Text Set #2: Calculus

Books

• The Humongous Book of Calculus Problems

W. Michael Kelley (author)

Grade Level: 12+

This book works through several calculus problems that cover every topic that is taught in a high school calculus and more. Textbooks can often seem dry, but this book provides comical commentary on every problem that makes math more approachable. I have used this book in my own calculus classes to provide another approach to confusing topics. This is a great book for anyone struggling in their calculus class or for those who would like some enrichment.

• The Cartoon Guide to Calculus

Larry Gonnick (author)

Grade Level: 12+

Gonnick is a mathematician who went to Harvard, but he has an excellent way of explaining the most difficult concepts in a simple way, especially in this book dealing with calculus. He uses graphics to clarify abstract topics, and he even throws in comedy to alleviate the stress!

• Calculus Made Easy

Silvanus Philips Thompson and Martin Gardner (authors)

Grade Level: 12+

Some people assume that calculus is hard because of its reputation, and it is hard if you are not mathematically prepared for this course. This book gives a short and sweet overview of the most important aspects in a high school calculus course. This could be a great book to assign for summer reading!

Websites

• Calculus-Help

http://www.calculus-help.com/tutorials

Grade Level: 12+

This is one of my absolute favorite sites for calculus! My high school calculus teacher showed me the first two videos listed about limits, and I still reference them today. This site is owned by the author of the Humongous books like the one mentioned above. The video tutorials are basic but accurate, which helps start people off on the right track when they are being introduced to the early topics.

• Calculus Derivatives and Limits Math Sheet

http://www.eeweb.com/toolbox/calculus-derivatives-and-limits-reference-sheet

Grade Level: 12+

Formulas need to be memorized, but a reference sheet can always be used to brush up those formulas that tend to be forgotten. This formula sheet has to be understood, so it may be a good idea for the teacher to walk through the sheet with your class. I have confidence that if students can read this formula sheet and understand it, they can read anything!

• Houdini's Escape

http://barzilai.org/cr/houdini.html

Grade Level: 12+

If students ever ask when they will use derivatives in real life, this is a great task to challenge them. Formulas are analyzed and derived to show students how calculus can be used to actually find the height of the concrete block Houdini should use to ensure he would not drown. This is a great project for a high school calculus class!

• Common Calculus Mistakes

http://mathmistakes.info/mistakes/calculus/index.html

Grade Level: 12+

A great way to assess students' knowledge is by having them point out the mistakes in people's work. This site gives intentionally wrong results so that people can try to find what went wrong. There are hints that can be seen by scrolling over the problem. The solution can be seen by scrolling over the answer section, and the correct piece is highlighted in blue. Students who really understand the material can really refine their skills with this site.

Articles

• Derivative of Area Equals Perimeter- Coincidence or Rule?

Rina Zazkis, Ilya Sinitsky, and Roza Leikin (authors)

Grade Level: for teachers

This article was written by questioning whether the fact that the derivative of the area of a circle is the circumference can be generalized to other shapes. Obviously, the derivative of the area of a square is not the perimeter of said square, but it is half of the perimeter. The authors provide several diagrams to show their findings, and I think this would be a fun read for math teachers at any level!

• Students' Exploratory Thinking about a Nonroutine Calculus Task

Keith Nabb (author)

Grade Level: for teachers

Have you ever wondered why points of inflection always seem to occur halfway between two critical numbers? Does this always happen? These questions elicit deep thinking about calculus and geometry together, and they can really enrich students' understanding. I think this article highlights what math is all about: asking questions and figuring out an answer using the skills you have acquired throughout your life.

• Understanding the Derivative through the Calculus Triangle

Eric Weber, Michael Tallman, Cameron Byerley, and Patrick W. Thompson (authors)

Grade Level: for teachers

Students often struggle with the fact that the derivative deals with rates of change. The authors of this article show that using the calculus triangle may help students learn what information we can learn from the derivative of a function. It is a way to show students that the derivative is more than just another function. It shows how two values compare to each other at any moment in time, and the triangle visually shows the slope between two close points.

• Delving into Limits of Sequences

Beth Cory and Ken W. Smith (authors)

Grade Level: for teachers

The formal definition of a limit may not be fully appreciated by students who are seeing it for the first time. Limits are the basis of all calculus, so it is important for students to know what limits are and how to find them. This article shows a way that two teachers have introduced limits to their calculus classes. Students arrive at their own conclusions by looking at several examples and non-examples. This is a great read for any calculus teacher!

• A Close Encounter with Infinity: Inventing New Mathematics

Keith A. Nabb (author)

Grade Level: for teachers

Apparently, I like articles written by Keith Nabb (see above article). This article focuses on another nonroutine problem about infinite series. Nabb was brainstorming ideas about how to enrich this fascinating topic, and he decided to ask students open-ended questions that allowed them to create series with certain characteristics or state why one could not be made. I think these types of questions could be asked at any level, and they help to assess true mastery of this topic.

Online Articles

• The Calculus Trap

http://www.artofproblemsolving.com/Resources/articles.php?page=calculustrap&

Grade Level: 12+

This article was very interesting to read because it obviously does not apply to every student, but it reveals the truth about the current high school curriculum. Students who excel through the curriculum quickly should not necessarily be pushed into calculus as soon as possible. There should always be more challenging problems that could enrich these students along the way.

• Masterful Teacher: How Calculus Became the Most Popular Class on Campus

http://www.edutopia.org/blog/masterful-teacher-jonathan-winn-calculus

Grade Level: for teachers

It is no secret that people need to be motivated to do something, and students are no exception to this rule. Jonathan Winn has found a way to get students excited about math, and his course is the most popular course on the campus of Crawford High Educational Complex in San Diego. He claims that it was very difficult at first to get students engaged, but he seems to know what can truly inspire students to want to learn math.

• Mathematical Firsts: Who Done It?

http://matharticles.com/ma/ma005.pdf

Richard H. Williams and Roy D. Mazzagatti (authors)

Grade Level: 12+

This is a fun read for anyone who is interested in the names of particular theorems or ideas. The authors take a look at several mathematical pseudoeponyms (Euler's formula, L'Hôpital's rule, and Pascal's triangle, to name a few). Each pseudoeponyms is accompanied by a description of why the person whose name appear in the title may or may not be the rightful owner of their discovery.

Reflection #10

It is no surprise that today's students are using the Internet for assignments and projects rather than books and other print sources. With all of this information at their fingerprints, they can easily find the president of Argentina or the gross domestic product of Kuala Lumpur in just a few seconds. However, this also comes at a cost thanks to the plethora of sources that are put online that are not credible. Students should learn how to sift through all of the hits on Google to determine what is factual and beneficial for their purposes. I liked the SAND method better than the ISSDAT method, mainly because it has fewer letters and they basically help students in the same manner. Research begins with a search, collecting as much information as needed. Then, students must analyze what they have found to sift through what is credible and most helpful for their learning. Finally, they have to note details such as the URL, author, date of publication, etc. I think this method should be reinforced each time students research something on the Internet because so many students do not realize that everything on the Internet may not be true, credible, or relevant to their particular assignment.

Motivation to Read

This article opened my eyes to the perception that adolescents have of reading. I am guilty myself of seeing reading as a primarily academic concept, even though I come in contact with a variety of media outside of the classroom. The questions that were asked in this study showed me how teachers can easily show students that they actually read more than they realize. By wording questions strategically, students can truly reflect on their interests and how they spend their time outside of school.

By asking students about their use of technology on a daily basis, teachers can not only see how they are using these technologies to read but also what technologies are used in their students' lives. I would say that most students have access to a television, computer, or phone, with many of them having access to more than one of these forms of technology. I would like to use this fact to have students interact in class electronically. From what I have observed, many high school students use Twitter on a daily basis. Twitter is a great way to search what is trending, and you can see what other people are saying about a certain topic. For example, one can search #quadraticequations and find what people are saying about this topic. When I just searched this, I saw one person say that it took him four days, but he now knows how to solve quadratic equations. He posted a picture of his work.

The best way to make students realize that reading is not as academic as they think is to reach out to where they are and allow them to read in ways that make sense for them. I would much rather read a description of the quadratic formula on an interactive website than try to memorize it from the textbook. By placing ourselves in our students shoes, we can find resources that truly interest out students. I want to focus on finding many different resources and allowing students to choose which resource they would like to use to learn more about topics in their course.

Web Resource #2

When I think of resources for calculus, I always think of calculus-help.com. I was first introduced to this site when I took calculus in high school. My teacher showed us the first video on the site about limits. I have watched several of the videos on this site, and they all have simplistic explanations of important calculus concepts. If I ever taught calculus, I would like to show my class some of the videos in class so the students can see another approach to these topics. The site even provides example problems that correlate to the AP Calculus AB exam. W. Michael Kelley created this site to help students feel more comfortable about calculus concepts, but he also wrote several other helpful books for other math classes. He has written The Humungous Book of Calculus Problems, The Humungous Book of Geometry Problems, The Complete Idiot's Guide to Precalculus, and many more books that can help anyone succeed in math.

All of the resources on this site are completely free, although you would have to purchase his books if you would like to have your own copy. Since the site was created by w. Michael Kelley, he has advertisements on the side for his books that could further help students learn math. Otherwise, the site is not commercialized. The site is appropriate for high school students, although many students who are not taking calculus would find the site fairly useless. While the site has excellent videos for topics like limits, continuity, and derivatives, there are no videos for integration, sequences, or series. These three topics are actually much more difficult for students to grasp than the ones that are addressed on the site, so I would like to see more videos posted in the future for these topics.

Text Set #1: Probability and Statistics

Books

• The Cartoon Guide to Statistics

Larry Gonick and Woollcott Smith (authors

Grade Level: 9 and up

This book makes statistics fun for anyone! The authors have created several situations that use play-on words to create comical stories that accurately present the material. This book could possibly replace the textbook.

• Focus in High School Mathematics: Reasoning and Sense Making in Statistics and Probability

J. Michael Shaughnessy, Beth Chance, and Henry Kranendonk (authors)

Grade Level: 9-12

NCTM published a series of books that help students focus on reasoning and sense making in various subjects. This particular text focuses on six investigations that show students how the skills they have learned in class can be applied to a real problem they might encounter themselves. It is a great resource for teacher to use in class with their students!

• Why Do Buses Come in Threes?: The Hidden Mathematics of Everyday Life

Rob Eastaway and Jeremy Wyndham (authors)

Grade Level: 9+

Students need to see math in real life in order to fully appreciate it for what it is worth. This book poses several questions and analyzes them mathematically to show its readers how people can use math as a tool to solve real problems. This is a great read for high school students and teachers alike!

Websites

• Probability

http://www.mathsisfun.com/data/probability.html

Grade Level: 7+

This site is great for showing complex topics in a very simplistic way. My favorite part of this particular page is the probability line because it shows a visual representation for the range of possible probabilities. The site also gives some practice problems that are multiple choice, and the solutions are provided!

• Gapminder

http://www.gapminder.org/

Grade Level: 9+

Teachers cringe when students always ask when they will use math in the real world. Well this site is great for these students! Gapminder shows interactive data that can be played over a timeline. For example, students can watch how CO2 emissions have increased in each country across the globe over the past 180 years.

• Statistics Online Computational Resource

http://www.socr.ucla.edu/SOCR.html

Grade Level: 9+

Students are comfortable working on computers in today's time, so having online resources can really engage the class as a whole. This site presents students with virtual experiments, games, and distributions that help students learn more about statistics in a fun, interactive setting.

• Math Goodies

http://www.mathgoodies.com/lessons/vol6/intro_probability.html

Grade Level: 7+

When students are confused about the difference between an experiment and an outcome, they can go to this site to help them refine these skills and test their knowledge. The site is visually appealing and includes a spinner and die that students can actually spin or roll to test what should be expected. There are even five questions at the end that can give immediate feedback.

• Braingle

http://www.braingle.com/Probability.html

Grade Level: 9+

This site is a great resource for students to use to practice applying probability to solve riddles. It provides a hint and answer for each riddle, so the students can choose how much they want to see before seeing the solution. These problems teach students to read carefully and analytically. Each riddle has a ranking for the level of fun and difficulty. It is a great resource for students to practice on their own!

Articles

• Fantasy Baseball with a Statistical Twist

Lori Koban and Erin McNelis (authors)

Grade Level: For teachers

This article shows teachers a great way to incorporate statistics in a project that is relatable to the students. The project has students analyze several different important traits of baseball players, and they must use this to inform their decision for their fantasy team. The article gives an excellent description of each step of the project, and the students should have a lot of fun doing these types of analyses.

• Rare and Exotic Probability Bugs

James R. Kennis (author)

Grade Level: For teachers

Students often arrive at the correct answer when calculating probabilities, although they may arrive at this answer using incorrect processes. This article analyzes a few of the most common mistakes that students can make in the thinking process, which teachers can anticipate when they are teaching basic probability concepts.

• Shooting Free Throws, Probability, and the Golden Ratio

Terry Goodman (author)

Grade Level: For teachers

Many people love watching and/or playing basketball, and students can use basic probabilities to analyze their free-throwing skills. The article shows many different approaches to presenting the data. For example, a Punnett square is used to show the probability of making both free throws, making the first and missing the second, and finally missing the first which means the second free throw does not matter. Functions are then created and graphed to show the probability of making a free throw given a certain percentage for each player. This is a fun way to get students to look at situations analytically to make improvements in the future!

• Analyzing Highway Speeding Data in the Statistics Classroom

Paul Laumakis (author)

Grade Level: For teachers

By the time students are taking a statistics course in high school, they are likely old enough to drive. Using a topic like this is a great way to show students how statistics can be applied in real life. The article looks at highway speeding data at two locations, and students should see the patterns that occur at both of these locations. They can then make decisions about how they should drive while they are at these locations, and they could even take this data to the police if necessary to show them how fast some people are driving through these areas. Statistics teachers should really consider giving this project for students to complete, not just for the practice of comparing two data sets.

Online Articles

• Stories vs. Statistics

John Allen Paulos (author)

http://opinionator.blogs.nytimes.com/2010/10/24/stories-vs-statistics/

Grade Level: 12+

This article shows the difference between posing hypothetical situations and analyzing real-life events. This highlights the fact that probabilities rare merely what you should expect out of a certain situation, not what is certain to happen. The author uses examples with which everyone should be familiar to show the relevance of statistics and probability. It is a good read for enriching those students in a statistics class.

• Probability of Playing College and Professional Baseball

http://www.hsbaseballweb.com/probability.htm

Grade Level: 9+

Many athletes have a dream of playing professionally and living a generous lifestyle. I am not saying that teachers need to crush their students' dreams, but I think that students need to know the reality of the dream for which they are reaching. This site gives a table of values from the NCAA with statistics about high school, college, and professional athletes. Students could possibly use sites like these to see where probabilities are found in their lives.

• Did You Know: High School Facts & Figures

http://www.stageoflife.com/StageHighSchool/OtherResources?Statistics_on_High_School_Students_and_Teenagers.aspx

Grade Level: 9+

Students often see statistics as a challenging subject that uses numbers as words. However, statistics give meaning to situations in real life, and this article helps students see how they compare to other teens in many different areas. This site shows how much students watch television, how students watch/read the news, how students spend their summers, and many more universal topics like these.

How to Teach Vocabulary to Students

The Bromley article for this week was a light read that was full of practical information for teaching new words to students. Math can be a difficult subject to learn and teach because students have to first understand what is being asked of them, a task that can be made easier if the teacher has the ability to effectively teach vocabulary. 

One of the first things I took from the article is that "English is three times larger in total number of words than German and six times larger than French." Many of the words are derived from other languages, and we tend to combine and fabricate new words daily. Despite the large size of the English vocabulary, I was shocked to read about how much simpler the English language is compared to other languages. Although there are several exceptions for word pronunciations, many English words can be phonetically broken down and spoken accurately.

However, one of the worst aspects of the English language in my opinion is the fact that words can have so many different meanings depending on the context in which the words are used. Product is a perfect example of this. In math this term is the result from multiplying two quantities together, but it might not have this specific definition when used in another context. Math is full of words that mean something very specific, and I think many students struggle to see the importance of using proper vocabulary when speaking and writing mathematically.

My favorite statement from this article: "It is more effective to teach fewer words well rather than several words less well." I just felt like this sums up everything that we have discussed in this article and in this class, as well. The whole purpose of teaching vocabulary is to increase reading comprehension, and this brings a whole new meaning to the phrase "less is more."

Building Strong Vocabularies

One of this week's readings talked about how to develop vocabularies in the classroom, a concept that I'm sure is important throughout any content. When I see a student who is frustrated in a math class, I try to ask them what exactly it is that is making them feel uncomfortable. The most common answer is, "I just don't understand what you are even saying!" This shows just how important it is for students to be able to talk the talk. I must do something as a teacher to help my students develop the proper vocabulary if they are to succeed in math.

The first strategy listed in chapter, Personal Glossary, is one that I fully intend on using in my classroom. I remember when I took geometry in high school, we had to keep spiral-bound index cards filled with definitions, theorems, and postulates. We were constantly being quizzed over vocabulary words! Once we had a firm grasp of these words, the problems suddenly became much easier to work when we could apply these theorems and postulates to solve problems.

Another strategy that could be used in conjunction with the Personal Glossary is the Verbal and Visual Word Association. This strategy reminds me a lot of the Frayer model. Students write the word, an example, a non-example, the definition, and possibly a picture to help them remember the word. This would be a great way to organize each individual word, and then the students can have all of these association arranged in their personal glossary.

Vocabulary is such a crucial piece of the learning puzzle, but I think focusing on developing vocabulary is an easy way to build confidence in students. Once they feel comfortable with vocabulary, they will hopefully feel more confident when approaching math problems.

Strategies for Developing Critical Reading

I enjoyed reading this chapter because I felt like I learned effective ways to get students to be critical of what they read. I really like the Polar Opposite strategy because there is no right or wrong answer, but students must support their reasoning behind their choice. However, I do not really see how this could be easily incorporated into a math class. Maybe I could have my students look at a student's work and have a Polar Opposite where "algebraically" and "geometrically" were my words. They would have to place where the student's approach lies on this range. 

Opinion-proofs are great for developing students' skills in showing support for their opinions. Again, there are no right or wrong answers if something is a true opinion. The reasoning fallacies on page 161 should be stressed when having students defend their answers, and I may even print them out and post them for the students to constantly see. There are multiple pathways that can be used to reach the correct answer in math, so having opinion-proofs can get students thinking about what the "best" way is to solve a problem. Other people may think better algebraically than numerically, but they should defend their reasoning when explaining their answer.

The Phony Document strategy reminds me of the Centaur display in the library on campus. I think math problems could be easily set up in a phony document, and students would have to read critically for the mistake in the reasoning shown. For example, all of the math interns have seen the famous "1=0" problem that "proves" that 1 is equal to 0. However, there is a flaw in the math that many people overlook. This strategy lends itself best to math, in my opinion, because students have to really read analytically in order to identify the flaw.

Tovani Chapters 5 and 6

I find myself relating so closely with the stories presented throughout chapter five. Sometimes it is quite difficult for me to think about how someone would think about a topic for the first time. For example, I had to teach the laws of exponents to my students two weeks ago, and it was a challenge to know how to explain this to them. I feel like this is a very elementary topic, but I have also been using these laws for many years now. The instructional focus guide sheet could help me determine what the main focus should be in assigning reading on topics in math. Not only should I be focusing on the main ideas of each topic, but I should also teach the students how to turn off their reciting voice and turn on their conversation voice. Reading a math textbook can be difficult because the text is so dense. I want to encourage my students to ask questions and work out the sample problems with the text so that they are really conversing with the author. Then, they can ask questions in class that help clarify and deepen their understanding of the material.

Chapter six gives several tools to help students hold their thinking to remember and reuse. I think it may be worth the time to show students how to mark up their textbook with sticky notes because they may seem overwhelmed at first when they are reading to learn the material. However, my favorite strategy in this chapter is show on page 82, the Quad-Entry Diary. I can see myself using this with many different topics in math. This seems to be a great way to get students to write things down on paper in an organized manner to help them remember what they have read about and/or discussed in class.

Books in the Classroom

The chapter that discussed why textbooks are not enough reminded me of our discussion in last week's class. In my college math classes, I often used my textbook as a resource, but I also looked online and other books to help me. More often than not, the other resources helped me more than the textbook. Textbooks are really not written for students, and I would like to provide other sources for my class that show different perspectives of math. Like Lisa mentioned last week in class, teachers often create better resources for their students than the textbooks.

The next chapter shows the importance of giving students a wide variety of readings. I really enjoyed reading about the different lengths, difficulties, and genres to which students should be exposed. I know from previous experience that it helps to break up challenging readings with easier passages that still gives the chance to learn important material. Building a classroom library can be intimidating at first, but I feel like it just takes time to build a library with articles and books that are relevant to students' lives.

I really liked how the book listed several resources listed by subject and difficulty that can be used to form a useful library for all students. I want to start looking at McKay's for some cheap books that can relate math to other subjects or to real-life topics. Hopefully having a library like this will help students see math as more of a tool that can help solve problems instead of a subject to be feared.

Reading Strategies and How to Use a Textbook

The chapter on reading strategies was a bit overwhelming at first, but I think there are several beneficial strategies that could easily be implemented into a math classroom. For example, I think the vocabulary tree would be an excellent tool to help students see the relationships between different functions (linear, quadratic, exponential, logarithmic, trigonometric, etc.). I would also like to teach my students to sketch their way through the text to help them visualize what certain mathematical concepts are really saying. In general, I think any type of map is beneficial in a math class because they help students see the bigger picture.

Reading a textbook can be difficult at times, especially a math textbook. However, chapter six lists some strategies that can help any student truly understand what is being presented in the text. I want to show my students how to use their textbook by giving them a textbook feature analysis. This will ask them to look at all aspects of the text to help them become independent readers. I also liked the Guide-O-Rama study guides discussed in this chapter. These can be beneficial for students and the teacher. Students can have a guide that tells them the facts on which they should focus, how the teacher thinks about what is being presented, and important questions they should consider. This strategy also gives the teacher the chance to ensure the students are focusing on the important parts of the text without being overwhelmed.

Web Resource - The Math Page

The Math Page is a great site for any high school math class! The site provides definitions, examples, and much more that allows students to interact with math in a fun way. Each subject is broken down into chapters so students can easily find where they are in class. The chapters provide students with the chance to work some problems on their own. Once they have attempted the problem, they can scroll over the red boxes to reveal the correct answer. I would really like my students to use this site because they can have immediate feedback as to whether or not they can reason through the problems correctly.

This site is completely free, and they even have a free app on the App Store and Android Market. I am not sure how much I would use this resource in class, but it would be a great way to supplement what is taught during the day when the students have questions at home. From what I have seen, there would be no need for parental supervision, but parents may want to sit in on the action since this site does an excellent job of breaking each step down.

Overall, I think this site can help those that still have questions when they leave school as well as those who really understand the material by providing easy-to-follow steps for each topic and challenge questions, too. I know I will be promoting this site for my students all year long!

Parallel Experiences and Real Rigor

Chapter 3 opened my eyes to the fact that teaching reading in math may actually help students in other disciplines read analytically. I am one of those people who say they are a math person because they do not like words, but I feel confident in my abilities to solve word problems in math. I think that modeling how to solve word problems for the whole class would be an easy way to show students what I experience when presented with these problems. They can see how I approach the problem, my thoughts throughout, and when and how I connect this problem to something with which I am already familiar. Graphs and charts also give students struggles. If I took the time to help them truly understand the information in the chart or graph, they may perform much better on standardized tests like the ACT as well as other classes like science or social studies.

I have vivid memories of my history class in high school assigning pages and pages of reading from the textbook each night, and thankfully one of my previous teachers had shown me how to take Cornell notes. This was the only way I knew how to tackle these readings, but I never thought introducing different texts that were more accessible could have made our lives so much easier. Creating text sets seems to be an interesting way to give students the chance to see math outside of the classroom. I have seen one teacher have a "Math in the News" bulletin board where students could bring in articles from the newspaper or a magazine to hang up for others to read. This helps promote reading in math as well as seeing connections to the real world. I feel like having engaging strategies like this bulletin board or text sets will make math seem less intimidating to those who feel nervous about coming to class.

Building a Community of Learners

I enjoyed reading this week’s readings because they all focused on building a strong environment in the classroom that helps supports and pushes all students to be better both academically and socially. The chapter in Subjects Matter looks specifically at how to build a community of learners. In order to be an effective teacher, one must create an environment where students feel comfortable discussing their thoughts in both an individual and group setting. Peers can be better teachers than the teacher in certain situations, which is why students need to have interactions with everyone in the classroom at some point during the day. It is crucial for teachers to get to know their students because once a connection between the teacher and student has been made, the teacher can then use this information to build lessons that directly relate to his students. The students would be more likely to take responsibility in the classroom, making them more independent learners. I may use the strategy mentioned on page 174 about having the student who I have helped on a particular problem explain that problem to others who have questions with that problem. This would give the original student the opportunity to solidify his knowledge, other students would still learn the material, and I could focus on other questions or problems students may be having. This chapter helped me see the benefits of having a classroom where students can learn from each other, ask questions without worrying about other people’s thoughts, and be actively engage in the learning process.