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

Archive for February, 2013

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.

Functions Bingo!- Just One of Multiple Instructional Strategies

H2- Honor Student Access to Content Material


Teacher-candidates use multiple instructional strategies, including the principles of second language acquisition, to address student academic language ability levels and cultural and linguistic backgrounds.

As a teacher, I use differentiated instructional strategies to target students’ different learning styles. I am also conscious of my academic language when presenting new vocabulary. In introducing new vocabulary, I used the principles of second language acquisition as a guide so as to meet students at different readiness levels.

For the last week we have been working through the algebraic concept of functions. To many of my students, this is a brand new concept. There are many different terms of vocabulary associated with functions and thus care was needed in teaching students this material. First, I had all student copy definitions into their math journals and phrase them in their own words. With each new day, we reviewed these terms through conversation and applied their meanings to new content and perspectives. In this way, I implemented the first of Barcroft’s Five Principles of Effective Second Language Vocabulary Instruction: Present new words frequently and repeatedly in input.

To differentiate the instruction by learning style, I used several different methods of instruction. One of the latest lessons I did consisted of students playing Functions Bingo! A few days ago we had a half day of school and as I was discussing the upcoming day with one of my students and suggesting we play a math game, he offered that we play bingo. As I thought about it, I found that bingo could easily be adapted to be an effective kinesthetic and visual way of reviewing input/output vocabulary associated with functions.

Bingo Functions   3by3 Bingo board

I gave each student the same bingo board (conventionally, in bingo, each person has a different board) and a different function (of the form: f(x)=3+x ). Students were given colored chips to place on their boards. In the front of the room, I had two dice: one red for negative numbers and one green for positive numbers. When I rolled both dice on the document camera, students were asked to determine the sum of the numbers and use the sum as the input of their functions. For example, if I rolled -1 and 3 students needed to determine the sum to be 2 and use it as the value of  in their function f(2)= … if the output of the function, given the specific input, was on their board, they could place a chip on that space. Once one student got a bingo, (three in a row/column/diagonal) I had all students dump their chips of their board and start again with a new function. The first student(s) to get three bingos won candy.

In this way, students were evaluating many functions at different input values through the context of a competitive, kinesthetic and visually stimulating activity. Additionally, with each dice roll, I used the words “Use this input and determine your function’s output, if you put this in, what comes out?” Thus, I frequently used functional vocabulary throughout the game.

Through the game of functions bingo, I have been able to introduce the concept of functions using multiple instructional strategies to meet students of different learning profiles and readiness levels. In the use of a verbal and collaborative game, I have also implemented principles of second language acquisition to address student language ability levels.

Math Riddles- Collaboration within the school!

E2- Exemplify collaboration within the school

Teacher-candidates participate collaboratively and professionally in school activities and using appropriate and respectful verbal and written communication.

As a teacher, I regularly engage in activities, meetings, and school events in a positive and professional manner.

IMG_20130205_150637One way that I have been able to communicate and interact with students outside of the classroom and in different grade levels is through the creation of “Math Riddles”. In articulating mathematical concepts in a fun and engaging way, I have given students a way to put their problem solving skills to use and see math as associated with fun! Each week I have posted a new math riddle outside the cafeteria with space for students to sign their names after presenting me with the correct solution. Rewards have been given to the first few students to answer correctly. At first the riddles were solved by student in my classroom, then, after a few weeks, several students from other classes began to solve them as well.

While this is just one way to reach out and connect with the student body as a whole, (I am also volunteering with the track team, regularly participating in staff events/meetings, and collaborating on lesson planning with other math teachers and in sharing resources) I believe it is an important source of connection.

In creating the math riddles, I am making math more accessible to students as well as making creating a specific reason for IMG_20130205_150657conversations with other students. I have learned to use games and riddles such as these to present math in a respectful and engaging way. Just this week, a student presented me with a math code that he had created, after looking it over I asked if I could use it as my next math riddle. How exciting to be able to not only challenge students, but use student designed material to do so!

In addition to presenting math in a new way, this supplemental material communicates my desire to be present in the school and be more than just a teacher confined to the classroom.

IMG_20130205_150644In the future I will continue to make my presence known at the middle school through additional math riddles, volunteer opportunities, and in collaboration with school staff and students.

Cultural Beauty in Education

Jewish School Visit

I recently had the opportunity to visit a private Jewish school and observe several math classes in progress. This was a fantastic experience and unlike any observation I have done in the past. Walking into the school was like entering a completely unique culture and community of of david

The school was for early-childhood through 8th grade and as such exhibited many characteristics of an elementary school with art and student work throughout halls and entryways. Signs and labels were written in beautiful handwriting or text in both Hebrew and English. The walls and floors were colorful and lively.

communityClassrooms could be described as what you might see in any elementary or middle school except for the added Hebrew phrase here and there. However, the culture and community in the classrooms were quite unique. Class size was considerably small which gave opportunity for a comfortable and familiar atmosphere. Some classes had as little as two students! In this case, the teacher was able to work directly with individual students, customize lessons, and present unique challenges.

Because of the fluid transition from elementary age classes to middle school, several elementary school characteristics were present in 6th and 7th grade classes. For example, one teacher had her students take their math textbooks and notebooks and circle up on the floor facing a floor whiteboard. In this less formal setting, the teacher proceeded to review vocabulary words and previously discussed material.

Another unique aspect of classes at this school is that class periods are much shorter (38 min) than the typical public school time block. This shortens the instructional time considerably, and yet the teachers are still expected to keep up with the state standards and timelines.  The reason for less time per period is the additional Jewish cultural classes taught during half of the day. Students at this school spend extensive time learning about and partaking in religious traditions, studying Jewish history, and learning Hebrew. In this way, students’ cultural backgrounds are deeply honored and provide the foundation upon which their education is based.

Because of the school’s strong religious and cultural bonds between curriculum and learning community, parent and familial support is very strong. Faculty and administrators of the school regularly inform, involve, and collaborate with family and community members in each student’s educational process. Social justice projects and multicultural field trips are peppered throughout the students’ academic career.

This school’s commitment to their faith and cultural roots allow faculty to create a beautiful harmony of family and education. Though the set of values may be different per school, I believe other schools may take a lesson from this and strive to create a similar culture of student value and community outreach.

Family and Community Involvement in Math Class

H4-Honor family/community involvement in the learning process


Teacher-candidates inform, involve, and collaborate with families/neighborhoods, and communities in each student’s educational process, including using information about student cultural identity, achievement and performance.


As a teacher, I seek out support for my students in all areas including their families. I recognize that students are greatly influenced by their families and communities both in life and academics. For this reason, one of my goals as a teacher is to interact with parents in a positive way, providing them with feedback on student learning and opportunities for ways for collaboration in learning.

At the middle school we recently began the second semester. As a way of introducing myself and initiating conversations with parents, I sent out a newsletter both electronically and in printed form. In order to ensure families received the newsletter, I asked students to have a parent or guardian sign the newsletter and bring it back to class. The class period that turned in the most newsletters (by percentage of class) received cupcakes!

Since I teach two different classes, algebra and 8th grade math, I sent out two different versions of the newsletter explaining the role I play in class as well as what students have accomplished over the last few weeks and what the game-plan is for the next unit.

2nd Semester Newsletter Algebra     2nd Semester Newsletter 8th Grade

These newsletters demonstrate an inclusive approach to students’ learning. In response to electronic copies of the newsletter, I received several confirmations by parents expressing appreciation for be included in their student’s learning. This opened the pathway of communication and allowed parents to view me as a support for their children.


In creating the newsletters, I was conscious of how I chose to present myself as an advocate for students’ learning. I also wanted to include evidence of students’ learning and give a specific example of student work and achievement (see newsletter attached for picture). I did this by integrating a picture of student work. In this way, I expressed my excitement of student accomplishment and appreciation of student work ethic.

Posting the picture provided an opportunity for students to be proud of their work and learning. By summarizing what the students had accomplished in the last few weeks, students could reflect on how much they had done. Additionally, by outlining the next unit, students can be confident in what they will be doing in the next few weeks. This provides parents and guardians the opportunity to discuss math topics with their students and investigate the extent of learning taking place.

As a teacher, I value family and community involvement, knowing that these are two great influences on the lives of my students. It is my goal to involve family and community members in the learning experience of my students and promote positive discussions on classroom work and achievement.


(Above picture not of actual students)

Technology in Math Class: Function Machines and Jeopardy!

P4-Practice the integration of appropriate technology with instruction


Teacher-candidates use technology that is effectively integrated to create technology proficient learners.

As a student teacher, I regularly utilize technology as a catalyst for learning and as a way of enhancing my instruction. In this way, students are given the opportunity to see and use technology as a learning tool.

In my 8th grade algebra class, we have begun a unit on functions. As a way of introducing the “big picture” concept of what functions are and how they work, I decided to show a short Youtube video created by several middle school math teachers.


This video is quirky and engaging visually (there are no words spoken). Explanations and definitions are given in written text along with humorous and cliché sayings. The end of the clip provides an opportunity for students to interact by guessing specific outputs and function rules. In this section of the video, I paused the video and asked for student guesses.

After the first introduction day, and in the second lesson, I used the game of Jeopardy to motivate student learning and provide an interactive way of reviewing key concepts associated with functions. I used a Powerpoint template to customize each question to fit our exact content and new vocabulary.

Jeopardy Functions!

Students were asked to form teams at their tables and then were given a question to answer in 1 minute. If the table answered incorrectly, then the question was bumped over to the next table. Because students had to determine answers to each question (just in case the first group got it wrong), all students participated in every question.

In both examples, technology was used as a way of enhancing the learning experience of students by providing opportunities for students to actively engage with the content.

In creating these learning experiences, I learned how technology can be used to instruct the class as a whole and yet engage individual students. This allowed me to gain experience in presenting math content using different methodologies other than direct instruction.

Through these two lessons, students were able to have new content presented in a lively and interactive way. The visuals and humor presented in the video and the group competition in the Jeopardy game alleviated some of the fear associated with new mathematical content and instead created a culture associating learning with fun.

As a teacher, one of my goals is to identify students’ fears and insecurities related to math content and provide transformative approaches to teaching so as to create a safe learning community. I plan to do this by using technology as a tool for instruction and student engagement, in this way, creating technology proficient learners.

Student Diversity Honored in Design of Application Activity

P-1 Practice intentional inquiry and planning for instruction.

Teacher-candidates plan and/or adapt standards-based curricula that are personalized to the diverse needs of each student.

As a student teacher, I create curricula based on standards and the readiness levels of my students.

Savings Problem

savings accountIn my 8th grade honors classes with Algebra students, we continued to work on recursive sequences using contextual problems to relate the concept to real life scenarios. I created a savings account activity and word problem where students were asked to determine how long it would take them to save $500 under different circumstances. Previous to this lesson, we had worked on recursive sequences for about 5 class periods and I had determined student readiness levels through formative assessments. In the activity, students were separated into 6 different groups and asked to solve one of two word problems. The two word problems were formulated for different readiness levels. The first problem was for those who were not quite proficient in finding explicit formulas and the second problem was for those who were proficient.

This activity was based on the standard:

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.

Students were asked in both problems to find the explicit formula of the savings scenario they were given and then find the term when the balance reached $500. Those who were not proficient received additional aid in interpreting the problem as well as steps to finding the explicit formula. The sequence they were given was arithmetic and thus was less challenging mathematically than the second geometric sequence. In this way, students in the first readiness group were able to focus more on interpretation and application of their skills rather than the more complex mathematics operations.

Creating this activity provided me the experience of using formative assessment and knowledge of students’ readiness levels to plan for instruction which challenges students appropriately. I was able to isolate specific learning objectives and provide the opportunity for diverse students to be individually and collaboratively successful.

This activity allowed students to build on their prior understanding of recursive sequences through contextual examples. Additionally, students were able to collaborate in their groups, creating a learning community, and using their peers as resources for their academic growth.

Finally, this activity is exemplary of future lessons that I will plan to challenge diverse students. While this lesson was generated to meet student readiness levels, in the future, I will use similar strategies to fit diversity in learning styles, learning profiles, and multicultural backgrounds.