Educational Aspirations

Reflecting on Effort

Last Monday my school started its third trimester marking period. As this new trimester begins students were given time to reflect on the past trimester. While the students brainstormed what to write I gave each one their personal file.

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For as long as I can remember teachers at my school have kept a file for every student in their class. This all-important file holds paper assessments, report cards and anecdotal notes taken throughout the year. This file is also what’s usually laid out during parent teacher conferences.

To the students surprise, I gave each one their own file for the reflection opportunity. Prio to handing out the files I made sure there wasn’t anything confidential in the files. Students were then asked to analyze all of their assessments and reflect on the second trimester. Students paged through their assessments and journal entries and filled out the sheet below.

I set aside about 30 minutes for students to look through their personal file and write their response. I wanted the students to analyze their own effort level. It’s interesting how students took on an ownership role as they took the file. They took this role seriously.

Business Meeting?

Some of the students took the entire amount of time while others needed more. When students finished they brought up their file and journal to discuss their views with me. I had a brief conversation with the students about their reflection and asked them questions related to their effort level. The students and I discussed how the statement below applies to what they produce in class.

Effective Effor

Although there’s room for improvement, I feel like the class is making positive strides in being able to reflect on experiences without solely looking at the grade. During one brief conference I asked a student whether they felt like effort in math class eventually leads to achievement. The student responded, “Not completely, but effort level impacts my overall grade.” Sometimes I find this to be a perception battle of grades/points vs learning experiences. Providing students opportunities to reflect can help balance this perception.

Class Math Discussions

conversation-01-01 — Making time for quality math discussions

A few years ago I remember my school district emphasizing the need to use more of a math workshop approach in the elementary classrooms. The school district even invited a math workshop specialist to present on all the different ways to set up groups and organize guided math. Some of the teachers gleaned the information and used parts of the model in their own classroom. The consensus was that some of the guided math approach was better than none at all.

As the years passed the idea of math workshop started to change. Teachers started to change the math instruction block to incorporate small group instruction. Whole group instruction still occurred, just in shorter bursts. The small groups consisted of around 5-6 students and rotated every 10 – 15 minutes. The groups didn’t meet everyday – that’s almost impossible. I remember barely making it through two rotations 2-3 times per week. The organization involved seemed overwhelming, but doable. This workshop model was modified depending on how the teacher organized their math class. After a couple of years the district changed it’s focus to emphasize reading instruction. One small part of the reading instruction is designed for students to share their understanding with others. After hearing about this type of model I decided to merge this type of model within my math classroom.

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As the district changed its initiatives my math model also started to change. Instead of fully devoting time to small group math practice, I decided to incorporate a math discussion within the teacher group for a portion of the time. Half of the time in the small group was used to work on direct problems associated with a standard, while the other time was set aside to discuss the math concept in detail. Over time the conversation started to eat up a larger potion of my small group time. This discussion component ended up becoming more formal after I found the conversations started to impact students’ understanding of math. The questions that I asked were often related to vocabulary or about a particular strategy that was used to find a solution. Students were given opportunities to answer the question and ask each other questions in the process. For the most part students were on task, but I’d have to reign in or rephrase responses as needed. I also found myself planning questions to intentionally ask during the small group time. I had to use some type of timer system to rotate groups at the right time. Most of all I felt like students were able to offer their input in a low-risk environment and discuss math while receiving some type of feedback from everyone involved. Also, students were starting to use some of our more formal math conversations in their written explanations. What I’m finding is that I need to be more intentional in creating opportunities for these classroom conversations to happen. They seem to open up additional learning opportunities that were closed off before. I feel as though slowing down the pace and delving deeper into math concepts has brought about this opportunity

Side note: I’ve also used this strategy with a whole-class discussion. Although it’s benefiting students I need to refine the logistics of using this strategy for the entire class. Also, I’ve experimented with Math Talks this year – definitely something that I want to explore a bit more in the next few months.

Connecting the Math Curriculum

A few months ago I informally asked a group of elementary students what they think of when they hear the word math. I heard many responses from the students. First and second graders focused on the words adding, subtraction, shapes and money. Upper elementary students emphasized multiplication, division, money (again!) and fractions. Often, the student responses were directly related to the last few units that were taught.

I find that the perspective of math changes as kids move grade levels. My own perspective of math has changed over time. I used to dread talking about fractions when I was in middle school. My perspective switched gears when I started to see the different uses of fractions outside of the classroom. When I started to see fractions as less abstract, my notion that they were evil started to dissipate. Events similar to this affected me and my teaching style during my first few years of teaching.

My first teaching job out of college started in an empty fourth grade classroom. I was placed on a team with a two veteran teachers. I remember being given a curriculum guide and told to teach math in specified units that were often separated into math concepts. I don’t believe there’s anything specifically wrong with this, but wonder now if the idea could use some tweaking. This type of unit lesson planning lasted throughout the year. During those units, the lessons were directly related to a particular standard and didn’t deviate much from that path. My team was extremely supportive, although we didn’t question the sequence or the curriculum. Once students took the unit test, the class moved to the next unit of study, which was generally a different math strand. For example, fractions were out and division was in. This process was repeated throughout the school year without revisiting past strands. Of course there was review, but the units didn’t seem connected in any way. As students moved through the units they often had a challenging time applying skills taught earlier in the year. The gap between the content in the units seemed to widen as the year progressed.

I bring this up because it relates to a book that I’m reading. Over the past month I’ve been participating in a math books study with some amazing educators. We’ve been discussing this book over GHO every other week. One of the passages that peaked my interest came from page. 74

“Structuring units – and – lessons within the units – around broad mathematical themes or approaches, rather than lists of specific skills, creates coherences that provides students with the foundational knowledge for more robust and meaningful learning of mathematics.”

As math educators plan units I feel as though the above is sometimes a missing component. Planning opportunities for students to discover how math concepts are connected can be a powerful learning tool. It also shows students that math is not defined as a checklist of singular concepts or “I can” statements. As students switch their mathematical lenses, they see the connected aspects of math, as I read on page 76.

“When they teach the sequence of lessons that they have prepared as a team, the teachers will continually ask students to switch the lenses that they use – from looking at a situation algebraically to exploring how it connects with geometry that they have been studying.”

This year I’m more intentional in planning lessons and activities that connect math strands. I follow the curriculum, but in addition to that I’m finding that activities and lessons that blend math strands gives students more opportunities to cement their mathematical understanding. Problem-based learning projects often lend themselves well to these types of lessons. Even showing students the sequence of the curriculum can prove beneficial as students see where they are starting and the expected finish. It also helps students to be able to view math beyond the abstract. That connectedness can bring a new appreciation and possibly a renewed math perspective.

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Better Student Reflections

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This year I’ve attempted to incorporate more student reflection opportunities in math classes. This reflection has taken on different forms. Reflecting on math practices is evident through classroom conversations and through student math journals I feel as though a heavy dose of student reflection can go a long way in having students build awareness of their strengths and areas that need bolstering. My focus this year has been geared towards students reflecting on their unit assessments. I want students to understand that reflecting on past performances and setting goals can help them improve going forward. I don’t believe all elementary students come to this conclusion all on their own.

At the beginning of the year I analyzed all the different methods to promote student reflection opportunities. The timing of student reflection matters. My classes generally have approximately 11 unit assessments throughout the year. Having formal reflection points after the assessments provide a number of checkpoints along the way. I decided to start by finding/creating student self-reflection templates.

My student reflection sheets have changed over time. The evolution of what was created can be found below.

1.)

Students identified corrected answers and showed reasoning.

At the beginning of the year students were asked to find problems that were incorrect and delve deeper into the reasoning. Students had to seek out why a problem was incorrect and explain how to find a correct solution. This offered little interaction between the student and teacher, although some students would come up to the teacher to get further clarification on specific concepts.

2.)

The above portion was added to the first and helped students focus on positive elements of their performance while still addressing areas of improvement.

3.)

One major area that I thought needed strengthening was in the perseverance department. The above section was added to help students find strategies that they could use when approaching a complex problem.

4.)

Around the third unit assessment I decided to merge a more standards-based grading approach. I had students identify which problems were associated with certain math strands. Students then analyzed those results to look at possibly setting a relative math goal.

5.)

By this time students were becoming more reflective learners. I liked what I was seeing and felt like students were benefiting from this reflection opportunity. I added more pieces that emphasized having a growth mindset – e.g. connecting to achievement. This was also the first time that I had students and teachers sign-off on the reflection. I was able to have short discussions with the students about their assessment.

6.)

fdaf — Identifying areas of strength/concerns

Around the fifth unit assessment I added a goal setting part to the reflection sheets. I haven’t tweaked the reflection sheet much since then feel the results are positive. Students are not as infatuated with the letter grade, but more focused on specific concept areas of the test.

I’ve used this sheet for the last few unit assessments. It’s in a Word format so feel free to edit it and make it your own. There’ll never be a perfect student reflection sheet but this has worked well for my students. Moreover, students are able to look back through their math journals and see their own growth over time.

Last week I passed back a graded unit assessment back to a group of fourth grade students. Each student took a peek at their paper, looked over their problems and grabbed their math journals. Students found a comfy place in the room to reflect on their results and set goals for the next unit. After students finished the reflection they brought the sheet up to me to discuss the reflection and next steps. We both signed-off on the reflection and the students move on to another activity.

This process of math reflection seems to help my students. It does take up around 30-45 minutes or so, but I feel like it’s time well spent.

How do you use student reflection in the math classroom?

Connecting Math Games and Computation

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I feel like the curriculum stars are in alignment. Many of my classes are exploring computation in some capacity. This rarely happens because of the scope and sequence of the curriculum at the elementary level. Computation is an interesting concept to explore in the classroom. I find students come to class with a variety of computation knowledge, although some of the background relates to procedures or tricks used to compute numbers. Other students have a conceptual understanding of the computation, but might be lacking in the procedural department. Either way, I find that students need more practice to become fluent with computing numbers. They also need to be able to distinguish and apply rules to problems e.g. signed numbers, fractions and order of operations.

Developing computational fluency can be found in a variety of forms, but as of late I’m finding games to be the most beneficial. Computation timed tests drive me nuts. I couldn’t stand that as a student and feel a bit embarrassed when they are assigneds. An alternative to this can be found using math games. Games provide low-risk opportunities for students to engage in math conversation and practice computation skills. This past week I was able to use one of these games with students in second and fourth grade.

The game involves using dice and strategy and computation skills. Students were given a game board and recording sheet. I pair the students using Michael‘s grouping spreadsheet and the students grab the sheet, dice and find a cozy place in the room. Students then roll the dice and fill in each line slot and match it with an answer on the game board. The game is over when all the slots have been filled. Click on the pictures to download a file of the game.

2nd Grade – Adding / Subtracting Multi-Digit Numbers

I first used the above game with second grade and then decided to use the same format with a fourth grade class.

4th Grade – Adding / Subtracting Signed Numbers

Both games seem to serve their purpose. Students are practicing their computation skills while using a variety of strategies to compute numbers. Students are also engaging in math conversations around computation and using vocabulary associated with computation. In addition to the game sheet, some students decided to grab a whiteboard and complete their computation there before transferring it to the game sheet. Hopefully these skills will develop into a deeper sense of computational fluency and cement as students progress through school.

Feedback Opportunities

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Last week I took part in a Twitter conversation about student feedback. The discussion evolved into why direct feedback is often more efficient than vague “good job” teacher responses. Most of us agreed that feedback and exploration is often the cornerstone in having students recognize misconceptions and build mathematical understanding. That feedback is so essential and is a vital ingredient in the learning process. The conversation had me thinking of the different feedback opportunities that exist in and out of the classroom.

Everyone receives feedback. In a school setting that feedback can come in the form of a supervisor, team, students and so many others. Beyond this, I receive feedback when I forget to shut my car door or leave the lights on. A loud beep blasts out of the speaker and I need to go back and fix the problem. When creating a document and I forget to save before exiting I receive a “do you want to save” message. Absolutely I want to save, and the feedback (or reminder) gives me an opportunity to do so. I could give other examples but the point is that feedback comes in forms that might not be associated with classroom use.

I feel as though giving students feedback is intentionally setting the stage for student improvement. What happens if the student doesn’t utilize that feedback? What good is the feedback if it sits on the paper? I feel like this happens more often than I would like to admit.

Giving students multiple opportunities to utilize feedback can lead to action. That action may lead students to make changes in how they approach problems and concepts. Although the teacher is one of the main feedback systems, it shouldn’t and isn’t the only option. While thinking of feedback, I started to brainstorm some possible feedback systems that can aid in the learning process. I can picture these systems being used to give feedback after some type of formative assessment or instruction.

Teacher – The teacher is one of the best feedback tools in the classroom. Fielding student questions, clarifying and anticipating next steps all play a role in how a teacher responds with feedback. Teachers all around the world offer feedback, so much that it becomes part of their daily lives. The feedback from teachers can be observed in written or verbal form.

Students – Peer editing and group work can be powerful. Of course, modeling and front loading needs to occur before this becomes an amazing tool. When students discuss answers with each other it opens up a door for feedback. Students can explain their reasoning and be critical friends in the process. Group work provides opportunities for students to become better at explaining their mathematical thinking and processes. Hearing how other students explain their thinking can lead students to an explanation that might not have been perceived before.

Math Classroom Conversations – Math class conversations can be beneficial to all involved. This also takes modeling before becoming a positive aspect of the classroom experience. Asking open-ended math questions and having students respond can lead students to ask additional questions. Feedback can be provided during this entire process while students construct understandings. Classroom conversations often involve some type of whole group question, group response and feedback.

Games – Math games can provide students with a low-risk opportunity to practice skills and show their understanding. I find that when students use math games they engage socially, think strategically and practice skills in the process. Board, card, dice and app games all provide feedback in different ways. Feedback is given in how the other students react to each other, how the answer is revealed and in the scoring element. Math games open up a door of possibilities and adds some competitiveness. Apps have helped revolutionize this idea. Kahoot and Socrative have gaming elements that provide students with additional feedback that can be used to inform instructional decisions.

Adaptive Software – No, this shouldn’t be the only method of feedback. Keeping that in mind, the feedback given through adaptive software can be be helpful to a point. Regardless of the adapted score or level, this type of feedback might not be tailored to the individual student. Although adaptive apps/software is a field that’s improving (as tech startups hire education professionals), this type of feedback isn’t as accurate as some of the other methods above.

How do you give feedback opportunities in the classroom?

Higher-Level Math Tasks

A few days ago I started reading Principles to Actions Ensuring Mathematical Success For All as part of a book study. As I was reading in preparation for our first session I came across a few ideas worth highlighting. Pages 18 and 19 discuss the four levels of cognitive demand in math classes. Along with expectations, these demands are often revealed in tasks or assignments that students are asked to complete.

The book describes lower-level demands as tasks related to memorization and procedures without connections. Memorizing rules/formulas and following procedures is often related to lower-level demands. Students often understand what’s expected when lower-level demands are required. Generally one answer or procedure is evident with this type of task. Worksheets that have students practice rote computation skills without words could fall into the lower-level demand category. Higher-level demands are procedures with connections and often require considerable cognitive effort to achieve. Anxiety is often a part of higher-level demands, although this may be because students don’t see these types of tasks as often.

After reading this page and looking at the different examples I started to reflect on how elementary math classrooms are organized. Math practice is needed, but students should also be given time to explore, discuss and make connections in a low-risk environment. I find more lower-level demands in math classrooms than higher-level, but an ideal ratio is challenging to ascertain.

So after reading pages 1-35 I decided to use an example of a higher-level demand activity with a fifth grade classroom. This particular class is learning about fraction multiplication and division. Students have learned in the past to multiply the numerators and denominators to arrive at a solution. To delve a bit deeper in their understanding I decided to use and adapt one of the tasks in the book. I first grouped the students into teams and gave each team 12 triangular blocks and a whiteboard marker.

Students were asked to show a visual model of 1/6 of 1/2. Some students knew the answer already but seemed unsure of how to show the answer visually. Many of the groups weren’t quite sure on how to approach the construction of the fractions. They understood the abstract and procedural but had a challenging time visualizing the fractions.

After seeing the students struggle a bit I’m glad that I decided to have them work in pairs. Students started to build models of 1/2 using the 12 triangles. Some of the groups came to a conclusion that two different sets of six triangles shows half. Then from there students started to think of what’s 1/6 of the 1/2. Students took out 1/6 but then debated on the value. Some groups said that the answer was 1/6 while others were confident that it was 1/12. Eventually the students decided that 1/12 was the correct solution.

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I went around the classroom and took some pictures of the different creations. Not everyone created the same type of model. This was a great opportunity to highlight some of the different models that arrived at the same solution.

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Afterwards, I thought that offering exactly 12 triangles helped but limited the choices for a visual model. The student models were somewhat similar as a result of the level of scaffolding. As students reflected on their actions in this activity I heard some interesting conversations. Students were aware of the procedure to multiply fractions less than one, but started to visualize the model through this activity. I thought this might be one way to introduce fraction multiplication at the fourth grade level. I also thought that this activity was well worth the time and I’m looking at incorporating additional high-level cognitive demand activities in the future.

Student Shape Books

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Last week I introduced one second grade class to Christopher’s Which Shape Doesn’t Belong book. After hearing about its success on Twitter I decided to use it with one of my classrooms. After downloading the pdf I displayed the images in front of the class and asked the students to think of which shape didn’t belong. Just about everyone in the class raised their hands. Students overwhelmingly decided that the unfilled shape didn’t belong. Students were ready for the next page of shapes when I saw a hand raise from the back of the classroom. That particular student said that wasn’t the only answer. Quite a bit of the class raised their eyebrows and their voices in saying that the unfilled shape was the answer. The student raising his hand said that the triangle doesn’t belong because it only has three vertices. Other students started to raise their hands with additional solutions. Through this process students started to find more solutions. The student input became contagious. I would sum up what happened during the next 10 minutes here. Words like vertex, diagonal, side, symmetry, and angles were starting to be part of our class conversation. I also was able to identify misconceptions and ask questions to think about their responses. This led to more student responses and questions. This conversation wasn’t planned but I felt like it was worth the time and fit in perfectly with my geometry unit. I was going to move to the second page of the book when our class ran out of time.

So the next day the class started the day off with page two of the book. Again, students found different solutions and the class continued the conversation. After a brief amount of time I introduced a shape book activity.

For this activity students were asked to create a personal shape book similar to Christopher’s book. In addition to creating a which shape book, students were asked to include particular shapes in their book.

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Students were given guidance on the first page. I explained the directions, what was expected for the assignment and answered a few questions. I included a formative assessment on the last page of the booklet. Students worked diligently in creating the initial parts of their books for the rest of the class. Most of the time was spent on the reasoning pages. The gallery below will show some of our progress from last week.

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I’m planning on having students share their books with the class next week.

Operations and Mazes

In a few weeks my fifth grade students will start their pre-algebra unit. Before delving into the unit students often need a reminder on how to use the order of operations with fractions and decimals. Half of today’s class was dedicated to reinforcing number sense and computation skills. At some point students will need to be able to use these skills along with maneuvering variables on both sides of an equation. I find that some students struggle with pre-algebra if they don’t have sound number sense skills. So today I ended up using an Illuminations operations activity.

I passed out the above sheet to each student then reviewed the directions. Students were paired and asked to find a spot in the room to work. Students were asked to hide their calculators and estimate one path that will lead to the largest number. Each group came up with their own path.

Trial and Error

Students were then asked to use a calculator to find the path that ends with the largest number. It was interesting to listen in on the student conversations. Here are a few of the statements that I picked up:

“If you divide the number it will decrease”

“Not really, if you divide less than one the number will increase”

“If you divide by a really small number than our number will skyrocket”

“But we can’t multiply by a number less than one”

“But we can multiply by a large number”

“Let’s just work with the multiplication and division paths, those will make the number jump”

“Let’s work sideways instead of making a path straight down. Gives us more opportunities to increase”

While listening to the students I decided to not intervene. It was insightful to hear how the different strategies were planned and executed. There were some student arguments and stonewalling. Eventually students had to defend their reasoning as groups needed to find a solution. Near the end of class students presented their final paths and the class calculated the total. Students soon started to realize that their answer would differ depending on if they followed the order of operations. This changed many of the answers as some groups completed each operation individually. In the end students all decided on one pathway to find the largest number. Students then informally reflected on this activity through a class conversation.

Solutions

Before sending the students on to their next class I mentioned to them the Pick-a-Path game website. The interactive component has more options and might be a decent supplemental activity. I’m hoping to see that a few students took the initiative to check out the site tonight. It might even be part of a classroom discussion tomorrow.

Sample Size and Reliability

Monday was my school’s first day back from break. The students had two weeks off and many students and teachers are still getting back into school mode. The teacher coffee machine ~~was~~ is still working overtime. The first day tends to ease students back into the concepts taught back in mid-December. One of the better ways to transition is to debrief with the students about their break. This is also an opportunity for students to make connections and reconnect with their peers.

After debriefing with the students about their break one of my classes delved a bit deeper into a data analysis unit. This class studied different types of graphs back in December. We explored stem-and-leaf plots, bar graphs, pie graphs and even took a look at box plots. One of the objectives of the lesson on Monday was to explore the relationship between sample size and the reliability of the results.

This lesson was actually adapted from a fifth grade Everyday Math lesson. Before class I decided to use different colored unifix cubs to represent candy colors. I’d prefer to use regular candy but we have so many allergies and a wellness policy that nixes the use of candy in the classroom. Anyway, I took 100 unifix cubes and split them up into 50 being chocolate, 30 cherry, 10 lime and 10 orange. I didn’t tell the students how many cubes there were or the color allocation.

Before digging into the manipulatives the class discussed why using sampling was important. Students discovered how sampling is much less time-consuming compared to surveying all people/objects. We then discussed how much of a sample is appropriate. Students were all over the place with their estimates. Throughout the conversation I was attempting to sett the stage for students to make some connections and find clarity on the concept through this activity.

Students were placed in groups of two for this activity. Each partner randomly chose five unifix cubes.

Random Sample

The groups then combined their cubes and documented their total. About 80/100 cubes were taken after all the students documented their total. Each group reported out their findings. Some groups had almost all chocolate while other groups had zero orange or lime. It was interesting to see how the students reacted as other groups reported out their results. It seemed like they wanted to question their own results. Students were then asked to make a prediction of the actual results based on the sampling. The class then combined the results of the groups and shared the results.

I brought the students to the back table in the classroom and dumped the cube container. We counted each color to see how accurate our class sample was to the actual result. Students then compared their group results to the class and then to the actual results.

I then gave the students an opportunity to reflect on the comparison as a class. Some groups were very close to the actual percentage while others were way off. I explained that this is part of the sampling process. Students were then asked to journal about their experience and the class will explore this topic in more detail later in the week.