Class Math Discussions

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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

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.


 

photo credit: Hyperlink via photopin (license)

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.

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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.

Students show 1/2 of 1/6 using triangles #sllearns #mathchat

A video posted by SL Math Replacement (@slmathreplacement) on

 

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.

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Click for pdf book

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.

 

Sample Size and Reliability

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.

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Setup before class

 

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.

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Better than what I expected

 

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.

Addressing Local Math Misconceptions

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Over the past week I had time to disconnect a bit and spend time with family. I was able to stay with relatives in another state and spent most of the time catching up with people I haven’t seen in a while. It was a great time to refresh and reflect on the past year.  While relaxing one relative in particular asked me about this “new” math that’s in the schools now. I was asked why schools are changing how they teach math and why it needed to change. Specifically she spoke of the different strategies used to compute numbers.  I’m assuming she meant the extensive use of the number line and compatible numbers.  I defended the reasons for a more conceptual understanding of mathematics, especially at the elementary level. Many of the “new” strategies help build that understanding and enable students in developing a foundational understanding of numbers. The relative was receptive and asked more questions related to this topic. I felt like her understanding of the topic became clearer as we discussed the use of  multiple strategies utilized to teach computation. This was a small part of our longer conversation, but the topic had me thinking about how to provide opportunities to address misconceptions. In particular, I thought how my conversation could apply to addressing math misconceptions in schools.

I feel like one of the more important issues with student misconceptions stems from a lack of addressing them. They tend pile up and build over time. I vaguely remember having a math teacher that asked if his students had any questions. I remember looking around and wondering if I was the only one in the class that had multiple questions. Unfortunately I kept my hand down even though I was lost. The teacher then quickly surveyed the room and seeing that no one had their hands up, moved onto the next topic. I found that the less I asked questions that less comfortable I was with the current concept and the process continued until I finished the class. Looking back, I’m sure there were other opportunities to address my misconceptions and questions; I just don’t remember any of them at the time of this writing. I didn’t learn as much as I should from the class, but I started to understand that I needed additional opportunities to ask clarifying questions.

In addition to including many opportunities to address misconceptions, the classroom environment plays a pivotal role in having students feel comfortable in offering input.  The strategies below can be used in a variety of settings. I’ve had success with the strategies, although some have been more successful than others.

Classroom Math Conversations

Classroom conversations can be a powerful strategy in gaining a better understanding of students’ viewpoints. Using open-ended math question and having groups respond to the class can offer opportunities for a healthy math debate. For example, I’ve seen some teachers use Always, Sometimes, Never with great success. Math Talks can also be an avenue in which classroom conversations can develop. Through these conversations teachers can glean important information and possibly misconceptions that can be addressed later or at that time. These types of math conversations, accompanied with anchor charts can document the classroom’s learning journey.  The anchor charts can then be revisited as students construct their understanding.

Formative Assessments

These types of assessments can take different forms. Some teachers prefer to use exit cards, while others use a quiz model. Formative assessments can be used via technology means and some may take the form of a paper/pencil quiz. Regardless of the form, the student’s response can give teachers an indication of understanding. In order for the teacher to give feedback the question needs to be appropriate. Students need to be given the opportunity to explain their reasoning or steps involved in solving problems. If not, the problem is wrong/right, and the teacher is unaware of where the mishap is occurring. Using written or verbal feedback to address the misconception can lead to a more in-depth conversation at a later time. Some students may need the reinforced conversations while others may not. I believe most teachers understand their students and at what level to scaffold feedback.

Journaling

Similar to answering a question in a classroom student group, journaling can provide students a low-risk venue to showcase their understanding. Through a prompt, math journaling can allow students to explain their mathematical thinking and processes in a written form.  Students often become more aware of their growth as the year progresses.  I find that students might not know that they have a misconception until it’s brought to light.  It’s the you don’t know what you don’t know dilemma. The concept of math journaling can be used for teachers to write feedback to individual students and ask questions that give students opportunities to reflect on their writing and math process. Allowing a bit of extra time to confer with a student after their math journaling process can be beneficial as teachers may want to review specific concepts with students.


All of these strategies above seem to go well with a heavy dose of teacher feedback and student self-reflection.  Through reflection, students can help internalize and address the misconceptions.

How do you address misconceptions in the math classroom?

 

Educanon and Formative Assessments

Educanon and Formative Assessments

Educanon and Formative Assessments

The second grade classrooms at my school reviewed subtraction strategies last week. Students were subtracting numbers on a number line and becoming more confident in using regrouping strategies. Based on pre-assessment results I felt like some of the students would benefit from additional enrichment. While talking with a few colleagues I revisited Educanon. I first heard of Educanon from Mary and I briefly used it last year. So I dusted off my username and password and logged into my account again.

A while back I created a subtraction video using Explain Everything. I turned off the microphone function (my dog was barking when I created this) and just used the pointer and drawing functions. The video was only around two minutes in length, but had 10 questions. I added a variety of questions, including multiple choice, fill in the blank and checklists. The last question asked the students to use a whiteboard to find the difference between two numbers.

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During class students were placed in two different stations. One station was Educanon and the other was working with base-ten blocks to subtract multidigit numbers. The stations rotated after approximately 10-15 minutes. All students logged in and finished the Educanon within the time period.

After class I was able to review the results. I felt like this data could be helpful for the teacher as well as the student.

Student answers

The next day I printed out the student results and compiled a reflection sheet.

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Click for file

Each student’s answers were stapled to the back of the reflection sheet. As a class we reviewed each question together and students filled out their specific sheet. Out of all the categories on the sheet, I thought the “Key Vocabulary/Concept” section stood out. Students started to develop an understanding of what type of skills were being addressed from each question. This was also an opportunity to emphasize certain math vocabulary words. At the end of this reflection session, students circled their effort level on this formative assessment.

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I thought this was a beneficial activity for a second grade classroom. Students are also starting to think more about their own mathematical thinking and learning. I’ll be using the data and student reflection in preparation for more challenging subtraction concepts later in the year.