2.0+Reflection+on+Navigating+and+Problem+Solving

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: 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.
 * 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.

This first reflection is a one page summary of two articles, "Problem Solving and Mathematical Beliefs" and "Navigating Classroom Change". Paste your reflection followed by your name. This is due Sunday 11:37pm.

CHECK YOUR EMAIL IF YOU HAVEN'T SINCE THURSDAY. DR B

Valerie Gipper I recall that as a secondary education mathematics student, my teachers would joke about the “drill and kill” routine. This routine entails getting students to learn mathematics by drilling the procedures, rules, and facts so heavily into them that they loose any interest in doing real problem solving. This seemed to be the case with the talented students interviewed for “Problem Solving and Mathematical Beliefs,” who saw math as nothing other than a means to a numerical, correct answer. Their emphasis on right answers as the point of mathematics was both astonishing and unsurprising to me. It was astonishing in the sense that even the particularly talented students had no concept of problem solving as a part of mathematics, but it was also unsurprising because of how most secondary math classrooms are structured. Despite my teachers joking about the “drill and kill” routine, it was prevalent at my school. This emphasis on fact and procedure memorization gives the impression that there is only one way to complete a certain type of problem, and only that one way will produce a correct answer. After a certain amount of memorization, students no longer encounter problems and only see exercises. Doing the exercises will certainly produce the correct answer and lead to the student passing the class, but after just regurgitating procedures all year, he or she could forget those memorized processes and be left with an A in the class but nothing in their brains.

Both articles called for a change in the teaching of mathematics in order to fill students’ brains as well as help them achieve good grades. In, “Navigating Classroom Change,” one teacher made an effort to change how his students saw mathematics. By giving them problems as opposed to exercises, he facilitated real mathematical thinking in his classroom. More importantly, the students realized that doing mathematics doesn’t mean running through a set procedure, it involves much more than that. Beyond practicing their problem solving, students were exploring the numbers, patterns, and mathematical relationships for themselves. This allows for two very important ideas to form. Firstly, students exploring math with each other, unhindered by a rigid lesson, gives them the chance to see that their teacher and math book are not the sum of all mathematical knowledge; they are able to see that they can do mathematics without the help of either and without relying on familiar procedures. Secondly, because the students found a conclusion on their own, they get a sense of “owning” the knowledge. It becomes theirs because they worked for it, rather than simply asking the teacher. For me, I find that knowledge I myself struggle to acquire sticks with me much longer than when someone simply shows me.

Rather than focusing on the correct answer by drill and kill methods, we should be focusing on the process itself of solving the problem. If we work to change our classrooms into spaces for student exploration, students will no longer see math as nothing other than endless computations. Instead of spending their summers forgetting memorized procedures, students will fill their brains with knowledge that they truly earned and carry it on to further their mathematical education. _

The article titled //Problem Solving and Mathematical Beliefs// discussed the idea of teaching students problem solving when in a math class, rather than simply getting an answer to a problem. Martha Frank, the author of the article, obtained interviews with students to see their thoughts about math. The statements the students came up with were the same ones we voted on in class. These students were under the impression of five common math myths which are (1) math was simply computation, (2) problems should only take a few short steps, (3) math was about getting the right answer, (4) the job of students is to receive the knowledge and show they received it, and (5) the job of the teacher is to give the knowledge and make sure the students received it. Frank’s quote from Wheatly exemplifies the whole point of why students need to learn problem solving (2-7). The skill of being able to problem solve will be helpful later on in life. It is important to look at problem solving as a problem and not an exercise, as this article has taught me. There is a distinct difference between problems and exercises (2-9). There is a place in the math class for exercises, however problems invoke problem solving. Many students when confronted with a problem will revert back to their original thinking, for example “the problem takes too long therefore it is not math.” This usually leads the students to giving up. Frank’s ideas for helping fix this train of thought is (1) start problem solving early, (2) be sure your problems are problems, (3) focus on solutions not answers, (4) students should frequently work in small groups, and (5) de-emphasize computation. These ideas proposed Frank to help improve problem solving were also the ideas of the author of the other article, //Navigating Classroom Change//, Lindsey Umbeck. She actually implemented these ideas into her classroom. She wanted to structure her room around the students creating the learning with unknown problems. Umbeck followed a model entitled The Launch-Explore-Summarize teaching model (3-5). This model is broken down into the three sections of the name. This helped the students create their own thoughts and solve the issue at hand without the teacher’s help. While working in their groups, the students would frequently ask for feedback and help and Umbeck had to make sure she never gave out answers or her thoughts on how they did the problem. This is important for the students to grow in their problem solving skills. The example of how Umbeck did this can be exemplified though the grape juice task (5-6). Umbeck completely changed the way she ran her classroom but this did not come without negativity. The issue with starting this type of math class with students in junior high is they have already had years of math experience. They harbor those same thoughts talked about in Frank’s article. Umbeck as well had issues with not giving them the right answers when they asked because that is how other math classes had been (5-1). In the end, Umbeck did experience success and her students were excited over their findings (7-1). Success is there and can happen, however total achievement is not guaranteed because there is always room for growth. Finally, Umbeck states her efforts can be seen through Shaw and Jakubowski’s proposed stages (7-4). These articles explored problem solving and getting students to think about how they solved math in an entirely different fashion. It helps the students in life and improves their thinking skills. Katey Cook

While reading the “Problem Solving and Mathematical Beliefs” article a few things stuck out to me. I agree that “the role of the mathematics teacher is to transmit mathematical knowledge and to verify that students have received this knowledge,” (2-5). It is important to //verify// that every student truly receives and comprehends the knowledge presented to them. The author then goes on to say, “Teachers verify that students have received knowledge by checking students’ answers to make sure they are correct,” (2-5). This quote seems to focus solely on the end result and not the computational process or necessary steps to get to the correct answer. Sometimes students arrive at the correct answer by sheer luck or complete guessing.

Why do educators distinguish between //problems// and //exercises//? (2-6) Either I’ve spent my whole mathematical career unable to distinguish between these two ideas, or I have been mislead by past worksheets and texts. I liked the problem-solving tips for success: 1. Start problem solving early, 2. Be sure your problems are problems, 3. Focus on solutions, not answers, 4. Students should frequently work together in small groups, 5. De-emphasize computation (pg 1). In order for this these to be successful we need to help spread the word among colleagues. In the case of tip number one, how can we stress the importance of introducing problems to elementary students? I feel that the work towards the solution is more important than the final answer.

In the article “Navigating Classroom Change,” the author implemented a “Launch-Explore-Summarize” model to teach in her mathematics classroom (2-5). This approach to teaching was different for me in that it recommended giving students very little instruction before they began work on a mathematical task (2-5). I am hesitant to try this approach because practice makes permanent. If any of the kids in groups approach a problem confident that they’ve found a way to solve a problem but really are practicing a common mistake it would look poorly on me later on. I’m glad this educator stuck with her new Launch-Explore-Summarize routine because her students became comfortable with it and expected her to continue with it (4-9). I’m happy she stuck to her plan because I’ve seen, first-hand while volunteering, a teacher who changed various daily routines and tried to implement new routines halfway through the Spring semester.

Something that remains in my mind after reading “Navigating Classroom Change” is the question why. I feel as though “why?” is overlooked so often in general questioning, or it’s tacked onto the end of an exercise to try and prod further, but most educators breeze over that part of a question when, in fact, it can be the most important part of the exercise. Tori Ward

The first article entitled, “Problem Solving and Mathematical Beliefs,” pointed out many interesting points of interest. One of those points was, “How can we get students to become better problem solvers?” However, the article calls first for a change in students thinking about math. I do not know how many times from classmates in high school, “I don’t get or I don’t like math,” which to me is sad because math is not that hard. I also really enjoy the responses I get when people ask me what my major is and I respond with simply, “math,” the looks and sounds of disgust I receive have now become expected.

The article goes on to describe a study done with students apart of Purdue University’s STAR (Seminars for the Talented and Academically Ready) program which is for middle school students. Students were observed and interviewed as a part of this study. A list of beliefs was compiled based on the examination of the data gathered, this list includes: (1) Mathematics is computation, (2) Mathematics problems should be quickly solvable in just a few steps, (3) The goal of doing mathematics is to obtain “right answers,” (4) The role of the mathematics student is to receive mathematical knowledge and to demonstrate that is has been received and (5) The role of the mathematics teacher is to transmit mathematical knowledge and to verify that students have received this knowledge. I think these beliefs that students have about math are based on very little experience with math. I also used to believe things similar to these but after continuing my education through high school and into college I now realize that these are not the case.

I think we as future teachers need to focus on how to help our students become better problem solvers and help to change their views about math.

The second article, “Navigating Classroom Change,” is mainly about how to change a classroom into a more productive classroom and the steps the author took to make the transformation. The first step was to “clarify my vision,” which included determining a clearer view of the changes that were wanted to be made. Next, “needing a new structure,” this step is about finding a teaching method that follows a path you would like to take. In this case, the “Launch-Explore-Summarize teaching method. Launch was a brief teacher led instruction introducing the activity. Explore is letting the students begin workings on their own. Summarize involves a class discussion of the different approaches the students took to solve the problem. The third step discussed, “establishing new classroom norms,” entails not simply implementing a new teaching method, but that one must prepare the students for the change. Finally, “using a variety of approaches,” meaning giving students different types of problems to solve in which there may not necessarily be only one way to solve.

I hope as a teacher to not fall into the norm mentioned in the beginning of the article of simply lecturing to my students and hoping they are learning something. I want an active classroom where the students engage together and learn from one another as well.

Kaitlin Froehlke

The two articles we have read so far holds important information that we as future educators need to recognize and acknowledge. The “Problem Solving and Mathematical Beliefs” article was based solely on the importance of problem solving skills. Middle school students were interviewed about mathematics on how they feel about the subject and their responses were typical to mine. In middle school I didn't want hard problems that forced me to think. I wanted the teacher to give me a lesson, show me a formula then I could use that to solve my homework. Quick and easy was how I liked my work and the students in this article feel the same way. This may be a way to get good grades but as far as problem solving skills and thinking from scratch this is rendered useless. The brain needs to know how to solve a problem without given any way about doing so. Problem Solving is what you do when you don't know what to do 33-7. This mode of thought is very critical in today's world for the ever changing environment. We need people who can think on their toes without any guided solution. This article has helpful suggestions on the development on mathematical beliefs that will be helpful in problem solving.

The “Navigating Classroom Change” article was based solely on how a young teacher took new actions to foster the mathematical environment the he envisioned. He fostered problem solving ides which forced his students to work together as team and for them to develop their own problem solving strategies in order for them to come up with the correct solutions. The Launch-Explore-Summarize teaching model 90-5 provided a way to structure how students participated in tasks. The launch process requires little work of the teacher. It just allows the teacher to give the introduction about the task. The Explore part was where the students actually began to work. The teacher made sure not to critique the work of the students so basically having faith in the students ability to think mathematically. 91-3. The summarize process is where the teacher emphasized key mathematical ideas that were discussed and helped students come to consensus. 91-5. In this structure the teacher is able to initiate a plan, follow through with it and see the end results. The teacher is able to instill important problem solving strategies and the students will be in engaged because they themselves found out the answer and can somewhat own that knowledge that is given to them. As a teacher we must still re-evaluate ourselves as future educators making sure that we: recognize a need for change, make a commitment to that change, construct a new vision for the practice, project ourselves and our classroom into that vision, take action and began to make changes, and to continuously reflected on those changes and compared our practices to our vision. 94-6 All this is saying is that you must take head to every little thing to do as an educator.

Both articles holds important information that we as future educators need to recognize and acknowledge. What I grabbed from both articles was that mathematics is not only computational it is finding out deeper meaning that can be used over and over again. You may forget how to solve a exercise but you cant forget how to solve a problem. It is a thought process versus and memory logger.

-FREDRICK MARTIN

In “Navigating Classroom Change” Lindsey M. Umbeck describes her vision and her methods for altering the way students deal with mathematics. This includes a student-centered approach, multiple entry points and solution paths, and worthwhile mathematical tasks. She incorporates all of these into a method called Launch-Explore-Summarize. Martha L. Frank, in her article “Problem Solving and Mathematical Beliefs”, looks at how to develop students into better problem solvers by examining students’ beliefs about mathematics and the subsequent implications for both problem solving and teaching. Both articles I read recognize that students’ current way of thinking about mathematics is a matter of concern and describe methods that the authors believe will result in meaningful change.

Umbeck and Frank refute the idea that doing mathematics means memorizing a set or rules/algorithms and using these to complete computational exercises. Although many students believe this, I completely agree with the authors that great care must be taken to develop students into problem solvers rather than being experts at “plugging and chugging”. Part of what deters students from learning true mathematics is their focus on the “right answer”, and I have certainly observed this. These articles stress that that “reasoning and justification” or “problem solving” are just as valuable, if not more so, as finding the correct answer. There is a difference between exercises and problems; there is also a difference between answers and solutions.

A vital component in both articles is the role of students working together in small groups, which I support to some extent. A focused small group environment often stimulates discussion and questions that would not arise while working individually. I agree with the articles that teachers can use small groups to create independent thinking and dependence on students’ own ideas instead of the teacher’s. Creativity is another skill that can blossom in this environment.

I think the biggest challenge that small groups pose is the tendency to get off-task, and this concerns me in that it could hinder learning. Thus, I believe teachers must learn to manage the group discussions in their classrooms in such a way that students remain focused on the problem at hand and keep working persistently to come up with a solution. According to Umbeck, the teacher goes through a challenging process that involves directing questions that arise back to students as well as listening. She states that teachers must continuously reflect and solve problems themselves in order to encourage this new mathematical environment. Frank encourages teachers to focus on solutions and “de-emphasize” computation.

I recognize that many of the misconceptions about math that Umbeck and Frank point out have influenced my own approach to mathematics as a student, too. In fact, it’s simply easier to think about mathematics in terms of computation and “right answers”; it removes the possibility of feeling inadequate because no solution can be found and also lets us avoid exerting too much effort. However, I believe that the way the readings encourage us to view mathematics will in the end be much more beneficial; it dares us as teachers to rise to another level and develop problem solving skills in students that will aid them for the rest of their lives, not just in mathematics class.

Mandi Mills

Problem-solving is a skill that as future teachers we must implement. In both articles emphasis was put on the importance of problem-solving and how it can alter the culture of your classroom. These articles have made me wonder, in my classroom, how will I incorporate problem-solving into my lessons.

The survey conducted in “Problem Solving and Mathematical Beliefs” shows that children are learning math only on the surface based upon the list of beliefs collected (2-1). As teachers, we have to change these notions that students have about math. Math isn’t simply computing numbers, and its goal is much more important than just finding “right answers”. So how do we go about showing students that there is much more depth to this subject? It starts with shifting the focus of school mathematics to problem-solving (2-7). I can relate to the student that, when presented with a difficult problem, says “I can’t do this” (2-10). I feel throughout my schooling, I have mainly been presented with exercises, so when a problem does arise, I don’t think I have the tools to solve it and give up. Many students, including me, lack the confidence to attempt a difficult problem when there is a possibility of getting a wrong answer. This is why we have to encourage students at a young age to work through those problems. Emphasis needs to be taken off generating a correct answer to prevent students from feeling like they wasted their time on a problem (3-1). When it comes to grading, have students record all of their problem-solving strategies and give the majority of the points for that work. In this manner, a student won’t be as focused on the answer, but the problem-solving techniques.

While reading “Navigating Classroom Change”, it dawned on me that this is what the hats and scarves problem was trying to achieve. The problem followed the Launch-Explore-Summarize teaching model (2-5). Reflecting on my past math classes, I can honestly say I have never come across a problem like hats and scarves before, which I find sad. With no mathematical procedures being implied, to solve the problem, I adopted the trial and error approach…which ended in error most of the time, but that is beside the point. The point is that this teaching model is wonderful. I explored many different mathematical techniques to try and solve the problem, but I was still learning in the process. I just can’t believe that my grade school teachers hadn’t adopted this teaching method to get students more involved in their math work.

Now that we’ve learned about how to shift mathematics focus to problem-solving, I’ve got a question. When you’re a teacher are you just going to teach in survival mode, or are you going to challenge your students by approaching the subject in a way that takes a little more effort? Decisions, decisions…

Hailey McDonell

Martha L. Frank’s article “Problem Solving and Mathematical Beliefs” is well titled, to say the least. Frank discusses the over use of computational mathematics in secondary classrooms and why students today struggle with problem solving. According to Frank, problem solving “is what you do when you don’t know what to do” on a problem. I almost used the word “exercises” there, but the author made the difference between these two words clear: in a math exercise, students know or are given an algorithm to apply in order to reach a numerical answer. Problems, on the other hand, take a little more time and thought. Problem solving can take between five and ten minutes a problem, and requires strategies such as working backward, trial and error, collecting data, making charts, and looking for patterns. Some beliefs from the article that I used to empathize with are that math is computational, the goal of math is to obtain “right answers,” and the role of the math teacher is to transmit information. I still agree that part of math is computation and finding right answers, but after reading this article I can see how a math teacher may have alternate roles than strictly transmitting information. The other article we read, “Navigating Classroom Change” by Lindsay Umbeck. This article is about how the author created her ideal classroom, and offers suggestions to how we may satisfy our own expectations as well. There are many conflicts when it comes to establishing your ideal classroom: attempting to manage your classroom, navigating the constant time crunch, and working around lack of support are all reasons for bland, generic math classes. A major part of Umbeck’s article explains one technique she uses, called “Launch-Explore-Summarize.” I am familiar with this strategy, not by name, but when a teacher provides little instruction, but instead allows students to collaborate on a problem and share their findings. I like this method because it provides students with a sense of ownership for their own learning. When students are able to take ownership over ideas and/or concepts there is a better chance this new knowledge will stick. In order to carry out this type of instruction, it is important that a teacher selects items that do not separate mathematical thinking from mathematical skills or concepts, ideas which capture students’ curiosity and that invite them to speculate and to pursue their own intuitions. Although this strategy requires the teacher to show more faith in their students to take control of their own learning, I agree that this is a more effective method than standing in front of the class and lecturing every day. I feel as though our professor assigned these readings together because they share a common theme; that being secondary students need a more involved mathematical education. Instead of drilling algorithms to our students we should offer more open ended math problems which require them to think deeply and reason amongst each other. I hope to implement this style of classroom management myself without failing to reach the standards set by the NCTM.

-Tim Hollenbeck

Problem Solving and Mathematical Beliefs was extremely insightful when it came to pinpointing the issue with why mathematics doesn’t interest many students. As stated in the article “Problem solving, not computation, needs to be the focus of mathematics instruction if we want our students to become good problem solvers” (3-9). This statement is monumental in changing the thinking our students have been exposed to and are used to. When we teach our students to solve problems, we also teach them autonomy and metacognition. Metacognition and autonomy are proven to help students succeed in the classroom and with any subject, not just math. Shifting the focus of our lessons will shift the thinking processes of our students. However just like problem solving should take no less than 5 min (3-7), shifting our focus and shifting our students thinking will also take time and come with its own challenges.

Any change in a classroom will take some time. Navigating classroom change is a great follow-up read to understand how changes might look and what other challenges teachers have encountered when undertaking such a task. New teachers are often found entering survival mode once and abandon the concept of their ideal classroom environment (1-2). It is important that we as teachers do not forget about our vision of an ideal classroom because our vision is where we are most effective in that we are comfortable knowing we are accomplishing what we want to accomplish. In order to work towards your ideal classroom it is advised to start small. This is so that a teacher can manage the change. The new norm in a classroom can be met with many different obstacles, many coming from the students themselves (3-6). As mentioned in the Frank article students think that mathematics is basically computation because that’s what most teachers focus on (3-9), but shifting that way of thinking may also be shifting them out of their comfort zones. When this happens students usually look to the teacher to provide them a solution to their problems but as demonstrated in the Umbeck article it is important to support students by following through with the effective strategy and structured activities selected, praising them on the positive outcomes, questioning, and guiding them. Although Umbridge doesn’t reach what she considers her ideal classroom completely it is apparent that she made great strides towards it. She also deliberately emphasizes the need to go through a period of reflection, comparison, and change (6-5). Denise Slate

In the article titled “Problem Solving and Mathematical Beliefs” students made it clear how they viewed mathematics (2-2). I can’t say that my views going through middle school and high school were much different. I hated when problems were not easily solvable, if I encountered a problem that I had not seen before I would give up. I only attempted to do problems that I knew the rules for how to solve. I also just wanted to know how to find the right answer. I didn’t care for the reasoning behind the solution; I only cared about the final answer. I saw the teacher as having the responsibility of transmitting their knowledge about math to me, and I (the student) needed to show that I had received that knowledge. It’s easy for me to identify the students’ thinking about math and the roles of the teacher and pupil, as I had a similar thought process when thinking about math. The article went on to discuss the difference between the definition of problem solving for educators and students (2-7). Wheatley sees problem solving as what you do when you don’t know what to do. While most students just see it has doing exercises. Students may be frustrated with this definition and learning how to really problem solve. The article goes on to state what the implications of problem solving are. Students need to begin to encounter problems early in their educational career. Educators need to be sure that their problems are problems. There needs to be more focus on solutions then on answers. I agree with this idea. Many times students just want the answer and do not care about how they got to it. We as educators need to emphasize the solution. The article titled “Navigating Classroom Change” went through a teacher’s journey to change the mathematics classroom’s norms and create a different atmosphere. Umbeck tried to employ this new vision after adopting survival tactics after finding herself at a place with a lack of resources and support to make her vision a reality. She created a vision that she found aligned with goals of the NCTM. She used a model called “Launch-Explore-Summarize” (3-5). She gave students background info that she thought was needed to understand the problem at hand. She then put the students in groups to work on the problem. She mentioned that students had some frustration with the problem because it was different from what they had expected from a math classroom. They couldn’t follow a known algorithm to obtain their answer. They had to find their own way to solve the problem at hand and give the solution. As a student I can understand the students’ frustrations with this. I cannot stand when a professor hands me an assignment and isn’t very clear in what they want. Umbeck made it seem like students adopted this new model of approaching mathematics and prescribed some steps to achieve her goals for her classroom (7-4). I believe that the most important steps of the process were committing to the change, taking action and implementing the vision, and reflecting on the results that she obtained. If an educator isn’t committed to a change in the classroom, he will not see the change that they want to see in their classroom. An educator also needs to take action and do something to make that change more than a vision. They also need to evaluate how the “change” goes and see if they are on the right track to make their vision a reality. -Mike Freeland

Reading both of these articles was very insightful and brought out a new light of how to teach mathematics, and how students think about mathematics. Learning math in my middle school and high school math classes it always seemed to be a step-by-step procedure or lists of rules used to obtain numerical answers, as mentioned in Problem Solving and Mathematical Beliefs. This is probably why math came so easy to me because all I would have to do was watch the teacher do one example and then imply that example with the other problems just putting in the different numbers, and getting the correct answer. The main focus was to get the correct answer, even though the teacher would always say show your work, they seemed never to pay attention to the work. I would have a friend that was not the greatest in math and he would always get the odd answers correct and sometimes the even answers. The way he would show his work for the odd answers is by doing the work that he thought was correct then looking in the back of the book and either multiply, divide, add, or subtract his answer to whatever number would produce the correct answer. Never once would he get marked down because he had some work shown and the correct answer.

This is a great reason on why it is important to have group work, like mentioned in Navigating Classroom Change, instead the tradition way of teacher explaining the new chapter with examples that they book has, giving out the assignment, and giving the rest of the class time to work on homework. Students need to be able to justify their reasoning on how they came to the conclusion they did. It is a huge change and will take some time to do, just like it did with Lindsay Umbeck, but hopefully this class and reading articles like this we will be able to learn from other teachers so that we don’t spend the first two years figuring it out. Those first two year students are missing out on the opportunity to learn mathematics which will help them in further classes and throughout their lives. This can be done by what Martha Frank mentioned that students need to be given more story problems early, before middle school, and often that make them struggle to figure them out and that can be solved in more than one way. I don’t remember when I started story problems in school, but they are the one things that I never liked doing in math, and always had a hard time with solving them even though they were never really challenging and all you needed to do was figure out what number went into which spot for the same equation that you have been solving for the past five questions. Bret Van Zanten

The articles assigned for reading both were intelligently written and contain integral details and thoughts that mathematics teachers should know and utilize. When I dove into “Problem Solving and Mathematical Beliefs” many of my initial thoughts on the way mathematics is taught were validated. Many college students now are under the distinct belief that they are no good at math and it is a skill they cannot master. This thought is so prevalent that even Lindsay Umbeck described it in her initial paragraph (1-1). I found Navigating Classroom Change to be especially interesting and insightful. As a teacher in training my biggest fear is falling into the lecture-listen format that I fell victim to throughout my own education. In my own experience I found this method alienated many students who needed the most help and bored the students who needed the least. Umbeck addressed this problem by implementing the Launch-Explore-Summarize method which I really gravitated towards. I think the idea of small purposefully chosen groups of students to think and talk through selected problems is especially interesting. I like the thought that students can gain understanding through meaningful discussions with their peers. Furthermore when the groups come back together with the rest of the class to summarize their findings it may spark a heated debates about which group is actually right. Whether the students come to the right solution or not is irrelevant, the fact that they are willing to debate about who is right and why shows that they are interested and engaged in the topic. Since everything I have read and heard about students success depends on their ability to be engaged in the topic, this method would appear to be a great approach. The second article by Martha Frank confirmed many of the thoughts I had prior to doing the actual reading. During my own career as a student from grade school all the way through my college courses I felt like my teachers as well as many of the students, if not most, held the same five beliefs that were talked about during the article. When forced into the paradigm students ability to think of creative and intelligent solutions is stifled. They are obligated to follow a set of rules they might not have even known they had been following simply because this process had been drilled into them since their mathematics education had begun. <span style="display: block; font-family: 'times new roman',serif; font-size: 12pt; text-align: left;">Overall these articles were a useful tool in figuring out how to run a classroom as a mathematics teacher. Whether these exact methods are used or whether they are tweaked to the individual educators style the main idea is sound and will be used again. Other than given further insight into a working classroom these articles also instilled a bit of fear into me. Will I be able to cultivate the atmosphere in my own classroom that I have been dreaming about for years? <span style="display: block; font-family: 'times new roman',serif; font-size: 12pt; text-align: left;">Christopher Cardon

<span style="display: block; font-family: 'times new roman',serif; font-size: 12pt; text-align: left;">With the reading of //Problem Solving and Mathematical Behavior//, I was surprised at the feelings of the studied middle school students outlined (2-2 through 2-6) because I shared a majority of those feelings. With the way mathematics is taught in the United States, it is easy to reach such conclusions because so much of the emphasis in classrooms //is// placed on getting the correct answers through some sort of operation or formula. This particular article is stating that our mathematical ability is limited to exercises rather than problems for this very reason. This leads me to question what should be taught in mathematics courses. <span style="display: block; font-family: 'times new roman',serif; font-size: 12pt; text-align: left;">Problem solving is very important because just teaching students a bunch of formulas does little good in the real world because the students will need to learn when they should apply their learning. However, I find having the formulas and equations is similar to putting tools in a toolbox. Also, doing “exercises” where I can be told that I was doing something correctly is what drew me to mathematics. In other courses, say English, my interpretation of a poem can be anything because it is my response and understanding to the poem, when I would prefer to have the correct interpretation. In math, if I correctly found the area of a triangle, that was correct regardless of what others thought was right. <span style="display: block; font-family: 'times new roman',serif; font-size: 12pt; text-align: left;">This brings me to the other reading, //Navigating Classroom Change//. It covers a math teacher who wanted mathematics to be a journey, a class utilizing “problem solving and making and evaluating conjectures” (1-2). While the author, Lindsay Umbeck, struggled, eventually she started making strides towards her goal. It is also interesting that she used the NCTM to help her reach that goal (3-3), showing the usefulness of such a resource. In fact, her method of teaching outlines the NCTM’s six principles of Equity, Curriculum, Teaching, etc. <span style="display: block; font-family: 'times new roman',serif; font-size: 12pt; text-align: left;">I am still hesitant to buy into such a philosophy as Umbeck, but I have to agree somewhat. When she discusses the sense of pride students feel for coming up with their own solutions (7-1), I think of my own experiences. In an education course, I was consistently pushed out of my comfort zone. This came from how we were required to participate in discussions on certain topics, but we were never given any answers. We were given readings, and then would discuss with our classmates. This student led discussion was awkward at first, but by the end of the course the class was more comfortable in truly exploring topics. The school work given also challenged us to fully explore topics and reflect on our own experiences. By the end of the course, I was mentally burnt-out, but the experience was extremely valuable. <span style="display: block; font-family: 'times new roman',serif; font-size: 12pt; text-align: left;">I still think about my experiences in mathematics and I have always enjoyed formulas and given methods of finding the “correct” solutions and that makes me hesitant to change. However, the merits of a problem oriented course are great, leading to benefits like a deeper understanding of true mathematics and greater problem solving which is more important in real world situations. The latter is much more important and maybe the first thing that needs to change is my own thinking.

<span style="display: block; font-family: 'times new roman',serif; font-size: 12pt; text-align: left;">-Marcus Edgette

Renegotiating the classroom in order to better engage students and allow them to participate in learning and practicing mathematics sounds like a great idea. It can however be very difficult to change the way math in the classroom has looked and has affected it’s students. As Lindsay Umbeck lays out the process of renegotiating classroom norms between student, teacher and their peers, she faces many difficulties (Umbeck, 2011). Students found the groups to be less structured and tasks more difficult when finding the “correct” answer wasn’t the end goal. It is very easy for a teacher to monitor group work and try and guide or nudge the students more toward a correct route.

The difficult thing can be to control your own advice or strategy and really listen to a student’s thinking and respect their thought pattern, allowing them to run with abstract ideas that may be more of a round-a-bout tactic, but will ultimately allow a student to engage with their own thinking in the mathematics.

Cooperative learning is an environment where students are expected to “form partnerships with one another - as they deepen their understanding of math…[paraphrased]”(Nebesniak and Heaton, 2010). Cooperative learning can be a scary implementation because it involves giving up control of the learning process. It shifts students and their peers the responsibility of encouraging and guiding one another, clarifying details, actively listening, and intentional involvement of all students. Students are to shown to have more confidence in the group setting. They are more likely to participate more quickly and diligently. They are more “…willing to try problems, learn from mistakes and help others.”(Nebesniak and Heaton, 2010). Allowing the students to take over and work on math together and around one another gives them a chance to tap in to their personal creativity and explore mathematical ideas, sometimes without even realizing it. They become more comfortable engaging in the math within the classroom and it gives them confidence to continue doing so outside.

-kyle d.