2.3+Mathematical+Tasks

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**

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 * completeness and timeliness of the entries;
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Provide a one page summary of "Mathematical Tasks as a Framework for Reflection". Paste your reflection followed by your name. This is due Thursday, Feb 2 by 2:00am

The most difficult part of reflection and mathematical tasks is figuring out what to focus on (1-2). As a teacher, you can have the students working on high level demand tasks, or low level (2-1,2).

I thought it was cool how it showed the teacher Ron Castleman doing the lesson in different ways. The first time he taught the lesson, he eliminated the challenge because the students just wanted to know the answer, they didn't want to think it through (5-3). However, the second time he realized the students' never really paid attention to the diagram he gave them (5-4). This time he wanted to keep the task at a high level and have the students solve the problem themselves (5-5). He asked them questions to lead them to an answer. The second time Ron taught this, he was happy that the students figured out how to do the problem on their own.

I think it is cool how all the teachers got together afterward and described in detail what they seen and what they learned (6-2). The article gives a few ways to help teachers observe what they do. One way is another teacher observing you and you get feedback at the end of the lesson, or videotaping yourself to watch your teaching later (7-3, 8-4). **(Nicole Parry)**

From this article, what stuck out to me the most was that teachers need to constantly reflect on their own teaching. More specifically, Stein feels that the mathematical tasks that teachers assign should be the focus of this reflection (1-2). Although I feel that considering the tasks we give students is important, I do not think it overweighs other issues—interpersonal relationships, building student confidence, and making math relevant to students, I do like what Stein offers on how to assess the tasks we give out students.

Stein uses Doyle as a source in suggesting that the tasks we give students is the most significant factor teachers can control in helping students become knowledgeable mathematics students: Doyle states that tasks are the “basis for student learning” (2-1). There is a useful model in Stein’s article on how tasks affect learning: first, tasks appear in the curriculum, then they are set up by teachers, implemented by students, and finally students learn (3-3). I think that teachers should aim for tasks which allows students to learn in this way, but also think that teachers should promote learning which helps students create their own problems.

Through examples of real teachers, Stein explores how problems need to be high-level tasks that students must have to answer for themselves—teachers cannot give clues that too easily give answers away (4-3). In addition, task decline is linked to “classroom management, too little or too much time, and not holding students accountable”. Staying clear of problems with these situations will help students excel.

Most importantly, teachers must be critical of themselves and this can be aided by colleagues. To ensure high-level tasks are being implemented, teachers can rely on their coworkers to observe their classroom (7-2). Good observation involves first assessing whether the task is high-skills level or not and once this is assessed (7-3), and once this is determined, looking at how tasks are implemented (8-3). Personally, I think watching a video of myself would be beneficial (which Stein does recommend) because taking a moment to see my own weaknesses is something I can always do, even if there is not another teacher free to make comments. -- Katie Pingle

This article looked at mathematical tasks dealing with fractions, decimals, and percents. It looked at lower level tasks versus higher level tasks, and the difference between the two, is how much of a challenge there is for the students. It showed the experience of two different teachers and how their activities affected the class. These activities showed the difference between lower level and higher level tasks, and the experiences showed that tasks in the classroom form the basis for student's learning (2-2). A big focus of this article relates back to one of the NCTM standards, and that is connections. We need to teach with the intent of helping students make connections, which means we should be using the higher level tasks. A higher level task still uses procedures, but it also helps build connections between mathematical ideas (3-2). The two variations of higher level tasks they had were procedures with connections, and doing mathematics. The experiences in this article relate to orchestrating discussion, because they talked about the importance of guidance, but also not helping too much or too little. I learned that reflection is important in my own classroom, because you look at Ron Castleman or Theresa Bradford's classrooms, and when they first did the activities, they offered too much assistance and guided students down a path that made a higher level task become a lower level task. Ron's lesson shows us that as teachers, we need to guide students down the best path that leads to them making connections, compared to leading down the path that solves the problem the quickest. The best way for students to make connections and understand the concepts, is to let them come up with their own correct way to solve a problem, because this uses their own thoughts, so it sticks with them better. As the teachers, we can only help so much, and we need to remember that the more we tell them, the less they gain from the assignment or problem. Our colleagues are there to help and support us as teachers. With their help, we can understand different mistakes we make, how to fix them, and how they affect the classroom (7-2). This plays an important role in any subject, but colleague observation is important in Math, because it allows you to get feedback from colleagues who understand the math, and they can focus on the reaching style. The observer should give their observations first, and let the observed teacher comment on those along with ask questions, because we are always trying to improve how we do our job (8-2). -**Josh Kaylor**

The article, “Mathematical Tasks as a Framework for Reflection”, introduces a framework which helps teachers to reflect on their classroom experience based on mathematical tasks used during the classroom instruction (2-1). Reflecting on the classroom experience is a way to make teachers aware of how they teach (1-1). A task is defined as a segment of classroom activity that is devoted to the development of a particular mathematical idea (2-1). These tasks form the basis for students’ learning (2-2).

The framework distinguishes three phases through which tasks pass: as they appear in curricular/instructional materials; as they are set up or announced by teachers; and how they are implemented by the students (3-4). The nature of a task often changes as it passes from one phase to another (3-5). The goal is to provide mathematical tasks that require students to think conceptually and that stimulate students to make connections to everyday life. The framework can give teachers insight into the evolution of their lessons and to use this insight as a catalyst to reflect on their own instruction and to discuss instruction techniques with colleagues (4-1).

The case of Mrs. Bradford was very interesting in that she set up a mathematical task that would challenge her students, but it ended up overwhelming them since they had not encountered open-ended tasks before (4-4). The students’ reaction caused Mrs. Bradford to overcompensate and lead the students to the answer. I can understand this phenomenon as a teacher, because students are conditioned to think that mathematical problems should be solved quickly and utilizing one method. Students get very frustrated when they have to extend themselves beyond simple calculations. This may be an outcome of our fast-paced culture in general.

Teachers are in a difficult position of balancing guidance without giving students the solution or solution path (6-4). It may take an entire class period to get through one mathematical task. Students must tap into their pre-existing knowledge to help them along the path (5-8) and to be held accountable for the work involved to proceed down the solution path.

This balancing act is something that teachers will fine-tune over time with self-reflection and/or supportive critique from their colleagues. I think the idea of videotaping a lesson would help a teacher look at herself and be able to analyze the classroom experience in private. If an atmosphere of support is available in the workplace, teachers can share their videos and critique each other. This would allow brainstorming of best practices so that all teachers can set up a classroom in which tasks are set up and implemented at high levels of cognitive demand (8-5).

As I mentioned before, our fast-paced technological culture has molded a generation used to getting information very quickly and easily. This paradigm is brought into the classroom by the student and applied to all subjects, including mathematics. Situations in which to apply critical thinking skills are lacking more and more in our everyday lives. Critical thinking skills must be __exercised__ in order for them to be developed. Teachers can set up the classroom tasks for these skills to flourish. These are the skills our students need to succeed in the global economy. **//(Susan Copeland)//**

__ Mathematical Tasks as a Framework for Reflection __

In this article, it defines a task as “…a segment of classroom activity that is devoted to the development of a particular mathematical idea” (269-1). The author describes the mathematical tasks framework to be designed to help students learn in an understanding way, so it’s not just memorization (if it is followed through in the correct way, though). It begins with the tasks as they appear in the curricular/instruction manual, then the tasks as set up by the teacher, then as implemented by students, and lastly that leads to student learning.

The article describes an experience where the framework was attempted by a teacher, but not followed through in the necessary steps needed. The teacher presented the class with an open-ended task that involved designing game boards for a fundraiser. Right away, she noticed the students were overwhelmed with all of the decisions that they needed to make, so she stepped in and guided them in the direction she felt was helpful. When all of the game boards were completed, though, they all ended up looking similar. This is because the students didn’t have enough background in critical thinking so they were stuck and didn’t know what to do, but more importantly didn’t know what to do with the fact that they didn’t have an immediate solution, or way to figure one out.

Another story told in the article was of another teacher that attempted a critical thinking task, and also did not have the luck as the previous teacher. However, this one tried it with his class and video recorded it, and then watched the video with other colleagues and teachers in order to get feedback. The other teachers noticed how much he guided his students, and almost spelled it out for them so they didn’t have to think about the problem (272-4). All he had to do was suggest where they should start and the rest was just applying a procedure without any understanding. After getting the feedback, though, he attempted the same problem with a different class. This time he only asked questions to help guide their thinking, without giving away the solution or how to do it. The students came up with their own solutions and reasoning behind it, and he made them explain how they got their answers so that they understood their own logic.

After these experiences with both of the teachers classrooms, there was a meeting held for them and others to share their stories with colleagues. The second teacher explained that “…it was easy to get so tied up in what //you did// that you lost sight of what students were learning from the experience” (273-1). Teachers need to look at all different factors in their classroom, including allowing sufficient time and holding students accountable for high-level thinking. It also states in the article that the framework is not intended to be used rigidly (274-2). It is simply supposed to be a guideline, or a tool for reflection.

The main focus of this article was to bring attention to the things teachers should or should not do that often gets overlooked. It’s not necessarily what you’re teaching; it’s about how you present the information and learning to observe the students, then adjusting lesson plans according to observations. Asking questions that guide their thinking, then making them explain how they tackled a task and found a solution is vital in successful teaching. After all, what better way to figure out how students learn the best, than to listen to what they tell you? **-(Shanna Thorn)** “Mathematical Tasks as a Framework” brings about the issues students face when certain procedures are placed in front of them and what these procedures are really asking of the students. The article defines a //task// as “a segment of classroom activity that is devoted to the development of a particular mathematical idea” (2-1). Each task, depending on how it is setup to how the task is implemented factors in to how the thinking of every student is engaged and if connections are being made (2-3). The author moves on to talk about how the relations between fractions, decimals and percents require making connections to and from each (3-2). Demonstrating the fluid movement between the three is considered to be a Higher-Level Demand. The illustration of the Higher-Level Demands as depicted in Figure 1 show that the task sets the students up with having to draw or show a diagram in which they are to exercise what the task is asking. These diagrams, or grids that were really asked in the task, guide the students to really think about what is being represented and require reasoning to their approach of the task (3-4). Theresa Bradford’s task of the tape roll toss is an example in which there was a lack of prior knowledge from the students to know what to look for or to be able to answer what was being asked promptly with ease (4-5). The students were confused with what to look for and did not know what to choose from all of the possible combinations of board game designs (4-3). This is a great example of a high-level task but without the proper guidance or without the students having the “proper” knowledge to solve the task. Ron Castleman’s task that was presented to his class had the opposite effect of Theresa Bradford’s task. Although when the task was first assigned the students were restless and did not know how to do what was asked, after a little prompting they solved it easily without requiring any real thinking (5-3). The second time around Ron presented the students with different instructions that brought student’s thinking on their own naturally. Paying attention to the relationship of the rectangular grid, the students were able to see what each column and row represented and could then take it from there. When Ron, Theresa, and their colleagues were brought together, they discussed what had gone wrong and how their tasks could have been setup for proper student engagement in the tasks. After reviewing their misconceptions about why the students did not easily find the path to completing the tasks, each of them came to the conclusion that they would need to find ways to help the progression of student thinking without giving them the answers (6-5). Even though this would be difficult to grasp, each of them admitted that it was easier to give them a rubric; even though this would supply the students to monitor what they had done and needed to do (6-6). I enjoyed the section of the article that talked about teachers observing teachers because this allows teachers to reflect on their own teaching while other teachers offer feedback as to what had happened and how it could have happened. The teachers may discuss what the tasks was really asking, how the students reacted to the task, and if the task was really implemented as a high-level task (8-2). Although the videotape that a teacher may record of themselves may be helpful, I feel that having someone else watch the tape or just having them in the classroom may be more beneficial to better feedback.

Implementing a high-level task effectively will offer students with further cognitive learning that is of the highest level (8-6). The execution of a high-level task must be precisely organized in order for students to fully gather the benefits the task is intended to provide. **(Ryan Sherman)**