Multiple Assessment Approaches

            I taught a unit on plane geometry to a high school Pre-Algebra class.  The class is made up of 9^th and 10^th graders and all students are below grade level.  Many of these students do not do homework or do not turn it in.  They are still developing their abilities as thoughtful, responsible students.  For this reason, a single approach to assessment would not give me a real view of student learning. 

            The first part of the unit was supposed to be a review of concepts learned in previous math classes.  Using the pre-assessment as a formative assessment, I was able to make a plan to focus each lesson in the unit based on student strengths and weaknesses.  For example, for our first learning goal, the results of the pre-assessment informed me that students were particularly weak in using proper notation when naming two-dimensional geometric figures so I was able to focus the lesson on using proper notation.

            I included daily warm-ups as an assessment.  These quick problems at the beginning of class consisted of a problem related to the learning goal that students should already be capable of completing, a review problem from the previous day that linked specifically to a new learning objective or a problem that I perceived students to have misconceptions with from the previous days lesson.  This assessment directed the lesson that followed.  I could see where students were having problems or if they needed a quick review of a particular topic.

            Another form of assessment was discovery learning projects.  This allowed students to show me what they can do by themselves.  I could circulate and see the students’ thinking about angles and see what they already know, how they approach a problem and how they could apply previously learned concepts to a new situation.  This information was valuable in determining how to proceed with the current lesson and allowed me to adjust future lessons to play to students’ strengths in mathematical reasoning.

            A more traditional form of assessment took place in homework.  In mathematics, homework is essential to provide practice and solidify concepts.  As a teacher, homework also allowed me to see if concepts were mastered, needed a quick review or if they needed to be retaught completely.  Reflecting on the lessons and the resulting homework grades, I could see which parts of my lesson were effective and which strategies I could modify in future lessons to garner greater student learning. 

            Homework was also used as summative assessment, assigning a grade to student performance.  However, homework corrections were allowed up until the day of the test so students could come in to get help.  These one-on-one sessions were also a great time to assess student understanding.  If multiple students came in for help on the same type of problem, I could assume that others in the class were having similar difficulties and could adjust my instruction to remediate these problems.

            Quiz and test reviews were another type of formative assessment.  Students completed the reviews and then brought them into class.  We graded them together and they worked on their own.  If there was a problem that many students missed, we would go over it as a class.  The class reviews are completely based on the assessment of student performance on their individual review worksheets.  The reviews were made up of problems nearly identical to those on the formal assessment and were closely aligned to the learning goals and objectives so that these could be measured.

            Perhaps the most useful formative assessment was in-class guided practice.  By watching students work through problems, I could pinpoint exactly where a mistake occurred and identify whether it was a procedural or comprehension problem.  Since it happened in a lesson, I could immediately change course and correct misconceptions before they became practiced habit in homework.  This is especially important in this class because frustration builds easily and if errors can be detected and corrected quickly it can cut down on misconceptions and reteaching time.

            My department uses common assessments so my quiz and post-assessment were pre-determined.  I used the same test for both pre and post assessment.  The time between the pre and post assessment and the lack of previous experience with the topic made me confident that this would not skew the data.  In fact when presented with the post-assessment, not one student noticed that they had seen it before.

Measurement of Student Learning

            The variety of assessments allowed for a deep view into student understanding.  I required student participation and required that students showed their work on problems so that the students could find their own mistakes and so that I could see where errors occurred.  It was important to me that students did not just memorize my steps and the procedure, but that they understood the topic and that they can use the information when they encounter it in a similar problem or in a future class. 

            The homework was aligned with the goals, the in-class work and the post assessments.  Students were given many opportunities to practice and to use the skills in multiple situations.  The diagrams looked different in each problem.  The problems required progressively more thought and integration of prior knowledge.  The quiz was created to measure student learning up to a point so that the students and I could see their progress and what they needed to study further. 

            The pre and post assessment was closely aligned with my learning goals and gave students the opportunity to show what they had learned in class while connecting to prior learning, including the coordinate plane.

Explain how your assessments authentically and appropriately measure student learning.

Adaptations

            One adaptation that I saw was necessary right away was with my ELL students.  In previous units I had observed that these students fell behind when a lesson was particularly language intensive.  The unit on plane geometry was not one of these units.  English words that are introduced are defined visually and verbally so it is easier to pick them up.  I did however provide Spanish notes for some lessons and grouped students so they could translate for each other if necessary.  I also paid close attention to the words that these students used and the errors they made so that they could be immediately corrected.  One ELL student performs exceptionally well in math so I encouraged him to help others.  I did keep a packet for one student to take with her to her ELL class as well so that she could get further help there.

            I broke the questions on the pre-assessment into learning objectives so that I could easily see where students were in regards to each objective.  I put each student into each objective as correct, partially correct or incorrect with explanations as to which part was right if they were partially correct.  From some of this analysis, I could see which misconceptions needed to be corrected or which pre-requisite skills were lacking for some students.  This helped me make decisions about how to proceed with the unit.

            My pre-assessment showed that some students had some knowledge of the topics already, but none had mastered them.  This class usually has a very diverse knowledge base, so the fact that the scores were all low actually made my adaptations easier.  The pre-assessments also showed that all students were lacking in the same areas.  This helped me decide which lessons needed to be basic and which topics we could explore more in depth.

            The first conclusion I made based on the pre-assessment was that I could move rather quickly through the first learning goal which was a review of points, lines, line segments, rays and planes.  The students were mostly able to correctly identify different figures, but did not name them correctly so the majority of that lesson was based on proper notation.  The lesson plan in the textbook covered a lot of topics, so I made a few possible end points for the lesson in case I needed to cover something in more detail.  We were able to get through the entire lesson in one day.

In the pre-assessment, I saw a high level of intuition for angles so I decided to do discovery activities for two of the lessons.  This allowed students to explore relationships between angles and come to their own conclusions about those angles.  They then had to prove informally that their hypotheses were correct.  This activity aligned perfectly with the learning goal that students can prove informally the relationship between the angles formed by a transversal crossing parallel lines.

I also saw a high level of intuition about symmetry, so I decided to do a challenge activity that utilized both graphing on the coordinate plane (a point of trouble with many students) and symmetry.  This would give them the needed practice of the coordinate plane while using their natural abilities at recognizing and completing symmetry.

The one learning goal that was split was transformations.  Some students showed mastery of recognizing transformations, some showed no ability and some showed some ability or intuition on the topic based on the names of the different transformations.  As I planned to teach that lesson I decided to make it interactive and explore many transformations so that all students would learn and be challenged.  I also wrote warm-up problems each day so that it could direct the lesson or review a topic if needed.

The adaptations made to the lessons based on the pre-assessment made it less necessary to adapt as I went.  The pre-assessment analysis was accurate in projecting the amount of time needed on each topic and the types of adaptations to make for student learning.  In other units, I have had to adjust the time spent on topics, but in this unit on plane geometry, there were very few changes needed after the initial changes made as a result of the pre-assessment. 

Alignment of learning goals, objectives and assessments

            I began with learning goals and objectives.  I looked over the unit and found the goals that aligned with common core standards.  I adapted these standards to fit my students and my unit.  I then developed specific, measurable objectives for each learning goal and created lesson plans, activities and homework around those objectives.

            The first learning goal was not specifically aligned with a state standard, but was necessary to complete the other goals.  Students were to identify and correctly name points, lines, segments, rays and planes.  I assessed this ability by working on note packets and graphic organizers in class.

            Learning goal two involved using supplementary, complementary and vertical angles to write and solve equations.  I wrote objectives involving each type of angle and found a worksheet that asked students to write and solve equations based on a diagram with these types of angles.  As I looked at the objective for vertical angles and the students’ performance on the pre-assessment, I decided to create a discovery learning activity for vertical angles. This discovery would also prepare them for a more complex discovery in the next goal.  I assessed students in class on their ability to write and accurately solve the equations and then gave them homework to assess their ability.  This homework had different types of angles and some problems that combined multiple types of angles. 

            The state standard that aligned with my learning goal 3 was that students could use formal arguments about the angles created when parallel lines are crossed by a transversal.  My students had not done this type of work before, so I created a highly scaffolded discovery learning activity based on the scientific method for students to form a hypothesis and prove that it is correct.  Their conclusions were shared in a group setting so that I could clarify any misconceptions that arose in the activity. 

            Learning goal 4 was similar to learning goal 3.  Students had to use their knowledge of triangles to write and solve equations for a missing angle.  I assessed students as they worked through online problems.  These problems asked for an equation and for the answer.  We worked through each example step by step.    Homework had many different kinds of triangle with different angles missing.  Students had to write the equation for each triangle and solve for the missing angle.

            The objectives achieved in learning goal 4 were extended to include polygons in learning goal 5.  Similar problems were used in class and in homework to assess student learning and application of knowledge.

            Learning goal 6 involved classifying quadrilaterals.  The objectives for this goal progressed from students simply using a venn diagram or algorithm to answer questions about quadrilaterals to students using the coordinate plane to draw and classify quadrilaterals.  In-class work and homework reflected that progression.

            The last learning goal was transformations.  The final objective in this goal utilized all student learning in the unit.  Students had to graph a polygon on the coordinate plane and then describe the transformation from one polygon to a congruent polygon. 

For each class period, I created a warm-up that reinforced and assessed understanding of a learning goal.  Problems from in class work and homework representing each learning objective were used in the mid-chapter quiz review, quiz and chapter test review.

Analysis

            I aligned a learning objective to each problem on the pre and post assessment.  When possible, I wrote out each step and figured out which learning objective aligned to each step.  I could then score each question and find which students achieved the learning objective.  I will then go through each assessment and place students as correct, partially correct or incorrect.  I can determine performance from their scores and the particular mistakes made or problems missed.

Formative Assessments That Measure Student Progress

            One formative assessment I will use to measure student progress is a daily warm-up.  This warm-up occurs daily and asks students to demonstrate mastery over a learning objective.  A question or problem that aligns with one of the learning objectives is used and students are asked to write the question down and show all work to solve it.  These warm-ups allow me to see which students can do problems on their own without help from me and which students are still struggling.  When they show their work, I can find any mistakes and correct them if necessary.

            Another formative assessment I used was a discovery learning activity involving parallel lines cut by a transversal.  Students were asked to use their knowledge about complementary, supplementary and vertical angles to write a hypothesis about the relationship between angles formed when a transversal cuts two parallel lines.  They then were asked to prove their hypothesis true using knowledge of angles.  Those who truly understood these kinds of angles could complete the activity easily with little assistance.  A little bit of extra help solidified the concept in the minds of those that were struggling.

Importance of supporting evidence

            It is important to collect evidence of student learning so that we can remediate as necessary.   It is impossible to keep in our heads the needs of every student.  If it is recorded it is easier to recall which students need help with a particular area.  If we do not keep records and keep supporting evidence, then we cannot reach every student in the best manner.

References

Wyoming Department of Education. (2012). Common Core State Standards for Mathematics. Retrieved, March 18, 2014 from http://edu.wyoming.gov/sf-docs/standards/final-2012-math-standards.pdf