Carri+Carver


 * Run Away with the Spoon: How to get students to think independently and critically **


 * Abstract **


 * How can teachers help students become more capable of figuring things out for themselves and thinking critically? **


 * In this project, students were challenged to solve real-world problems on their own. Efforts were made to increase student confidence, independence and persistence through the growth mindset and reforms in assessment practices. **


 * About the Author **

Ms. Carri Carver is a mathematics teacher at Woodward Academy. During her eight years in the classroom, she has taught Geometry, Algebra 2, and Precalculus. Prior to teaching, Ms. Carver was the director of a non-profit organization, which operated free after-school programs in Title I schools in an impoverished area of South Carolina. Ms. Carver graduated magna cum laude with a B.S. in mathematics and a B.A. in Music from Furman University in 1998. She earned her M. Ed. in Mathematics Education from Georgia State University in 2009. Ms. Carver is passionate in her pursuit of ways to teach mathematics so that it is engaging and relevant to students.

About Me
I am a mother, teacher, partner, sister, daughter, friend, and colleague. I enjoy sci-fi/fantasy books, outdoor activities, running and exercise. I am devoted to my three children and to being the best teacher that I can be. I hope that my life's work will have a meaningful and positive impact on the lives of others.

Though I never planned to be an educator, since my first days in the classroom, I have found teaching to be both challenging and fulfilling. It truly is my calling. I am never bored, and I never wonder why I am doing what I am doing. I enjoy getting to know my students as individuals, and I am passionate about continually improving the teaching and learning that takes place in my classroom and at my school.

Takeaways / Big Ideas
I learned a tremendous amount about assessment while participating in the E.E. Ford Assessment Cohort. As part of the cohort experience, I read two books on assessment (see References: Chappuis and Marzano) and numerous articles. As I read about various assessment reforms, I experimented with them in my classes. The table below summarizes the areas of teaching and learning that were most affected by what I learned in the cohort. I have included resources as much as possible to illustrate what I did in my classroom.

//**A Summary of my Takeaways from the Assessment Cohort **// Are the students learning what I am teaching? ||
 * = **1. Assessing Student Learning: **
 * **Before ** || **After ** ||
 * Students completed homework and classwork that was not graded. I provided answers, but I did not know how individual students were performing. I assessed student learning informally by asking questions in class and by noticing the questions students were asking about homework. I often did not know until I graded assessments how well students had learned the material. || Checking in frequently with students to see what they have learned has become habitual for me. When students complete classwork and homework, I often ask them to tell me how many they got right out of the total. I record this though it is not a grade. I encourage students to track their progress and to try to improve each time.

Students use hand signals such as thumbs up and thumbs down to let me know how they are doing. I now know when my students are ready for assessments, and I am confident that they will do well. || What is graded? How is the final grade assigned? ||
 * **2. Assigning Grades:**
 * **Before ** || **<span style="font-family: 'Arial','sans-serif'; font-size: 16px;">After ** ||
 * <span style="font-family: 'Arial','sans-serif'; font-size: 16px;">Every assessment was graded and counted in a student’s overall grade. This included pop quizzes, homework quizzes and other assignments such as calculator activities. || <span style="font-family: 'Arial','sans-serif'; font-size: 16px;">Rather than using the threat of pop quizzes and homework quizzes to force students to learn, I focus on getting students ready for assessments. The students and I both track how they are doing as they prepare for the assessment.

<span style="font-family: 'Arial','sans-serif'; font-size: 16px;">In my AR project, I focused on independent critical thinking. I graded all of the work the students did, but I did not average the grades. Instead, I based the grade that went in the grade book on a student’s best work. This encouraged students to do their best on everything, but it did not punish them for failed attempts at critical thinking. See below for a complete description of the AR project. || <span style="display: block; font-family: Arial,Helvetica,sans-serif; font-size: 120%; text-align: center;">What do I want the students to learn? Do students know what they are supposed to know? ||
 * **<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">3. Learning Targets: **
 * **<span style="font-family: 'Arial','sans-serif'; font-size: 16px;">Before ** || **<span style="font-family: 'Arial','sans-serif'; font-size: 16px;">After ** ||
 * <span style="font-family: 'Arial','sans-serif'; font-size: 16px;">I thought that students would figure out what I wanted to them to be able to do from the problems I assigned. || <span style="font-family: 'Arial','sans-serif'; font-size: 16px;">Students need to know exactly what is expected of them. The learning targets need to be explicitly stated rather than implied. It will take time for me to fully implement this in my classes, but I was pleased with my experiments in this area. See the resource below for an example of how I communicated learning targets to my students on a practice quiz.

<span style="font-family: 'Arial','sans-serif'; font-size: 16px;">//Practice Quiz with Learning Targets:// || <span style="display: block; font-family: Arial,Helvetica,sans-serif; font-size: 120%; text-align: center;">Should students be allowed to redo assessments? ||
 * **<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">4. Redo Policy: **
 * **<span style="font-family: 'Arial','sans-serif'; font-size: 16px;">Before ** || **<span style="font-family: 'Arial','sans-serif'; font-size: 16px;">After ** ||
 * <span style="font-family: 'Arial','sans-serif'; font-size: 16px;">I never allowed students to redo assessments. Occasionally I allowed students to correct assessments for partial credit. || <span style="font-family: 'Arial','sans-serif'; font-size: 16px;">I allowed my Geometry students to revise proofs after I graded them. Since a proof question is an assessment of reasoning skills, it is not essential that the proof be done correctly the first time or in a limited amount of time. I want the students to be proficient at reasoning so repeated attempts are not problematic. In fact, I am encouraging the students to hone their skills. See my proof rubric below.

<span style="font-family: 'Arial','sans-serif'; font-size: 16px;">I do not advocate a blanket redo policy, but it can be used effectively in certain situations. || <span style="display: block; font-family: Arial,Helvetica,sans-serif; font-size: 120%; text-align: center;">How do I let the students know what they need to do to improve? ||
 * **<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">5. Feedback: **
 * **<span style="font-family: 'Arial','sans-serif'; font-size: 16px;">Before ** || **<span style="font-family: 'Arial','sans-serif'; font-size: 16px;">After ** ||
 * <span style="font-family: 'Arial','sans-serif'; font-size: 16px;">I expected students to determine how they were progressing by checking their own homework. During classwork, I went around to individual desks to make sure students were working. When I graded assessments, I just marked incorrect answers. || <span style="font-family: 'Arial','sans-serif'; font-size: 16px;">I regularly go to each student’s desk to see if things are going well. I put smiley faces on papers that have correct answers. I give guidance to those that need it. This has drastically improved student effort on ungraded classwork.

<span style="font-family: 'Arial','sans-serif'; font-size: 16px;">When students complete practice problems, I communicate what acceptable levels of performance are. For instance, if there are 10 questions, they need to get 8 or more correct in order to be ready for the next assessment.

<span style="font-family: 'Arial','sans-serif'; font-size: 16px;">As part of the proofs rubric for Geometry (see below), I incorporated a shorthand system for giving students specific feedback about a step in a proof. This enabled students to make corrections without me telling them the answers. I would like to expand this system in order to give students better feedback on all types of questions.

<span style="font-family: 'Arial','sans-serif'; font-size: 16px;">//Proofs Rubric with Feedback System://

<span style="font-family: 'Arial','sans-serif'; font-size: 16px;">I realize that students need training in self-assessment and goal setting. I need to train them to analyze their strengths and weaknesses. See the resource below for a self-assessment format that helped students analyze their performance on a quiz.

<span style="font-family: 'Arial','sans-serif'; font-size: 16px;">//Self-Assessment Form with Learning Targets:// || <span style="display: block; font-family: Arial,Helvetica,sans-serif; font-size: 120%; text-align: center;">Are all of the students engaged? Is everyone learning? || <span style="display: block; font-family: Arial,Helvetica,sans-serif; font-size: 120%; text-align: center;">How do I tailor my teaching to meet the needs of each student? ||
 * **<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">6. Classroom Discussion: **
 * **<span style="font-family: 'Arial','sans-serif'; font-size: 16px;">Before ** || **<span style="font-family: 'Arial','sans-serif'; font-size: 16px;">After ** ||
 * <span style="font-family: 'Arial','sans-serif'; font-size: 16px;">Only students that voluntarily answer questions were engaged in class discussions. When I asked an essential question about the lesson, only one or two students answered. I expected the others to learn the answer by listening to the discussion. || <span style="font-family: 'Arial','sans-serif'; font-size: 16px;">Quick Writes allow all of the students to answer the key questions in a unit. In a Quick Write, every student writes an answer to a discussion question. Students share answers with each other in pairs or in groups. Sometimes students turn in their Quick Writes so that I can check them for understanding. At other times, I read them as they are written in class. A Quick Write gives every student an opportunity to show understanding of a key concept. ||
 * **<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">7. Differentiation: **
 * **<span style="font-family: 'Arial','sans-serif'; font-size: 16px;">Before ** || **<span style="font-family: 'Arial','sans-serif'; font-size: 16px;">After ** ||
 * <span style="font-family: 'Arial','sans-serif'; font-size: 16px;">Prior to assessments, I reviewed every concept with the whole class. || <span style="font-family: 'Arial','sans-serif'; font-size: 16px;">In preparation for an assessment, I provide a menu of review options for students based on how they have performed on previous quizzes or practice problems. Students that struggled in a particular area are steered toward an appropriate activity. Often these activities present the material in a different way or utilize technology. For students who have mastered the concepts, I provide enrichment activities.

<span style="font-family: 'Arial','sans-serif'; font-size: 16px;">//Example of Differentiated Review Plan://

<span style="font-family: 'Arial','sans-serif'; font-size: 16px;">//Activity that uses technology and presents material in a different way:// || <span style="display: block; font-family: Arial,Helvetica,sans-serif; font-size: 120%; text-align: center;">Am I really assessing critical thinking and reasoning? Are students held accountable for doing critical thinking? ||
 * **<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">8. Critical Thinking: **
 * **<span style="font-family: 'Arial','sans-serif'; font-size: 16px;">Before ** || **<span style="font-family: 'Arial','sans-serif'; font-size: 16px;">After ** ||
 * <span style="font-family: 'Arial','sans-serif'; font-size: 16px;">I asked students to complete activities that required critical thinking on a regular basis. Sometimes I graded these activities in order to get students to complete them; however, the grades were often poor. I felt like I was not being fair to the students. On other occasions, I asked students to do the activities, but then I went over the answers. I did not assess whether the students were able to do the thinking on their own. When I asked reasoning questions on assessments, I had already gone over the answers in class many times. || <span style="font-family: 'Arial','sans-serif'; font-size: 16px;">I use exit tickets to hold students accountable for learning something from activities. Students must try to demonstrate what they have gotten out of the activity prior to leaving class. I am able to determine if the activity was successful.

<span style="font-family: 'Arial','sans-serif'; font-size: 16px;">//Graphing Logs Activity with Exit Ticket://

<span style="font-family: 'Arial','sans-serif'; font-size: 16px;">During an analysis of my Geometry assessments, I discovered that what appears to be critical thinking on my assessments really is not because I have already provided all of the answers in class. Reasoning is being reduced to regurgitation. My entire AR project is focused on assessing students on their critical thinking and reasoning skills in a fair way. ||

= My Action Research =

=AR Overview=

//**<span style="color: #000000; font-family: Arial,Helvetica,sans-serif; font-size: 120%;">Major FAIL in Precal **//

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">During the fall semester, I gave a take-home assessment to my 11th grade Precalculus students. The students had over a week to complete the assessment. One of the problems was a real-world problem about roller coasters that required use of a graphing calculator and some trial and error to find a suitable solution. I provided instruction on utilizing the graphing calculator; however, I decided not to work an example in class. I wanted the students to figure it out for themselves. I thought the problem was an excellent and reasonable application of what the students had learned in the unit.

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">Only two students out of fourteen completed all parts of the application problem. The overall grades on the take-home test were 1 A, 1 B, 1 C, 5 D’s, and 6 F’s due to performance on this one problem. In my journal at the time, I wrote:

//**<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">“Are these students really incapable of applying what they have learned to a non-routine problem? Why can’t they do ANYTHING by themselves? What am I going to do about this?” **//

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">I truly believe that ALL students can solve problems and think critically in mathematics. This is not something that only gifted students can do. I chose to work on this for my Action Research Project because I refuse to give up on these students, and I refuse to let them give up on themselves.

=AR Question=


 * <span style="color: #000000; font-family: Arial,Helvetica,sans-serif; font-size: 18.66px;">How can I help my Precalculus students become better at solving problems they have not seen before? At figuring things out for themselves? **

=AR Process=

//** Target Group **//

My AR project focused on my Precalculus with Algebra class during spring semester 2013. All sixteen of the students are in the eleventh grade. The level of the course is Enriched College Prep, which means it is considered an advanced course; however, it is the lowest level of Precalculus at Woodward Academy. The course covers Precalculus topics in three semesters rather than the more typical two. Students come into the course from Algebra 2. Three of the students were placed in the course because they excelled in College Prep Algebra 2; these students “moved up” to the Enriched Prep level. The remainder of the students did not excel in Algebra 2 Enriched Prep and were placed in Precalculus with Algebra because it moves at a slower pace than the other Precalculus course //<span style="font-family: Arial,Helvetica,sans-serif; font-size: 130%;">s. //

//**<span style="font-family: Arial,Helvetica,sans-serif; font-size: 130%;">About the students **// -All juniors -Enriched College Prep Class (i.e. advanced level) -7 boys, 9 girls -3 moved up to this course -13 had difficulty in advanced/enriched Algebra 2 -Grades for fall semester of Precalculus: 3 A’s, 8 B’s, 4 C’s -1 student new to class in January -8 White/Caucasian, 4 Black/African-American, 1 Jamaican-American, 1 Mexican-American, 1 South-Asian American, 1 Indian

//** Guiding Principles **//

The following beliefs guided me as I crafted a plan for helping my students become better at solving problems on their own and figuring things out for themselves:


 * 1) Students need practice figuring things out for themselves.
 * 2) Students will improve with practice.
 * 3) Students must be held accountable for figuring things out for themselves.
 * 4) Students need to have the freedom to fail.
 * 5) Students need guidance on how to be more successful problem solvers.

//** Important Considerations **//


 * // What exactly do I want the students to be able to do? //

I wanted my students to be able to apply concepts they had mastered to real-world problems without outside help. I required the students to engage in higher order thinking including application, analysis, and evaluation.


 * // How can I give the students practice opportunities while keeping the problems novel? //

I believed that my students needed practice in order to become better at solving problems on their own; however, I could not give them the same type of problem more than once. After the students became aware of how to apply a particular concept to a real-world problem, then the opportunity to “figure it out on their own” was gone. I decided to have a focus on solving real-world problems that spanned several units of Precalculus including exponential equations, logarithms, and sequences/series.


 * // How can I hold students accountable? //

I felt that my students had to be held accountable for trying to solve the real world problems on their own and for doing their best work each time. I thought that all of their attempts should be assessed so that the students and I would know how they were doing. I decided to collect and assign a score to each problem solving attempt. I also realized that I could not help the students figure out the problems if I wanted them to do it by themselves. I had to refuse to answer questions!


 * // How do I assign a grade? //

Though I felt strongly that each problem solving attempt should be collected and scored, I also felt that the students had to have the freedom to fail. Solving an authentic problem without assistance is challenging for anyone. I was asking the students to take a major academic risk by grading them on this. I decided to assign one overall quiz grade for all of a student’s problems solving attempts. I would assign the final grade by looking at all of a student’s scores on problem solving and deciding what his or her level of performance was by the end. This process is similar to what Marzano refers to in his book, __Formative Assessment and Standards Based Grading__, in which all of the formative scores are considered (but not averaged) in determining a student’s summative score or final grade (P81).


 * // How do I assess student performance? //

In order to assess student performance the way that I have described, I had to score each problem solving attempt using the same rubric and grading scale. I utilized the 4.0 grading scale as presented by Marzano (P48). This rubric is presented below in the table on the left. E <span style="color: #000000; font-family: Arial,Helvetica,sans-serif; font-size: 12pt; line-height: 115%;">ach time I assigned a problem solving task, I gave the students the rubric and defined which parts of the task were level 2.0, 3.0, and 4.0.The table on the right explains this process.

cess at 2.0, 3.0, and 4.0.
 * ** Score ** || ** Grade ** || ** Content ** ||
 * 4.0 || A+ || Independently achieves complete suc

// For this task: // || // For this task: // ||
 * 3.5 || A || Independently achieves complete success at 2.0 and 3.0 with partial success at 4.0. ||
 * 3.0 || B || Independently achieves complete success at 2.0 and 3.0.
 * 2.5 || B- || Independently achieves complete success at 2.0 and partial success at 3.0. ||
 * 2.0 || C || Independently achieves complete success at 2.0.

// For this task: // ||
 * 1.5 || C- || Independently achieves partial success at 2.0 with major errors at 3.0. ||
 * 1.0 || D || With help, complete success at 2.0 and some success at 3.0 ||
 * 0.5 || D- || With help, some success at 2.0 but none at 3.0. ||
 * 0.0 || F, 40% || Even with help, no success ||


 * //<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">How will I give feedback? //

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">The students needed to know how they were doing with the problem solving and how to improve. I provided the students with a tracking sheet where they could track their progress and analyze their performance. I asked them to analyze what they could do to improve on each task. I also encouraged the students to correct their work though the corrections were not assessed.


 * //<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">What are the characteristics of successful mathematical problem solvers? //

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">Through my research, I found that answers to this question are highly complex and not definitive. Entire books have been written on this subject, yet it is not clear why some individuals are better at problem solving than others. There is not a clear cut method for helping an individual become a successful mathematical problem solver. Inadvertently I had chosen to do my research on a challenging subject.

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">I did learn that there are certain factors that can affect a person’s ability to solve mathematical problems. These factors can be divided into cognitive and affective aspects (Boekarts, Seegers, and Vermeer 1995, P242). Cognitive factors include knowledge base (such as mathematical concepts, rules and processes) as well as executive control or metacognition (P242). Affective factors are based on a student’s beliefs and emotions. An affective factor may positively or negatively affect a person’s problem solving ability. Some of these affects are listed below: (242-243)

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">Expectation of success <span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">Persistence || <span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">Anger while working <span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">Expectation of failure <span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">Frustration ||
 * **<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">Positive affects ** || **<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">Negative affects ** ||
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">Confidence

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">I chose to investigate the affective factors that were influencing the problem solving abilities of my students. I had informally identified that confidence was a challenge for some of my students as was persistence. I hoped to increase the positive affects and decrease the negative affects by emphasizing the growth mindset as presented in Carol Dweck’s book Mindset (Avineri, Belledin, et. al., P71).

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">I did not focus on the mathematical knowledge base of my students because I had not identified their performance on traditional assessments as problematic. (See fall semester grades above.) Their difficulty seemed to lie specifically in applying concepts to novel situations. I also chose not to focus on mathematical problem solving processes known as heuristics because this had not been proven effective in past studies (Swing, Stoiber, and Peterson P413).

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">I found that another productive avenue for improving student performance in problem solving lies in executive control or metacognition (Boekarts, Seegers, and Vermeer 1995, P242). I hoped to address this by training and encouraging my students to engage in the “Seven Habits of Highly Successful Problem Solvers,” which I loosely adapted from Costa and Kallick’s “Describing 16 Habits of Mind.”

//**<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">Seven Habits of Highly Successful Problem Solvers **//


 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">1. Persistence **
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">If at first I did not succeed, I tried something else.


 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">2. Staying on task **
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">I stayed on task and focused for the entire time.


 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">3. Quality **
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">I did my best work. It is neat, organized, legible and clear.


 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">4. Communication **
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">I communicated clearly by showing my work and explaining what I was doing.


 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">5. Application **
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">I tried to apply concepts that I have been learning in Precalculus.


 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">6. Accuracy **
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">I checked my work for errors. I believe it is correct and accurate


 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">7. Using time wisely **
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">I worked efficiently. I did not waste time.

//** Action Plan: Jan 22, 2013 - March 28, 2013 **//
 * 1) <span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">Pre-Survey and Introduction to the Project
 * 2) <span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">Task 1: Growth and Decay
 * 3) <span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">Task 2: Coffee Cooling
 * 4) <span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">Task 3: Case of the Cooling Corpse
 * 5) <span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">Corrections to Tasks 2-3
 * 6) <span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">Final Task: Choice of 3: Rice on a Chessboard, Lillies on a Pond, or The Big Melt
 * 7) <span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">Post-Survey and Open-ended Feedback Survey

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">*Emphasis on figuring things out independently and the growth mindset permeating classes daily.

//**<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">Pre- & Post-Survey **//

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">To begin my AR project, I gave the students a pre-survey utilizing the Likert scale. Students were asked to Strongly Agree, Agree, Disagree, or Disagree Strongly with 30 statements. The statements were related to the following topics that I found relevant to my AR project:
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">Attitudes toward math
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">Positive Affects for Problem Solving
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">Negative Affects for Problem Solving
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">Independence
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">Growth Mindset

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">Through the implementation of AR project, I hoped that I would see improvement in these areas.

//**<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">Introduction to the Project **//

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">I introduced the students to the project after they took the survey. I explained the 4.0 grading scale and rubric. I tried to convey how the final grade would be based on a student’s performance on all of the problem solving tasks but would not be an average. This system of grading was foreign to the students. I later learned that many of the students experienced anxiety about the grade they would earn for the problem solving tasks. I also told the students that I would not be helping them complete the tasks. Throughout my conversations about the project, I tried to instill the importance of the growth mindset, which includes embracing challenge and believing that progress is possible (Dweck).

//**<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">The Problem Solving Tasks **//

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">Six, 50-minute periods were devoted to problem solving tasks over the course of two months. During this time, three units of Precalculus were completed: Exponential Equations, Logarithms, and Sequences and Series. The students had to work alone on each task (except the first task on which I allowed a partner), and I did not provide assistance.

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">After the first two tasks, I incorporated the “Seven Habits of Highly Successful Problem Solvers” in order to help the students with their work habits. Though most of the students were working for the entire class period, many had very little to show for their efforts. I could not believe how little some students accomplished in the allotted time. I was also displeased with the quality and organization of the work that was turned in. I hoped that the work habits rubric would help. I felt that most students were trying their best, and I was impressed with their investment in the project; however, a few of them needed guidance on how to work alone effectively.

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">After the third task, I allowed a day for the students to correct their graded work on the first three tasks. I told the students that the next task would be the last and that a grade would be assigned afterward. For the last task, I provided a choice of three problems. Though only one problem was required, ten of the students completed an extra problem in order to prove to me that they had grown in their problem solving abilities.

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">-Exponential modeling <span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">-Compound interest formula <span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">-Continuous interest formula <span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">-Using graphing calculators or logarithms to solve <span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">-Horizontal asymptotes || || <span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">-Exponential modeling <span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">-Exponential regression <span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">-Percent error <span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">-Using graphing calculators or logarithms to solve || || <span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">-Exponential modeling <span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">-Horizontal asymptotes <span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">-Logarithms || || Rice on a Chessboard || <span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">An artisan sells an Indian king a chessboard for the price of 1 grain of rice on the first square, 2 on the second and so on. How much does this cost? How does it compare to the world’s rice production in a year? || <span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">-Geometric sequences and series <span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">-Dimensional analysis <span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">-Logarithms || || <span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">Lillies on a Pond || <span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">The water lilies on a pond are doubling in size each day. When will the pond be smothered? How can the pond be saved? || <span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">-Geometric sequences <span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">-Dimensional analysis <span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">-Logarithms || <span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">See Final tasks above. || <span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">The Big Melt || <span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">The perennial ice cap may be shrinking due to global warming. Considering data from the past 30 years, can you model the shrinkage of the ice cap? || <span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">-Exponential modeling <span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">-Exponential regression <span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">-Logarithms <span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">-Horizontal asymptotes || <span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">See Final tasks above. ||
 * ** Task ** || ** Brief Description ** || ** Mathematics concepts ** || ** Files ** ||
 * Task 1: Growth/Decay || <span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">How can you use the compound interest formulas to model population growth and decay? Should the rate of population growth/decay be continuous? Why or why not? || <span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">-Exponential equations
 * Task 2: Coffee Cooling || <span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">Commercial coffee brewing pots keep coffee at such a high temperature that is unsafe to serve. This led to the famous McDonald’s lawsuit in which a woman was awarded $2.7 million. How long should restaurants let coffee cool before serving it? || <span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">-Exponential decay
 * Task 3: Case of the Cooling Corpse || <span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">Someone has murdered the coroner. Can you help Agent 008 find the time of death and solve the murder? || <span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">-Exponential equations
 * Final Task Choice:
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">Final Task Choice:
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">Final Task Choice:

//** Additional Resources **// || ||
 * ** Item ** || ** Description ** || ** File ** ||
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">Tracking Form || <span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">Students used this form to track their progress on the problem solving tasks. (Marzano, P81-89) || [[file:Record sheet for tasks.docx]]
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">Pre- and Post-Survey || <span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">This survey was given before and after the project. It was a google form that students completed online. || [[file:Precal with Algebra_ Anonymous Survey.pdf]] ||
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">Open-ended Feedback Survey || <span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">After the project was complete, students responded to these open-ended questions. This was also a google from that students completed online. || [[file:Precal Project Feedback survey.pdf]] ||
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">Seven Habits of Highly Successful Problem Solvers || <span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">Students used this rubric to self-assess their work habits while working on the problem solving tasks. || [[file:7 Work Habits.docx]]

AR Data Samples
//**<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">Student Performance **// <span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">The table below shows the summary statistics for each problem solving task. The slide show below tracks the performance of each individual student on each task. The bar graphs in the slide show also show each student’s average score for the tasks, and the score that each student received for a final grade. I determined the final grade by looking at each student’s best two scores. For an analysis of this data, see the Data Analysis section below. media type="custom" key="22858714" width="120" height="120"

//**<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">Student Attitudes **//

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">The slide show below compares the results of the pre- and post-survey. The survey was comprised of 30 questions about student attitudes toward math, positive affects for problem solving, negative affects for problem solving, independence, and the growth mindset. The slide show analyzes the change from pre-survey to post-survey for each question. For a summary of these results and an interpretation, see the Data Analysis section below. media type="custom" key="22858684" width="120" height="120"

//**<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">Student Feedback **//

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">After the problem solving tasks were completed, the students completed an open-ended survey that asked the following questions:
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">How did you feel while working on the above tasks?
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">What did you like or appreciate about working on these tasks?
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">What didn't you like about it?
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">What did you learn about yourself?
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">Do you have any additional comments or feedback? Please share!

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">In the table below, excerpts of select student responses are shared. To see all responses, open the pdf file. For an analysis of the student responses, see the Data Analysis section below.

//**<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">Excerpts of Student Responses to Open-Ended Survey **//


 * **<span style="color: #000000; font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">Question **

<span style="color: #000000; font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">How did you feel while working on the tasks? || **<span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">Excerpts of Select Responses **

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">- I enjoyed the working alone on the tasks. The subjects of them were not boring which made them more fun to complete.

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">- I didn't like it. I am good at math, but I don't do well on things on my own. …I am not good at and do not like critical thinking. I got frustrated quickly….

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">- At first, I was really frustrated but by the last task I felt a lot more sure of myself and I thought it was much easier. ||
 * <span style="color: #000000; font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">What did you like or appreciate about working on these tasks? || <span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">- I liked that these tasks forced us to think on our own.

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">- It kind of felt empowering to figure out precalculus for myself, even if I was also confused and frustrated at the same time. These tasks were not like problems we've done before, so you really need to be comfortable with your own stored knowledge. That's hard for me to do, but I really do think these tasks pushed me to another level of thinking…..

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">- I liked that they were all interesting (like the mystery one). It wasn't just boring numbers so it kept me engaged. ||
 * <span style="color: #000000; font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">What didn't you like about it? || <span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">-We got no help at all during the task…..

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">- Well, it was so incredibly infuriating at first. I honest to God felt stupid.

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">- That we have a quiz grade being based on us working on stuff that we haven't learned ||
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">What did you learn about yourself? || <span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">- I learned i am very capable of figuring things out for myself….

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">- I learned that I can be very independent and do not to be spoon fed new information by a teacher.

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">- I can do things without help

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">- I've realized that I can ultimately complete these more challenging problems without the help of a teacher if I really try to organize concepts in my head. ||
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">Do you have any additional comments or feedback? Please share! || <span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">- These assignments have actually helped me in my other classes because I can answer my own questions most of the time now!

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">- For a pre-cal class, I would have to say that these problems are actually really innovating methods for students to challenge themselves with. Most of the problems are real world issues which destroys the assumption that math courses like this are not practical. I enjoyed being a part of this project, even if I wanted to rip my hair out of my head at times. || <span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">To read all student responses, download this file:

AR Data Analysis
//**<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">Analysis of Student Performance **//

//**<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">Did my Precalculus students become better at independent problem solving? Did the AR project help my students become better at figuring things out for themselves? **//

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">The graphs to the right compare student grades in the fall on a problem solving task to the final grades students earned in the spring for completing the problem solving tasks in the AR project. I realize that this is a comparison of apples and oranges. The original problem in the fall was graded with a completely different rubric, and grades were out of 100 points. However, the contrast between the two graphs is powerful. Nearly 80% of my class earned a D or an F the first time I asked them to do independent problem solving. After the AR project, not one student earned a D or an F. All of the students were able to experience some level of success at independent problem solving. From this vantage point, I believe the project was successful.



//**<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">Did the students improve with practice? Is the improvement statistically significant? **//

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">The scatter plot to the right shows the formative scores on each of the five tasks. Linear regression suggests that there is a slight positive slope to the data, but the correlation is weak. This result is not statistically significant. Though all of the students earned a passing grade overall for the tasks, the data does not demonstrate that the class improved over time or from one task to the next.

//**<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">Did the students perform better on the final task(s)? **//

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">For the final task, I gave the students a choice of three problems. They were required to do one of the three. Ten students completed two of three problems in order to demonstrate their growth in independent problem solving. All but one student worked on either Final Task 4: Rice on a Chessboard or Final Task 5: Lillies on a Pond. Only one student completed Task 6 so it does not appear in this analysis. The student did earn the maximum score of 4.0 on Task 6: The Big Melt.

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">Comparing the first three tasks to the final two does reveal some statistically significant results. The beta test results show that students did improve from task 1 to 5, from 3 to 4, and from 3 to 5. These results are statistically significant. Here is some more evidence that the AR project helped the students to become better problem solvers.

//<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">**What were the results** ////<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;"> **of my grading reforms?** //

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">I chose to assign one final grade for a student's body of work on the problem solving tasks. In the end, most of the students had five formative scores for problem solving. I looked at the best two scores and averaged them; this became the final grade. I rounded up when the average was in between two scores on the rubric. If time had allowed, I would have liked to have had students whose grades were in between two levels on the rubric complete an additional task. Marzano suggests that when the appropriate grade to assign is unclear that the student should be asked to provide more evidence of learning.

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">The final grades resulting from this process are drastically different than what students would have earned if I had averaged all of their problem solving scores in the traditional manner. According to averaging, the highest score would have been a 2.75, which I would have rounded up to 3.0. By my method, several students earned a 3.5. When averaging, the median score was a 2.2, which would have rounded to a 2.0. The median via my methodology was a 3.0. This is a difference of a letter grade or 10 percentage points. This is a major difference.

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">I believe the methodology I used was fair to the students and in alignment with my goal of helping them to become better at problem solving. I wanted them to get better with practice and for their grades to reflect how far they had come. I also wanted them to embrace the challenge and the opportunity to grow. In the open-ended survey, many of the students expressed how much they had learned about themselves and how they had grown. I think this system of grading gave them the freedom to grow.



//**<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">How did student performance on problem solving tasks compare to performance on traditional assessments? **//

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">Seven of the student earned the same grade on the problem solving tasks as they had in the course during the same time period. Four students earned a higher grade on the problem solving tasks than in the course. I believe that these students were more engaged and invested in the AR project than they were in traditional classroom activities such as homework and note-taking. I believe this suggests that for less motivated students real-world problem solving can positively affect performance. Three students earned a lower grade on problem solving than in the course. Two of the three had A's in the course. It was challenging to earn an A on the problem solving tasks; however, several students did. I believe some students earn an A in my class due to a very strong work ethic and significant memorization of concepts and procedures. I applaud these students for their efforts, but I need to hold them accountable for critical thinking as well. An A in Precalculus should be an indication that a student that has engaged successfully in higher order thinking.

//**<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">Analysis of Changes in Student Attitudes **//

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">The bar graph below summarizes the changes in student answers from the pre-survey to the post-survey. There were improvements in all five categories; however, there were declines in nearly every category as well. The only area that experienced completely positive changes in student attitudes was independence. After the project, the students were more inclined to agree that it is important for them to figure things out on their own and that they are capable of doing this. There were net gains in answers that indicated a student had the growth mindset as well. Unfortunately, there were net losses in the other three categories. An unintended consequence of the AR project was that students reported more negative affects for problem solving and less positive affects. In fact, this result was the exact opposite of the result I had hoped for. The biggest change was in frustration levels. The students found the problem solving tasks very frustrating. Students also experienced more anxiety and worry due to being assessed on problem solving.

//**<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">Analysis of Student Feedback **//

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">I analyzed student responses to the open-ended survey by looking for themes. In the graphics below, the size of the rectangles or text is proportional to the number of times a theme appeared in student responses.

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">To see student responses by theme for each question, click the circle beside each question. The graphic will change. media type="custom" key="22836854" width="42" height="42"media type="custom" key="22837808"

AR Conclusions
<span style="font-family: 'Arial','sans-serif'; font-size: 16px;">Overall I believe the project was a success. My students did become more capable of independent problem solving even though their attitudes did not reflect this. I hope to expand the focus on independent problem solving and critical thinking so that it spans the entire year that I teach the students Precalculus. I think that I can implement similar projects in my other classes as well. I truly want my students to be able to reason mathematically and solve problems; this project has helped me to figure out how to hold students accountable and how to assess their work.

<span style="font-family: 'Arial','sans-serif'; font-size: 16px;">Though the project was successful, there are many opportunities for me to improve upon my process. In the discussion below, I outline the major areas of the project that I hope to improve along with ideas about how I might proceed.


 * **//<span style="font-family: 'Arial','sans-serif'; font-size: 16px;">Rubric //**

<span style="font-family: 'Arial','sans-serif'; font-size: 16px;">The rubric that I adapted from Marzano did not work as well as I had hoped. The premise of the 4.0 scale is that students cannot succeed at level 3.0 without first succeeding at level 2.0. This was not true for the type of work that my students were doing. On several occasions, students successfully completed level 3.0 but made errors at level 2.0. The rubric did not allow room for this possibility.

<span style="font-family: 'Arial','sans-serif'; font-size: 16px;">I still need a rubric that I can use to score all critical thinking and problem solving assignments. I like the idea of having separate scores for calculations, communication, reasoning and other aspects of the task. This would enable students to do well on some parts even if they made errors on other parts. This would also help students pinpoint which aspects they most need to work on.

<span style="font-family: 'Arial','sans-serif'; font-size: 16px;">After the first couple of tasks, I realized that the feedback I gave students as I graded each task was not often relevant to the next task. Since no two tasks were alike, the feedback on one task would rarely translate to the next one. A better rubric would also give students better feedback as to where they went astray.


 * **//<span style="font-family: 'Arial','sans-serif'; font-size: 16px;">Mathematical knowledge and errors //**

<span style="font-family: 'Arial','sans-serif'; font-size: 16px;">Issues with prior mathematical knowledge did interfere with the ability of some of my students to successfully complete the tasks. At times, careless errors prevented students from arriving at reasonable solutions. I hope that a better rubric will help minimize the impact errors have on a student’s overall performance. I have also considered checking student work at critical junctures to prevent the entire task from going awry. Students could also be trained to use technology to double check their work. I used Excel spreadsheets to help me when I graded some of the tasks. The students could learn this skill. Finally, a system for providing predetermined hints to students who are struggling might help students make more progress.


 * **//<span style="font-family: 'Arial','sans-serif'; font-size: 16px;">Difficulty level of tasks //**

<span style="font-family: 'Arial','sans-serif'; font-size: 16px;">A major weakness of my AR project was my inability to ensure that the problem solving tasks were equally difficult. I knew when I planned the project that this would be a challenge. I endeavored to craft tasks that made similar intellectual and mathematical demands on the students, but I cannot prove that they did. Unfortunately, one explanation for the positive results of the AR project is that the last task might have been the easiest. For the purposes of research, this issue is important; however, from the standpoint of helping my students become better at critical thinking, this is not a major concern. I want my students to engage in critical thinking; it is okay if some tasks are easier than others!


 * **//<span style="font-family: 'Arial','sans-serif'; font-size: 16px;">Attendance //**

<span style="font-family: 'Arial','sans-serif'; font-size: 16px;">Student attendance presented a challenge during the AR project. Two students still have an incomplete for the problem solving grade. One student was only present for one of the six days that were devoted to the tasks. Another student was absent so often that when she did attend she did not have the prerequisite skills to complete the tasks. As I move forward with making problem solving tasks a permanent fixture in my classes, I will need to have a method for dealing with student absences. A student cannot be excused from critical thinking because he or she was absent.


 * **//<span style="font-family: 'Arial','sans-serif'; font-size: 16px;">Growth mindset //**

<span style="color: #000000; font-family: 'Arial','sans-serif'; font-size: 16px;">From the pre- and post-surveys, I learned that students think they have the growth mindset. They know what the growth mindset is, and they can answer the questions in such a way as to appear to have the growth mindset. However, the answers to the open-ended survey revealed that some of the students really have the fixed mindset. They think they know what they are good at and what they are not good at. They are not very open to change nor do they really like challenges. The actions and reactions of these students also convey the fixed mindset.

<span style="color: #000000; font-family: 'Arial','sans-serif'; font-size: 16px;">I still believe that students will be more successful if they have more of a growth mindset. I need to find a better way to assess student mindset, and I need more strategies for changing student mindset. I did find that student answers to open-ended questions are more revealing that the Likert scale. I could ask students to analyze their own statements to see what mindset they reveal.


 * **//<span style="color: #000000; font-family: 'Arial','sans-serif'; font-size: 16px;">Executive control and metacognition //**

<span style="color: #000000; font-family: 'Arial','sans-serif'; font-size: 16px;">My efforts to change the affective factors related to problem solving were not successful. This is the one area of the project that was an utter failure. At this point, I do not see how to reverse this trend. I regret that my students experienced more frustration due to the project, but I think it was worth it and so do they! They need to be challenged to do critical thinking. I am considering strategies to help students that are struggling including hints and a review at critical junctures. I also hope that I can reduce student anxiety about being graded by better explaining the grading process and by using the process more frequently.

<span style="color: #000000; font-family: 'Arial','sans-serif'; font-size: 16px;">As I continue this work, I intend to focus more on teaching students executive control and metacognition strategies. I began this work with the work habits rubric, which I called “The Seven Habits of Highly Successful Problem Solvers.” I believe this is a productive avenue for me to pursue in my research, and I believe that this would greatly benefit my students.

Lit Review & Resources
<span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">Avineri, Tamar, Christine Belledin, Julie Graves, Richard Noble, Maria Hernandez, Donita Robinson, & Dan Teague. (2011) Issues of Equity for Advanced Students. In M. Strutchens & J. Reed (Eds.) Focus in High School Mathematics: Fostering reasoning and Sense Making for All Students, (pp. 65-83). Reston, VA: The National Council of Teachers of Mathematics.

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">Boekaerts, M., Seegers, G., & Vermeer, H. (1995). Solving math problems: Where and why does the solution process go astray? Educational Studies in Mathematics, 28(3), 241-262.

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">Chappuis, Jan, Rick Stiggins, Steve Chappuis, & Judith Arter. (2012) Classroom Assessment for Student Learning: Doing It Right-Using It Well. Upper Saddle River, NJ: Pearson.

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">Costa, Arthur, & Kallick, Bena. Describing Habits of Mind. Retrieved February 2, 2013 from http://www.instituteforhabitsofmind.com/resources/pdf/16HOM.pdf.

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">Dweck, Carol. (2008) Mindset. New York: Ballentine Books.

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 12pt;">Grattoni, Christopher. (2007) Spatial Skills and Mathematical Problem Solving Ability in High School Students. Retrieved January 7, 2013 from http://www.sesp.northwestern.edu/docs/masters/ 1469752174482341f7902e2.pdf.

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Reflections
<span style="font-family: Arial,Helvetica,sans-serif; font-size: 120%;">The E.E. Ford Assessment was a transformational experience for me. I relished the opportunity to learn about assessment and to reform my classroom practices. The biggest challenge for me now is to try to integrate all that I have learned into what I do each day. I have learned that there is no magic formula for successful teaching and learning. Instead, teachers, parents and students have to work together to figure out what works.