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

Posts tagged ‘learning styles’

P2- Practice Differentiated Instruction

P2- Practice Differentiated Instruction

Teacher-candidates apply principles of differentiated instruction, including theories of language acquisition, stages of language, and academic language development, in the integration of subject matter across the content areas of reading, mathematical, scientific, and aesthetic reasoning.

This means that as a teacher, I construct my lessons by student interest and readiness, carefully integrating new vocabulary and academic language. I have done this by creating an engaging activity in which student learn new concepts around quadratic formulas. In their reflections on the activity, students were able to use their newly acquired vocabulary with the language function of “describe”. This lesson, as well as the student reflections, gave students the opportunity to develop fluency of the academic language surrounding quadratic functions such as parabolas, projectile motion, and vertex. In order to integrate the theories of language acquisition, this activity used principals 3 and 4. The exit ticket limited the forced output during the initial stages of learning new words as well as limited the forced semantic elaboration during the initial stages of learning new words.



The student work sample demonstrates how students have used the new vocabulary and language function to show their understanding. The rocket portfolio packet demonstrates how students were given the opportunity to choose their role in the group activity. In this way, the lesson was differentiated by student interest. The lesson was also differentiated by individual readiness as I created the collaborative groups to be mixed ability leveDifferentiation 3Differentiation 2

As I created this activity, I learned how to engage students in math content and inspire conversation around quadratic equations in a safe learning environment. Students were able to learn the real-life applicability of quadratic equations by shooting a rocket and using an equation to describe its height. In the future, I would like to build on student reflections, by giving them personalized feedback.

O2- Appropriate Challenges in Math

O2- Offer appropriate challenge in the content area.


Teacher-candidates plan and/or adapt curricula that are standards driven so students develop understanding and problem-solving expertise in the content area(s) using reading, written and oral communication, and technology.

This means that as a teacher, I design and adapt my instruction, based on the standards, to give students multiple pathways of success. My instruction challenges students to use reading, written and oral communication, and technology in order to problem-solve and demonstrate their understanding.

In concluding and reviewing the topic of transformations in my 8th grade algebra class, I designed a lesson plan which gave students multiple ways of articulating their knowledge, while challenging them to work with transformations using multiple perspectives.

Transformations Lesson 5

Specifically, this lesson had the following objectives. Students will write the learning target in their journal (read/write), use the learning target to remain on task during group work (reflect/problem-solve), discuss the learning target and why it is important during the closing discussion (oral communication), and finally, demonstrate proficiency by completing the exit ticket (graphing, written communication).

Throughout the unit on transformation, you-tube videos and online graphing sites where used compliment instructional materials.


After teaching this lesson, I learned about the differences of my students’ learning styles and the methods they prefer in demonstrating their understanding. In the future, I plan to use similar strategies for effective lessons on other topics. In addition, I will use give students more freedom of choice (differentiation by interest) in demonstrating their understanding.

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.

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.

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.