H5 – Applications of Learning Beyond the Classroom

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 the teacher helps students make connections between the mathematics taught in the classroom and the applications in the world around them. Students should be able to see the connection between the math and their surroundings, essentially answering the question “Why do I need to learn this?”

The evidence presented is a collection of activities which help students make connections between the mathematical concept and the world in  which they participate. Three units stand out as particularly relevant to my students. The first, was the use of polynomials, students were exposed to polynomials as they are used at Pixar animation and the United States Navy Office (USNO) to calculate the location of the moon and on a given day. The second application is for studying complex numbers. I provided students with access to a link which explains how trigonometry and complex numbers relate to spring systems in engineering. Finally, the most important is how exponential functions and the use of logarithms. On sample is to model how the human body decays drugs over time and another sample is how earthquakes are rated using the Richter Scale. For both, we discovered some shocking outcomes using mathematics. Making content relatable to students improves engagement and improved engagement increases learning opportunities.

Links to Evidence:

Each of these samples came from my own curiosity of how the content relates to the world around me. Generally, my students are interested in space, art, science, computers and engineering. In creating this content and hooking the students into participating in the activities for the real world application according to their interests, I have help students articulate the purpose for the content within their immediate future. For me, I learned about many applications of these tools too. For the Pixar Animation information, I contacted Tony DeRose, a Research Group Lead at Pixar Animation. I learned how to bring the world of mathematics into my classroom directly from the industry leaders themselves. With the creation of the Richter Scale Activity, I have become familiar with the common misunderstanding about how the Richter Scale actually works. Interestingly, a one point increase on the Richter Scale is NOT 10 times the previous energy, but rather about 27 times more energy. With Drug decay, students learned that theoretically, a drug will NEVER leave the body, its concentration just decreases.

Students benefit from the application because of their immediate use and interest in the topic. When students have some applications for the work they are doing in Mathematics, they become more interested in exploring more about the topic. I have become successful if I have interested one student to pursue a STEM career and they have used the tools learned through the application lessons in my class.

One area of weakness in helping students realize the potential for this topic is having them research the applications themselves. While I am truly interested in the matter, students will most benefit if they are able to do the research themselves and make the connections. One barrier is my fear that they will not be able to find inaccessible content because the mathematics is too advanced for their understanding. I could promote the learning by encouraging these students to do a project in which they find the application of these tools within the world and talk with an industry expert about the application of these tools themselves and ask questions to build understanding. This may pique their interest even more and teach them about the applications of learning beyond the classroom.


A Day in the Life of a Student

Today’s goal was to follow the school day of a sophomore student throughout their day. To protect the identity of this student they will be called MW. The goal was for me to observe five different classrooms during a six period day. Within each class, I was looking for teaching strategies and tools to use in my own classroom. At the end of the day, I asked MW some questions about their day. While the student interview was only a case study, MW provides several helpful suggestions to improve their learning experience.

The morning classes include Math, English and Biology. Within all of these classes, I noticed good responsive listening. Teachers would ask student questions and then summarize what the student said and repeated the comment to the class so all could hear. In English, the instructor was careful about making smaller comments to encourage conversation in class. The activity in English was to analyze a short story, a topic which can be shallow or deep depending on the students’ understanding. The instructor contributed to the classroom discussion as well to prompt students to analyze the short story more deeply. At the end of the classroom discussion, the teacher make a more sophisticated analysis than the students were able. This exposed the students to higher expectations. The teacher used OneNote to distribute materials to students and students were able to access copies of the teachers journal.

In math class, students started with an entry task to summarize their homework and clarify questions. The class activity was to debrief a challenging problem from the homework and clarify understanding of the basic elements of trigonometric ratios. Students had questions to prepare for a quiz the next day. The activity included a worksheet where students needed to use prior work to plot data and discover a relationship of a sine wave. The end of class, summarized their understanding of “accuracy” and careful procedures for getting more reliable results.

Biology was a very busy class, the lesson was very engaging and highly differentiated. One comment about the lesson that I particularly enjoyed was the presentation of the entry task. The day’s objective was clearly posted on the board and students were able download a copy of class activities from OneNote. Student’s shared out the Initial Thinking questions by popcorn method, students would choose others to “keep the conversation going.” The instructor was clear about the importance of understanding the Carbon Cycle and other biological systems and stated that this would be tested on the End of Course Assessment (EOC). The class activities included a dice game and students would mimic a carbon molecule in the journey throughout the day. Students seemed to enjoy the activity and learned about the many processes of the carbon cycle. Students were able to articulate what happens to a carbon atom as it moves from the sun to plants and then into animals and back into the atmosphere. The activity was followed by a debrief so students could articulate their learning. After the debrief, students watched a video that repeated the material again. This instructor was very smart about providing many learning opportunities for important content. The class ended with an exit slip that informally assessed students learning from the day. Students were instructed to make a carbon cycle as homework to use to prepare for their EOC test in the spring.

After lunch, MW was schedules to attend Current World Problems, Spanish and Health. Mostly, this class period was work time for students to finish a group project about government systems since 1500. MW shared with me the group project and the required elements for a grade. Before work began, students the teacher had a student present a “SHIELD” which allowed the student to comment about their past present and future. The intention of the exercise if for the class to know each other better. Students can share about their interests and goals. Next the teacher gave instructions of work time, he was very clear about presenting a product at the end of the period and made suggestions for groups to produce exemplary work (i.e. Review each other’s papers, get organized and then work together to produce a product). Essentially, the teacher anticipated potential shortcomings and took preventative measures to work through the project. We also talked about some of the functionalities of the grading system to optimise its use.

Next was Spanish in which students entered the class and listened to the instructor through an immersion lecture. The teacher spoke little english when teaching the material, but frequently broke into english to emphasize important points. He continued to discuss why immersion was used and why he believed in this philosophy for language acquisition. Most of this period was direct instruction and lecture based. Students did not respond frequently and when they did, it was brief.

The last class was Health where there was a lot of individual work time. Some of the lesson was direct instruction where students had access to a OneNote document to help follow the lesson and take notes. The period was an introduction to nutrition and involved the explanation of a project. The teacher clearly displayed the objective and the required elements of the project so that students knew what was expected of them throughout the unit. Many students who were in previous classes were also in this class. It was interesting to note the difference in classroom dynamic because of these students. Those who were quiet in the morning, we not more rambunctious and hard to get back on task. Teachers should consider this when teaching.

Finally, after the day, I asked MW some questions about their day. This student liked the biology lesson because there was an activity where she could move. She mentioned this was uncharacteristic of her typical day, but when she moves around, she claims she remembers more. Typically, she enjoys her math class because there are clear instructions and she tends to work hard. MW is also in a robotics class (which I did not attend) but this takes up a lot of her time. Sometimes teachers don’t understand that this is a challenging workload. Along these lines, MW claims that she becomes overwhelmed when projects are overlapping and she does not have the tools to manage this better. MW recommended teaching students in advisory time management and note taking skills. She also would like a space after school to get some work done and manage her schedule. Throughout the day following the student, I realized that the school has very high expectations for the students and teachers don’t often repeat information. This school expects students to understand verbal instructions the first time, which is a challenging skill for many people, even adults.

Bloggary #5: Writing Workshop for Math

Daniels, Zemelman and Steineke (2007) suggest teachers implement a writing workshop in classes providing critical feedback for emerging writers. Writing in a math class may seem foreign to an outsider, but writing can help students articulate their mathematical understanding. In fact, mathematical language is highly technical and best practices include direct writing with clear concise explanations and supporting graphics. The style of writing is very different from expository or novels and requires great attention to detail.

High school students taking Advanced Placement (AP) classes are exposed to a style of writing called Free Response Questions (FRQ’s). For math classes, many students are not previously exposed to FRQ writing which requires a direct style supported by evidence and facts. Educators wait until junior and senior level classes to present FRQ style writing which is unfortunate because the thought required to write such documents is invaluable. Teaching students to write FRQ’s using the Writing Workshop would provide students with tools to exceed standards on AP exams. Math writing tends to be a little bit more focused than expository writing since students are limited by their content knowledge. However, if teachers take a “toolbox” approach to the learning, students can access many previous skills. The “toolbox” approach is where skills are learned and then stored in a student’s toolbox as a collection to access when approaching challenging problems.

Implementation of a writing workshop would not be challenging for an instructor. FRQ’s are already written with grading materials easily accessible for teachers and provided by the College Board (the testing agency for AP tests). Since all previous tests are published and accessible online, students will have no problem selecting a topic of their choice to write a full response. Teachers can select specific criteria for grading the writing aspect while the published solutions can be used for the accuracy of the mathematics. Features math teachers may look for include; precision of language, supported claims (words, pictures or other) and organization of thought. In my classroom, I envision a workshop where each student works on up to five FRQ’s at one time and uses partner grading or teacher conferencing to work through many technical writing issues. Students would be presented with a model and brief writing instruction and their work packet would be graded for specific writing elements to help students articulate their argument and write clearly.

Since teaching a topic is a true indicator of understanding, the project could be extended by having students write their own FRQ’s and create a grading rubric for future students. Students could set goals for what they hope to gain from their writing, possibly understanding a challenging concept more deeply. Students need not be limited to the domain of calculus, but rather they could use their creative license to write a question for students in lower classes. This intense focus and peer review helps students engage in the material for a longer period of time which increases the chances of comprehending the content (Borich, 2014). Since the AP exam expects content mastery, the writing workshop technique can help students engage in challenging material for longer.


Borich, G. D. (2014). Effective Teaching Methods: Research-Based Practice (8th ed.). Upper Saddle River, NJ.: Pearson Education, Inc.

Daniels, H., Zemelman, S., & Steineke, N. (2007). Content-Area Writing: Every Teachers Guide ( ed.). Portsmouth, NH: Heinemann.

Bloggary #2: Struggling Readers

“Math textbooks have the highest content load per sentence of all the secondary textbooks” (Barton and Heidema, 2002; Daniels and Zemelman, 2014, pp. 189). It is no wonder that students are struggling, textbooks present complicated reading beyond the level of understanding for most students. One of the major challenges with mathematics is being able to determine what is important and what is not important as well as being able to visualize the concepts more concretely. Steve Zemelman (2014, pp. 277) discusses an encounter with a student who is struggling to understand the dense description of a falcon preying on rabbits as it relates to physics. Zemelman provides help to the students by having students turn the words into mental pictures, ultimately decoding the situation. In many cases, students struggle further by failing to accept help from a teacher due to a poor teacher-student relationship. In return, students often feel that it is acceptable to give up, not even making an effort to find a solution.

Building relationships with students is one of the main reasons I wanted to become a teacher.   The impact a teacher can make on a student’s life is enormous. Motivating students to be ambitious and work to resolve problems is not only a skill for success in school but also in the work place and as a contributing member of society. Students can learn this by having a community-oriented classroom.  Teachers should 1) create a trusting atmosphere where it is safe to take risks, 2) organize learning so students can help one another, 3) provide students opportunities to take on classroom responsibility, 4) facilitate connections between class and student life, and 5) use engaging content to help student fall in love with the subject (Daniels and Zemelman, 2014, pp 205-206). Further, with a community-based classroom, students are interdependent encouraging each other to accomplish common goals.

Returning to individual reading strategies of dense texts, I think the modeling strategy of Think-Alouds during direct instruction helps students sift through complicated material. Since math and science curriculum is tested on a one size fits all, it is challenging to amend the text specifically to student needs.  More important, however, is to help students think critically through challenging texts and word problems. Even with a B.S. in Mathematics, I often struggle to grasp the concept of a high school word problem when I first read it. I take myself through a checklist of what is needed and what can be discarded and document everything I need. By demonstrating this to students, they can acquire the same skills I have developed over the years. The challenge with displaying knowledge in math is that students are required to have good understanding of the topic, before they can articulate a solution. If a student knows what is happening and how to reach a solution, they can then comprehend all of the important parts of a problem.

By building relationships with students, teachers can help provide skills for general problem solving and decoding text. Another strategy teachers can use to present material is by providing an article and then posing very general questions regarding the situation. By taking a literary approach, students can integrate their outside knowledge and problem solve before the math is presented. In an eighth grade class in Chicago class, Jacqueline Sanders presents students a simple question with a complicated answer, “Where does the money from your job go?” (Daniels and Zemelman, 2014, pp. 262) Students were forced to read though several articles, websites and tax code books to use math in determining where the money from a job goes. Students could choose the level of difficulty of their sources, but ultimately they used math and reading skills to sort through both important and trivial facts to answer their question. By removing the textbook, Ms. Sanders was able to provide a different type of math instruction.

In all, students need assistance in decoding meaningful reading. The reading does not need to be from a textbook either. Students should have applicable outlets to display their learning and reading. Finally, by modeling the thought process when reading through high content reading passages, students can learn to decode and comprehend what they are reading in order to apply it to the content area.



Barton, M. L., & Heidema, C. (2002). Teaching reading in mathematics: A supplement (2nd ed.). Alexandria, VA: ASCD.

Daniels, H., & Zemelman, S. (2014). Subjects matter: Exceeding Standards Through Powerful Content-Area Reading (Second ed.). Portsmouth, NH: Heinemann.