decent understanding of mixed numbers and where they fall on a number line.

On Thursday and Friday I introduced students to a fraction block activity. Students were given a sheet with fraction parts. Each block was split into a certain amount of equal square parts.

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Each student was given an envelope to put their pieces in once they were finished with the activity. Students cut out each block and were asked to put them in order from least to greatest value. Students were able to complete the task.

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We then had a conversation about quarters, halves and wholes. I then gave each student the card below.

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Students placed the A block near the top of their desk and started comparing the different blocks. The class completed the first question together.

I then gave students time to work on the rest of the problems. Students were then given time to use trial-and-error to find which blocks worked for each problem. I went around to the different table groups and asked students questions about their strategies. Students ended up matching the squares with other shapes to determine what was a quarter, half, almost a half, and what happens when you combine shapes. After about 10 minutes the class reviewed the sheet and found that some problems could be answered with multiple solutions. Students put the sheets in their envelopes since we ran out of time.

The next day students completed some more challenging half-sheets involving their blocks.

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Students struggled a bit with this as they had to look at A as half instead of one whole. This changed the value of all of the other blocks. I allowed students to work in groups for about five minutes and then independently for another five. This gave them an opportunity to gain another perspective and a different strategy. Afterwards, I reviewed the possible solutions with the class.

Next week I’m taking this activity one step further and using the blocks without markings. I’m borrowing this idea from Graham’s post on defacing manipulatives.

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Students will complete similar half-sheets, but without the evident markings. I’m looking forward to seeing how students’ strategies change and the math conversations that follow next week. Click here to download the activity that I used.

Reasonable Solutions

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I believe teaching multiple grade levels within the same day has value. Being able to observe how students think about numbers and the strategies that they use over time gives teachers a different perspective. It also shows some of the linear progression of math skills and strategies. I found this especially evident as I read through Kathy Richardson’s book during July. I currently serve as a math teacher for students in grades 2-5. I get to see how students progress over time and what tends to trip them up. I also see the problems that emerge when students start to rely on tricks and formulas before having a deep understanding of a particular concept. One thing that I also continue to observe is that students sometimes struggle to be reasonable with their estimates. Part of that may be due to an over-reliance on algorithms and the other part may relate to exposure. Students aren’t given (or take the) time to reflect and ask themselves whether the answer truly makes sense or not. This tells me that students are relying on a prescribed process or algorithm and reasonableness comes second.

In an effort to move towards reasoning, I’ve been using Estimation 180 on a daily basis. I feel that the class is become better at estimating and their justification has improved. Making sense with number puzzles also seem to be helping students create reasonable estimates and solutions. Basically, students are given a story that has blanks.

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Students are then are given a number bank. Sometimes too many numbers are in the bank.

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Then students have to justify why they picked each answer. This can be completed in verbal or written form.

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Usually I have students explain their reasoning with a partner. The class has completed a number of these types of making sense with numbers puzzles. I can say that students are now looking more closely at the magnitude of the actual numbers before estimating or finalizing an answer. That’s progress and I’m confident that students are more willing to use that strategy along their math journey in the future.

We’re Back in Business

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Students are scheduled to enter my school tomorrow morning. It’s been a whole two weeks since I’ve seen my students. Tomorrow, routines will be reestablished, backpacks will be filled, students will be chattering about their break, and students and teachers alike will get back into school mode after a brief hiatus. As tomorrow approaches, I’m reflecting on what my classes have accomplished and what still is on the plate ahead of us. I spent a good amount of time yesterday planning out the next week of instruction and it confirmed my anxiousness to know that the school year is over half-way completed. As I look over the next few months, I’m finding curriculum pacing guides, standardized testing, school performances and field trips all impact my instruction to a certain degree. This happens every year and it has me thinking of what time I truly have left with the kids. I’m also aware that these next few months directly impact students in meaningful ways. For some, this will be my last year with a group of fifth graders that I’ve seen since they were in second grade. I want to ensure that I make the most of that time remaining. That doesn’t necessarily mean speeding through the curriculum. I’m hoping to gives students opportunities during the next few months to make connections, reflect and set goals. As we all come back tomorrow, I want to communicate the following to my kids:

1.) The learning experiences that you’ll encounter in the next few months are intentionally designed for you to make meaningful math connections. Perseverance will be key in helping you create these connections. You might find that you don’t understand a particular concept when we introduce it. That’s okay. Learning is a process and we’re all in this together.

2.) Group projects, individual assignments and standardized tests are on the calendar and will be approaching in the next few months. Keep in mind that I believe you’ll will show your potential on all of these. The scores and marks will help teachers and your parents have a better understanding of your strengths and areas that might need to be bolstered. Also keep in mind that the scores are a number and don’t represent who you are as a person.

3.) Let’s celebrate a milestone. We’ve worked hard and have made significant progress since September. Each student in here has made gains and I want us to reflect a bit on our success. There’s more to accomplish, of course, but reflecting on our past growth can also encourage us to move forward with additional confidence.

I’d like to communicate this to all my classes at some point tomorrow. I won’t necessarily read off a script, but I feel like flushing it out on here is a decent starting point.

It’s time to get back into the routine of setting my alarm clock to wake up extra early. I’ll be joining the trove of educators heading back into their schools this week. I’m looking forward to tomorrow.

Random Sampling

Before break my students tackled the challenging topic of random sampling. I feel like it’s challenging because some students tend to view their opinion as one that applies to other people around them. It can be a tough concept for students to wrap their heads around. When I introduce this topic students have many questions. Usually they follow along the lines of …

why can’t you ask everyone?
who determines if the random sampling is accurate?
how many people do you need to ask?
is their always bias involved in random sampling?

Some of these questions are more challenging than others. Some I don’t even approach and let students make their own determination. In the past, I had students create questions and ask a random sampling of students. Students would then create charts and indicate whether they truly sampled the students fairly. For the most part the activity hit the objective, although the sampling available at my school was minimal. Students were able to ask questions about our school and students within. Issues came up because of the lack of age groups and diversity.

Last Monday I participated in #msmathchat. The conversation surrounded the topic of teaching about data and statistics. Elizabeth sent out the Tweet below.

A2: Love this interactive from Scholastic looking at data sampling & how it changes based on sample https://t.co/7htgOwnjSl

— Elizabeth Raskin (@elizraskin) December 20, 2016

I saved the Tweet for later as my students are in the midst of their data unit. I looked at it later that evening and thought I could immediately use it with my kids. I put together a template that students could use as they progressed through the site.

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The next day students started at the skate park activity and used three random sampling techniques. Afterwards, students were able to see the how their actual results compared to the entire population. Students then moved on to complete the rest of the scenarios. For the most part students started to change the way they asked the questions to get a better estimate. This was a better activity than what I’ve used in the past. The students responses to the last question brought a better insight to how students perceive random sampling. I believe they’re making headway. I’m hoping that the class can reflect back on this activity after break and they can take the benefits of that experience moving forward.

Two Different Camps

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My school has six days of school left before break. Between now and then I’ll be giving a unit assessment to my fifth grade crew. We’ve been studying angle relationships for the past few weeks. To be honest, it’s been a great unit but it’s also been challenging. There’s been a good amount of struggle in this unit. It’s the good type of struggle. Right now I feel like students are in one of two camps.

One camp is focused on the measurement and precision component. When given a question about angles they want to take out a protractor and start measuring. They want to be precise and get an exact answer. I’d say that some in this camp perceive this type of geometry as a measurement skill, rather than a looking at it as a problem associated with angle relationships.

screen-shot-2016-12-13-at-6-38-09-pm — Where’s my calculator?

The other camp is all about looking at the angles and the relationships that exist. They’re at the point of not even bothering to use their protractor. They also look at the lines, rays and line segments that make up the construction of a shape.

Getting both of these camps on the same page has been an interesting adventure. Both have positive aspirations and have been showing a tremendous amount of effort. I believe it’s important for students to use mathematical tools to solve problems, but that’s not what this unit is about. For so many years students have been asked to be specific and precise when calculating and finding math solutions. This is still the case, but students are now asked to use their understanding of angles and shapes to come to conclusions.

We had a classroom discussion last week about this very issue. I asked students to put away their protractors and calculators. They were asked to identify specific shapes and describe the characteristics of them in detail. The class then explored the different polygons on the Illuminations site. Click on the image to visit the actual site.

Students were allowed time to play and create connections. The focus of the exploration was targeted towards sum of the angles in polygons. The students in the first camp started to put their protractors away while the students in camp two looked at how the angle measurements changed when the triangle was stretched. Looking back, this was such an important period of time. Afterwards, students were given time to review angle relationships without using a measurement tool. They were using their prior knowledge of shapes and relationships solve problems. This was a bit of shift. So, I decided to build upon the first task and added a reasoning component.

screen-shot-2016-12-13-at-6-56-59-pm — Getting camp one and two on the same page

I’ll be grading the task above tonight. Including an “explain your reasoning” component added a bit for vigor to the task. Based on the class conversations I heard today I’m thinking that students looked at precision as well as angle relationships while tackling the problem. After grading them at some point tonight, I’ll review the results with the kids tomorrow.

Coding and Positive Changes

Our school is in the midst of the Hour of Code. This year more than ever, I feel like there’s more of presence of how technology, coding and the curriculum are connected. This is due to a number of factors. A new superintendent, technology coaches and additional teachers are all playing a positive role with this connection.

This year I intentionally looked for ways to incorporate coding into my math classes. In the past, the coding was fun and beneficial, but it felt as though it was disconnected from the actual scope and sequence of the curriculum. It was great during the Hour of Code, but then the whole idea faded once school hit winter break. While searching for curriculum connections, I came across Brian’s fantastic blog. I started to find direct curriculum connections that I could use for the Hour of Code. The two different videos that I used are below. Both were used for a fourth and fifth grade classroom.

Both were great in connecting basic coding and measurement skills. It was interesting to have kids use their schema, as well as trial-and-error to find out how to calculate the area and circumference. I gave students an overview of the Scratch blocks and let them figure out the solution.

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I feel like this was useful as Scratch helped reinforce skills that we’re exploring in class. I look forward to incorporating it a bit more as this week progresses.

Side note: Earlier in the day one of our technology coaches sent the elementary teachers a Google Doc of different coding QR codes (first and third) that can easily be used with an iPad. This information is available for all teachers to use as needed. Some teachers need a starting point and this may provide just that. This is one of the positive changes that I noted above.

What’s your strategy?

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My fourth grade crew has been exploring fractions for the past two weeks. Students have been making some amazing connections between what they’ve learned before and what they’re currently experiencing. Last year the same group of students added and subtracted fractions with unlike denominators. The process to find the sum and difference was highlighted and that’s what students prioritized. That was last year. Although the process was and still is important, this year’s focus in on application. How do students apply their fraction computation skills in different situations? That takes a different skill set. Being able complete a simple algorithm doesn’t necessarily help students read a problem, identify what’s needed and find the best solution. More so, I feel like the application and strategy piece trumps the actual algorithm process at this stage.

So, I brought out a fraction recipe problem from last year.

screen-shot-2016-02-26-at-5-26-13-pm Similar to last year, students had to change the recipe based on the amount of muffins needed. Unlike last year, I didn’t introduce the fraction multiplication or division algorithm. I had students work in groups and document their strategy to find a solution.

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Students had to indicate whether the number of muffins increased or decreased, by how much and how to change each ingredient. The group conversations were fantastic. Groups had a brief conference with me to discuss their strategy once they arrived at a solution.

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The conversations that occurred during my 1:1 meetings with student groups were beneficial. Students took what they wrote as a strategy and elaborated with different examples. I’m thinking that students will write in their math journals about their experience tomorrow. I’m assuming that this will also help transition students towards understanding why the fraction algorithms work.

screen-shot-2016-12-01-at-6-32-08-pm — Can you tell that I like my new stamps?

Estimating as Part of the Process

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My fourth and fifth grade classes explored fraction models this week. I enjoy teaching about the concept at both of these levels concurrently. I can see the linear progression of skills associated with fractions and the different perceptions of fractions. My fourth grade crew is finding equivalent fractions while my fifth graders are multiplying/dividing fractions. Both groups are finding success, but I’m also seeing similar struggles. Students are fairly consistent with being able to convert mixed numbers to fractions and combine fractions. Issues still exist in being able to estimate fraction computation problems and determining which operation to use while completing word problems

This year I’ve been focusing in on making sure students are using estimation strategies. This is especially important when dealing with fractions and eventually decimals. Unfortunately, I tend to find that time spent on the process (algorithm) trumps the reasonableness (estimate) from time to time. Part of this is due to past math experiences and time management. After the last assessment on fractions, I started to look for additional ways to incorporate estimation within my fraction unit. I came across Open Middle last year and I’m finding their fraction resources to be a great addition. Both, my fourth and fifth graders completed a few different Open Middle fraction problems this week.

I’m finding that students are estimating a lot more when they are involved in these types of activities. The tasks I use from OpenMiddle emphasize the need to estimate first and calculate second. These types of puzzles are interesting for students. They are low-risk, but yet have a high ceiling. I also found this to be evident with an activity that I found out of this book. I can’t say enough good things about the ideas and resources found within that resource.

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Students had to find the missing numerator, denominator or variable. In both, the Open Middle and Make it True activity, student worked in groups of 2-3. I gave them about 10-15 minutes to collaborate. The sheet below was adapted from the book above.

screen-shot-2016-11-18-at-9-39-39-pm — Fifth graders worked on this for 10-15 minutes. Class discussion followed

They shared ideas, estimated and came to a consensus on what the solution should be. I had the student groups write their answers on the board and the class discussed all the different solutions afterwards. The class conversation incorporated a decent amount of review and also gave an opportunity for students to ask for clarification. I’m looking forward to having more classes like this. The class conversation component that occurs after a collaborative effort is starting to become an even more valuable piece of my math instruction.

Self-Reported Grades

The first trimester grading period ended about a week ago. Soon, students will receive their report card grades and teacher comments. The majority of teachers in my school have been carefully crafting the right words to be placed on the report card. These comments often communicate how the students are learning compared to the standard, possible struggles, and next steps to improve their learning journey. The report cards are usually sent home via backpack and most students gravitate towards the letter grade that is at the top of the report card. My school isn’t standards-based so that letter grade is often a place of emphasis. The rest of the report cards components come secondary. I’ve noticed this trend for years. This year I’m changing up this process to help students understand and reflect on their own learning before they receive their actual report cards. I decided to create an activity based on Hattie’s self-reported grades influencer.

In preparation for this activity I filled out each report card with comments that I thought were appropriate. These comments mentioned the scope and sequence of math skills explored during the trimester. They also communicated what students could bolster during the second trimester. I left the actual grade portion of the report card blank. I also left the MS, LS, AC and NI blank. These were for students to fill out.

I gave each student their partially filled out report card and student file. The student file contains all of the unit assessments for the first trimester. Students were also asked to use their math reflection journal during this activity. This tended to help empower the students as they were given all the tools needed to fill out their own report card. Before students started to assess themselves I decided to review what the MS, LS, AC and NI meant.

screen-shot-2016-11-13-at-7-21-40-am This took the most time, but I feel like it was worthwhile as students were connecting how particular math skills fit within certain learning goals. They started to analyze their unit assessments, journal and reflection sheets to determine whether they mastered the skill or not.

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After students filled out their report card I met with them 1:1 for about five minutes. We had a productive conversation regarding where the student assessed themselves. Sometimes the students were right on point, while other times they were very critical of their own performance. The process of reviewing their own performance brought a new meaning to the actual report card. Some students also asked questions about the comments and asked that certain items to be taken out or added.

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When the report cards come out I find the students have a few different reactions. Some students shove the report card into their backpack while others critically analyze their results in preparation to answer questions from their parents. In an instant, the amount of effort and time spent in crafting the right words can easily be ignored or highlighted. I’m thinking that this activity will help students to start to see their report in a different light. Self-assessing takes time, but this is an activity that I plan on using during the second and third trimesters.

Decimals and Number Lines

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My third grade class is learning about decimals. Students have been identifying place value positions up the hundredths place. So far students have been successful in decomposing numbers into expanded form and using base-ten blocks to compare decimals.

Comparing decimals between the tenths, hundredths and thousandths proved challenging. I was finding that some student were perceiving that a larger number indicates a greater value (0.1 compared to 0.09). I asked students to place decimals on an actual number line. This was where we ran into a few problems. There was a disconnect between comparing decimals with symbols and comparing them on an actual number line. Students understood how to use the greater, less than and equal sign but became confused once hundredths were introduced. After running into this issue multiple times, I was starting to find that some students could compare decimals, but didn’t understand where to place them on a number line. Then maybe they didn’t understand the value in relation to a number line?

The next morning I ate my breakfast and paged through Teaching Student-Centered Mathematics, a book that I’ve been using this year. This resource is a gem and I highly recommend any middle school or even upper elementary teachers to add it to their inventory. After reviewing a few a few different options I came across an activity from NCTM (page 151) that was placed in the book. I thought this might be a worthwhile activity for my third graders. The project asked students to place decimals on a number line between 0 – 1. The project also asked students to explain why they placed each number in a specific location. I thought this might be a good way to assess whether students can translate their value of a decimal to a number line. That morning I asked students to use dice, create a number line and explain why they picked each point on the line.

I collected the projects and had to reevaluate whether to proceed with the next lesson. Some students knocked it out of the park with some fabulous answers, while others needed some work. Regardless, it was the high-quality feedback that I was looking for and I was able to quickly address misconceptions. That was an #eduwin situation.

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Next week students will be adding and subtracting decimals. That should be interesting as my students use the partial-sums and traditional algorithms.