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

Posts tagged ‘Differentiation’

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.

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.

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.