2.2+Reflection+3+Orchestrating+Discussion

The readings reflections have two main purposes: 1) to hold you accountable for careful reading of and reflection on the readings assigned in class; and 2) to provide you with a record of what you've learned and thought about as a result of the readings.
 * Readings Reflections**

The readings reflections will be evaluated using the following criteria:
 * completeness and timeliness of the entries;
 * comprehension of the main ideas of the readings; and
 * depth and quality of integration of the ideas with your own thinking.

Submit your readings reflection **before** reading anyone's on the Wiki page and then paste it into the existing reflection page for that current reading. This is due by Wed 9/21 by 11:30pm.

No discussion is required but feel free to post if you want to. This is not for extra credit, just if you want to note something. After you post your paper, go back through the 4 articles we have read thus far. There are many references made to other papers within these articles. Are there any of these references that you would like to read and have a copy? What I would do is a "divide and conquer" where only a portion of the class would read certain papers and then share what was in the paper, deciding if it is a worthy reference. Email these references to me (you perhaps can copy/paste directly from the article). Email them by Thursday 8pm.

The authors of “Orchestrating Discussions” recognize that while discussions can promote conceptual understanding of mathematics through thinking, reasoning, and problem solving (2-1), they also can present certain challenges to teachers, such as how to react to diverse responses and how to value both the students’ thinking and the mathematical ideas (2-2). To this end, with the NCTM standards in mind (2-1), the authors have developed a method that they believe can result in effective and worthwhile discussion in the mathematics classroom, The Five Practices Model (3-3). The first step in this model is to anticipate different ways the task can be solved (3-5). Teachers may want to refer to their colleagues and to research ahead of time (4-1). This is a very wise idea (tell me why it is wise) and can help direct the focus of class. Secondly, teachers should monitor student responses (4-2). For example, they could circulate through the classroom to pay attention to students’ thinking and strategies as they work (4-2). This includes watching, listening, and asking questions of them (5-1). I think that this can appear relatively simple but prove to be complex; as a teacher I want to assist them without controlling them. The third step involves selecting students to share their work with the class (5-3). This could be the most difficult part of the entire process, especially because we know from experience that in the middle grades students are acutely aware of what their peers think of them. Thus, I believe it is important not to harshly criticize their ideas or make them publicly uncomfortable We focus on the mathematics and NOT the person. Additionally, it would be a good idea for all students to have a chance to share at some point. The fourth practice is to sequence the presentations (6-3). I believe this step is crucial to understanding because the order in which students present their solutions can greatly influence students’ understanding of the key mathematical ideas. Therefore, it is vital to sequence them purposefully (6-3). If the solutions are presented in an illogical order, I think students could form inaccurate perceptions about the mathematics involved. Finally, the authors maintain that it is the teacher’s responsibility to connect solutions to other solutions and to key mathematical ideas (7-4). Ideally, the student presentations should build on each other to strengthen comprehension of the mathematical ideas rather than emphasize the differences (7-4). I think the way the teacher manages the connection process is likely the most influential piece of this instructional method; without solid connections, students could leave class more confused about mathematics than when they entered. Most of these practices in the Five Practices Model involve planning ahead, but there is adjustment to be made as discussion proceeds (7-6). I think that the more time the teacher spends familiarizing him- or herself with the mathematical ideas and possible approaches, the more prepared he or she will be to handle approaches that were not initially anticipated. Is this something you could work with now? Reflect on specifics of these ideas. The authors conclude that these five practices give teachers a “roadmap” to effectively orchestrate discussions where they can reliably shape the interaction into mathematical understanding (8-1). They maintain that this is a model for a successful whole class discussion that not only builds on student thinking but also advances the teacher’s mathematical goals (8-2). I agree with the authors that a significant challenge facing classroom discussion is how best to balance encouraging student thinking with developing mathematical ideas. It seems to be a pendulum that swings too far in either one direction or the other. Say more of what you mean by this pendulum. Overall, I believe the Five Practices Model has the potential to be used very effectively in the middle school mathematics classroom to face these difficulties. What difficulties? In fact, I would be very interested in implementing this approach myself at some point in my teaching experience. It may soon happen!

Mandi Mills

The main purpose of this article was to discuss the importance and effective use of classroom discussion. I agree when the author talks about discussion giving students opportunities to share ideas, better understand the material and so on (549-1). I think it is very beneficial to have students learn from one another, and I also believe the art of teaching someone helps solidify ideas for the “teacher” as well.

The five practices model (550-3) lists five steps, if you will that teachers can follow in order to have an effective classroom discussion. The first is about anticipating what solutions to a certain problem students may have (550-5). In doing so, teachers can get a jump start on possible problems that may arise. I think it is very important to try to anticipate as many solutions as possible, this saves time in a class discussion because the teacher will already know if this is a correct solution or not. Why is it important to have these solutions? Is it just about saving time? Next, monitoring which involves not simply watching and listening but getting involved with students and listening to what they have to say (551-2). I also believe it is important for students to feel heard and to ask them why they think their solution is correct. On the other hand if it is not correct we can gently guide them in a different direction. The third of five practices is selecting, which involves deciding which students will present the problem to the class (552-3). Fourthly, sequencing is determining the order of selections in a way that will benefit the whole class discussion (553-3). In this way teachers can clear up any misconceptions that any students may have. How do they do that? Finally, connecting the problem (554-4) to larger or key mathematical ideas, which I believe is important because students always want to know what problems in class have to do with anything and with this step we can show them. Not sure that was what the authors meant by connecting the problem to key mathematical ideas.

Overall I think class discussions are great tools for teachers to use. because.... I also think the five practices model is something I could use in my classroom. because..

Kaitlin Froehlke

This article really opened my mind to the extent teachers must go though to ready themselves and the classroom for discussion. Before reading this article, I felt the teacher had less of a role and the students took the main role in classroom discussion simply because of some of the other articles we have read. Before we stated how students need to come up with their own solutions and problem solve with out the direct guidance of the teacher. The authors of the article did not discount this at all; in fact that was still the basis of their idea. However, for me anyways, they shed new light on how the teacher is to play a role in this interaction. Good connection to previous ideas.

Each of the five practices the authors discussed really seems to fit into what we have been discussing class and reading about in other articles. Such as? The first, anticipating, is something that didn’t come to mind when I thought about getting ready for a discussion. Teachers must anticipate as many possible ways as they can which their students could solve a problem, both the correct ways and the incorrect (4-1). In is important the instructor puts their mind in the eyes of a middle school age student. If they are able to do so, they will be able to see how these students are trying to connect what the y already know from previous lessons and implement it into the problem at hand. In addition, because the teachers have anticipated what the students will do, they can in turn anticipate some questions to ask which will help their students obtain the “why” in how the solution may or may not work (4-2, 5-2).The principle of monitoring struck a chord with me. I look back to my middle school days and think about the group work we did and then tried to think where the teacher was during all of this. Most of mine were at their desk, talking with teachers who were passing in the hallway, or taking a moment to get ahead on another class. (This previous piece digresses a bit with a "reflection sparked by the reading" vs "reflections on the ideas in the readings themselves.") The idea of monitoring is pertinent to the growth of the students. By walking around and asking your anticipated questions, teachers can see who has obtained a solution and who to have to explain in the discussion (4-3). Monitoring is not just walking around but engaging with your students in their solving of the problems (5-3). Being able to select a variety of solutions to showcase is important for students ability to problem solve. The principle of selecting, which leads right into sequencing, allows the students to see a variety of both right and incorrect ways of coming to a solution. It is up to the teacher to choose the solutions to showcase and the order which will help the students better understand (7-1, 2). For example, the authors talked about showcasing a common way of doing the problem incorrectly which may work out in this one case, however will not work all the time (7-4). The final principle and, in my opinion, the most important principle, is the idea of connecting. Allowing the students to look through the eyes of their peers helps them evaluate accuracy and efficiency to apply to their problem solving skills (7-6). Being able to look at a problem and have a selection of ways to approach a solution for it is a vital skill. There is more to the connection component than accuracy and efficiency. I know when I was in middle school, when I came to a solution, I stopped. If I were to have had the benefit of connecting with the others in my classroom, I would have been able to grow with my ability to approach problems differently. ( Digressing again which okay as long as these are not your only reflections, which they aren't )

Katey Cook

The article “Orchestrating Discussion” discusses the task of building on and validating student responses all the while making sure it is relating to the mathematical topic at hand. (2-1) The authors first want the educator to realize that when giving the lesson it is vitally important not only to decide what is important information for the students to grasp but also how to get them to grasp the information without it being hand fed to the students via the common lecture format. (2-2). We also learn via this article five basic practices for keeping the discussion on the right path.

The first of these practices is to anticipate the solutions some students may come up with. (2-5) This part of the plan requires extensive planning on the part of the educator. The educator must think like his or her students in order to come up with responses they expect to receive. Having an idea of what responses may surface will give the educator an idea of how to tailor the lesson as well as proceeding questions to keep the students train of thought on target. This step in the process may be time consuming but it is vital to managing the classroom. (3-2)

The second part of the five part process is in my opinion the most important. The teacher must monitor the student’s progress as it is happening (4-2). During this process the teacher can learn a better understanding of how the students got to the solution they have. Furthermore the teacher can have preplanned questions for the students to get them thinking about the problem in another way. They can show the students that although they might be on the right track there are other ways to solve the problem. These questions can give the students time and ideas on how to improve or revise their solution. (4-4)

Selecting the students or groups of students to present their answers or solutions to the rest of the classroom is also an important part this method. For instance if a group comes up with an idea that is generally right and shows how they came up with the idea in a way that is easily represented and explained this group might want to be selected to present first. By selecting them first it will allow other students to learn from their presentation and revise their own. Conversely a teacher might want to select a presentation with obvious flaws in it so that the rest of the class may see and learn from this usually common mistake. (5-4) Sequencing which presentations go in what order can also help in the students understanding of a particular unit. There are multiple ways in which to sequence the presentations however I really stuck to displaying the method used by the majority of the classroom first. (6-1) By displaying an alternate method next you validate the majority of the classroom and a true and meaningful discussion can start to take shape. (6-3)

All of these steps are great and helpful to get students involved in mathematical thinking and discussion however without connecting it all, the lesson has been lost. The teacher must probe and hint to the students during their discussions the similarities and differences between their approaches. (6-6) During this time the students can also see the fastest and easiest way this problem was solved. (6-6) When the students see and explain these differences and similarities the teacher can then reveal what the students already knew…. The lesson they intended them to learn all along.

Chris Cardon

The main purpose of the “Orchestrating Discussions” article is to get across the main practices that constitute a model for effectively using student responses in whole-class discussions that can potentially make teaching with high-level tasks more manageable for teachers. 549-1 The article says discussions that focus on cognitively challenging mathematical tasks, namely those that promote thinking, reasoning, and problem solving, are a primary mechanism for promoting conceptual understanding of mathematics. Okay, you directly quoted a line and failed to reference that. Not good. This would be called plagiarism so paraphrase OR indicate using quotation marks and include page number. I feel these discussions can truly help the understanding as well. It is with these cognitive tasks that students will grasp the concept of what it means to do mathematics. Give me an example of what you are saying here. These discussions able the students to share ideas, clarify understandings, develop convincing arguments regarding how and why things work, develop a language for expressing mathematical ideas, and learn to see things from other perspectives. 549-1 Good you have a page number but include quotes since it is a direct quote again.

Teachers must be able to determine how to orchestrate a discussion built from a diverse set of responses. In other words teachers must decide what aspects of a task to highlight, how to organize and orchestrate the work of the students, what questions to ask to challenge those with varied levels of expertise, and how to support students without taking over the process of thinking for them and thus eliminating the challenge. 550-2 The model that effectively uses student response in whole class discussions which can make teaching with high-level tasks more manageable is called the five practices model. The five practices are anticipating student response to challenging mathematical tasks, monitoring students work in engagement with tasks, selecting particular students to present their mathematical work, sequencing the student responses that will be displayed in a specific order, and connecting different students’ responses and connecting the response to key mathematical ideas 550-5. The article goes into depth what each practice is about. Say a bit about each practice so that I know you comprehend what these practices are about.

In this article the main idea is that using the five principles can make discussions of cognitively challenging tasks more manageable for teachers. By giving teachers a roadmap that they can follow before and during whole class discussions, these practices build on student thinking while continuing to advance the teachers mathematical goals for a lesson. I must know how to orchestrate a classroom discussion in order for mathematical thought to be present in my classroom as a future educator. I don't think that was the main point of understanding how to orchestrate classroom discussion.

Fredrick Martin

In one of my recent classes our teacher relied heavily on class discussion. Nearly every class, twice a week, we would be given a topic, some time to work or discuss it with our neighbors or group members, then assemble as a class again and share our conversation or ideas with the class. I thoroughly enjoyed that class, and was blown away by that professor’s style of classroom management and the effect it had on my learning. First, I wondered how this teacher could so efficiently hear our thoughts and incorporate them into the lesson; it was so cool, how does she it every time!? I often got over it by the time I got home, but still, I was very impressed. How could she decide what task to highlight, organize and orchestrate the work of all her students, know which questions to ask, and support students without taking over the process of thinking for them (550-2)? “I wish”, I often thought to myself, “that I could run a classroom similar to this. But I’m a math teacher, I don’t really get a choice, I need to stand up and lecture to my classes…” After reading this article I now have hope that I may, some day, run a class comparable to the one I liked so much myself. It will not be hard, now that I know the five practices of orchestrating class discussions:


 * 1) Anticipate student responses
 * 2) Monitor students’ work
 * 3) Select particular students to present their work
 * 4) Sequence the student responses into specific order; and
 * 5) Connect different student’s responses to key mathematical ideas.

Looking back I can see my former professor doing all five of these for our education class, and now I know they are applicable to mathematics classrooms as well. I hope not to depreciate the value of lessons by giving students too much or too little support, or by over-directing them, but I am confident with proper planning and practice, classroom discussions will greatly aid the education of my students.

Tim Hollenbeck

In “Orchestrating Discussions,” Smith, Hughes, Engel, and Stein present the Five Practices Model for teaching math at the middle school level. This model encourages teachers to plan ahead in terms of which order certain facets of a mathematical problem should be presented and which common misconceptions should be dispelled quickly. After thinking through a problem completely, a teacher can best determine in which direction to steer classroom conversation. This involves incorporating student ideas, both expected and unexpected, into the lesson, while keeping everyone engaged and on the right track. By allowing students to explore concepts, we play the role of supporter; we avoid, “taking over the process of thinking for them thus eliminating the challenge,” as discouraged by the NCTM (2-2).

A majority of our previous readings have been about the benefit of allowing students to think through problems either on their own or with each other; however, this article builds constructively on those ideas by outlining what needs to be accomplished //after// students have had a chance to explore their own thoughts. We, as teachers, must anticipate what the students will be thinking in order to plan ahead affectively so that class discussions will be comprehensive and accessible to all. The authors note that this is only possible if we shift our decision-making, about what to discuss and when, to the planning phase of the lesson (2-5). By deciding the order of how correct and incorrect methods should be presented ahead of time, we can know which students should speak first in order to best benefit the whole class. Of course, this would require us to monitor the students as they work and keep track of how different students are choosing to solve the problem. I thought that the table (figure 3) that the authors describe would be a clever way to keep the class’s thoughts organized (4-1).

Once the discussion is actually underway, the sequence that we have chosen for a particular concept should allow for student ideas to develop from one another (6-6). Rather than presenting student ideas in an unorganized heap, we must choose which ideas to make central and when. As a result, the student ideas generated in that lesson should be the building blocks for conceptual understanding. One point that the authors mentioned briefly that I feel deserves more attention is keeping track of how often specific students are chosen to present their answers (5-1). If one struggling student is consistently chosen to present his or her answer to clear up a misconception, the student may feel that his or her ideas are always wrong. Similarly if one particularly advanced student is always chosen to present the most efficient method of solving a problem, then other students who are proud of their correct answers may feel slighted because their method was not discussed. At the middle school level, where self-esteem is not always a constant, we must do our best not to make any student feel singled out.

As teachers, we must walk the thin line between too much and too little direction and explanation. By incorporating student ideas into over-arching conceptual discussion, we allow students to explore their own thoughts as well as understand how their thoughts fit into the larger mathematical picture. This article was really helpful to me in terms of thinking about my future classroom. The authors gave concrete suggestions and examples that I feel would enhance my teaching abilities and benefit my future students.

Valerie Gipper

One of the main points of the article was to give a model for discussion that would build on students’ thinking while ensuring that mathematical ideas remain at the center of the discussion (3-3). This model is guided by 5 practices which are listed on pg. 550 (3-4). I think that the anticipating practice will be very difficult. I tend to see solutions to math problems in one way. As teachers we will need to anticipate many possible angles that our students will approach problems.

For the monitoring practice teachers are to walk around the room and ask preplanned questions that are derived from anticipating how the students will solve the problem (5-2). I think it is good to be prepared and have preplanned questions, but what do you do if you didn’t anticipate the solution the students give you?

I thought it was interesting how they made a point to carefully select the sequence of presentations of solutions of students (6-5). In the past I thought teachers just picked a student who got the right solution, and had them write it up on the board and explain their steps. It was nice to see the strategy of showing concrete ideas (drawings) and then showing more abstract solutions (7-2). I think this would be a good way of sequencing presentations of a solution, in that it gives students exposure to what a visual representation, and then moving on to more abstract concepts.

Overall I think that this model is applicable to the mathematics classroom. The teacher needs to be really dedicated to this model and improve their skills within each step to “perfect” it in their classroom. If a teacher decides to use this model, they need to employ it regularly so their students gain confidence in communicating their knowledge with others.

Mike Freeland

The focus of //Orchestrating Discussions// is to help teachers organize their classroom discussion to its top possible potential. The authors of this article find that properly executed discussion on challenging mathematics problems can further student understanding and help them to learn the language of mathematics, among other things (2-1). Also mentioned are the pitfalls that discussion can fall into, namely becoming more like show and tell than a fully implemented discussion (3-3). To avoid this, the Five Practices Model is described.

The Model outlines strategies for a teacher to use to keep discussion as beneficial as possible. These involve anticipating student’s answers, close observation of the class, planned student presentation and organization, and connecting student’s ideas to key mathematical ideas (3-4). Throughout the rest of the article, all five parts of the Five Practice Models are applied to a real-world lesson.

The goal that the authors have in mind is trying to provide teachers with a method to control discussion (8-2). There is also the claim that by utilizing their method, teachers can make sure that the mathematical ideas that they want to impart on their students can occur (8-3).

I find it difficult to embrace such a method, but what they are trying to accomplish would be a step towards reaching the NCTM’s Communication Standards. This method would make sure that students are able to explain their reasoning behind their mathematical processes, work and discuss with their peers, and it is much more interactive than direct instruction.

Marcus Edgette

It is to my understanding that the authors of this article wanted to give advice on how to run a classroom so to speak. They do this by giving us five guidelines. These guidelines are: Anticipating, Monitoring, Selecting, Sequencing, and Connecting (2-4). These five guidelines make are supposed to make it easier on a teacher to focus on key topics and describes in detail how to do it. The first rule is that in the planning stage of our lesson we anticipate how students might solve a problem. This could take collaboration with other teachers, past experience and research on what the outcomes are supposed to be. Next is the monitoring recommendation, which puts forth the notion that monitoring student discussions is not just about listening and watching it should involve asking thought provoking mathematical ideas (4-2). The third regulation given is that of selecting. Selecting according to the article is being particular about which students we would select to present to the class. This is so that we can maintain a discussion that is guided towards the main principles that we as teachers have selected as important. In relation to being selective you must also select a sequence so that it makes it easier to transition to the next mathematical concept and focuses on achieving the goal of the lesson. Connecting takes on the issue of how to help students connect the lesson with mathematical ideas, build on the students ideas and create ideas by critiquing their peers work.

This article has helped me tremendously in terms of what creates the kind of discussion that I as a teacher would like to foster. It has a main focus about what we as teachers need to do if we want discussion to fare well in our classrooms. Much of it relied heavily on planning, to which I concur. In the little bit of teaching experience I have it has shown me that to be unprepared can be extremely detrimental to the flow of discussion. It also relies heavily on the idea that we as a teacher must be extremely involved so that we can do what is necessary to adjust and direct the student discussion to where we want it to go especially for unexpected solutions. I like that it’s asking us to prepare for the amount of preparation we will have to do and how it will affect our student’s perceptions.

Denise Slate

Whole-class discussions can be very effective for students, but it takes a lot of preparation from the teacher to get to that point. In “Orchestrating Discussions”, the benefits of classroom discussions (1-1) and the steps to having a successful discussion are explained (2-4).

Discussions give students the opportunity to learn from their peers, discover new mathematical strategies, and develop a deeper mathematical understanding (1-1). Students set the tone for discussions, and it is the teacher’s responsibility to ask questions that encourage students’ critical thinking skills. The challenge arises of how much or how little support does a teacher give to assist a student in his thinking process (2-2). From a student’s stand-point, having to answer a question on the spot can be nerve-wracking and makes it hard to think straight. As a teacher, it is difficult to watch your student struggle, but providing an answer causes a “decline in the cognitive demands of the task” (2-2). Incorporating cooperative learning into classroom discussions can eliminate the anxiety that direct questions may cause. Students will have time to talk over answers with teammates and get other opinions as well.

The five practices model acts as a template for whole-class discussions (2-4). A teacher anticipating different mathematical techniques is the first step (2-6). More effort goes into orchestrating classroom discussion than direct instruction. Anticipating is more work for the teacher but you are giving your students the opportunity to see things from different perspectives. Monitoring student discussion is the next practice in the model (3-3). The teacher can monitor by walking around the classroom and asking students questions that spark new ideas. The teacher can then select a few student solutions to demonstrate her mathematical goal (4-4). I feel it is important to vary which students are selected. This goes back to the article on student confidence. Selecting a quieter student, as opposed to the straight A students, will help raise his/her confidence. Sequencing follows and requires preparation from the teacher (5-6). How the teacher presents the solutions depends on what she is trying to achieve with her lesson. Lastly, connecting allows students to draw similarities and differences from the approaches (6-6). Students mathematical ideas will be strengthened from these discussions if the teacher has orchestrated them effectively.

Direct instruction seems so simple compared to whole-class discussions. The amount of preparation involved to have a successful discussion is substantial, but well worth it if you want your students to go beyond regurgitating equations.

Hailey McDonell

The five practices model (550-4) is a great way to come up with a lesson plan for teachers. There are certain things that we need to take in to consideration with each of these practices that they didn’t really touch on in the article.

With anticipating (550-6) what it is that students are going to come up with on the task given at hand it is not the easiest thing to do. Yes we may be able to think of the majority ideas that they may come up with for the correct answer, but coming up with the incorrect (551-1) answers that they may think of is not so easy. Students have a wild imagination and can think of the craziest things, and in previous experience, I have seen people come up with random numbers or ideas that they believe are formulas that brake mathematical rules.

When monitoring students (551-3) and selecting (552-4) I can easily see teachers always going towards the same students every time to see if they are coming up with, and it being the advanced students in the class. This way they know that they understand the task at hand and are able to present their solution to the class. One of the ideas that the author mentions, that I believe is a must do, to keep track of which students present their work, so that all students have the opportunity to share their thinking publicly (553-1). With monitoring you will have to circle around the students with an open mind, because some of them will come up with the correct solution that you didn’t think of and you won’t want to persuade them from their thinking to a way that is similar to yours with the questions (552-2) to direct their way of thinking to a solution that you came up with.

Sequencing the order of which student should present (553-6) their thinking is something that is going to be different for each class. I believe that this is something that may take a few days with each class to know what the best way is, and it is important to keep in mind that it can be different for each lesson too (554-4). The author did mention a great starting point on using the majority of the students used (554-1), even if this is the wrong way. Which can be a hard one to choose because you don’t want to pick on a student, or point them out because they didn’t do it correct.

Drawing connections between the solution and their own work (554-6) is probably me the most important part of the whole lesson, especially for those that did do it correctly. They can be able to find things that they did do correct and hopefully where it was that they went wrong. It also shows to students that there is not just one way on coming up with the solution.

Bret Van Zanten

“Orchestrating Discussions” is a collaborative article by four professors of education at various universities across the United States. Their goal is to present a model of five practices for successful whole-class discussion where students are engaged and the teacher has control over the overall discussion. These five practices are- 1) anticipating student responses to challenging mathematical tasks; 2) monitoring students’ work on and engagement with the tasks; 3) selecting particular students to present their mathematical work; 4) sequencing the student responses that will be displayed in a specific order; and 5) connecting different students’ responses and connecting the responses to key mathematical ideas (550-5).

The first practice is to anticipate the various responses students can give to a single task. This practice is helpful for a teacher in multiple ways. First, it requires the teacher to solve to problem themselves with different methods of approach and identify common errors that could occur (551-2).

The second practice is to monitor students’ work and engagement when presented a task (551-3). One strategy to accomplish this is to continually circulate around the classroom while students are working. In my ED 4060 class, we’ve been discussing classroom set-ups to determine which type would personally work best. The set up I like the best would easily allow me to float around the room and monitor student work.

The third practice is to select particular students to present their work (552-4). After floating around the room it is possible to identify which students followed which paths to get to their answer and then show the whole class that there are multiple paths to a correct solution or that the is an incorrect path. I am in favor of using this technique because I want the whole class to be able to evaluate themselves and identify why a certain path is incorrect.

The fourth practice is to sequence students’ responses in order to gain the most out of every contribution (554-1). If I were to use this technique, I think I would use the example given and present the strategy that the majority of the class used in order to “validate the work that students did” (554-1).

The fifth practice is to connect the various things discussed in class together and link them to mathematical ideas (554-6). This practice helps bring closure to a discussion. Instead of leaving the discussion with separate approaches, the whole class understands the range of approaches and how they “build on each other to develop powerful mathematical ideas” (554-6).

These five practices are all interconnected and are a great model to, at the very least, test out as an instructional method. This strategy helps teachers “more effectively orchestrate discussion that are responsive to both students and the discipline” (555-1).They also highlight the fact that many problems presented to middle school students have multiply ways to end at the correct answer, which is a valuable way to learn because the students can identify mistakes and take ownership of their own learning. Tori Ward