2.0+Reflection+on+Navigating+and+PS

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 Monday, Jan 16 before noon.

The article, “Problem Solving and Mathematical Beliefs”, addresses the question of how the predominant mathematic beliefs of students can be realigned by a change in our nation’s classrooms to emphasize the importance of problem solving.

To implement a change in beliefs, teachers must understand the current mathematical belief system of their students. The article details the five predominant beliefs (2-2 through 2-6) of a sampling of mathematically talented middle school students (1-4). Where did the students get these beliefs? The study noted that these beliefs were developed slowly throughout a student’s mathematical career in the classroom (2-4).

The beliefs were not a surprise to me. I asked my mathematically talented middle school son for his beliefs about math. He said math was “calculations”. When I pressed him further on the role of the teacher, he said “to help me solve the problem”. As I let him read the five beliefs in the article, he agreed with all of them. When I questioned him on utilizing mathematical problem solving techniques on real-life complex issues, he stated that “there wasn’t any time in class to do that sort of activity”.

The article further details five methods that teachers can implement to change their classroom culture to one that promotes problem solving skills (2-6 through 2-10).

As I read this article, I visited my own mathematical belief system and how it affects my current teaching of mathematics at WMU and KVCC. My belief system formed through my educational experience is similar to that of these students. There is comfort in teaching the way I was taught. For my students, there is comfort is learning the way they have always learned. I am evaluating this new learning paradigm and trying to implement changes in my classroom to push my students to be more independent thinkers and problem solvers. I must understand and implement my changing role as a teacher in today’s classroom.

The article, “Navigating Classroom Change”, details one instructor’s mission to change her classroom culture from one of teacher-centered to student-centered (2-2). The strategy was to structure an entire lesson around a complex and unfamiliar problem so that students learned new concepts as they developed solution strategies (2-2). The Launch-Explore-Summarize teaching model was implemented (2-5). The components are detailed (2-6 and 2-7, 3-1 through 3-5) in the article.

In noting her experience in changing her classroom culture, the author had to overcome the initial rejection of this method by her students (3-7 and 3-8) and her own insecurities about letting go of some control of the class (3-9). My experience as an instructor has been to make the students comfortable and offer assistance when requested. It is difficult for me to let the students struggle (they will be mad at you for awhile), but I have realized that most students will come to a reasonable solution if left to depend on themselves and their fellow students.

This methodology of problem solving is applicable to the real world work environment which many students will find themselves in. Working 14 years in industry myself, I was frequently thrown into new job responsibilities and needed to rely on my intuition, my knowledge/skill set, the resources available, and my coworkers to move forward. The boss didn’t always know everything and many times was too busy with his own responsibilities.

What one discovers as a teacher is that if you let your students struggle on their own, they will reach success. They will be engaged and proud of their accomplishments (4-1). Supporting and promoting those academic victories leads to more active engagement by the students and slowly changes the classroom culture (4-3). **//(Susan Copeland)//**

In order for students to learn better problem solving strategies, they have to change their beliefs about mathematics (1-1). Many students that I have seen get easily frustrated, then give up. This is not a good attitude to have when trying to solve a problem. The students that they interviewed believed that mathematics is computation, so it is all about the addition, subtraction, multiplication and division(2-2). The problem with this is when you look at it like that, then there is no reasoning, no struggle, and it means that there is no problem solving. These students did not think that non-routine problems, like proofs math writing assignments, were part of the normal curriculum and should be considered extra credit (2-3). This means that anything that they could not follow a step-by-step process that was using the basic computation skills should not be considered a routine problem.
 * //Reading Reflection #1//**

The students also felt that doing mathematics is either right or wrong (2-4). This attitude strongly hinders the problem solving we wish to teach students. We need to teach students that it is not the answer being right that matters, as much as knowing the right process to get to the right answer. This will make problem solving easier, because in many problems, there is more than one right answer, so they need to know the process to get to one of those answers. These students would probably think that problem solving is not mathematics, because the researchers say problem solving is what you do when you do not know what to do.

Teaching problem solving needs to start early, because middle school is too late for them to change their ways of thinking (3-4). We need to focus on students using strategies rather than step-by-step procedures, because strategies are useful in problem solving (3-5). Math teachers need to make problem solving the primary subject taught, not computation, because after computation if taught, problem solving does not seem like math.

The second article is about navigating classroom change. The issue with trying to teach mathematics, is many students view math as memorization (2-1). This makes it hard to teach them new ideas, new processes, or new ways of thinking. It is hard to maintain a good learning community and help students, while also following your goals and avoiding the rote memorization. The key to the interactive, involved, and student-centered classroom is teaching lessons that are challenging, allow for multiple solutions, and push students outside their comfort zone (3-2).

As teachers, we need to keep the combination of computation and problem solving together, and teach them both (3-3). This is important because it keeps students going in the direction we want them to. It is important for students to feel ownership in their tasks, because this will hopefully lead to them remembering things better (3-6). If they feel ownership, they are also much more likely to work harder on those subjects.

Changing the classroom is not an easy process, nor will it happen right away (3-7). It will take work, patience, and determination. An effective way to bring in new methods of teaching, is to use group work, so that students do not feel like they are venturing into new waters all by themselves. Groups allow students to toss out different ideas, and the teacher gets to see how different people interact with others. It also allows students to use their strengths without exposing their weaknesses too much.

Their are several things needed to change a classroom: recognizing a need for change, a commitment to change, a new vision for the teacher's practice, projecting yourself into the classroom, taking action, and reflecting on changes and visions (7-6). **Josh Kaylor**

I found the central argument of “Problem Solving and Mathematical Beliefs”, by Martha Frank, to be how building better problem solvers is rooted in first changing the beliefs students have about mathematics and how to help students build problem solving strategies through implicational techniques. To begin, Frank first reveals what her study of “mathematically talented middle school students” concluded of the beliefs these students had of mathematics (1-3): “mathematics is computational” (2-2), “mathematics problems should be quickly solvable in just a few steps” (2-3), “the goal of doing mathematics is to obtain the right answer” (2-4), “the role of mathematics students is to receive mathematical knowledge and to demonstrate that it has been received” (2-5), and lastly “the role of the mathematics teacher is to transmit mathematical knowledge and to verify that students have received this knowledge” (2-6).  I liked Frank’s argument on the different between a problem and an exercise because it really laid out the answer to why students hold the above beliefs. As discussed, problems cannot be solved easily and requires actual thinking (2-8) whereas an exercise should be solvable through procedures given to students (2-9). Because many students are familiar with exercises instead of problems, this is why they do not see mathematics the way they should. Frank argues that students who are not use to ‘problems’ could face the following when given one: refusal to try (2-10), over-simplifying the problem into an exercise (2-11), frustration from confusion leading to lack of confidence (3-1), or “real progress in a solution” (3-2). Changing how students view and partake in math cannot be envisioned as easy; Frank offers some implications in striving towards this goal. One of which is start problem solving early (3-6). As we go into Secondary teaching, we cannot have much control on what happens in early education, so this will be difficult for us but I wonder how much of the damage of not having adequate problem solving in early years can be unwound in later years. Frank also suggests focusing on solutions (3-8) which I especially agree with because the right answer without explanation could have come from the paper of the student sitting nearby—we want our students to have to ‘think’. One other major lesson from Frank I appreciated was encouraging group work (3-9); like Frank, I agree that students should learn to verbalize their ideas and not be so dependent on their teachers—after all, we only have them for one year.  After reading Frank’s article, I was left with questions and red flags about how creating this classroom actually happens and what struggles will occur. I was pleasantly surprised to find Lindsay Umbeck’s “Navigating Classroom Change” to answer many of these concerns. Umbeck described how she was caught up in a teacher-dependent classroom and saw the need for change (1-2). To guide her, she used Gaye Williams’ ideas that “tasks should allow for multiple entry points and solution paths” and also “students should be pushed to find a new procedural approach to a solution” (2-2). Umbeck also turned to “FigureThis! Math Challenges for Families” (a resource I am interested in exploring for real problems for students) (2-4). Umbeck followed the teaching model Launch-Explore-Summarize (2-5), which I found rather straight-forward although there were a couple ideas I found especially valuable: setting up “launch” by reviewing critical information (2-6) (something that would be difficult to not “give away” the answers and techniques to solve), explore involving “synergize insights” so students had to use each other’s ideas (2-7) something difficult because students may regret what other students think, and finally, summarizing which involves students reflecting on what other groups decided upon which also could yield students to only agree with what their group came up with (3-4).  As I mentioned, the first article left me with some red-flags and getting students to dive into problem solving was one of them. Umbeck experienced this (3-5) and suggests reminding students there is not just one right way to do the problem but to do their best as well as encouraging them to try will be helpful (3-1). She also discussed the issue I pondered of how students would lack the confidence to learn mathematics through problem solving which Umbeck credits “affirmative feedback about trying ideas” (3-4). Also, Umbeck settles the question posed in Franks article about what to do when students transform a problem into an exercise with asking further questions to students so they realize they are missing a bigger piece of the problem (3-5). One of the best ways a teacher can stay involved in the problem solving process seemed to be realizing his/her role during group work including: visiting groups regularly, participate in some discussion, ask questions which “clarify and refocus on productive pathways”, sharing excitement and not providing hints to indicate correctness” (2-3). By learning to let go of some control (2-7) and following these implications, I believe teachers will be surprised like Umbeck was by students’ problem solving potentials. -- //**Katie Pingle**//
 * From Theory to Practice: Frank and Umbeck's Solutions to Helping Students Become Problem Solvers**


 * __Helping Students Become Critical Thinkers__**

The article “Problem Solving and Mathematical Beliefs” addresses the fact that the beliefs and views of mathematics among middle school students now are not as they should be. The way the students think about mathematics and how it should be done is more about getting “right answers” (2-5) and solving an exercise as fast and efficient as possible, rather than fully understanding what the problem is asking, and critically thinking about the best way to solve it.

There was a study done in which five questions were asked to a group of four middle school students (2-2 through 2-6) to better understand how students thought about mathematics. It was discovered that they believed mathematics was just a series of memorized algorithms including addition, subtraction, multiplication, and division (2-2). This means the students didn’t look at a problem and study it, to understand what it was asking and what was needed to be done to solve it. They saw certain key things that pointed them to specific rules to use, and guidelines to follow. When the rules led them to an answer that was “wrong”, they got easily frustrated and gave up (2-4).

The article also states that students thought of mathematics as a sort of “package” that was obtained by listening to the teacher, doing homework exercises, and reading the textbook (2-5). Then, when tested on the material, the student “understands” it if they produce correct answers, and they don’t if they get it wrong.

In order to become better mathematics students, Martha Frank suggests that students need to change their beliefs about mathematics. Currently, they believe that if you don’t know what to do when approached with a problem, then you don’t understand it and you just plain don’t know it (2-7). However, for the students to become better mathematicians, they need to learn how to solve a problem when they don’t know what to do. Being good at math should mean that you are able to work out a solution when it is not initially clear what is to be done (3-6). The article outlines five implications in which teachers can follow to help the students work on their mathematics skills.

The first implication states that students should not be first encountering problems by the time they hit middle school, which they need to have seen difficult problems with unclear solutions before then (3-5). That is because by that time, the students have already got their beliefs set and embedded in their thought process, making it difficult to change. Another implication says to focus on solutions, and not answers (3-7). That way they aren’t only concerned about which rules to follow and whether they get the question right or wrong, they pay attention to //how// a problem should be solved. This would help them in the future, when facing a problem that they are unsure of the answer. Another implication suggests that students frequently work together in small groups (3-8). They can help each other reach solutions together, and it teaches them that the students can figure problems out without needing the teacher to tell them exactly what to do and if they are right or wrong.

The second article, titled “Navigating Classroom Change” discusses the way in which this particular teacher changed their classroom practice from teacher-centered to student-centered. She wanted to do this by introducing a new and unfamiliar problem to her students, giving them the necessary information needed (but not too much so that they can make sense of the problem and reach their own solution) then letting them figure it out for the most part (2-2).

The teacher did so by following a Launch-Explore-Summarize teaching model in the classroom (1-5). This was done so by giving the students the problem to work on in small groups, letting them explore the different solutions to the problem, then talking as a whole class and discovering how everyone solved it and figuring out which was the best way to do it.

It is understandable that the students had a difficult time with the new style of classroom learning that the teacher was trying to implement, but that was to be expected. She said at first that the students had a tough time figuring out exactly what was needed to be done, and what way to approach the problem was best (4-3). This was due to lack of confidence, and the fact that it required the students to use critical thinking, rather than go down a checklist of rules to find an answer to the problem.

After they got through the Explore phase of the teaching model, next was to Summarize with the entire class to find what others did and figure out the main idea of the problem (4-9). This part of the exercise is very important, because it allows the students to collaborate together to work towards a main goal. It lets them reach the big idea of the lesson together, so hopefully they can remember the concept better since they came up with it on their own.

The teacher also encouraged the students to elaborate on their solutions and explain how they reach answers better by sending value signals (5-1). This helps the students be forced to go over step by step how they came up with solutions, helping embed the process in their mind and actually allowing them to be able to explain it to other students who may be having more trouble (5-3).

Altering the practice of a classroom to become more student-centered is shown to be a difficult task, but also a very rewarding one. It allows the students to become better critical thinkers since they have to come up with solutions on their own and are forced to articulate how they reached their conclusions. The process not only allows them to better understand mathematics and how to think about it, but it helps provide a way of thinking for problems in the future they may come across. -**Shanna Thorn**

Reading Assignment #1

In the first reading assignment, //Problem Solving and Mathematical Beliefs//, the emphasis on problem solving has been put into two categories: successfully changing the curriculum (1-1) or successful implementation of instructional approaches (1-2).

The students’ view of what mathematics //should// be is quite skewed from how I view mathematics today. They believed that mathematics is computation memorization (2-2), which is surprisingly accurate. I viewed my mathematics when I was in middle school and part of high school as implementing a certain formula or algorithm whenever I recognized the exercise that needed it.

The students viewed that mathematics problems should be able to be solved in just a few steps (2-3). This may be true for simple addition, subtraction, multiplication, and division, but once students move into higher math classes where the problems become more comprehensive, the problems take longer to solve. Students also believe that mathematics is defined by being right or wrong (2-4). I believe that there is more to mathematics than being completely right or wrong because the process in which you achieve your answer, whether right or wrong, is more important than the answer itself. Students often feel that the problem is worthless and become discouraged unless they come out with the right answer (2-4).

Wheatley defined problem solving, as I would think of it, as: “Problem solving is what you do when you don’t know what to do” (2-7). Too often do students focus on doing the exercises, the questions that only require a quick “computation”, than the problems, the questions that require more thought and time to complete.

Implications were made that teachers should follow to better the minds of their students and mold them into better problem solvers. Teachers should implement problem solving early so that the student can grow and improve over time (3-6). Focusing on how to get to the right answer is more important than the answer itself (3-8). Mathematics should be a collaborative activity instead of students only listening to their teacher (3-9). Students need to become better in problem solving than know how to do computations and algorithms and the instruction of the teachers directs how the students learn and develop as learners (3-9).

The second article, “Navigating Classroom Change”, talks about how the classrooms should be focused more on the students rather than the students (1-2). Helping students work together and organizing your classroom into a student-centered classroom may be an effective way that students can engage in learning and interact with each other.

Umbeck talks about how the teacher can form groups for the students to work together and then hold class discussions about how each group went about with the problem given. By not giving the students much information or tools about how to go about the problem, the students must work together and come up with their own method to executing the problem. Also, Each group could ask questions and share their solutions with the rest of the class (3-4). The role of the teacher while the groups are still working independently would be to push the groups to look for the big ideas and to come up with a way to use their problem solving skills (3-4). This is known as the Launch-Explore-Summarize teaching model.

Elaborating on how the solutions were reached and discussing their methods, the students may gain insight on their first hand experiences on how to problem solve. Umbeck describes how a great tool the teacher can use is to give affirmative feedback to the students (3-4). Letting the students know that it is okay to feel uncomfortable when trying new ideas is a good way for students to step out and try new things. Knowing that change is needed and taking action on this need, letting your ideas becoming known in the classroom, and always interacting with the students are key to transforming your classroom into an effective classroom. **(Ryan Sherman)** I learned that math teachers should put an emphasis on problem solving instead of computational solving. I learned that students believe different things about math: math is computation, math problems should be quickly solvable in just a few steps, the goal of doing math is to obtain right answers, the role for the math student is to receive math knowledge and to demonstrate that it has been received, and the role of the math teacher is to transmit math knowledge and to verify that students have received this knowledge (33-2). Although students believe this is what math is, these articles say something else. These articles say that the point of math is to learn problem solving techniques instead of just computation. To get the student engaged in the math. I would like to try this when I become a teacher, although I would need some practice because I have had no experience with this. I really like how the teacher went from being at the front of the class teaching, to the observer while the students figured out problems on their own (90-2). One of the articles explains things we can do to get the students to become better problem solvers: start problem solving early, be sure your problems are problems not exercises, focus on solutions and not answers, students should frequently work together in small groups, and de-emphasize computation (34-6). It was cool in one article that even though the students were problem solving, they all had different ideas on how to get the solution. Overall, I learned that there are many different options when teaching math and some are better at getting certain desired results. **(Nicole Parry)**