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

Archive for January, 2013

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.

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Teacher Observation #2: Ms. K

Teacher Observation #2: Ms. K

12442583961705917736smiley eyes.svg.medSpecial Education Teacher Ms. K is a true example of teaching through organized instruction and creating a safe learning environment.

Ms. K has an established classroom routine each day. The schedule is written on a board near the front of the room:

1.  10 Minute Warm-up Exercise

2.  Vocabulary Review

3.  Lesson

4.  3 Minute Break

5.  Individual Practice

6.  3 Minute Break

By strictly following this schedule, Ms. K’s students can be at ease throughout the day, knowing what is ahead and what they are to be doing at all times. The vocabulary words, learning goal, and examples given during the “lesson” period of the day are all to be written down in their personal journals. This expectation is the same every day so that students are familiar with what they need to do in order to be successful. These journals, along with all worksheets and warm-up exercises, are kept in personal files that are set out on their desks prior to each class period and then collected at the end. Collecting the students’ work each day eliminate the possibility of forgetting or misplacing important documents in between periods or leaving items at home. In this way, the focus is on academic learning and students are able to devote their energies to the subjects being taught without worrying about paperwork details. 

Organization and classroom management meet through Ms. K’s ticket jar system. When students are demonstrating positive behavior in staying on task, Ms. K gives them a ticket which they then write their names on and place in the ticket jar. On Friday’s Ms. K draws a ticket from the jar and awards the winner with a prize. The more positive behavior displayed, the more chances students have of winning. Rewards become large-scale (class wide parties) when entire class periods demonstrate positive behavior and excellent study skills. Classes that accomplish this get a star on the chart in the back of the room. The class with the most stars at the end of the month wins a party!

Along the same classroom management theme, Ms. K creates a safe and learning conducive environment with her classroom decor, atmosphere,  and posting of classroom expectations. Soft Christmas lights line the wall space above her large window on the far side of the room. Seasonal decor embellish plain tables and counter spaces as well as empty windows. Colorful posters and student work are neatly displayed on most walls. As Ms. K has gotten to know her students, she has asked their preferences on noise levels and light brightness. As a result, some classes have music playing softly during warm-up exercises and individual work times. Other classes have the lights dimmed during individual work time. Accommodations like these are class specific and only changed when students can come to agreements on what they want their learning spaces to be like.  Students continue to take ownership for their environment by determining their own classroom expectations and then writing them on posters which hang on the front wall. Expectations like, “Listen to others” and “Ask Questions” are phrased positively and encourage students to engage in the classroom.

Ms. K’s organization and classroom management techniques have given her students the freedom and comfort of learning in a safe space that promotes their specific learning preferences. I hope to implement similar strategies in my own classroom so as to encourage students to take ownership of their own learning and promote academic growth.

The Classroom as an Environment for Learning: A Holistic Approach

H-3 Honor the classroom/school community as a milieu for learning. When the classroom and school community are respected as places of learning, student success is promoted and learning is uninhibited by unnecessary restrictions.

In the lesson plan presented below, students are encouraged to work collaboratively within groups of 3-6 students to solve for recursive equations given a sequence. Each group was given a different sequence and asked to solve using dry erase markers directly on their tables before then copying their work into math journals. In this way, students teach each other by discussing and writing their work for each other to see. Having completed their assigned problem, groups were then asked to review the work of at least two other groups and writing the others’ results in their personal math journals. Finally, one group was selected to demonstrate their work and rational in front of the class.

Recursive Review Lesson

IMG_20130117_113030Students were uninhibited by the tables as work spaces since they were able to use as much space as needed. Students were also able to quickly erase mistakes and make corrections. When the groups were satisfied with their own work, they were asked to leave it on the tables for other groups to review.  In reviewing the work of others, students were able to look to the entire class of peers as a resource for learning. They could learn from the work of others, ask questions, and then take notes in their personal math journals.

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Finally, one group of students was asked to come forward and present their work in front of the class. Though their work was observed by several groups, the specific sequence they were given was at a level of greater difficulty and importance. In this way, all students were able to benefit from a more instructional time of learning (peer to peer instruction). The teacher, instead of directly instructing, led the presenting group through a series of questions about how they had solved their problem and why they were able to perform the operations they did. Not only did the students presenting have the opportunity to grow in communication and further understanding of their own work, but the entire class benefited from their work without having to struggle through it individually.

 

Through this lesson, I learned the power of peer instruction and collaboration. The teacher need not be the only source of learning. When students are in an environment which promotes peer communication and creativity in displaying their learning, they are able to thrive and grow as learners.

In the future, table tops will be continued to be used as resources for displayed work and examples. Additionally, I plan to continue selecting students to present in front of the class to demonstrate their work. I will especially look to encourage less confident students to do so (in content they display proficiency). I believe this to be one way to promote a healthy learning community where all students are valued.

Teacher Observation 1

 

Teacher Observation Reflection

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Setting: Middle School Math Classroom

The teacher I am working directly with, we will call him Mr. T, demonstrates excellent teaching techniques and classroom management methodology.

Mr. T primarily uses differentiated instruction to enable the students to take ownership and responsibility for their own learning. The format of his class period might start with a 10min instructional period where he provides students with new material, then 15-20min of collaborative work on assigned problems (while he formatively assesses and answers questions while working with struggling students). After this, he would have 10min for students to demonstrate their work to the class on the projector, and finally the remaining time will be spent getting feedback (exit slips) and answering or asking questions. This format is student centered and very much follows a differentiated instructional approach.

Mr. T pushes students to take ownership of their learning by insisting and reminding them to ask questions, to see him during lunch for further clarification, and come talk to him after class if necessary. He teaches them to ask specific questions, slow the teacher down, and “manage the teacher” instead of allowing the teacher to “manage the student”. Furthermore, he sets high expectations for them. They are to bring their own supplies, (i.e. calculator, ruler, pencil) in this way, preparing for their learning. Students are also required to take notes and write key terms and important ideas in their math journals. By requiring students to practice good note-taking, Mr. T is preparing them for their future high school classrooms. He will sometimes stop the “math lesson” to discuss a reason for the classroom norms and how they will help the students in their future lives. For example, Mr. T has said, “Ladies and gentlemen, I see a lot of people looking down at their notebooks or out the window and Student1 is answering one of my questions. Now, here, the expectation is that when someone is talking, you look at them. You do this to not only hear what they are saying, but to show them honor. In this way, you communicate your respect for them and what they have to say.” Discussions like these are a proactive approach to classroom management. In this scenario, Student1 was talking, and while the other students were not talking over Studen1, they were not paying attention to what was being said. Mr. T uses opportunities like these to reiterate the classroom norms and to encourage positive behavior. Additionally, Mr. T takes the time to provide positive feedback when students are demonstrating the desired classroom behavior. For example, the other day a very shy student (Student2) raised her hand to ask a question during an instructional segment of the lesson. Mr. T stopped and thanked the student for asking the question. Later, as Student2 was working during a collaborative work time, Mr. T made it a point to go tell the student that he was proud of her. When she asked why, he simply said he was proud she had been brave enough to ask him a question.

So then, Mr. T uses his teaching time and classroom norms to create a positive environment of where the students are responsible for their own learning.