Key Theoretical Beliefs

The first, and perhaps most important, theoretical belief that I hold is that of preparing assessments to test across the entire spectrum of multiple intelligences.  The Theory of Multiple Intelligences is a concept developed by Howard Gardner in 1983 to describe the different ways that people can excel in cognitive tasks.  As a child I did very well in school and I was often complimented by instructors.  However, while I was more than able to excel in study and enjoyed being social, I knew that I was less skilled socially than my peers.  I soon realized that social skills were just another way to “be smart” and that I lacked that particular skill.  I came to believe later, that people are all just about the same, we just have different skills.  When I read the research of Mr. Gardner, I knew that he was describing what I’d recognized as true in the world around me as early as middle school. 

The Theory of Multiple Intelligence has impacted the creation of the performance assessment I intend to use during my student teaching.  The performance assessment, which will call upon students to create a graphic organizer in which they sequence the steps involved in the process of using a chosen methodology to solve a linear system of equations problem, allows more visual students to excel.  The detailed, sequential steps that the students will describe will allow students not only to demonstrate the depth of understanding they’ve reached concerning their chosen methodology, but will also allow them to practice scientific documentation.

Walking hand-in-hand with the Theory of Multiple Intelligences is the idea of differentiated instruction.  Differentiated instruction is the idea that each student learns and expresses that learning in a different way.  The concept of differentiated instruction can be applied in my future mathematics class in various ways.  For example, I might use students’ interests in an attempt to better connect them to the curriculum, by showing them how they might see parabolas in basketball and how parabolas and basic physics apply to a bank-shot.  Another way I can use differentiated instruction in the classroom is to use a wide range of instructional techniques.  Though the focus, and consequently results, of different instructional methods are used to draw attention to different aspects of the same learning, a teacher who believes in differentiated instruction would be sure to include direct instruction, inquiry-driven instruction, and other instruction methodologies to ensure that all students are gaining an equal opportunity to properly access learning.

Relationship Between Key Theoretical Beliefs and Assessment Plan

Due to my mentor teacher using a “boxed” curriculum, I had less control over what was in the test than I have had in the past.  I had some limited freedom in which to modify the contents of the test and have done so in an attempt to use the Theory of Differentiated Instruction on my pre and post tests.  In order to include elements of Differentiated Instruction without overly modifying the test, I added two word problems to the test.  These problems, while historically being seen as more difficult, will go a long way to reaching concrete thinkers.  While abstract thinkers are perfectly happy performing algebraic operations without the need to connect the operations to reality, concrete thinkers need some context in which to perform successfully.  Giving such students word problems, gives them something more “tangible” or “real” and will enable them to access the test more readily.

The third theoretical belief that I hold is that of Socratic questioning.  Socratic questioning is a process of disciplined question asking.  This process helps students to think about their thinking and cultivates the ability of students to think critically.  This technique is exceedingly useful when exploring, through dialogue, topics that are very broad and difficult for new learners to access.  It is a useful way to learn what preconceptions students bring to the classroom regarding learning and is a useful way to bring the students to understand whether these assumptions are correct or not and why.  Speaking specifically of my future mathematics class, I intend to use Socratic questions to help students navigate through problems, rather than just giving them the answer.  Modeling this process of question asking will show them how to get through problems that would otherwise leave them unable to succeed. 

It is my intention to use Socratic questioning in my performance assessment and during class work.  The students at the school of my student teaching placement have had little practice with the scientific documentation process.  The performance assessment that I’ve made for them will allow them to practice this skill, but it is highly likely that the students will, having internalized the learning themselves, skip large steps or fail to explain clearly the process of solving linear systems of equations problems using their chosen method.  Instead of saying, “You need to elaborate more on how you got to this step.” I can say, “If I did as you’ve written here, I would get this answer, which is obviously incorrect.” or “Do you think someone who didn’t know how to do this problem would understand what you’ve done in this step?”  This This process will mirror the Socratic questioning process and slowly guide the students into the Zone of Proximal Development where they can be successful in the performance assessment.

During class work, I can use the Socratic questioning process in many ways.  One solid example of how I might use it in my class would include saying something along the lines of, “Ok, so we have some x’s over here and some over there.  We also have some constants here and there.  What should we do next?” instead of “Now, you have to subtract 6x from this side of the equation to move it over to the other side of the equals sign.”  This process of requiring the students to guess at the next step, will ensure that they aren’t passive receptors of knowledge, but actively engaged in their own knowledge construction.  This verbal scaffolding and engagement will assist in ensuring that students are internalizing the information and allow them to work their way through algebraic processes with many complicated steps.

Obstacles to Assessment Plan

As I briefly described above, my mentor teacher is asking that I use a boxed curriculum to teach the students.  While I have some freedom in how I prepare and deliver lessons, and even some limited freedom in changing the content of assignments.  I will not be creating any worksheets or tests wholly from scratch.  This is both a benefit and a drawback.  While I will not be required to spend inordinate amounts of time creating materials, I will have little control over what learning objectives the assignments hit.  I will have to balance the learning objectives that I as a teacher feel the students need while also making sure that they are able to do the worksheets and tests included in the boxed curriculum. 

Additionally, my mentor and I share very different ideals of a well-regulated classroom.  Until only very recently, I’ve envisioned my classroom as a well-oiled, military machine, running on well-practiced procedures and routines, very much like Wong described in his book, “The First 100 Days of School”.  However, having gained some classroom experience and had a bit of training, I’ve come to realize that classroom management styles do not wholly spring from the teacher in the school, but must come from a merging of the needs of the students and the needs of the teacher and harmonize with the school culture.  This will place a bit more strain on me as the teacher, since I will be all-but required to adapt more Democratic teaching techniques.  Techniques that I had historically rejected as being impossible for me to pull off in light of the fact that they seem to call upon aspects of teaching that I consider weak spots in myself.  However, this forced modification in my classroom management style will afford me the opportunity to strengthen those weak areas and grant exposure to alternate management styles that will very likely serve me well in the future, based on the circumstances.  Thus, while it may be an inconvenience in the short run, it may be a benefit in the long run.

Relationship Between Assessment Plan and Student Needs

Regardless of the rules of the classroom, math teachers all want the same thing for their students.  Namely, they want their students to grow to be literate in the language of mathematics.  This foundation in math will allow them to be able to succeed in university and any career they might choose in the future, especially those of a scientific nature.

In order to ensure that students are able to grow into those mathematically demanding careers, the goal of my teaching -- the journey that I want to take my students on -- culminates with students having a solid knowledge of operations involved in every aspect of basic math as well as how those operations are mathematically justifiable and how the operations, solutions, and proofs relate to coordinate planes and how they are applicable to the lives of the students in the real world.  Assessments in my class will be tools for me to use to check the progress of the class toward these ends.  They give me precious insight into how effective I am as a teacher, specifically which methods work with which class or individuals, and which students are succeeding and which students need assistance to excel.  Assessments will also allow me to target students in need for specialized remediation, giving them the support they need to reach the point described above.

Armed with a knowledge of mathematics students will be in a position of maximizing their potential success.  The role of assessment in reaching my visionfor my students and maximizing their potential, were I forced to make a metaphor, would be that of a GPS.  GPS’s tell you not only where you are, but they also show you how to get where you’re going.  Assessments and tests will allow me to see where the students are and what will be necessary to get them to where I want them to be.

Description of Pre-Assessment

           The first question on the pre-assessment was designed to get students to consider the cases in which a linear system of equations would have no solution.  This is most commonly true when the lines given are parallel to each other and do not intersect.  Even though this test was included in the boxed curriculum, I decided to leave this question first because it was relevant to the information I believed the students should internalize prior to the unit finishing and because it was a relatively easy question that would allow the students to “ease” into the test.

              The second and third questions were review questions.  The students had studied solving linear systems of equations prior to starting this unit, but it was unclear to me how deeply they had internalized such information, had they done so at all.  This information was useful to me as I had assumed the students had some knowledge of this method and needed to know which students would need some additional time grasping the operations involved.  Additionally, these questions were important because they allowed me precious insight into whether or not the students had actually connected the concept of the operations they had studied to how those operations and solutions related to the Cartesian Plane. 

              Questions 4 and 5 of the pre-test required the students to use the substitution method to solve linear systems of equations.  Students were expected to do very poorly on this section of the test since the operations involved the students had not studied before, are quite complex, and are considerably abstract in nature.  That being said, merely from using the term “substitution” in the instructions above the problems, a few students were able to intuit the first step or two involved in the process.  This was valuable information as those students did very well in the unit and required more challenge than other students.

              Questions 6 and 7 of the pre-test required the students to use the elimination method to solve linear systems of equations.  Students were expected to do very poorly on this section as well, since the operations involved the students had not studied, are most complex, require relative mental acrobatics to justify mathematically, and have proven to be most difficult for middle school students to grasp.  Unlike the substitution questions, students were not able to intuit the initial processes in this method via the wording of the instructions.

              Questions 8 and 9 of the pre-test tasked the students with solving a linear system of equations after first deriving the equations from a given word problem.  Students had seen word problems before in differing contexts, but had not done word problems with linear systems of equations prior to this test.  Students were not equipped with the steps involved in distilling equations from sentences and thus performed poorly, as expected, on this section of the test.

              The final question on the pre-test, question 10, asked the students to solve two linear inequalities and graph them on a Cartesian Plane before labeling the inequalities’ solution set.  Students were expected to know what was going on in this problem, having previously studied inequalities, but were not expected to be able to complete the problem successfully.  This expectation proved true as most students were able to get partial, but not full credit on the question.

Examples of Pre-Assessment and Learning Goals

              In the 4th and 5th questions on the pre-test, students were asked to solve a set of linear equations, using the substitution method.  The ability to perform this operation successfully relates to the 5th learning objective of the unit, which states, “Students will be able to solve a linear system of equations using the substitution method.”  If students are able to solve this problem using the substitution method, it would prove that not only had they learned how to solve linear systems of equations, but also how to apply the substitution method.  Incidentally, this would also demonstrate having internalized the process of algebraic manipulation as well as an ability to mentally categorize and track a great many different parallel operations, since the substitution method requires the use of long equations with potentially many variables.         

              The 8^th and 9^th questions on the pre-test task the students with first constructing two linear equations as gleaned from a paragraph of text describing a situation, then using their preferred method to solve the problem.  This process relates directly to the 7^th learning objective of this unit, which states, “Students will be able to solve word problems using linear systems of equations.”  Word problems are uniquely valuable as proving that a student has internalized a great deal of information across many different categories of learning.  Firstly, they show that students have become familiar with the process of distilling information from a paragraph of text, which historically has proven to be a difficult task for students in middle school.  Additionally, they show that the students, in correctly setting up the equations, understand the relationships the variables have to values.  This implies a certain level of relational understanding and deductive reasoning.  They then show, in the solving of the problem, that the students have mastered a preferred technique to solve linear systems of equations.  These will all be key skills for students in their future.

Relationship of Pre-Assessment to Subject and Grade Level

              The class of my student teaching placement consisted of a mix of 8^th and 9^th grade students.  The school streams students by ability level, not grade level, and therefore has a few classes that displayed a cross section of the school in terms of age.  The school streaming students by ability level rather than age was of benefit to me as a teacher, because I was more able to assume certain levels of understanding and ability going into the unit.  It would have been inappropriate for me to expect the students to perform operations that are typically studied in higher levels of math, but also I was justified in making certain assumptions about what the students would be able to do going into the unit. 

              When looked at abstractly, the pre-test covered 5 integral topics of study.  Each of the topics is considered by mathematics teachers, NCTM, and Oregon State as necessary for the proper learning of Algebra 1 and is represented as such by inclusion in the state and professional standards write-ups.  Of course, students are expected to understand what a linear system of equations is, but beyond that, students are expected to know the 3 means of solving such problems.  The 3 methods of solution include through graphing, through the use of the substitution method, and via the application of the elimination method.  In response to the necessity of understanding of these methods, the test checks for student understanding through the presentation of 2 problems addressing each type of solution method.  Each of the 3 ways to solve these problems represents 3 of the 5 main points covered by the pre-test.  The 4^th is student ability to solve word problems.  Lastly, the test checks for student understanding of the concept of inequalities and of the concept of solution sets pertaining to inequalities.  For the reasons described above, I feel that each section of the pre-test is educationally justifiable to use in Algebra 1 as well as in the grade level of the students in the class.

Description of Post-Assessment

              The pre and post-test that I used are identical.  The first question asked students what a problem with no solution would look like.  The 2^nd and 3^rd question asked students to solve a system of linear equations problem using the graphing method, the 4^th and 5^th questions asked students to use the substitution method, while the 6^th and 7^th required students to use the elimination method.  The 8^th and 9^th problems explored students’ ability to solve linear systems of equations word problems and the 10^th problem checked student understanding of techniques surrounding linear systems of equations as applied to inequalities.

              The whole test was out of a potential total of 28 points.  There were 10 questions, 9 of which were worth a possible 3 points and one of which was worth just a single point.  The first question on the exam was worth a single point only.  However, every following question on the exam was scored on a 3 point scale.  If the student showed some work, but was unable to solve the problem, and was not even close to a correct answer, they would be awarded a single point.  If the student showed their work and was somewhat close to a correct answer, they would be awarded 2 points.  If the student showed their work and got a correct answer, they would be awarded a full 3 points.  Getting a correct answer, but showing no work was worth only 2 points.  This is a system of my own design that I modified from that of a math instructor that I had when I was in middle school, which I then passed by my mentor teacher prior to its implementation.  My mentor teacher thought that it was a fair way to grade and gave their approval.  I feel that this is a good system due to the fact that it tends to be forgiving of simple mistakes, but requires a few correct answers in order to get a higher grade.

Examples of Post-Assessment and Learning Goals

              Questions 2 and 3 of the post-test ask students to use the graphing method to solve the problems given.  These questions relate to learning objective number 3, which states, “Students will be able to use the graphing method to solve linear systems of equations.”  Students did reasonably well on this section going into the unit, as they had studied it to a limited degree prior to my starting teaching, but still improved a great deal on the post-test.

              Question 10 of the post-test had students graph two linear inequalities and label their corresponding solution set.  Students were able to do this to some degree on the pre-test, but did much better on the post-test.  All students were able to get at least 2 out of 3 points on this section of the test.

Relationship of Post-Assessment to Subject and Grade Level

              When looked at abstractly, the pre-test covered 5 integral topics of study.  Beyond being required to understand what a linear system of equations is, students are expected to know the 3 means of solving such problems.  The 3 methods of solution include through graphing, through the use of the substitution method, and via the application of the elimination method.  In response to the necessity of understanding of these methods, the test checks for student understanding through the presentation of 2 problems addressing each type of solution method.  Each of the 3 ways to solve these problems represents 3 of the 5 main points covered by the pre-test.  The 4^th is student ability to solve word problems.  Lastly, the test checks for student understanding of the concept of inequalities and of the concept of solution sets pertaining to inequalities. 

Having researched the grade level standards, I can say that each section of this test addresses standards required by Oregon State at the Algebra 1 level.  The standards addressed and their descriptions can be found in the “Goals and Objectives” section of the work sample.

Description of Formative Assessment

              On the order of once per class meeting, I will be giving the students a packet of worksheets designed by the makers of the boxed curriculum I will be using.  The packets are comprised of 4 pages.  The first two pages of the packets are guided notes.  They introduce the new topic through gradual steps that build on previous learning in order to ease the students into the new topic with relative ease.  The guided notes in this section are useful in class as example problems.  The guided notes section also contains algebraic justification for any new operations the students may be learning.  The second two pages of the worksheet are homework problems that the students can begin in class and finish as homework.  The homework problems given are well ordered as they go from easier problems at the front to harder problems at the back.  This challenges students to think as they gain confidence in their ability to perform the required task.

              These packets are also very useful in preparing students for the pre and post tests.  Since the same people that made the tests make the packets, the pre and post-tests, the questions on the homework are exactly the same in form and execution as those found on the tests.  This easy flow from homework to test reduces any “jarring” students may experience from receiving a test in a different format from homework they’ve grown accustomed to doing and places them in the best possible position for success.  An example of a formative assessment can be found below.

Examples of Formative Assessment and Learning Goals

              The packets used in class were designed and chosen for use pursuant to Oregon State mathematics standard 8.1.4, which states that students will be able to “use linear functions and equations to represent, analyze and solve problems, and to make predictions and inferences”.  The packets used in class covered the substitution method, the elimination method, and word problems relating to linear systems of equations problems.  Therefore, one can see that the formative assessments used in class directly address the ability of students to solve linear systems of equations problems through various means and therefore address the standard.

              Having discussed how the packets used in class address the standard, let’s discuss how the packets then address the individual learning objectives of the unit.  I decided to teach one lesson on each learning objective.  One lesson was done on the substitution method, one on the elimination method, and et cetera.  The packets were designed to facilitate this kind of teaching.  Each packet focused on one way to solve linear systems problems.  Thus, each packet reinforced learning done in class and addressed the learning objective for that day.  Through this readily-accessible breakdown of discrete lessons, learning objectives were clearly addressed in the formative assessments, which were also aligned to the state standard.

Relationship of Formative Assessment to Subject and Grade Level

           In addition to being aligned to learning objectives and state standards, this assignment is relevant to this grade level because it is assessing students on their ability regarding Oregon State mathematics standards for algebra learners.  Students were offered at least one class period to do these packets.  During the course of the unit, the packets were such that the students were able to finish some in class while others required additional time at home in order to finish.  Students had access to my assistance during class and after school in addition to their textbooks and notes in order to ensure success.  Additionally, the students had a large amount of experience with these packets having used them in lower level math classes, and consequently knew their format well.

              Due to the students being well acquainted with the format of the packets and the associated expectations of the packet-makers, I found that they had little problem with accessing the worksheets and there were no difficulties launching the activities.  The only problem that I had integrating these packets into my daily lessons was the diction the packets used.  The packets tended to explain operations in very verbose mathematic terminology and therefore were occasionally difficult for the students to understand.

              In order to be sure that the students had used the packets to successfully meet the standards, I collected the packets and graded them.  I was at that time able to see problems the students were having with the operation at hand and even provide feedback to them regarding any mistakes.  I was convinced throughout the unit, through viewing the students’ performance on the worksheets, that the students had each reached an acceptable level of ability regarding the individual operations involved.

Description of Performance Assessment

              The performance assessment I decided to include in this unit began as an idea that I had while reading a teacher research article.  The idea was to have the students, using a sequencing graphic organizer, create a visual depicting a chosen method and how to apply it to solve a given problem.  The point of this project was not to solve the problem, however.  The point of this project was two-fold.  First, I wanted the students to show me that they had mastered at least one form of solving linear systems of equations and understood the process very deeply.  Additionally, I wanted to offer the students a chance to practice the skill of a technical/scientific documentation process.  The students were required to make a graphic organizer and describe the steps involved in great detail, as if they were describing the process to someone who doesn’t understand how to do Algebra.  This process resembles research in that regard as any experiment done must be documented in such a way that someone unfamiliar with the research involved can reproduce the experiment.  This project offered students with a more visual intelligence a chance to excel and offered students a chance to use color and categorization, which are skills they may not get a chance to practice every day at school.

              In launching this assignment, I first chose a website that had an array of sequencing graphic organizers.  I decided to give the students some class time to research the graphic organizer they would prefer to use.  The students had about 20 minutes of class time to look through the organizers offered and choose one, however, most students chose to use a flowchart rather than use a graphic organizer that was presented on the website I had chosen for them, I believe this was due to the fact that I had shown them an example of my own making that was in the form of a flowchart.  I did not require the students to make the flowchart on a computer, but I think that should I use this activity in the future I will do so, due to the fact that some students just have naturally messy handwriting and I’d rather not have to take neatness points off for students’ natural writing ability (or lack thereof).  I gave the students 5 days to complete the assignment.  The only requirements involved were that the students completely detail the process, have visuals/diagrams/graphs, and add color as a tool to increase reader understanding.  The students did very well on this assignment scoring a B average on the assignment.  Some projects were more in depth than I had anticipated and displayed a level of professionalism that I had not hoped to expect.

              To score the assignment, I gave the students a scoring guide that graded across 5 categories.  The categories included “Completion”, “Mathematical Concepts”, “Explanation”, “Diagrams and Sketches”, and “Neatness and Organization”.  “Completion” described the portion of the assignment being graded for timeliness of submission and whether or not the assignment was complete.  “Mathematical Concepts” described the score the student would receive on whether or not they appeared to actually understand the process involved to the level required of them.  “Explanation” detailed the students’ success at fully and thoroughly detailing in writing how the steps in the process worked as they went through the flowchart.  “Diagrams and Sketches” detailed the students’ success at using diagrams, sketches, or graphs to improve the reader’s understanding of the material.  Finally, “Neatness and Organization” described the score the student would receive regarding the readability of the graphic organizer they submitted.  Each of these categories offered a potential score of 4 points for a total of a maximum of twenty points on a perfect assignment.

Examples of Performance Assessment and Learning Goals

              Due to the focus of the performance assessment being to get the students to show that they had deeply internalized one of the 3 methods for solving linear systems of equations problems, this assignment directly addressed the above-described standard, specifically, this assignment addresses mathematics standard 8.1.4, which states that students will be able to “use linear functions and equations to represent, analyze and solve problems, and to make predictions and inferences.”  If students can show that they can explain in great detail, as if to someone who doesn’t know how to do algebra, the process of solving a linear system of equations problem, they show that they have mastered that method of solving these problems to the point that they could even teach the process to another. 

              Given that the students were to choose their own preferred method of solving linear systems of equations problems, this assignment addressed different learning objectives, keeping in mind that each different methodology was covered as a learning objective on different class meetings.  However, this assignment would satisfy student learning as required by state standards and befitting no less than 1 of the unit learning objectives.

Relationship of Performance Assessment to Subject and Grade Level

              This assignment is appropriate for this grade level because beyond addressing the standards developed for the level it does not require the students to perform any activities beyond a level common to 8 and 9^th grade students.  The assignment did not require the use of a computer, though some students used them to exemplary success.  It required only for the students to spend some time thinking about and writing about the method they chose.  Additionally, the students had a large amount of support in doing the assignment.  I was available to them before or after school or both for no less than 3 school days.  I also told the students that I would be perfectly happy to review their assignments before submission should they desire.

              Since the assignment didn’t require the students to perform above a level described by the standards for their grade level and because the students had my support throughout the execution of the assignment, this assignment was a good fit for their ability level while offering them a challenge and an opportunity to improve their skills in the appropriate areas.

Uses of Assessment Data

           Feedback Plan to Students

           In order to ensure student progress toward mastery of the learning goals, I was sure to grade and hand back formative assessments during the next class meeting.  Skinner said that the closer, chronologically, the reinforcement to the behavior the more effective it is in modifying future behavior.  It was with that in mind that I graded and handed back assignments as soon as possible in order to ensure that students got meaningful reinforcement on their work.  During the longer class periods, I had the students peer-grade the homework.  I knew that this would open up the possibility of students correcting mistakes on their paper before handing them in, but I didn’t mind because it aided student learning.  If students corrected the mistakes they made on their papers quickly before handing them in after we’d peer-graded them, they would be afforded the benefit of facilitating the necessary modifications in the students’ mental pathways such that they would remember not to make that error in the future.

              If students were incorrect on a spoken assertion in class, I was careful not to say that they were wrong in front of other students as it would only work to raise their Affective Filter, shutting down new learning for a time.  Instead, I asked if students agreed or disagreed by a show of hands.  If there were some students who disagreed I would ask them what they thought was correct and why they disagreed with the original student.  This served to facilitate a more debate-like dialog and eliminated students worrying about being “right” or “wrong” as per the teacher’s judgments. 

Feedback Plan to Parents

              The only opportunity I was afforded to speak directly to parents was during parent/teacher conferences.  I was able to sit in on a few conferences and discuss the progress of the students during my student teaching work sample unit.  Since the students did very well, including those with some behavior issues, the conferences were largely very positive.  I believe these conferences were both successful for me professionally and most informative as I learned a great deal about how to couch constructive criticism in a positive light.

              Beyond meeting the students’ parents face-to-face, the grades that I gave students were posted to the internet, and later included in the teacher’s overall grades for the term.  This was a form of indirect communication, since the grades and comments would go home to the students’ parents and due to the grades being available to the parents online.

              Due to the students in this class doing so well, I did not feel it was necessary to have a discussion with any of the parents of the students in my class, including the one with whom I had such a vehement personality conflict early on.  She rebelled against my teaching midway through the unit and perhaps intentionally bombed the mid-unit quiz (I say potentially because she later got an A on the unit test).  But, after I pulled her in after school for a counseling session with my mentor teacher she did much better in class and therefore felt it unnecessary to include her parents.  Should she begin to do poorly in class or display signs of further insubordination, I might consider a phone call to her mother.

              Impact of Assessment Data

During the course of my practicum teaching, I used mostly direct instruction teaching methodology.  I did use some written warm-ups, some think-pair-share, and even a couple constructivist activities, but for the most part I used Direct Instruction to deliver new content quickly.  I was able to deliver new content and have the students begin to use the target operations themselves over a longer period of time such that they could incorporate more practice in class with my support.  Through gaining more scaffolded practice of the algebraic operations, students had the best chance of improving.

Though it is hard to compare teaching methodology at this time, given my lack of experience teaching, let alone teaching mathematics with constructivist methodology, I can say that the methodology I employed, while traditional, was successful.  I can say that every student improved on every learning goal, though the amount of improvement is at times less than I would have liked to see.  Though it certainly would’ve taken longer to accomplish, I wonder if using a constructivist methodology as opposed to direct instruction would cause the students to internalize more of the math.  I also wonder if that time tradeoff would be worthwhile considering that the constructivist methodology would almost assuredly require more time in class to do the same amount of work. 

              I think that in the future, using more of a mix of direct instruction and constructivist methodologies would be of more benefit to the students.  The next time I teach this unit, I will be sure to have at least one activity a day that requires the students to take a constructivist approach to knowledge acquisition and see if that has a positive impact on their overall learning.