The paths to "Eureka" moments: Teaching Mathematics in Secondary Education

Posts tagged ‘Summative Assessment’

O1- Standards Based Curriculum

O1- Offer an organized curriculum aligned to standards and outcomes.

Teacher-candidates align instruction to the learning standards and outcomes so all students know the learning targets and their progress toward meeting them. This means that I design curriculum based on the standards and use learning targets to anchor student expectations. In doing this, I encourage students to take ownership of their own learning reflect on their progress towards meeting those standards.

In the last month of student teaching, my algebra and 8th grade classes have been going through a year-end-review. In order to give students the opportunity to self-reflect on their own understanding of key topics in their respective curriculum, I designed the instruction to cover 1-2 standards each day with an exit or entry ticket specific to those standards. As the weeks progressed, students were given a chart in which they could label the topic/standard covered, record the graded exit ticket score, and then reflect upon their own level of confidence in that topic area. Finally, the chart has a column for action steps in which students can write down one way they will improve in this standard/topic area if necessary.

Two example exit tickets are can be found in the following links.

Algebra: Transformations Exit ticket          8th Grade: Exit ticket Rational and Squared Numbers


The learning targets given during these review days were directly related to the standards. For example, in the exit ticket given for algebra students, the learning target was, “I will graph functions and describe them as transformations of the parent function.”

The standards for this lesson were, “Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.” Therefore, by asking the students to write the learning targets in their journals, complete the lesson activities, and then conclude the lesson with the exit ticket and chart, students were made aware of the standards and given the opportunity to reflect on their progress toward meeting them.

In creating this review strategy, I learned the power of encouraging students to take responsibility for their own learning and education. Students were able to get feedback on specific skills and competencies, reflect on how their scores relate to their confidence levels, and then take a course of action toward further progress.

In the future, I hope to implement this strategy as a typical review exercise and give students specific action steps to choose from so that they can grow in the areas they struggle in. In this way, I will give students further opportunity to be responsible for their learning and be less dependent on the teacher.

Student Choice! Stations Lesson

H1-Honor student diversity and development.


Teacher-candidates plan and/or adapt learner centered curricula that engages students in a variety of culturally responsive, developmentally, and age appropriate strategies.

As a teacher, the content I teach must be accessible to all students of diverse learning profiles, readiness levels, and interests.

On the fifth day of our Algebra unit on functions, I decided to do a stations activity. The stations were designed to allow students to work on various ways of working with functions. The activity enabled students to choose the areas where they felt they needed additional practice.

Functions day 5

The stations were:

1)    Domain/Range: Worksheets with multiple ways of representing given information.

2)    Input/Output: Worksheets with many different functions and problems where students are asked to find either specific inputs or outputs.

3)    Graphing: Students are given several related functions (shifted along either the x or y-axis) and asked to graph them on different coordinate planes. (See graph paper)

DomainRange Worksheet     function_output      graphing_coordinate_plane 

As with other workdays, students were encouraged to work collaboratively and use dry-erase markers on their tables to demonstrate their work. “Go-to” people were designated at each station as peer leaders to whom students could direct questions before asking me.

After the stations activity a differentiated quiz was given to all students. The quiz had two versions based on student readiness level. The only difference between the quizzes was the complexity of the math involved; the function content was the same. Prior to giving the quiz,  I explained why I was giving two different quizzes. “Those who showed an understanding of functions (based on pre-assessment) receive a quiz with more complex math as well as functional notation. This is to challenge each student, not to label one group “smart” and another “dumb”. All students received the same type of questions, just different levels of math complexity.”

Functions Quiz 1   Functions Quiz 2

This lesson was planned so as to be learner centered. It allowed students to work with the content in a variety of ways and from multiple perspectives. Students were able to work collaboratively- challenging each student in a developmental way. Finally, the quiz was given in such a way as to give each student the opportunity to succeed and demonstrate their academic knowledge.

In creating this lesson I was able to grow in my understanding of how to differentiate instruction and assessments. I focused on individual student readiness levels and was able to formatively assess where students had strengths and weaknesses. By giving students the choice of which stations to work at, they were responsible for their own learning and quiz preparation.

In the future, I will continue to strive to make my lessons student-centered, differentiating my instruction to meet students where they are at developmentally.

Practicing Standards-based Assessment

P-3 Practice standards-based assessment. 

Teacher-candidates use standards-based assessment that is systematically analyzed using multiple formative, summative, and self-assessment strategies to monitor and improve instruction. As a teacher, I use inquiry-based formative assessment as well as quizzes in the midst of units to determine student readiness levels. Using  the data and feedback to inform my instruction,  I am able to create lessons that target the standards based content in which students struggle. 

The following quiz was given to a class of 8th grade algebra students to assess their understanding of recursive sequences.

The standard specifically addressed was:

A1.7.C Express arithmetic and geometric sequences in both explicit and recursive forms, translate between the two forms, explain how rate of change is represented in each form, and use the forms to find specific terms in the sequence.

After reviewing the quizzes and discovering areas where students struggled through student inquiry and exit tickets, I was able to plan several lessons of review using different methodologies of teaching (see previous post: LINK for specific lesson). The three sample quizzes are representative of the spread of achievement in the class.  Student A demonstrated mastery of the subject in the first quiz and was challenged with more difficult problems in following lessons. Student B was not proficient in the quiz content, but demonstrated good problem solving techniques. And Student C struggled with the content in the initial quiz.

Attached are both quiz results: Student A Student B Student C

After teaching these lessons where we reviewed definitions, examples, and worked collaboratively to expand our learning, a second quiz was given.

The results of the second quiz compared to that of the first, demonstrate the success of intentional planning for specific topics of instruction. As evidenced by the three examples of student work, students were able to better communicate their analysis of the sequences and summarize their understanding of recursive equations. Student A continued to demonstrate mastery, Student B demonstrated proficiency, and Student C made the largest leap in academic achievement by demonstrating mastery.

Through this experience, I learned that lessons are effective when taught based student need. If I had not quizzed the students at the beginning of the week, I would not have been able to tailor my instruction to their needs, but would have perhaps taught concepts they had a firm grasp of, while neglecting areas they may have struggled in.

Students also benefit from inquiry and planning for instruction. On the one hand, students are able to see their academic progress which will motivate for future learning. Additionally, class time is spent learning and growing in areas where students are not proficient. This creates an efficient learning time without reviewing topics students have already mastered.

As demonstrated by the above evidence, I practice standards-based assessment through regular formative assessments (inquiry based observation and quizzes) and timely summative assessments.

TI: Informed by Standards Based Assessment (Philosophy of Assessment)

Assessment Philosophy

Assessment models, feedback, great teaching, and differentiation each help motivate students to learn.

Assessment models provide standards for both students and teachers to live up to. This provides structure and consistency on which students can depend for feedback on their progress.

Feedback is a tool by which teachers can convey student success and area of improvement. Using this tool correctly can challenge students and provide them with opportunities for learning.

In order to properly utilize the tool of “Feedback” one must implement great teaching techniques. If used inappropriately, a student’s self-esteem, motivation, and ability perception can be harmed.

It is for this reason that teachers must implement differentiation in the classroom. Not all students are the same. They differ in learning styles, personality types, and in ways of communicating. Thus, great teachers must strive to provide multiple methods of assessment so as to effectively chart their learning, and provide future challenges.