14 thoughts on “Plans for Improvementing Mathematical Thinking”

1. Plan for improving mathematical thinking

   I can’t help but start by thinking about the Dan Meyer video that we watched in class and a few of his other videos that I have watched since. I think he is spot on with his analysis of what needs to change in order to improve mathematical thinking and mathematical problem solving skills amongst our students. Students in this generation, with the technology advancements, and the information at their fingertips, are losing the ability to think deeply about much of anything. This of course includes mathematics. It is not that our students don’t know how to problem solve, because they are great at resourcing, which is a very important problem solving skill, but when they exhaust their resources, many don’t have the ability to think independently.
   In my classroom, not much is different. In fact, one might be able to argue that it is even worse. I teach at West Valley High School in Yakima. This school is the most affluent public high school in Yakima and with that I have noticed that a lot of our students come in with the attitudes of entitlement and that everything will be given to them. This doesn’t help the cause of improving mathematical thinking, because it encourages the students belief that if I don’t understand something, then somebody else will take care of it for me.
   There is much that needs to be changed in student performance in this area, however as in anything, we can’t expect immediate drastic change. Most change occurs over many, many small steps. So my goal, following Dan Meyer’s lead, is to make small intentional changes to problems that force students to really think about the problem that is assigned rather than just grab the information and plug it into the formula that will spit out a meaningless answer. Students need to be challenged to think about what information is needed, and to develop a plan on how to solve the problem. Then go and gather the necessary information to solve the problem.
   There are a few ways that I plan to measure the change in student performance. One is just by listening to their comments. Since this is going to require more thought, they are more than likely start by complaining. Another is to listen for the questions that they are asking. I suspect that questions should move from, which formula should I use? to, does “this” or “that” matter? Also, over time (repeated practice) I should be able to see individual students progress on their level of solving problems like these. They might at the beginning struggle with determining the important information, but later progress to a level where now they struggle with determining the solution.
   In conclusion, we have a significant problem in math education. Our students are struggling to think mathematically, and we as educators are struggling to provide an avenue for students to improve this skill. We, as teachers, need to respond, and one way that we can do that is by stripping away much of the given information and really force students to think about the problems and to devise a plan to solve them.

2. The above comment is from Aaron

3. My setting is in a high school Geometry class. My geometry class has a variety of different students from grades 9-11. These students from 9th grade are in an advanced track because Geometry is traditionally a class for students in 10th grade.

   It is often the case that students are either more Algebra or Geometry minded. Either the case, it is beneficial for students to see the connection between the two, so that transitioning back into an Algebra course after completing Geometry seems more familiar and less foreign. In my experience throughout an Algebra II/Trigonometry class, much of the year was devoted to relearning concepts that should have been taught in an Algebra I class. If these ideas were reinforced in a Geometry course, there would be a lot less wasted time. Using the rule of four will incorporate more Algebra skills in a Geometry setting than what is normally experienced. Also, incorporating the rule of four in projects and assessments will increase the rigor in a mathematics course.

   To make the shift from standard exercises with a step by step guide to solving the problem to making connections between visual, numerical, symbolic, and verbal representations of the problem, the change in student performance is substantial. We are actually asking students to think! Instead of working on exercises after taking notes, students will be working in depth on various projects and explaining mathematical ideas using the Rule of Four.

   To make this transition, I suggest teachers implement more projects and class activities rather than the transfer of information that happens in a traditional teacher-led class. In this setting, students will be more responsible for their learning and in turn will be more motivated to learn. Using the rule of four in these projects will help students understand the material more deeply and allow them to represent math in four different ways. I think using the rule of four to teach students is a great idea because it incorporates more than just one or two ways to view the problem. Using multiple representations can be very beneficial for students because not everyone learns the same way. When teachers assign projects, they should encourage their students to explain concepts using the rule of four to ensure maximum comprehension. If students can use the rule of four to explain the material, it is a good indicator of their knowledge.

   Student performance will be measured based on classroom activities and projects. Teachers can phrase questions on classroom activities, projects and tests so that student responses must follow the rule of four. Specifically stating that the response should include visual, verbal (written), numerical, and symbolic representations will guide students.

4. The classroom setting for this assignment a 7th grade Math/Science class that is taught in a daily 120 minute block. The math curriculum being used is the Connected Mathematics Project (CMP).
   The need for the educational change considered in this response comes from the 2008 Final Report of the National Mathematics Advisory Panel published by the U.S. Department of Education. Among the “main findings and recommendations” mentioned in the executive summary, the report states that proficiency with fractions “seems to be severely underdeveloped” (pg. xvii). Furthermore, they suggest that the areas of whole numbers, fractions, and particular aspects of geometry and measurement comprise the “critical foundations” of algebra. In addition to these observations, the Panel recommends on pg. 11 of their report that “use should be made of what is clearly known from rigorous research about how children learn, especially by recognizing… the mutually reinforcing benefits of conceptual understanding, procedural fluency, and automatic [italics added] (i.e. quick and effortless) recall of facts…”.
   As a result of the findings above, there is a need for students to develop proficiency in the areas of whole numbers, fractions, and problem solving. The report defines proficiency to mean that “students should understand key concepts, achieve automaticity [italics added] as appropriate (e.g., with addition and related subtraction facts),…and use these competencies to solve problems” (pg. xvii). Consequently, I would like to focus on the area of automaticity of basic math facts (addition, subtraction, multiplication, and division) when using fractions.
   The first instructional change that I plan to implement is to disallow students from using calculators and, in response, require them to develop automaticity in basic math operations of whole numbers and fractions. The 2008 Report also states that calculators have limited or no impact on calculation skills, problem solving, or conceptual development. Furthermore, the Panel cautions “that to the degree that calculators impede the development of automacity [italics added], fluency in computation will be adversely affected” (pg. xxiv). Secondly, I plan to use a daily math facts practice sessions following the method of Donald B. Crawford, Ph.d. His method essentially exposes students to no more than 4 math facts per day and specifically aims for automaticity through timed practice and quizzes. His research suggests that automaticity is effectively developed through brief exposure to a few topics under time constraints. Subsequent lessons create a cycle of learning new facts in conjunction with reviewing prior facts so that students are continually reviewing and improving their automaticity.
   Measuring student performance will be achieved in the following way. Initially, a pre-test will be given to the students to assess their overall readiness for 7th grade. After the cycle of learning/review math facts begins, daily scores from quizzes will be kept to monitor who is developing automaticity and who needs individualized support.

5. I plan to improve the mathematical thinking of my students in my classroom, which is in a portable building about a block from the main high school building, through problem solving. I teach 2 periods of algebra I, 2 periods of geometry, and 1 period of pre-algebra. There are 2 computers in my classroom and a SmartTM Board on which I present my daily lessons.
   I want to engage my students, motivate them to want to learn mathematics and better prepare them for the End-of-Course Exam. I would like to incorporate problem solving into nearly every lesson. Holt McDougal is the curriculum we use for algebra and geometry at Prosser High School. It has a good problem solving handbook in the textbook and problem solving worksheets for every lesson found in its accompanying “Chapter Resources” workbook. The example problems and the homework problems of these Holt resources need only a little tweaking to make them open-ended. By solving real-world, relevant problems, students will be engaged and see a reason for learning the mathematics of the day.
   The two most noticeable changes in student performance will be more group-work by the students rather than individual seatwork and a louder (which will be a big change for me), more active classroom where students will be discovering, sharing, discussing, debating, and enjoying the mathematics.
   I plan to include problem solving in my daily lessons. I plan on beginning each lesson with a motivating, relevant problem that the students need to solve to introduce the mathematical concept at the center of the daily lesson. I’m hoping this will also keep them engaged in the mathematics and help them find reason and purpose in learning how to use mathematics to solve problems. I also want to include a few interesting and relevant word problems to be solved on the daily homework assignment.
   I plan on measuring this change in student performance through end-of-unit assessments. I will include open-ended word problems on assessments. Last year, the majority of my students left the word problems blank on the assessments. I’m hoping that receiving daily practice with solving them in groups work will give my students the confidence to try to solve them on their own. An increase in assessment scores, as well as fewer word problems left blank, will assure me that I am on the right track in helping my students improve their problem solving abilities.
   From: Suzanne Colgren

6. The classroom setting for this instructional issue is high school Algebra with students of grades 9-10. The need for educational change is that, as a whole, students in the United States are not performing well when it comes to solving problems that require them to make connections. This needs to change enough to where students are able to communicate that they have a deep enough mathematical understanding to not only do mathematical procedures but also perform when it comes to problem solving.

   As a prospect math teacher I think this is important to make the material relevant to all students. Math is a subject that students commonly claim to dislike. Students also usually come up with the common a question, asking when they will possibly use the content they are covering. If a teacher is able to make content relevant, students will have a use for it and be more likely to work harder at it because they will be interested. It is impossible to make this content relevant and interesting to all students if you do not know them. If it is not relevant to all then it is not really equitable.

   To make material more relevant and stimulating I plan to take regular, state mandated curriculum and made an interesting lessons plan out of it. With creativity and extra effort I think mathematics can be made much more interesting than procedural problems in books. Many teachers may think that in order to whip through the state curriculum they need to fly through class and stick to what is in the book. I plan to use the book as a tool in the future but also build on what the books provide and produce active lessons to do with the students as well.

   When delivering content to my classroom I will make sure to engage them. There are many research based teaching practices that can be used that are shown to keep interest in the classroom. These include allowing for group work, discovery learning, and active participation. It will take extra creativity on my part to establish interesting lessons for my students so they want to pay attention but it is vital that I do. Once I have formed interesting lessons I will deliver them in a clear, concise manner so students know exactly what they do. Sometimes when teachers are over wordy with instruction it is confusing, and confusion leads to frustration. People are usually not motivated when frustrated. I am not saying that every single lesson will always be super fun for every student. Although it will be my goal to have this be true. I also think that even if a student is not interested sometimes, if you can keep them active and excited enough to look forward to the next good lesson, then you are doing your job as a teacher.

   Another thing I plan on using when delivering content in my classroom is the rule of four. I will explain the content using symbols, pictures (graphs), numbers, and words. This will not only help the students see the material in different ways several times but also benefit different types of learners. Students will also need to explain their mathematics back to me using the rule of four. I believe it will increase their mathematical thinking because they will have to think more in depth to be able to do this. I will measure change in student performance based on this rule of four. If they can effectively use it, they have probably gained a deeper mathematical understanding. I would also evaluate and take the scores they got on standardized tests into consideration.

7. My setting is a Geometry class with about 32 students. These students are from a low income community and have a wide range of math skills. The students at this high school do not want to take geometry classes; they fear the class and they take it only because is a requirement for graduation. The school’s test scores in math have been low for the past ye. The last two years the scores decrease instead of getting better. My plan is to use SBG system to improve the performance of the students. The Standard Base Grading is a way to measure what the students know and be able to do. The data collected in quizzes, test, and EOC will help me improve my lesson plans to make them more effective.

8. Improving the Mathematical Thinking of Students

   Algebra 1 is a very challenging course to teach in my district, as I am sure is the case in many other districts. Currently, it is the lowest level class that we offer at the high school, other than for those that qualify for special education. Being that it is the lowest level offered, students who do not pass the class will take it again the next year, and will continue to take it until they pass. Because of this, there tends to be a larger number of lower achieving students in Algebra 1, who do not possess strong mathematical thinking skills, than at any other level of math.
   Algebra 1 is at the core of every branch of mathematics. Therefore, it is important that students have a strong Algebra 1 base and an even stronger ability to think mathematically. The curriculum that my district currently uses emphasizes memorization of skills for specific problems; however, it does not put emphasis on problem solving. It is my opinion, that mathematical thinking is the basis of all problem solving. If I can encourage problem solving, I will be encouraging mathematical thinking.
   An instructional change that I feel would be beneficial for my Algebra 1 classes would be to implement more opportunities for problem solving. The chance that a student is going to encounter a situation in his adult life where he needs to factor a quadratic is pretty limited. However, throughout their lives, my students are going to be faced with many situations where they will have to pull from all areas of their knowledge in order to solve a problem. They may even have to research before they are able to solve the problem.
   Problem Solving is a lifelong skill. In order to emphasize this skill and expose students to it more often, my idea is to use one class period a week to work on an activity that involves problem solving. In order to make it relevant to what we have been working on in class, I incorporate those current concepts and skills into the activity. This plan has two benefits. One benefit is students will be able to make a connection between the concepts from class and a real life situation where those concepts can be useful in solving a problem. The second benefit is that students are using their problem solving skills on a more consistent basis. The more skills are practiced, the more natural they become. If students can find the value in problem solving it will likely become a skill that they use more often, which encourages mathematical thinking.
   To measure the change in mathematical thinking, I will observe the discussions that are taking place as students try to work through the problems each week. Most likely in the beginning they will be a little apprehensive. I hope through repeating the process weekly, they will become more comfortable and open to discussion. Through my observations, I will be able to determine whether this is the case or not. Another method I can use to measure performance is the end product. I will be able to see what methods students are using to solve the problems, what issues they encountered along the way, and how they overcame those issues. Both of these methods will help me to measure the change in performance when it comes to mathematical thinking. In the long run, I hope that students will become more efficient and more confident in thinking mathematically.

9. It has become evident that I feel more comfortable teaching in much the same manner in which I was taught. Perhaps it is because it was the teaching method I responded to, or perhaps it just happened to be the teaching method of teachers I admired. Regardless of the reason, I now find it is affecting my teaching.

   Teaching high risk students is a challenge. One major challenge is their lack of foundational skills, especially in mathematics. Students have an alarming low understanding of concepts, like multiplication, that I learned in 3rd grade. I can’t say that my method of remediation, memorizing multiplication to 12 by 12 and then the traditional algorithm of multiplication for larger numbers, was very helpful. Which is sad considering the importance of multiplication for later concepts of division, fractions, decimals, percents, ratios, fractions, proportions, etc.

   I was taught that if I had mastery of my multiplication through 12 by 12, I would be ok in life. I believe it would be more advantageous to have mastery of the number sense and multiplication of 100 by 100. I believe that by using simple mental math tricks, students will be able to achieve this task.

   Instead of teaching the old algorithmic right-to-left multiplication I will teach students to memorize their multiplication tables 5 by 5. This will provide a nice foundation for the following finger multiplication and mental math tricks or what we in the math field call the distributive property. For the rest of their multiplication to 10 by 10, I will teach them finger multiplication. After that, it will be a combination of the distributive property, left-to-right, criss-cross method, and factoring method.
   Student performance will be measured with a pretest and posttest over a three week period and again at quarter and semester. Problems will be random, created from a worksheet creator like Kuta pre-algebra software. Pretest and posttest will be identical for each student, however additional test will be random.
   Since I will be introducing a variety of methods to teach multiplication, it would be wise for me to document my perceived effectiveness in a daily reflection. My approach, delivery, and attitude of each method could affect its effectiveness.

10. I work at a community college. The class is MPC 090, which is a remedial class that uses a prealgebra book. I don’t have state standards, the standards I follow are the 40 questions types that the students are required to answer on the final for the class. This is a night class (7-9:30 TTH) with 20 to 30 students. They vary in ages from 18 and older. They also vary greatly in their ability to do basic math without a calculator. There is a need for change because if they are not given specific examples of how to do each problem they cannot solve them. I plan on implementing change in the way I deliver a lesson. I am going to teach lessons as if I were setting down to work out a set of problems myself, and think aloud the thought process that I go through to solve the problems. I plan on measuring the change by monitoring their questions in class and how they solve the practice problems as well as comparing their quiz and tests scores with previous classes’ scores.

11. Through the use of a tool given to my department in the last couple of years, I plan on improving the mathematical thinking of my students via adherence and incorporation of “Mathematical Habits of Mind” and “Habits of Interaction.”

    Examples of these are:

    (1) Private Think Time

    (2) Making Connections

    These are examples of types of behaviors and patterns of thought that I want my students to develop. As math teachers, we do these automatically, and I want my students be able to do them just as well.

12. Go to http://gplb.edublogs.org/2011/07/17/plans-for-improving-mathematical-thinking/ for a more detailed plan.

13. Math 524 Course Syllabus
    Method and Materials for Secondary Mathematics

    Ax2 + Bx +C in Context
    Outcomes
    Students can visualize the whole context of the Ax2 + Bx +C. They can move among the different representations, and they can use winplot to graphic. Students already know how to use WinPlot.

    Materials
    1) Activity 1. Lecture way to factor, even square completion and complex numbers. Quiz and Test. Self evaluation rubric

    Procedure

    This class will be in the computer lab because students will need the winplot software in order to do the graphics they need.

    Time Activities Purpose
    30 min 1. Professor handouts to students Activity 1
    2. Students work with a partner. Students make generalizations
    20 min Students share their answers Students double checks their knowledge

    Assessments

    We will measure the skills to move among the different context of Ax2 + Bx +C. They are algebraically, numerically, and graphically. In addition, students will evaluate their self learning.

    See additional information
    estradaa.edublogs.org

14. I can’t find the comments for Using Mastery-Learning and Planning for Remediation so I will put my link here:
    http://elenalala.edublogs.org/transforming-graphs/