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#MichEd in the Classroom

#MichEd in the Classroom

A couple weeks ago, I had the opportunity to work with Brad Wilson and #MichEd, an effort to connect Michigan teachers and students and highlight the amazing things going on around Michigan schools. If you want to know more about them, head here: http://miched.net/. You can also check out more pictures from his visit by clicking on the picture above.

This experience has, by far, been one of the best I’ve had in the world of education. Brad visited the classroom that I am currently teacher assisting in to highlight what new teachers are bringing into the classroom. The lesson that I taught was a hands-on activity in which students were to create their own hubcaps, focusing on maintaining rotational symmetry. This served as an introduction to the concept, forcing students to use manipulatives in order to understand the ideas behind rotational symmetry. Students used paper plates, angle rulers, and wooden shapes to construct artistic hubcaps that were rotationally symmetric. They learned about what it means to have rotational symmetry, what an angle of rotation was, and the order of rotational symmetry. My students gained so much from this activity. They were having fun, talking about mathematical concepts, and working collaboratively to achieve their goal. (Plus, we got some pretty awesome new decorations for the classroom out of it too.)

The main reason Brad was there, both in my classroom and later in a discussion with other Grand Valley teacher assistants, was to understand what new ideas future teachers were bringing to the table and how we were implementing them in the classroom. One of the main ideas in the lesson that I taught was working with manipulatives. Were they necessary in building an understanding for my students? Why did I choose to use them? I truly believe that students need to construct their own understanding. Too much of mathematics education of the past has focused on teachers spewing out information and hoping students would absorb it. This isn’t how people learn math. That’s not how people learn anything. Instead, we need to be guides in students’ education. Throughout my lesson, students were constructing their own meaning on what rotational symmetry meant. They had guidelines to follow, but were forced to make decisions on how to place shapes in order to maintain the symmetry they desired. Some students split their hubcap into 8 slices and made each slice rotationally symmetric. Some students made 8 slices but made every other slice rotationally symmetric. And they could argue why it worked. The student who made all 8 slices identical only had to rotate their hubcap to the next slice to reach a point of symmetry. The student who skipped slices had to rotate two slices. But, still, they both had rotational symmetry. It’s these ideas that students develop THEMSELVES that are important. I could tell them what rotational symmetry was. I could show them examples. But, that doesn’t teach them how to DO mathematics. They aren’t DOING anything.

During our discussion with Brad, all of us GVSU students discussed how we have seen math change. We’ve been lucky enough to have the opportunity to be a part of the movement toward better math teaching, and better teaching in general. When we were in school, we were part of the “old” ways. We were set there to absorb information that our teachers spit out. We were sponges. Which also meant that we could wring ourselves out after a test and forget everything we learned. Until college. That’s when it changed for all of us. We had professors and classes that changed almost everything we knew about math education, what DOING math really was. We were using manipulatives for the first time since elementary school. We were constructing pictorial representations. We were explaining our thinking. FOR THE FIRST TIME. As 21-year-old college mathematics majors, we were doing math for the first time. Why? Why didn’t our teachers teach us in the way we were learning now? So much has changed since we learned math that we had to learn it all over again. And, it’s for the better. We’re taking this new idea of doing math into classrooms with us. We’re teaching our students what it means to do math and not just absorb information. We’re teaching them how to be problem-solvers, how to work together. We’re teaching them how to construct their own learning and discover ideas. We’re teaching them how to explain their thinking, develop conceptual understandings, and really DO math.

Brad is creating a podcast with all of the things we talked about over at #MichEd in the near future, so I won’t give everything away. But, we’re focusing on the future and we’re focusing on the students. We’re doing what’s best for them, not what’s easy, and not what our teachers of the past have done. There are so many great teachers in Michigan flooding our schools with great ideas, innovative strategies, and a passion for teaching students. This next generation of Michigan teachers feels the same.

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Student voices should be the loudest ones in your head.

Student voices should be the loudest ones in your head.

Too often, teachers are making choices about instruction and content delivery without truly thinking of their students. But, why is that happening? Shouldn’t all teachers be basing their lessons almost entirely on the needs of their students? Student voices should be the loudest ones in your head as you’re planning.

This podcast, by #MichEd host Brad Wilson, highlights what students want, what they’re looking for in school, and what makes them learn best. The majority of students discussed hands-on learning. This type of learning keeps students engaged, interested, and willing to do the work. They need the stimulation in order to retain and understand content. Students also talked about making choices. They want to make choices about topics they work with, pacing of content, peers they are in a group with, and how they demonstrate their knowledge. Isn’t this what teachers have been recently trying to push in the field of education? Why haven’t we just been listening to students?

One of the main points that I found so interesting is that students know how they learn best. As teachers, we shouldn’t take this away from them. We want our students to be skilled in metacognition and self-awareness. If our students understand themselves enough to know how they learn, then we need to let them learn in that way. Now, I know its near impossible in a classroom to let each student learn exactly how they want everyday, but the alternative is not to teach how we want to everyday instead. Teachers should be taking into account the learning styles of students and giving them the opportunity to learn in the way they learn best as often as possible. If the majority of your class consists of hands-on, visual learners, the majority of your lessons should be hands-on and visual. Give students options. They are capable of making decisions and should be encouraged to do so. The podcast highlights many students who know exactly how they learn, and who even noted that other students learn differently from them. If students have choices of how to receive content, we can engage so many more students than by forcing them to learn how we think they should learn.

After all, our whole job is for them. Teaching is all about the student. Their voices should be the loudest we hear.

Let’s talk about the good, not the bad.

Recently, I’ve found myself, along with the teachers I’m surrounded by, focusing more on the behavioral issues with my students than on their education. My students seem to just want to misbehave. Maybe it’s the time of the year. Maybe it’s the extra long winter. Or, maybe, they’ve just had enough of everyone yelling at them. 

It’s not so much that the teachers want to get these kids in trouble. It’s that the students have made it nearly impossible to teach. The entire staff has pretty much reached the end of their patience with the students. But, clearly, the “trouble” approach hasn’t gotten very far. In the past two weeks, I’ve been trying to reward the little things with my students. Like all classrooms, I have the students who are just always problems. Granted, these kids are usually my favorites. They just have too much energy for a classroom. 

Instead of scolding these students for talking, I’ve been trying to subtly reinforce their good behavior. Many times, this means that I pull them aside during work time and tell them that I appreciate how they’ve been doing. One student who is probably one of the most talkative has been showing more effort toward his work and participating more in class. During warm-ups this past week, he was asking questions about the material and if he was doing the work right (which he was). This is impressive because, since I’ve been there, I don’t think he’s done one full warm-up problem. I told him that I really appreciated that he had been working so hard with the material and paying more attention. He wouldn’t admit that he had been trying, but it was clear to me. 

I’ve done this with a few other students, both with similar behavioral issues and without. I think that it’s so important to keep reinforcing students for their hard work, good behavior, and success in the classroom. I find that this works so much more to control student behavior. Students are much more likely to behave when they’re being reinforced for good behavior. Especially if they know how much the teacher cares about them. No one wants to disappoint anyone who cares about them, so students will work hard to do their best for you. So, I’ve been trying really hard to give positive reinforcement to all my students. In some cases, I may be the only person who tells a student that they’re proud of them. And that’s a big deal. I’m so proud of all of my students, all for different reasons. I think they deserve to know that. 

Ask Questions!

Ask-The-Right-Questions-

One of the most important things I’ve found in teaching so far is the importance of asking meaningful questions to deepen mathematical understanding. I want my students to be mathematical thinkers and to be able to discuss their thinking with me and with their peers. I think that so much of mathematics education has focused on answers and procedures. Instead of that, I want to instill in my students the idea that the journey to get to answers, the thinking, is what doing math is all about. That’s what is important. 

At my latest observation, I asked my coordinator, Jon, to focus on the questions I was asking students and how they can influence the thinking of my students. I wanted to analyze how well my questions were actually inspiring thinking within my students and if I could be more effective at this. 

One of the main realizations we had was how often I was starting questions with “why?” That was even the whole question a lot of times. Though this is a good start to the way I want students to be thinking, it comes off as a bit aggressive. Jon talked about how there are genuine questions we can ask as teachers and how these are often more effective. For example, we ask a lot of questions like “what is the distance formula?” for repetition purposes, but these aren’t genuine questions. We know the answer. Instead, a genuine question would be “how did you think about that?” We don’t know how our students’ thought processes work and having them explain it is what I would want my ideal student to be able to do– to explain, in detail, how they thought about the math involved. 

Another idea we came across was the issue of how to reword questions when students are not responding well. When my students struggled with the answer, it was hard for me to reword it in a different enough way to get them to understand. Then, for time management purposes, I would often give them or lead them into answers that I wanted them to discover or illustrate to me themselves. Having ideas for multiple questions before the lesson will help me to get students to understand what I’m asking. This will take a lot of planning, but will definitely be worth it. 

My final “grow” moment from this lesson was about addressing student misconceptions. When students come up with ideas that are incorrect, but common (and understandable), I have a hard time knowing how to respond. I know that saying “no” in any part of my response is something I don’t want to do. This has a lot to do with my opinion that students who hear the word “no” will shut down and not listen to any of the rest of my response. For example, in this particular lesson, I had asked students which side was the hypotenuse and how they knew. One of my students responded and said “I know that side is the hypotenuse because it’s the one that’s diagonal.” My response in the moment was “yeah, but, actually, it doesn’t always have to be. We know it’s the hypotenuse because it’s across from the right angle.” I immediately regretted saying the “yeah, but…” part because I don’t want my students to be confused. Jon and I talked about saying something like “that’s really interesting! Let’s look at this,” showing them an example of something that disproves their idea. In this way, we are not shooting students down, not giving them incorrect ideas, and clearing up misconceptions (that much of the class shared.) 

Now what? 

In the future, I want to be conscious of the ideas that we discussed. Firstly, I want to change my question starters and begin asking more genuine questions. I think my students will really respond better to questions about their thinking, rather than “why?” Why questions seem to be looking for the mathematically correct answer of why, which isn’t my goal at all. So, I want to focus more on the goal I have and which questions seem to be getting there. Second, I want to plan, plan, plan. I want to plan multiple questions that address the big ideas of the lesson and how to get students to be able to answer them. This will take practice and experience, but I’m enthusiastic to start trying. Last, I want to spend a minute or two actually addressing misconceptions. I don’t want to just tell them why something is wrong, I want to show them. I want my students to see exactly why the hypotenuse is not always the diagonal line in order to better understand and retain that information. I think all of these things will help me to increase the mathematical thinking of my students. ‘Cause, after all, it’s the journey that’s important, right? 

 

Mathematical Discussion

After reading the Five Practices for Orchestrating Mathematical Discussion, I began to analyze my own and other teachers’ teaching practices. The Five Practices are as follows:

  1. Anticipate—anticipating student thinking and response
  2. Monitor—monitoring student responses for ones that will direct discussion
  3. Select—selecting student work to create a mathematical discussion
  4. Order—place student work in logical order
  5. Connect—make connections throughout the discussion

One thing that I think my CT does really well is anticipating student thinking. She can always tell what questions her students will ask, what they will grasp easily, what concepts they will struggle with, and what strategies they will use to problem solve. On assessments, she is almost always spot on when describing which problems will be difficult for students and which problems they will make. I think a lot of this comes from knowing your students, individually, and as a group. For example, we just finished a unit on functions and on their test today, they had to use a table to create a linear equation, something they did in this unit and the one before it. Even before they began the test, I knew that few of my students would get a completely right equation. As a whole group, my students struggle with deciding which piece of information is the slope and which is the y-intercept. In this particular problem, the slope was negative. Even though a question asking whether it was positive or negative directly followed (and many got right), they still missed the negative part in their equation. From working closely with my students the past weeks, I have learned so much about their strengths and weaknesses to help me in anticipating their work in the future.

The second practice is to monitor. In my class, we are extremely lucky. My first two classes consist of special education students or low-medium performing math students. Though there are some negatives to this, the benefits are enormous. For one, we have four teachers in the classroom during these classes- my CT, the special education teacher, myself, and my partner. This means we have the ability to closely monitor each student. Between the four of us, we know the commonalities between students, and can help reach individual questions that would not help progress a mathematical discussion. In my class, my CT doesn’t often have students share their work, so this is where her involvement in the Five Practices ends. I begin teaching my unit on Monday and I’m planning on implementing more discussion to really help students build their understandings.

The third practice is to select and the fourth is to order. I think these go hand in hand. One of the best examples I’ve seen of this is in my middle school math education course with Lisa. Often, we were challenged to complete problems in more than one way. We used pictorial representations a lot and were often required to construct more than one on assignments. As we worked in class, Lisa would walk around, ask us about our thinking, and keep track of which methods were particularly of interest. She would then ask specific people to share their work, in a specific order. Some were incorrect, but developed our mathematical thinking. Others were correct and were usually set up in a way so that we saw the most common methods first and progressed to more unique ones. We would discuss the math behind each and work with each other to see why it worked. In this way, Lisa facilitated our discussion and ensured that we saw examples of many people’s work so we could develop our understanding. I like that many times she would show us examples that were very common. In this way, I could understand more fully and begin to feel comfortable with the material. As we moved into more eccentric ways of solving, I could use our prior discussion to assist me. I think this would work so well in the classroom. The common methods would validate thinking for most students and then expand their understanding as we moved into different methods.

The final practice is to make connections for your students. Often times, too often I think, teachers fail to explicitly state connections. We have already developed this network of math and don’t realize students need our help to do the same. So, math can start to seem like a bunch of unrelated, difficult topics for students, when, in fact, all math is built upon connections. Even in one discussion, multiple methods of solving one problem can seem disconnected to students. When my kids were learning about solving systems of linear equations, we learned the substitution and the elimination method. After learning both, my CT mentioned that they would both be on the assessment and a student asked why. They had no idea why they were connected and why they would be on the same test, or even why you could choose between the methods to solve. Some students had made the connection, but some hadn’t. Teachers need to be explicit about these connections, even if they seem obvious to us. The connections could be what solidifies a student’s understanding.

Overall, I think these Five Practices are brilliant for implementing productive and meaningful mathematical discussion. Too often, teachers forget that discussion is often the part of instruction in which students solidify their learning. With my upcoming unit, I want to ensure that students are seeing each other’s work, explaining their own work, making connections, and discussing mathematics.

Math is Thinking

Yesterday, I had the opportunity to attend a professional development day with the 8th grade teachers in our district. The day was run by a woman, Carrie, from Math in Focus, which is the textbook that the math classes in the district use. It’s based on Singapore Math, which strives to teach students mathematical concepts deeply and with meaning.

We began our day discussing the successes and challenges thus far in using the Math in Focus text, as this is the first year the district is implementing it. The successes included the notion that ALL students, top and bottom and everything in between, were rising up in ability, the visualizations in the text made it easier to teach topics, connections were made between ideas, parents were able to learn alongside their children, etc. Challenges included the requirement of critical thinking, pacing due to snow days and other interruptions, the high difficulty level of topics and problems, and the frequent use of story problems. We addressed each of these issues and discussed how the challenges might be alleviated and the successes continued. 

Next, we focused on the four instructional strategies that Math in Focus is based on–gradual release, math is thinking, concrete-pictorial-abstract pedagogy, and visualization. All the teachers in the district were unanimous on which they were most comfortable with (gradual release) and which they were struggling with (math is thinking.) Math is Thinking is focused on critical thinking strategies to problem solve. The teachers were having issues with this due to a lack of time, a lack of focus from students, and the students’ uneasiness to attack problems that involved critical thinking. Carrie wants teachers to explicitly note to students that they don’t care what they say, as long as they’re thinking. Answers can be wrong. Questions can be “dumb.” But, students must be thinking about the math involved. Some questions that Carrie uses in teaching to encourage this are: “what is different?,” “what is the same?,” “where’s your entry point with this problem?,” “what do you notice?,” and “where’s your thinking at right now?” Each of these questions give students the opportunity to think critically about the problem at hand and make connections with their prior knowledge. 

Following this, we were to watch Carrie give a demonstration lesson to one of my CT’s classes about functions and their multiple representations. We all constructed a three-columned paper with the headings: Student Action and Thinking, Teacher Action and Thinking, and I Wonder… We were to use this to take notes during the lesson and guide our observations about specific student and teacher actions, focusing on the four instructional strategies mentioned earlier. Some of my main takeaways are listed below: 

  • Use partnered discussion for students to talkabout their thinking. Ensure that everyone is participating. Carrie used phrases like “what are you supposed to be doing right now?” or “this is your job, to figure out what a function is from a friend.”
  • Ensure that students are using visualization. Use phrases like “can you picture it?” when students give an answer. “Can you picture a function passing the vertical line test, what does it look like?” 
  • Students often say “I think I know the answer.” Get them in the habit of saying “I know the answer.” 
  • Giving students choices can increase engagement and motivation. Even if it’s as simple as choosing between guided practice problem one or two, students will enjoy the choice. 
  • Highlight same and different aspects for students between previous day’s problems, warm-up, modeled problem, guided practice, and homework. Ask them for the same and different parts to help them make connections. 
  • Use self-check often. Have students give a thumbs up or down on how confident they are with the material being presented. In this way, they evaluate themselves and you get an assessment of their progress. 
  • Make sure that independent practice is done completely alone. They don’t get help from other students or you. They need to practice their individual ability with this material. 
  • Only give homework if you’re completely confident they can complete it independently. If the class is struggling, think about giving homework on the previous day’s material or no homework instead. 
  • Make sure students clarify what they are saying. Don’t assume that they mean what you think they’re trying to say. Get them to say it.
  • Conduct a reflection or closure piece in every lesson. Make sure students grasp the main ideas and feel like the lesson has been completed.
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    My notes from during the lesson

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Pythagorean Theorem with Jellybeans

Pythagorean Theorem with Jellybeans

As I began planning for my unit on the Pythagorean Theorem, one important element kept striking me as vitally important–getting my kids up, moving, engaged, and interested. Eight-graders have a hard enough time sitting still and paying attention as it is, even without the added bonus of having to do it while learning math. Since the Pythagorean Theorem can be applied so widely to real-world scenarios, I wanted to utilize this privilege to its full extent.

 

This activity, found on the website of the National Council of the Teachers of Mathematics, is perfect for my students. The activity allows students to develop the Pythagorean Theorem through discovery of their own learning. It encourages students to test out scenarios, form conjectures, test these conjectures, and, ultimately, form the Pythagorean Theorem. It has the added bonus of students being able to play with candy. I believe that my students will enjoy the activity, and will greatly benefit from discovering the mathematics on their own. The Pythagorean Theorem is a vital concept throughout algebra, so students need a developed understanding of the ideas at hand.

 

Pythagorean Theorem with Jellybeans also incorporates a lot of the CCSS Standards for Mathematical Practice. Firstly, students must be able to make sense of the problem of finding a relationship between the squares drawn on sides of a right triangle and persevere in solving this problem. Second, students must reason abstractly. They will move between contextualizing variables to decontextualizing information fluently to represent a physical relationships in the form of a rule or conjecture. Third, they will critique the reasoning of others and develop arguments about their conjectures. They will be able to provide input on the conjectures of their group members and be able to make conjectures of their own. Fourthly, students will use the tools given to them to model the relationship mathematically. Fifth, they will use these tools strategically (and not eat the Jellybeans until they’re finished). Sixth, they will attend to precision to ensure that the conjecture they construct is accurate. Seventh, they will find a pattern between the squares on a right triangle and make use of this structure by developing a universal rule. Finally, after testing their conjectures, they will express reasoning in repeated regularity.

 

Rarely does an activity touch on all of the SMP, but I think this example is a great one. I’m excited to be able to try this in my class and (hopefully) see a change in the level of engagement and excitement in my kids as a result.