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

Posts tagged ‘Teaching Strategies’

Beg, borrow and steal?

As a new teacher, I’ve been told I should beg, borrow, and steal all materials I can get so as to not “reinvent the wheel”. I have found that this year, I have reinvented the wheel countless times, taken the wheel apart (kept what I wanted and trashed the rest), borrowed quite a bit, and screen-shot/stolen much!

Currently, I am in love with this 2nd-year teacher’s work on foldables in math journals. She is extremely creative in making the journals not only colorful and interactive, but also great study guides for students! With foldables, students are given self-made vocabulary, math facts, questions, and definitions all in a delightfully organized way! Ugh, someday I will be cool like her. For now, I will steal her ideas to help jazz up the end of the year.

Here is a lesson I am planning on using tomorrow in my Math 8 class: Exponent Rule Book

I’m pretty stoked, and I hope my students will see this as a helpful tool as they “study” for their unit test on Friday.


Zombie Apocalypse! Math games

Today I played a game I’ve re-named as Zombie Apocalypse!

I was inspired by this blog Math = Love who in turn found it on this blog: Out Rockin Constantly

My students loved the game! It helped them use some of their skills on evaluated expressions with exponents and develop new skills by seeing patterns and problem-solving. Everyone was involved as they did their own work on their personal whiteboard and the class quickly became a room of zombies with a few lone survivors.

The game was amazingly simple and can be adapted to any topic. Read the description by Nathan Kroft in Out Rockin Constantly for more details!

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.

H5- Honor student potential for roles in the greater society

H5- Honor student potential for roles in the greater society

Teacher-candidates prepare students to be responsible citizens for an environmentally sustainable, globally interconnected, and diverse society.

This means that teachers should teach in such a way so as to guide students to be responsible people in diverse society. In the math classroom, this includes thinking critically so as to solve real-world problems (such as environmental sustainability).

As a math teacher, there has been numerous times in which my classes have engaged in problem-solving exercises and activities. As a warm up before starting a unit on algebraic functions, I used a youtube video about the Enigma Machine. This machine was used in World War II to encode and decode messages by the Germans. The video, along with explaining the mathematics surrounding the machine, provided historical background and global facts surrounding the machine and its uses. The clip also explained how different countries had attempted to crack the code so as to intercept messages.

Four-rotor German Enigma cypher machine, 1939-1945.

This short film allowed students to think of mathematical topics as globally interconnected- a universal language. Following the films, students worked collaboratively to solve coded problems (functions with inputs and outputs) and determine the code (function rule). In working collaboratively, diverse students gained insight from multiple perspectives and were encouraged in their mutual respect for the skills of others (responsible citizenship).

Through using the video as inspiration and encouraging group work, I learned the power of collaboration among students and effective ways of motivating student learning. In the future, I would like to use this video as an inspiration for students to write more about what they learned and express their understanding in written reflection.

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.

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.