2.7+Reflection+8+Building+On

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 Sun by 11:49 pm.

The authors of this article explored the idea of building within the classroom. Building refers to one of two things. Either students respond to one another’s comments or they use one another’s comments to expand their own thinking mathematically (1-2). As teachers, we can make sure students follow this idea od building in three ways, (1) stating whether they agree or disagree with the statement at hand, (2) provide new evidence or insight to the idea presented, or (3) draw a conjecture to from the ideas presented (2-1). When I first began reading this article and read these ideas for students, all I could think of was how hard it was going to be to actually implement this into a classroom. I look back to my own classroom experiences and I know this idea of building was not followed. However, as I continued to read, I saw the role of the teacher and what they did and said made all the difference to making sure this happened.

First off, in the example the article talked about, David (the teacher) had his students go up to the overhead and present their ideas and, most importantly explain how they came to that conclusion (2-2). I think this is one of the most vital steps to getting students to build on one another. When a student has the ability to show what they have done, the other students will be able to visually see and critique what was done. After they students explained their approach to solving the crowd-estimation problem, David asked a vital question of “What do you think about this group’s method?” This question, and others like this, shows students the expectation of building within the classroom (4-2). The idea of the teacher asking these types of questions really helped me to think it is possible for the idea of building to be brought into the classroom.

When David created the “competition” of the two different ideas, it was a great way to get students to think. He basically summarized the two differing ideas but didn’t state which was the better one (4-4). In my opinion, that is the most important part of his summarizing. David is not pointing out one explanation and saying, “this is better, why?” He is creating the atmosphere where the students are encouraged to explore within their groups and their own thoughts as to which one is the better idea. This allowed the students to reflect and expand the ideas presented, which is one of the basic ideas of building within the classroom.

I have heard about wait time before, read about it in a previous education class, and we saw it in class with one of Cathy’s videos. It was nice to see here the two types of wait time: (1) after posing a question and (2) after a student makes a response (4-8). In my own experience, when a question is asked in a classroom or when a student makes a response, the teacher usually jumps in right away as to avoid long awkward pauses. I can see how many students stop their thinking when they know the teacher is just going to either validate what has just been said or not.

All in all, this article made me realize the importance of the teacher’s role. I have heard the idea of building in the classroom before but was always wary as to how to implement it. Not only did this article show how building should work but also it did a great job at giving insight as to how a teacher should facilitate the building. Katey Cook

In, “Students Building on One Another’s Mathematical Ideas,” Sherin, Louis, and Mendez describe a classroom environment where students are not only thinking through mathematical concepts for themselves, but also considering and evaluating one another’s ideas. In order to achieve this kind of environment, the authors keep in mind two goals. First, students are responding to each other’s ideas rather than just stating their own, and second, students use one another’s ideas as a basis for thinking and learning about mathematics. (1-2)

In order to envision what this kind of environment would look like, the authors take us into David Louis’s middle school classroom, where Louis creates opportunities for students to discuss mathematics. His students are not only stating their ideas but building one each other’s ideas, like Jeff (3-7), and this is because Louis has structured his class and teaching style with this kind of discourse in mind. The authors note three key points in his teaching that create foster an engaged class. First, Louis extends his “wait time,” after he poses a question or after a student poses an idea (4-8), because this allows students a chance to think about what was just presented. After each student has a chance to gather his or her thoughts and formulate opinions, the class discussion will benefit because there will be more ideas to consider, if each student contributes. Second, the class may break into small groups for discussion where Louis feels that it is necessary (4-9). By giving students a chance to think out loud and practice presenting their idea in the relative privacy of a small group, students will feel more prepared to contribute to full group discussion. Additionally, students get the chance to build off of one another’s ideas in their individual groups, which means that the ideas presented at the whole-group level will be more conceptual (3-6) and more carefully thought out. The third technique that Louis employs to get students to participate is self-evaluation (5-2). When asked to reflect upon their own contribution to each discussion, students get the idea that it is important to contribute to classroom ideas. These evaluations really shift the focus of the class from solution-oriented to discussion-oriented.

What this article really got me thinking about is the accessibility of student ideas in the classroom to other students. In a teacher-center environment, when a teacher presents an idea, the perceived authority and “all-knowing-wisdom” of the teacher may dissuade a student from dissenting with or adding to the idea. Some students may feel the confidence to comment on what the teacher presents with every slide, but the vast majority will remain silent for fear of having their idea look foolish next to the “all-knowing” teacher’s idea. By creating an environment based on students’ ideas, Louis is creating a classroom where no idea goes unquestioned or unexplained. Because students feel more comfortable disputing an idea that came from their peer, who has roughly the same mathematical ability, student ideas and opinions can flow freely. It is this free flow of ideas that is essential in the building process, but this free flow can only occur when the ideas are accessible to students. Classroom ideas must not be the un-questionable facts of an “all-knowing” educator, but instead the malleable ideas of peers.

Valerie Gipper

In an article entitled “Students’ Building on One Another’s Mathematical Ideas,” authors Miriam Sherin, David Louis, and Edith Mendez introduce us to a discussion model to use in the mathematics classroom. This model encourages students to engage in discussing and defending their mathematical logical concepts which attends to NCTM standards for being able to defend mathematical logic to their peers. The majority of the article focuses on the concept of building. Building has two goals inside of it: getting students to respond to one another’s ideas, and that students use each other’s ideas for approaching mathematics (1-2).

One of the reasons they installed this discussion technique in this classroom was to get students to do more than just state their own solutions (2-1). There are three ways that students build on one another’s ideas. The first is by simply agreeing and disagreeing. The second way is allowing students to provide new evidence for someone else’s idea. The third way is by getting info from another’s idea, and using it to create their own conjecture (2-1). In the article we were given a few examples of the building. They used a crowd-estimation problem. As students give possible solution ideas they continue to build on each other’s ideas, and make small changes for how they believe they can enhance the accuracy of the estimate (2-4).

These different ways of “building” are divided into different levels. Agreeing and disagreeing are very simple for students to do (2-5). Providing new evidence is another level, which is above agreeing and disagreeing (3-3). Creating new conjectures is the third level and demands the most from the student (3-6).

Later the article goes on to the teacher’s role in utilizing the building model for discussion in a mathematics classroom. The teacher needs to ask questions that force students to think about and respond to student’s ideas (4-2). Teachers need to ask questions such as: “what do you think about that idea?” Another thing a teacher can do to enhance the discussion is to summarize student’s ideas to refresh student’s memory on what their classmate has just said (4-3). It is also important for teachers to utilize small-group discussions and use wait-time to ensure everyone gets an opportunity to participate in discussion (4-8 and 4-9).

I think the Building model for discussion could be very useful for our math classes in the future. It allows everyone to participate, and by managing wait-time and other devices the teacher can make sure each individual is engaged in the discussion. This model also adheres to the NCTM standard about getting students to discuss mathematical ideas and use logical arguments to defend it. I also think it is important to note that the teacher needs to create an environment that doesn’t allow students to put each other’s ideas down. The agree or disagree option is very useful to ensure that students aren’t being put down, but rather just a statement of disagreeing. If a teacher is to utilize this model it needs to be the regular discussion format, it needs to become the norm for classroom discussion.

Mike Freeland

Mirian Gamoran Sherin, David Louis, and Edith Prentice Mendez spent a few years in a middle school classroom trying to create a place where students build mathematical understanding by talking together about what their peers think. They then wrote the article “Students’ Building on One Another’s Mathematical Ideas”.

The authors begin with presenting a problem that is given to the eighth grade class. The problem deals with a crowd estimation from an arial view picture. The teacher asks his students to estimate the number of people at a rally. He then asked a few groups to present their work on the overhead (187-6). Afterwards, he asked the class “What do people think about this group’s method?” (187-7). This gave the class a chance to begin building upon each other’s ideas.

They say there are different types of building; there is 1) agreeing or disagreeing, 2) providing new evidence, and 3) creating a new conjecture (187-5),(188-2),(188-6). The most simple way to build upon another students idea is to either personally agree or disagree to what they say to the class (187-5). This sounds so simple, it’s something we as humans do everyday; we decide if the words coming out of someone else’s mouth sound realistic or not. I’ve seen this occur in my pre-internship. I will ask a student to work through a warm-up problem for the whole class, and once we finish I ask everyone if they agree that this is correct or not.

Then, the next level of participation is not only responding to an idea, but giving further reasoning on why someone is right or wrong (188-2). Once two methods of estimation were presented to the class they were all able to chose which one the felt was more accurate and why. This, along with their prior knowledge on estimation, gave them the ability to give further reasoning.

The last type of building is when a student creates a new conjecture (188-6). This level of understanding is after a student gets a gut feeling and after hearing other’s ideas on the matter. They come to their own complex conclusions as to why things are the way they are.

In the crowd estimation problem the teacher asked some focusing questions like “What do you think about this method?” and “What do you guys think about what Robert just said?” (189-1). These questions opened the students up to conversing together. The teacher then brought the class together to summarize the problems and come to a consensus (189-4).

Tori Ward

In the “Students Building” article, the main theme of focus is the concept of building. In this article the teacher is envisioning a community where, “The teacher is not alone in the position of clarifying and questioning student methods. Rather ,students take an active role by commenting on and critiquing one another’s mathematical ideas” 186-1. Building involves goals that, “Students respond to other students comments rather than just state their own ideas and that students use one another’s ideas as the basis for thinking and learning about mathematics” 186-1. Having the ability to use outside references beyond your own knowledge makes the individual’s thought process much higher. When you can explain someone else’s way of thinking shows you that you can use alternative paths as a way of finding out solutions to a problem. The articles lists, “Categories of building” 187-1, that, “Students can build on other students ideas during whole class discussions” 187-1. These categories are, “Students state whether they agree or disagree with an idea that has been raised, students provide new evidence for, or insight into, someone else’s idea and students draw on a classmates idea in creating their own conjectures” 187-1. The main goal of these categories here makes the student contribute in whole class discussions that build on other students ideas.

The article says, “The focus on building develops a classroom community in which students ideas are valued as an integral component of learning and moves the class to thinking deeply about mathematical ideas” 188-8. I feel this focus is a good cause. It is with moving the class to thinking deeply about mathematical ideas where students begin to understand the concepts of mathematical thought. Students can then challenge themselves to try on more difficult situations that await them in the future. In order for building to occur the teacher must, “Find new ways to use students thinking to promote learning” 189-1. The two strategies used in the article by the teacher were, “Asking students to respond to other’s ideas and focusing students ideas for further discussion” 189-2. Asking students to respond to each other ideas forces students to, “Have opinions about their classmates ideas” 189-2 which furthers discussions and ,”Offering students ideas for debate provided an important and valuable focus for continuing the discussion” 189-6 which forces connections with ideas that deepens mathematical thought.

It is important that we make a classroom applicable for all students to build. The article lists this as doing, “Prolonging waiting time for students to speak, breaking the class into smaller groups for discussions, and asking students to reflect on their participation in discussions” 189-6. It is important for students to have enough time to think so that they can, “Have time to formulate their own ideas which seems to increase students comfort levels and ability to build on their classmates ideas” 189-8. Small group discussions enable, “All students to have an opportunity to discuss mathematical ideas and would be better prepared to contribute once the class reconvened” 189-9. Self assessment, “Encouraged students to value their peer’s contributions in future discussions” 190-2. All these applications make the building process much easier to accomplish in a classroom to grow. As a future secondary educator I must formulate my classroom in a way that students can build on other students’ strategies to come to a proper reasoning for their solution. I gained knowledge from this article to make students value the voices of their peers for a better classroom community which creates success among the students.

Fredrick Martin

The article “Students’ building on one another’s mathematical ideas” is about this concept of building. Building involves two related goals: (1) students respond to other students’ comments rather than just state their own ideas and (2) students use one another’s ideas as the basis for thinking and learning about mathematics (186-2). Building seems to be a way to get students involved with each other and participate in whole class discussions.

We go on to learn about three categories of building, (1) students state whether they agree or disagree with an idea, (2) students provide new evidence for someone else’s idea and (3) students draw on others ideas to make their own conjecture (187-1). I think building sounds like an interesting idea. I think it is important for students to listen and learn from each other and building seems to support that idea.

The role for teachers is slightly modified than the traditional role. With this idea of building teachers need to facilitate whole-class discussions and promote learning from one another. The role of the teacher includes a few key ideas, (1) asking students to respond to other’s ideas, (2) focusing student’s ideas for further discussion, and (3) creating opportunities for all students to build (189-1). I think for me the third idea is the most important to me. I am afraid of having class discussions that are dominated by the same people; one way to rectify this as mentioned in the article is to allow small group discussion time. This way everyone has an opportunity to talk with at least a few of their classmates and hopefully the whole class at some point.

Also, this idea of wait time is mentioned again in this article, (189-6). At first, I was not so sure about this idea but the more I read about it the more I believe it would actually work. Here two types are mentioned, (1) after posing a question to the class and (2) after a student presented an idea, that is before asking for someone to respond. I have experienced teachers using this technique and I feel that is encourages students to say more, to describe an idea better and perhaps more clearly. It seems to be human nature that not hearing immediate feedback causes people to continue an idea and to explain it more in detail, wanting to get that response.

Finally, we learn about how David using a self assessment task after a group discussion (190-1). I must admit I really like this idea, I think it helps the students to be responsible for how they participate in class discussions. I believe they also learn that participation is expected of them and they are also expected to participate in some way. Kaitlin Froehlke

At first the idea of building on student responses sounded like just another way to get students to think about math through their own eyes. This is a benefit of allowing students to share and build their ideas, but the definition of “building” in this case is given early on, and begins to go in a different direction than I expected… not to say that is a good direction.

The author starts out by mentioning that in Dave’s classroom, students sharing their ideas and solution strategies aloud are common, and he continues to push for building? I feel as though getting students to discuss their thought process to the class could be tricky enough, but now he wants students to challenge or elaborate on their classmates’ ideas as well? Fat chance. But once Dave mentions how he expects students to build, it seemed less farfetched. Three ways to build:

(1) Students state whether they agree or disagree with an idea that has been raised; (2) Students provide new evidence for, or insight into, someone else’s idea; and (3) Students draw on a classmate’s idea in creating their own conjectures.

I thought the Crowd Estimation Problem was an interesting one. It is a problem with more than one solution, so many students should feel comfortable starting the conversation by sharing their thought(s) with the class. That is exactly what Tina did. Shortly after one method was presented classmates could agree or disagree. No matter what their response was, the needed to elaborate on it. WHY do you agree or disagree? This is a very open way to invite students to share thoughts.

Agreeing or disagreeing with a statement is easy for students, elaborating on why you feel that way is not as black and white. You can’t just repeat what your classmate said, so new support is often offered. “Offering new support for someone else’s idea is a valuable way for students to contribute to whole-class discussions (188-3).” This is also where students can compare and contrast related mathematical ideas, this often pushes the discussion to a more advanced conceptual level (188-5).

One particularly advanced concept was offered by Jeff. This response was almost unreal to me, not because of the strength or complexity of his thoughts, but I feel as though students may have more of a tenancy to simply try to understand their classmates’ explanation and agree with that, rather than come up with a third, possibly more accurate, way to measure the population themselves.

These three methods do offer many opportunities for classmates to jump into the conversation, by agreeing, offering something new, or elaborating on existing ideas. These three levels of participation or justification vary in complexity as well, perhaps this justifies Jeff coming in with such a strong idea: a complex thinker building off of his classmates ideas.

<span style="font-family: 'Times New Roman','serif'; font-size: 16px;"> The focus on “building” develops a classroom community, in which students’ ideas are valued components of learning and move the class to thinking deeply about mathematical ideas (188-7).

<span style="font-family: 'Times New Roman','serif'; font-size: 16px;">How to facilitate: Begin by asking students what they think about the first method offered. After students decide they agree or disagree, ask them what they agree with, and perhaps why they agree, if you believe that answer needs justification.

<span style="font-family: 'Times New Roman','serif'; font-size: 16px;">Two times for a pause: After asking class a question, giving them time to come up with a solution, and then again after a student explains their answer, allowing classmates time to digest the new idea. I already reflected on “wait time” and the benefits of small group discussion, so I will refrain from stating redundancies. Finally, I thought the reflection sheet was interesting; although not every student may get a chance to contribute as much as they could to the conversation, they may reflect on what they did add to the lesson, and learn how to better contribute to the next one.

<span style="font-family: 'Times New Roman','serif'; font-size: 16px;">- Tim H.

The main topic of the article, “Student’s Building on One Another’s Mathematical Ideas”, is to encourage students to have an opinion about mathematical ideas that are presented in the classroom. There are two goals to aim for when trying to establish ‘building’ among students. The first is having students respond to others' comments, and the second is the student incorporating other student’s ideas to strengthen their own mathematical idea (1-2). I feel to make building successful in any classroom, norms have to be established at the very beginning of the school year. This way, students will know what is expected of them, and the goal would be to get all students to participate in class discussions, not just a select few students. When norms aren’t instituted, students may not feel a need to participate. We most definitely want our students to get the most out of these discussions.

Building comes in three categories that focus on getting students to elaborate on ideas: stating whether you agree or disagree, providing new insight, and drawing on classmates ideas to create own conjecture (2-1). David presented the crowd-estimation problem to encourage building in classroom discussion. The students demonstrated agreeing or disagreeing by responding to one another's ideas. I think this is a good way to get students’ opinions without making them feel uncomfortable. When I was in middle school, I did not like to speak my opinion, but agreeing or disagreeing takes some pressure off. By providing new insight, students are able to offer support for someone’s idea (3-5). I like this a lot because having students support one another creates a friendly classroom environment which is good for learning. When students use classmates’ ideas to build their own conjectures, they are making connections which causes the discussion to become more advanced (3-6).

As teachers, we play a big role in the building model. It is our job to ask students to respond to others’ ideas, focus student ideas, and create opportunities for all students to build (4-2). It is very important to show students that their opinion is important because this will encourage all to speak their minds. We need to be careful on how we react to students’ ideas. Responding enthusiastically to some conjectures and not to others may discourage a student from participating. To have building present in a classroom, the teacher needs to be student-centered because it is their ideas that are the core of discussion.

Hailey McDonell

Miriam Gamoran Sherin, David Louis, and Edith Prentice Mendez set out to “develop a middle school classroom in which students talk about mathematics”, but in order to make this happen, they discovered they needed to find ways for teachers to encourage students to engage in mathematical discourse (186-1). They share a strategy called building//.// This idea has two aspects: that students both //respond// to one another’s mathematical ideas and //use// these ideas for learning and thinking (186-2).

The authors then identify three different types of building (187-1). Through the crowd-estimation problem in David’s class and the discussion that followed, they point out specific interactions that display these. The first, agreeing or disagreeing, is definitely the most basic (187-5). Students interact with other students’ ideas simply by stating if they agree or disagree; I think all students should be able to do this easily, since it’s not hard to form an opinion about a mathematical idea. The next level of building involves providing new evidence (188-2); students not only express agreement or disagreement, but they bring in additional support to consider. The last category is “creating new conjectures from classmates’ ideas” (188-5), and this is the most sophisticated way for them to engage. Students actually create their own mathematical ideas, partly based on the others’ suggestions, and this is definitely an indication that they are developing mathematical connections. I think this is great because, unlike the first two categories, in which students merely //respond//, this last category shows a progression to truly //using// ideas of others in a new way. This reflects the true nature of mathematics, taking a previous idea and investigating it further from a new perspective.

Through this process of building, we may wonder what our role as the teacher is. The authors maintain that it is to facilitate and enable building and student discussions (188-10). They offer several suggestions for how to accomplish this. First, ask students to comment on others’ ideas (189-1). We must be careful that comments do not become negative or disparaging, though. Next, focus or summarize these ideas to set up debate or discussion (189-2). Last, it is essential to create opportunities for all students to build (189-5). This last strategy could use wait time (think of Cathy) or small-group discussions (similar to ideas from many other articles) or written self-assessment by students. The idea of written self-assessment is not one I would generate myself, but I think it is an excellent design for discovering what students are truly thinking and learning. Having students evaluate their involvement in the discussion increases their awareness of the learning that actually took place and how they were a part of this.

I believe it is essential for students to be able to respond to and use others’ mathematical ideas if they are to develop an understanding of mathematics. A good discussion incorporates many ideas, responses, and strategies, and the end result can be a mathematically engaged classroom and significantly deepened understanding for students. I don’t think it’s a good idea to allow students to stop at the agreement/disagreement stage, though, even though that is an important piece of building. As teachers, we need to push students to further develop their thoughts so that they are not only responding, but also utilizing and connecting mathematical ideas. Then, and only then, we will experience successful and meaningful class discussions.

Mandi Mills

The article’s main focus is a concept they call “building”. With using such a concept, the authors hope to have student discussion based on other student’s responses to mathematical idea and have that form the basis for mathematical learning. (1-2) In depth, they looked at how a student can respond to other students ideas. These three ways include having a student agree or disagree with another student’s response, give further evidence for a student’s idea, or create their own conjectures based off what a student has heard. (2-1) They also mention in the first paragraph of the article that they find importance in having such responses because this creates a community of active learners. (1-1)

There is then an example of a real lesson where the three ways a student can respond as outlined above took place. I find the students responses to be the important factor to note. Students are capable of forming their own hypothesis based off their knowledge, past experience, and the discussion that the article is promoting. For instance, the student, Robert, disagreed with the use of a method of solving the crowd problem. (2-4) Further in the article, a student, Jeff, contributed with the idea to use an averaging method to be more accurate in their estimate. (3-5) Such an observation took active participation mixed with some thought going into the mathematics behind the problem.

So with students contributing to learning, the article then goes into the teacher’s role in the learning process.The teacher’s role in this scenario is to keep discussion moving in the direction he wishes for it to take and find ways to keep students thinking about the problem at hand. (4-1) The four ideas the article concentrates on are those of asking and allowing students to respond to their peers, helping students stay focused on relevant information, allowing as many students to actively participate as they can by use of small discussion as well as the larger group discussion, and have students reflect on how they contributed to the classroom to allow for them to see how their own learning is affected by their participation. With the teacher there to facilitate and use these strategies, the “building” that the authors of the article want to occur was able to be met.

-Marcus Edgette