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.

GVSU Math in Action

Image

Today, I got to attend GVSU’s Math in Action conference, both as a volunteer and as a pre-service teacher. My first session was with Karen Novotny, who helped design the Adventures with Mathematics books. We went through stations of activities directed toward high school students. These activities focus on students DOING mathematics and working on deeper, meaningful problems. I really enjoy the idea of students working through games and activities that are aligned with the common core, require deep thought, are complex, and use meaningful applications of mathematics. 

The second session was with Tara Maynard, a high school teacher who has recently starting using the flipped classroom in her math classes. She has video lectures that students watch at home and she has activities and discussions while in class. I really love the idea of having a flipped math classroom. I think so many students could benefit from having the extra teacher interaction in class and the opportunities to ask questions. This also leads to more meaningful understandings of mathematics through discussion and deeper, complex problems. The only reservations I have about this is the applicability to all schools. Tara teaches in a school with a 1 to 1 ratio of students to iPads, which means that each child has the technological means to be successful in a flipped classroom. In my field placement, my students would not have this opportunity. I’m not even sure that all of my students would have any means of watching a lecture at home. So, though I love the ideas behind it, I really would be hesitant to implement it in situations where students do not have access to the technology needed. 

After lunch, I attended a session led by John Golden and some of my teacher assisting peers about a math competition they helped create at Grand Valley. It’s for middle school and high school math teams and is modeled after one through GRCC. Since I don’t teach yet, I was more interested in the types of mathematics and questions they presented to these teams. All of the questions were complex and written so well. They really required thought and innovation. I want to use these types of questions for my students. I want them to think and see mathematics in a new way, in a way that is creative and fun and requires some thought. 

The last session I attended was called Beautiful Mathematics and was all about the way that math is beautiful. Abe Edwards, the presenter, showed us a few different examples of how beautiful that math can be. One was of a Sierpinski triangle. Google it. I was so impressed by these kinds of mathematics. It just goes to show how we can get our students to be fully interested and impressed by the mathematical world. We want our kids (and ourselves) to do math for math’s sake, not as a means to an end. Math is beautiful and useful in its’ own right and these things can prove this. Math is beautiful. Such a great end to a great day. 

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.
  • Image

    My notes from during the lesson

Link

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.