1.4: College Culture and Expectations
-
- Last updated
- Save as PDF
Questions to consider
- What language and customs do you need to know to succeed in college?
- What is your responsibility for learning in college?
- What resources will you use to meet these expectations?
- What are the common challenges in the first year?
College Has Its Own Language and Customs
Going to college—even if you are not far from home—is a cultural experience. It comes with its own language and customs, some of which can be confusing or confounding at first. Just like traveling to a foreign country, it is best if you prepare by learning what words mean and what you are expected to say and do in certain situations.
Let’s first start with the language you may encounter. In most cases, there will be words that you have heard before, but they may have different meanings in a college setting. Take, for instance, “office hours.” If you are not in college, you would think that it means the hours of a day that an office is open. If it is your dentist’s office, it may mean Monday through Friday from 8 a.m. to 5 p.m. In college, “office hours” can refer to the specific hours a professor is in her office to meet with students, and those hours may be only a few each day: for example, Mondays and Wednesdays from 1 p.m. until 3 p.m.
“Syllabus” is another word that you may not have encountered, but it is one you will soon know very well. A syllabus is often called the “contract of the course” because it contains information about what to expect—from the professor and the student. It is meant to be a roadmap for succeeding in the class. Understanding that office hours are for you to ask your professor questions and the syllabus is the guide for what you will be doing in the class can make a big difference in your transition to college. The table on Common College Terms, has a brief list of other words that you will want to know when you hear them on campus.
| Term | What It Means | Why You Need to Know |
|---|---|---|
| Attendance policy | A policy that describes the attendance and absence expectations for a class | Professors will have different attendance expectations. Read your syllabus to determine which ones penalize you if you miss too many classes. |
| Final exam | A comprehensive assessment that is given at the end of a term | If your class has a final exam, you will want to prepare for it well in advance by reading assigned material, taking good notes, reviewing previous tests and assignments, and studying. |
| Learning | The process of acquiring knowledge | In college, most learning happens outside the classroom. Your professor will only cover the main ideas or the most challenging material in class. The rest of the learning will happen on your own. |
| Office hours | Specific hours professor is in the office to meet with students | Visiting your professor during office hours is a good way to get questions answered and to build rapport. |
| Plagiarism | Using someone’s words, images, or ideas as your own, without proper attribution | Plagiarism carries much more serious consequences in college, so it is best to speak to your professor about how to avoid it and review your student handbook’s policy. |
| Study | The process of using learning strategies to understand and recall information | Studying in college may look different than studying in high school in that it may take more effort and more time to learn more complex material. |
| Syllabus | The contract of a course that provides information about course expectations and policies | The syllabus will provide valuable information that your professor will assume you have read and understood. Refer to it first when you have a question about the course. |
Activity
The language that colleges and universities use can feel familiar but mean something different, as you learned in the section above, and it can also seem alien, especially when institutions use acronyms or abbreviations for buildings, offices, and locations on campus. Terms such as “quad” or “union” can denote a location or space for students. Then there may be terms such as “TLC” (The Learning Center, in this example) that designate a specific building or office. Describe a few of the new terms you have encountered so far and what they mean. If you are not sure, ask your professor or a fellow student to define it for you.
In addition to its own language, higher education has its own way of doing things. For example, you may be familiar with what a teacher did when you were in high school, but do you know what a professor does? It certainly seems like they fulfill a very similar role as teachers in high school, but in college professors’ roles are often much more diverse. In addition to teaching, they may also conduct research, mentor graduate students, write and review research articles, serve on and lead campus committees, serve in regional and national organizations in their disciplines, apply for and administer grants, advise students in their major, and serve as sponsors for student organizations. You can be assured that their days are far from routine. See the Table on Differences between High School Teachers and College Professors for just a few differences between high school teachers and college professors.
| High School Faculty | College Faculty |
|---|---|
| Often have degrees or certifications in teaching in addition to degrees in subject matter | Most likely have not even taken a course in teaching as part of their graduate program |
| Responsibilities include maximizing student learning and progress in a wide array of areas | Responsibilities include providing students with content and an assessment of their mastery of the content |
| Are available before or after school or during class if a student has a question | Are available during office hours or by appointment if a student needs additional instruction or advice |
| Communicate regularly and welcome questions from parents and families about a student’s progress | Cannot communicate with parents and families of students without permission because of the Federal Educational Rights and Privacy Act (FERPA) |
The relationships you build with your professors will be some of the most important ones you create during your college career. You will rely on them to help you find internships, write letters of recommendation, nominate you for honors or awards, and serve as references for jobs. You can develop those relationships by participating in class, visiting during office hours, asking for assistance with coursework, requesting recommendations for courses and majors, and getting to know the professor’s own academic interests. One way to think about the change in how your professors will relate to you is to think about the nature of relationships you have had growing up. In Figure 1.8: You and Your Relationships Before College you will see a representation of what your relationships probably looked like. Your family may have been the greatest influencer on you and your development.
"The relationships you build with your professors will be some of the most important ones during your college career."
In college, your networks are going to expand in ways that will help you develop other aspects of yourself. As described above, the relationships you will have with your professors will be some of the most important. But they won’t be the only relationships you will be cultivating while in college. Consider the Figure on You and Your Relationships during College and think about how you will go about expanding your network while you are completing your degree.
Your relationships with authority figures, family, and friends may change while you are in college, and at the very least, your relationships will expand to peer networks—not friends, but near-age peers or situational peers (e.g., a first-year college student who is going back to school after being out for 20 years)—and to faculty and staff who may work alongside you, mentor you, or supervise your studies. These relationships are important because they will allow you to expand your network, especially as it relates to your career. As stated earlier, developing relationships with faculty can provide you with more than just the benefits of a mentor. Faculty often review applications for on-campus jobs or university scholarships and awards; they also have connections with graduate programs, companies, and organizations. They may recommend you to colleagues or former classmates for internships and even jobs.
Other differences between high school and college are included in the table about Differences between High School and College. Because it is not an exhaustive list of the differences, be mindful of other differences you may notice. Also, if your most recent experience has been the world of work or the military, you may find that there are more noticeable differences between those experiences and college.
| High School | College | Why You Need to Know the Difference | |
|---|---|---|---|
|
Grades |
Grades are made up of frequent tests and homework, and you may be able to bring up a low initial grade by completing smaller assignments and bonuses. | Grades are often made up of fewer assignments, and initial low grades may keep you from earning high course grades at the end of the semester. | You will need to be prepared to earn high grades on all assignments because you may not have the opportunity to make up for lost ground. |
|
Learning |
Learning is often done in class with the teacher guiding the process, offering multiple ways to learn material and frequent quizzes to ensure that learning is occurring. | Learning happens mostly outside of class and on your own. Faculty are responsible for assigning material and covering the most essential ideas; you are responsible for tracking and monitoring your learning progress. | You will need to practice effective learning strategies on your own to ensure that you are mastering material at the appropriate pace. |
|
Getting Help |
Your teachers, parents, and a counselor are responsible for identifying your need for help and for creating a plan for you to get help with coursework if you need it. Extra assistance is usually reserved for students who have an official diagnosis or need. | You will most likely need help to complete all your courses successfully even if you did not need extra help in high school. You will be responsible for identifying that you need it, accessing the resources, and using them. | Because the responsibility is on you, not parents or teachers, to get the help you need, you will want to be aware of when you may be struggling to learn material. You then will need to know where the support can be accessed on campus or where you can access support online. |
|
Tests and Exams |
Tests cover small amounts of material and study days or study guides are common to help you focus on what you need to study. If you paid attention in class, you should be able to answer all the questions. | Tests are fewer and cover more material than in high school. If you read all the assigned material, took good notes in class, and spent time practicing effective study techniques, you should be able to answer all the questions. | This change in how much material and the depth of which you need to know the material is a shock for some students. This may mean you need to change your strategies dramatically to get the same results. |
Some of What You Will Learn Is “Hidden”
Many of the college expectations that have been outlined so far may not be considered common knowledge, which is one reason that so many colleges and universities have classes that help students learn what they need to know to succeed. The term, which was coined by sociologists, 7 describes unspoken, unwritten, or unacknowledged (hence, hidden ) rules that students are expected to follow that can affect their learning. To illustrate the concept, consider the situation in the following activity.
Activity
Situation: Your history syllabus indicates that, on Tuesday, your professor is lecturing on the chapter that covers the stock market crash of 1929.
This information sounds pretty straightforward. Your professor lectures on a topic and you will be there to hear it. However, there are some unwritten rules, or hidden curriculum, that are not likely to be communicated. Can you guess what they may be? Take a moment to write at least one potential unwritten rule.
-
What is an unwritten rule about what you should be doing before attending class?
_______________________________________________________________ -
What is an unwritten rule about what you should be doing in class?
_______________________________________________________________ -
What is an unwritten rule about what you should be doing after class?
_______________________________________________________________ -
What is an unwritten rule if you are not able to attend that class?
_______________________________________________________________
Some of your answers could have included the following:
Before class: Read the assigned chapter, take notes, record any questions you have about the reading.
During class: Take detailed notes, ask critical thinking or clarifying questions, avoid distractions, bring your book and your reading notes.
After class: Reorganize your notes in relation to your other notes, start the studying process by testing yourself on the material, make an appointment with your professor if you are not clear on a concept.
Absent: Communicate with the professor, get notes from a classmate, make sure you did not miss anything important in your notes.
The expectations before, during, and after class, as well as what you should do if you miss class, are often unspoken because many professors assume you already know and do these things or because they feel you should figure them out on your own. Nonetheless, some students struggle at first because they don’t know about these habits, behaviors, and strategies. But once they learn them, they are able to meet them with ease.
Learning Is Your Responsibility
As you may now realize by reviewing the differences between high school and college, learning in college is your responsibility. Before you read about the how and why of being responsible for your own learning, complete the Activity below.
Activity
For each statement, circle the number that best represents you, with 1 indicating that the statement is least like you, and 5 indicating that the statement is most like you.
| Most of the time, I can motivate myself to complete tasks even if they are boring or challenging. | ||||
| 1 | 2 | 3 | 4 | 5 |
| I regularly work hard when I need to complete a task no matter how small or big the task may be. | ||||
| 1 | 2 | 3 | 4 | 5 |
| I use different strategies to manage my time effectively and minimize procrastination to complete tasks. | ||||
| 1 | 2 | 3 | 4 | 5 |
| I regularly track my progress completing work and the quality of work I do produce. | ||||
| 1 | 2 | 3 | 4 | 5 |
| I believe how much I learn and how well I learn is my responsibility. | ||||
| 1 | 2 | 3 | 4 | 5 |
Were you able to mark mostly 4s and 5s? If you were even able to mark at least one 4 or 5, then you are well on your way to taking responsibility for your own learning. Let’s break down each statement in the components of the ownership of learning:
- Motivation. Being able to stay motivated while studying and balancing all you have to do in your classes will be important for meeting the rest of the components.
- Deliberate, focused effort. Taking ownership of learning will hinge on the effort that you put into the work. Because most learning in college will take place outside of the classroom, you will need determination to get the work done. And there will be times that the work will be challenging and maybe even boring, but finding a way to get through it when it is not exciting will pay in the long run.
- Time and task management. You will learn more about strategies for managing your time and the tasks of college in a later chapter, but without the ability to control your calendar, it will be difficult to block out the time to study.
- Progress tracking. A commitment to learning must include monitoring your learning, knowing not only what you have completed ( this is where a good time management strategy can help you track your tasks ), but also the quality of the work you have done.
Taking responsibility for your learning will take some time if you are not used to being in the driver’s seat. However, if you have any difficulty making this adjustment, you can and should reach out for help along the way.
What to Expect During the First Year
While you may not experience every transition within your first year, there are rhythms to each semester of the first year and each year you are in college. Knowing what to expect each month or week can better prepare you to take advantage of the times that you have more confidence and weather through the times that seem challenging. Review the table on First-Year College Student Milestones. There will be milestones each semester you are in college, but these will serve as an introduction to what you should expect in terms of the rhythms of the semester.
|
First-Year College Student Milestones for the First Semester |
||||
|---|---|---|---|---|
| August | September | October | November | December |
| Expanding social circles | Completing first test and projects | Feeling more confident about abilities | Balancing college with other obligations | Focusing on finishing strong |
| Experiencing homesickness or imposter syndrome | Earning “lower-than-usual” grades or not meeting personal expectations | Dealing with relationship issues | Staying healthy and reducing stress | Handling additional stress of the end of the semester |
| Adjusting to the pace of college | Learning to access resources for support | Planning for next semester and beyond | Thinking about majors and degrees | Thinking about the break and how to manage changes |
The first few weeks will be pretty exhilarating. You will meet new people, including classmates, college staff, and professors. You may also be living in a different environment, which may mean that a roommate is another new person to get to know. Overall, you will most likely feel both excited and nervous. You can be assured that even if the beginning of the semester goes smoothly, your classes will get more challenging each week. You will be making friends, learning who in your classes seem to know what is going on, and figuring your way around campus. You may even walk into the wrong building, go to the wrong class, or have trouble finding what you need during this time. But those first-week jitters will end soon. Students who are living away from home for the first time can feel homesick in the first few weeks, and others can feel what is called “imposter syndrome,” which is a fear some students have that they don’t belong in college because they don’t have the necessary skills for success. Those first few weeks sound pretty stressful, but the stress is temporary.
After the newness of college wears off, reality will set in. You may find that the courses and assignments do not seem much different than they did in high school (more on that later), but you may be in for a shock when you get your graded tests and papers. Many new college students find that their first grades are lower than they expected. For some students, this may mean they have earned a B when they are used to earning As, but for many students, it means they may experience their first failing or almost-failing grades in college because they have not used active, effective study strategies; instead, they studied how they did in high school, which is often insufficient. This can be a shock if you are not prepared, but it doesn’t have to devastate you if you are willing to use it as a wake-up call to do something different.
By the middle of the semester, you’ll likely feel much more confident and a little more relaxed. Your grades are improving because you started going to tutoring and using better study strategies. You are looking ahead, even beyond the first semester, to start planning your courses for the next term. If you are working while in college, you may also find that you have a rhythm down for balancing it all; additionally, your time management skills have likely improved.
By the last few weeks of the semester, you will be focused on the increasing importance of your assignments and upcoming finals and trying to figure out how to juggle that with the family obligations of the impending holidays. You may feel a little more pressure to prepare for finals, as this time is often viewed as the most stressful period of the semester. All of this additional workload and need to plan for the next semester can seem overwhelming, but if you plan ahead and use what you learn from this chapter and the rest of the course, you will be able to get through it more easily.
Don’t Do It Alone
Think about our earlier descriptions of two students, Reginald and Madison. What if they found that the first few weeks were a little harder than they had anticipated? Should they have given up and dropped out? Or should they have talked to someone about their struggles? Here is a secret about college success that not many people know: successful students seek help. They use resources. And they do that as often as necessary to get what they need. Your professors and advisors will expect the same from you, and your college will have all kinds of offices, staff, and programs that are designed to help. This bears calling out again: you need to use those resources . These are called “help-seeking behaviors,” and along with self-advocacy, which is speaking up for your needs, they are essential to your success. As you get more comfortable adjusting to life in college, you will find that asking for help is easier. In fact, you may become really good at it by the time you graduate, just in time for you to ask for help finding a job! Review the table on Issues, Campus Resources, and Potential Outcomes for a few examples of times you may need to ask for help. See if you can identify where on campus you can find the same or a similar resource.
|
Issues, Campus Resources, and Potential Outcomes |
|||
|---|---|---|---|
| Type | Issue | Campus Resource | Potential Outcome |
| Academic | You are struggling to master the homework in your math class. | The campus tutoring center | A peer or professional tutor can walk you through the steps until you can do them on your own. |
| Health | You have felt extremely tired over the past two days and now you have a cough. | The campus health center | A licensed professional can examine you and provide care. |
| Social | You haven’t found a group to belong to. Your classmates seem to be going in different directions and your roommate has different interests. | Student organizations and interest groups | Becoming a member of a group on campus can help you make new friends. |
| Financial | Your scholarship and student loan no longer cover your college expenses. You are not sure how to afford next semester. | Financial aid office | A financial aid counselor can provide you with information about your options for meeting your college expenses. |
Application
Using a blank sheet of paper, write your name in the center of the page and circle it. Then, draw six lines from the center (see example in the figure below) and label each for the six areas of adjustment that were discussed earlier. Identify a campus resource or strategy for making a smooth adjustment for each area.
Common Challenges in the First Year
It seems fitting to follow up the expectations for the first year with a list of common challenges that college students encounter along the way to a degree. If you experience any—or even all—of these, the important point here is that you are not alone and that you can overcome them by using your resources. Many college students have felt like this before, and they have survived—even thrived—despite them because they were able to identify a strategy or resource that they could use to help themselves. At some point in your academic career, you may do one or more of the following:
- Feel like an imposter. There is actually a name for this condition: imposter syndrome. Students who feel like an imposter are worried that they don’t belong, that someone will “expose them for being a fake.” This feeling is pretty common for anyone who finds themselves in a new environment and is not sure if they have what it takes to succeed. Trust the professionals who work with first-year college students: you do have what it takes, and you will succeed. Just give yourself time to get adjusted to everything.
- Worry about making a mistake. This concern often goes with imposter syndrome. Students who worry about making a mistake don’t like to answer questions in class, volunteer for a challenging assignment, and even ask for help from others. Instead of avoiding situations where you may fail, embrace the process of learning, which includes—is even dependent on—making mistakes. The more you practice courage in these situations and focus on what you are going to learn from failing, the more confident you become about your abilities.
- Try to manage everything yourself. Even superheroes need help from sidekicks and mere mortals. Trying to handle everything on your own every time an issue arises is a recipe for getting stressed out. There will be times when you are overwhelmed by all you have to do. This is when you will need to ask for and allow others to help you.
- Ignore your mental and physical health needs. If you feel you are on an emotional rollercoaster and you cannot find time to take care of yourself, then you have most likely ignored some part of your mental and physical well-being. What you need to do to stay healthy should be non-negotiable. In other words, your sleep, eating habits, exercise, and stress-reducing activities should be your highest priorities.
- Forget to enjoy the experience. Whether you are 18 years old and living on campus or 48 years old starting back to college after taking a break to work and raise a family, be sure to take the time to remind yourself of the joy that learning can bring.
Get Connected
Which apps help you meet the expectations of college? Will you be able to meet the expectations of being responsible for your schedule and assignments?
- My Study Life understands how college works and provides you with a calendar, to-do list, and reminders that will help you keep track of the work you have to do.
How can you set goals and work toward them while in college?
- The Strides app provides you with the opportunity to create SMART (Specific, Measureable, Attainable, Relevant, and Time bound) goals and track daily habits. These daily habits will add up over time toward your goals.
What can you do to develop your learning skills?
- Lumosity is a brain-training app that can help you build the thinking and learning skills you will need to meet learning challenges in college. If you want to test your memory and attention—and build your skills—take the fit test and then play different games to improve your fitness.
How can you develop networks with people in college?
- LinkedIn is a professional networking app that allows you to create a profile and network with others. Creating a LinkedIn account as a first-year college student will help you create a professional profile that you can use to find others with similar interests.
- Internships.com provides information, connections, and support to help your career planning and activities. Even if you are not planning an internship right away, you may find some useful and surprising ideas and strategies to motivate your approach.
Footnotes
- P.P. Bilbao, P. I. Lucido, T. C. Iringan and R. B. Javier. (2008). Curriculum Development .


3.4E: Double Integrals in Polar Coordinates (Exercises)
-
- Last updated
- Save as PDF
Terms and Concepts
1. When evaluating \(\displaystyle \int\int_R f(x,y)\,dA\) using polar coordinates, \(f(x,y)\) is replaced with _______ and \(dA\) is replaced with _______.
- Answer
- \(f(x,y)\) is replaced with \(f(r\cos \theta, r\sin\theta)\) and \(dA\) is replaced with \(r\,dr\,d\theta\).
2. Why would one be interested in evaluating a double integral with polar coordinates?
Defining Polar Regions
In exercises 3 - 6, express the region \(R\) in polar coordinates.
3) \(R\) is the region of the disk of radius 2 centered at the origin that lies in the first quadrant.
- Answer
- \(R = \big\{(r, \theta)\,|\,0 \leq r \leq 2, \space 0 \leq \theta \leq \frac{\pi}{2}\big\}\)
4) \(R\) is the region of the disk of radius 3 centered at the origin.
5) \(R\) is the region between the circles of radius 4 and radius 5 centered at the origin that lies in the second quadrant.
- Answer
- \(R = \big\{(r, \theta)\,|\,4 \leq r \leq 5, \space \frac{\pi}{2} \leq \theta \leq \pi\big\}\)
6) \(R\) is the region bounded by the \(y\)-axis and \(x = \sqrt{1 - y^2}\).
7) \(R\) is the region bounded by the \(x\)-axis and \(y = \sqrt{2 - x^2}\).
- Answer
- \(R = \big\{(r, \theta)\,|\,0 \leq r \leq \sqrt{2}, \space 0 \leq \theta \leq \pi\big\}\)
8) \(R = \big\{(x,y)\,|\,x^2 + y^2 \leq 4x\big\}\)
9) \(R = \big\{(x,y)\,|\,x^2 + y^2 \leq 4y\big\}\)
- Answer
- \(R = \big\{(r, \theta)\,|\,0 \leq r \leq 4 \space \sin \theta, \space 0 \leq \theta \leq \pi\big\}\)
In exercises 10 - 15, the graph of the polar rectangular region \(D\) is given. Express \(D\) in polar coordinates.
- Answer
- \(D = \big\{(r, \theta)\,|\, 3 \leq r \leq 5, \space \frac{\pi}{4} \leq \theta \leq \frac{\pi}{2}\big\}\)
12)
13)
- Answer
- \(D = \big\{(r, \theta)\,|\,3 \leq r \leq 5, \space \frac{3\pi}{4} \leq \theta \leq \frac{5\pi}{4}\big\}\)
15) In the following graph, the region \(D\) is bounded by \(y = x\) and \(y = x^2\).
- Answer
- \(D = \big\{(r, \theta)\,|\,0 \leq r \leq \tan \theta \space \sec \theta, \space 0 \leq \theta \leq \frac{\pi}{4}\big\}\)
Evaluating Polar Double Integrals
In exercises 16 - 25, evaluate the double integral \(\displaystyle \iint_R f(x,y) \,dA\) over the polar rectangular region \(R\) .
16) \(f(x,y) = x^2 + y^2\), \(R = \big\{(r, \theta)\,|\,3 \leq r \leq 5, \space 0 \leq \theta \leq 2\pi\big\}\)
17) \(f(x,y) = x + y\), \(R = \big\{(r, \theta)\,|\,3 \leq r \leq 5, \space 0 \leq \theta \leq 2\pi\big\}\)
- Answer
- \(0\)
18) \(f(x,y) = x^2 + xy, \space R = \big\{(r, \theta )\,|\,1 \leq r \leq 2, \space \pi \leq \theta \leq 2\pi\big\}\)
19) \(f(x,y) = x^4 + y^4, \space R = \big\{(r, \theta)\,|\,1 \leq r \leq 2, \space \frac{3\pi}{2} \leq \theta \leq 2\pi\big\}\)
- Answer
- \(\frac{63\pi}{16}\)
20) \(f(x,y) = \sqrt[3]{x^2 + y^2}\), where \(R = \big\{(r, \theta)\,|\,0 \leq r \leq 1, \space \frac{\pi}{2} \leq \theta \leq \pi\big\}\).
21) \(f(x,y) = x^4 + 2x^2y^2 + y^4\), where \(R = \big\{(r,\theta)\,|\,3 \leq r \leq 4, \space \frac{\pi}{3} \leq \theta \leq \frac{2\pi}{3}\big\}\).
- Answer
- \(\frac{3367\pi}{18}\)
22) \(f(x,y) = \sin (\arctan \frac{y}{x})\), where \(R = \big\{(r, \theta)\,|\,1 \leq r \leq 2, \space \frac{\pi}{6} \leq \theta \leq \frac{\pi}{3}\big\}\)
23) \(f(x,y) = \arctan \left(\frac{y}{x}\right)\), where \(R = \big\{(r, \theta)\,|\,2 \leq r \leq 3, \space \frac{\pi}{4} \leq \theta \leq \frac{\pi}{3}\big\}\)
- Answer
- \(\frac{35\pi^2}{576}\)
24) \(\displaystyle \iint_R e^{x^2+y^2} \left[1 + 2 \space \arctan \left(\frac{y}{x}\right)\right] \,dA, \space R = \big\{(r,\theta)\,|\,1 \leq r \leq 2, \space \frac{\pi}{6} \leq \theta \frac{\pi}{3}\big\}\)
25) \(\displaystyle \iint_R \left(e^{x^2+y^2} + x^4 + 2x^2y^2 + y^4 \right) \arctan \left(\frac{y}{x}\right) \,dA, \space R = \big\{(r, \theta )\,|\,1 \leq r \leq 2, \space \frac{\pi}{4} \leq \theta \leq \frac{\pi}{3}\big\}\)
- Answer
- \(\frac{7}{576}\pi^2 (21 - e + e^4)\)
Converting Double Integrals to Polar Form
In exercises 26 - 29, the integrals have been converted to polar coordinates. Verify that the identities are true and choose the easiest way to evaluate the integrals, in rectangular or polar coordinates.
26) \(\displaystyle \int_1^2 \int_0^x (x^2 + y^2)\,dy \space dx = \int_0^{\frac{\pi}{4}} \int_{\sec \theta}^{2 \space \sec \theta}r^3 \,dr \space d\theta\)
27) \(\displaystyle \int_2^3 \int_0^x \frac{x}{\sqrt{x^2 + y^2}}\,dy \space dx = \int_0^{\pi/4} \int_{2\sec\theta}^{3\sec \theta} \,r \space \cos \theta \space dr \space d\theta\)
- Answer
- \(\frac{5}{2} \ln (1 + \sqrt{2})\)
28) \(\displaystyle \int_0^1 \int_{x^2}^x \frac{1}{\sqrt{x^2 + y^2}}\,dy \space dx = \int_0^{\pi/4} \displaystyle \int_0^{\tan \theta \space \sec \theta} \space dr \space d\theta\)
29) \(\displaystyle \int_0^1 \int_{x^2}^x \frac{y}{\sqrt{x^2 + y^2}}\,dy \space dx = \int_0^{\pi/4} \displaystyle \int_0^{\tan \theta \space \sec \theta} \,r \space \sin \theta \space dr \space d\theta\)
- Answer
- \(\frac{1}{6}(2 - \sqrt{2})\)
In exercises 30 - 37, draw the region of integration, \(R\) , labeling all limits of integration, convert the integrals to polar coordinates and evaluate them.
30) \(\displaystyle \int_0^3 \int_0^{\sqrt{9-y^2}}\left(x^2 + y^2\right)\,dx \space dy\)
31) \(\displaystyle \int_0^2 \int_{-\sqrt{4-y^2}}^{\sqrt{4-y^2}}\left(x^2 + y^2\right)^2\,dx \space dy\)
- Answer
- \(\displaystyle \int_0^{\pi} \int_0^2 r^5 \,dr \space d\theta \quad = \quad \frac{32\pi}{3}\)
32) \(\displaystyle \int_0^1 \int_0^{\sqrt{1-x^2}} (x + y) \space dy \space dx\)
33) \(\displaystyle \int_0^4 \int_{-\sqrt{16-x^2}}^{\sqrt{16-x^2}} \sin (x^2 + y^2) \space dy \space dx\)
- Answer
- \(\displaystyle \int_{-\pi/2}^{\pi/2} \int_0^4 \,r \space \sin (r^2) \space dr \space d\theta \quad = \quad \pi \space \sin^2 8\)
34) \(\displaystyle \int_0^5 \int_{-\sqrt{25-x^2}}^{\sqrt{25-x^2}}\sqrt{x^2+y^2}\,dy\,dx\)
35) \(\displaystyle \int_{-4}^4 \int_{-\sqrt{16-y^2}}^{0}(2y-x)\,dx\,dy\)
- Answer
- \(\displaystyle \int_{\frac{\pi}{2}}^{\frac{3\pi}{2}} \int_0^{4} \big( 2r\sin \theta - r\cos\theta\big) \,r\,dr \space d\theta \quad = \quad \frac{128}{3}\)
36) \(\displaystyle \int_0^2 \int_{y}^{\sqrt{8-y^2}}(x+y)\,dx\,dy\)
37) \(\displaystyle \int_{-2}^{-1} \int_{0}^{\sqrt{4-x^2}}(x+5)\,dy\,dx+\int_{-1}^1\int_{\sqrt{1-x^2}}^{\sqrt{4-x^2}}(x+5)\,dy\,dx+\int_1^2\int_0^{\sqrt{4-x^2}}(x+5)\,dy\,dx\)
- Answer
- \(\displaystyle \int_{0}^{\pi} \int_1^{2} \big( r\cos \theta + 5\big) \,r\,dr \space d\theta \quad = \quad \frac{15\pi}{2}\)
38) Evaluate the integral \(\displaystyle \iint_D r \,dA\) where \(D\) is the region bounded by the polar axis and the upper half of the cardioid \(r = 1 + \cos \theta\).
39) Find the area of the region \(D\) bounded by the polar axis and the upper half of the cardioid \(r = 1 + \cos \theta\).
- Answer
- \(\frac{3\pi}{4}\)
40) Evaluate the integral \(\displaystyle \iint_D r \,dA,\) where \(D\) is the region bounded by the part of the four-leaved rose \(r = \sin 2\theta\) situated in the first quadrant (see the following figure).
41) Find the total area of the region enclosed by the four-leaved rose \(r = \sin 2\theta\) (see the figure in the previous exercise).
- Answer
- \(\frac{\pi}{2}\)
42) Find the area of the region \(D\) which is the region bounded by \(y = \sqrt{4 - x^2}\), \(x = \sqrt{3}\), \(x = 2\), and \(y = 0\).
43) Find the area of the region \(D\), which is the region inside the disk \(x^2 + y^2 \leq 4\) and to the right of the line \(x = 1\).
- Answer
- \(\frac{1}{3}(4\pi - 3\sqrt{3})\)
44) Determine the average value of the function \(f(x,y) = x^2 + y^2\) over the region \(D\) bounded by the polar curve \(r = \cos 2\theta\), where \(-\frac{\pi}{4} \leq \theta \leq \frac{\pi}{4}\) (see the following graph).
45) Determine the average value of the function \(f(x,y) = \sqrt{x^2 + y^2}\) over the region \(D\) bounded by the polar curve \(r = 3\sin 2\theta\), where \(0 \leq \theta \leq \frac{\pi}{2}\) (see the following graph).
- Answer
- \(\frac{16}{3\pi}\)
46) Find the volume of the solid situated in the first octant and bounded by the paraboloid \(z = 1 - 4x^2 - 4y^2\) and the planes \(x = 0, \space y = 0\), and \(z = 0\).
47) Find the volume of the solid bounded by the paraboloid \(z = 2 - 9x^2 - 9y^2\) and the plane \(z = 1\).
- Answer
- \(\frac{\pi}{18}\)
48)
- Find the volume of the solid \(S_1\) bounded by the cylinder \(x^2 + y^2 = 1\) and the planes \(z = 0\) and \(z = 1\).
- Find the volume of the solid \(S_2\) outside the double cone \(z^2 = x^2 + y^2\) inside the cylinder \(x^2 + y^2 = 1\), and above the plane \(z = 0\).
- Find the volume of the solid inside the cone \(z^2 = x^2 + y^2\) and below the plane \(z = 1\) by subtracting the volumes of the solids \(S_1\) and \(S_2\).
49)
- Find the volume of the solid \(S_1\) inside the unit sphere \(x^2 + y^2 + z^2 = 1\) and above the plane \(z = 0\).
- Find the volume of the solid \(S_2\) inside the double cone \((z - 1)^2 = x^2 + y^2\) and above the plane \(z = 0\).
- Find the volume of the solid outside the double cone \((z - 1)^2 = x^2 + y^2\) and inside the sphere \(x^2 + y^2 + z^2 = 1\).
- Answer
- a. \(\frac{2\pi}{3}\); b. \(\frac{\pi}{2}\); c. \(\frac{\pi}{6}\)
In Exercises 50-51, special double integrals are presented that are especially well suited for evaluation in polar coordinates.
50) The surface of a right circular cone with height \(h\) and base radius \(a\) can be described by the equation \(f(x,y)=h-h\sqrt{\frac{x^2}{a^2}+\frac{y^2}{a^2}}\), where the tip of the cone lies at \((0,0,h)\) and the circular base lies in the \(xy\)-plane, centered at the origin.
Confirm that the volume of a right circular cone with height \(h\) and base radius \(a\) is \(V=\frac{1}{3}\pi a^2h\) by evaluating \(\displaystyle \int\int_R f(x,y)\,dA\) in polar coordinates.
51) Consider \(\displaystyle \int\int_R e^{-(x^2+y^2)}\,dA.\)
(a) Why is this integral difficult to evaluate in rectangular coordinates, regardless of the region \(R\)?
(b) Let \(R\) be the region bounded by the circle of radius \(a\) centered at the origin. Evaluate the double integral using polar coordinates.
(c) Take the limit of your answer from (b), as \(a\to \infty\). What does this imply about the volume under the surface of \(e^{-(x^2+y^2)}\) over the entire \(xy\)-plane?
For the following two exercises, consider a spherical ring, which is a sphere with a cylindrical hole cut so that the axis of the cylinder passes through the center of the sphere (see the following figure).
52) If the sphere has radius 4 and the cylinder has radius 2 find the volume of the spherical ring.
53) A cylindrical hole of diameter 6 cm is bored through a sphere of radius 5 cm such that the axis of the cylinder passes through the center of the sphere. Find the volume of the resulting spherical ring.
- Answer
- \(\frac{256\pi}{3} \space \text{cm}^3\)
54) Find the volume of the solid that lies under the double cone \(z^2 = 4x^2 + 4y^2\), inside the cylinder \(x^2 + y^2 = x\), and above the plane \(z = 0\).
55) Find the volume of the solid that lies under the paraboloid \(z = x^2 + y^2\), inside the cylinder \(x^2 + y^2 = 1\) and above the plane \(z = 0\).
- Answer
- \(\frac{3\pi}{32}\)
56) Find the volume of the solid that lies under the plane \(x + y + z = 10\) and above the disk \(x^2 + y^2 = 4x\).
57) Find the volume of the solid that lies under the plane \(2x + y + 2z = 8\) and above the unit disk \(x^2 + y^2 = 1\).
- Answer
- \(4\pi\)
58) A radial function \(f\) is a function whose value at each point depends only on the distance between that point and the origin of the system of coordinates; that is, \(f (x,y) = g(r)\), where \(r = \sqrt{x^2 + y^2}\). Show that if \(f\) is a continuous radial function, then
\[\iint_D f(x,y)dA = (\theta_2 - \theta_1) [G(R_2) - G(R_1)], \space where \space G'(r) = rg(r) \nonumber \] and \((x,y) \in D = \big\{(r, \theta)\,|\,R_1 \leq r \leq R_2, \space 0 \leq \theta \leq 2\pi\big\}\), with \(0 \leq R_1 < R_2\) and \(0 \leq \theta_1 < \theta_2 \leq 2\pi\).
59) Use the information from the preceding exercise to calculate the integral \(\displaystyle \iint_D (x^2 + y^2)^3 dA,\) where \(D\) is the unit disk.
- Answer
- \(\frac{\pi}{4}\)
60) Let \(f(x,y) = \dfrac{F'(r)}{r}\) be a continuous radial function defined on the annular region \(D = \big\{(r,\theta)\,|\,R_1 \leq r \leq R_2, \space 0 \leq \theta \leq 2\pi\big\}\), where \(r = \sqrt{x^2 + y^2}\), \(0 < R_1 < R_2\), and \(F\) is a differentiable function.
Show that \(\displaystyle \iint_D f(x,y)\,dA = 2\pi [F(R_2) - F(R_1)].\)
61) Apply the preceding exercise to calculate the integral \(\displaystyle \iint_D \frac{e^{\sqrt{x^2+y^2}}}{\sqrt{x^2 + y^2}} \,dx \space dy\) where \(D\) is the annular region between the circles of radii 1 and 2 situated in the third quadrant.
- Answer
- \(\frac{1}{2} \pi e(e - 1)\)
62) Let \(f\) be a continuous function that can be expressed in polar coordinates as a function of \(\theta\) only; that is, \(f(x,y) = h(\theta)\), where \((x,y) \in D = \big\{(r, \theta)\,|\,R_1 \leq r \leq R_2, \space \theta_1 \leq \theta \leq \theta_2\big\}\), with \(0 \leq R_1 < R_2\) and \(0 \leq \theta_1 < \theta_2 \leq 2\pi\).
Show that \(\displaystyle \iint_D f(x,y) \,dA = \frac{1}{2} (R_2^2 - R_1^2) [H(\theta_2) - H(\theta_1)]\), where \(H\) is an antiderivative of \(h\).
63) Apply the preceding exercise to calculate the integral \(\displaystyle \iint_D \frac{y^2}{x^2}\,dA,\) where \(D = \big\{(r, \theta)\,|\, 1 \leq r \leq 2, \space \frac{\pi}{6} \leq \theta \leq \frac{\pi}{3}\big\}.\)
- Answer
- \(\sqrt{3} - \frac{\pi}{4}\)
64) Let \(f\) be a continuous function that can be expressed in polar coordinates as a function of \(\theta\) only; that is \(f(x,y) = g(r)h(\theta)\), where \((x,y) \in \big\{(r, \theta )\,|\,R_1 \leq r \leq R_2, \space \theta_1 \leq \theta \leq \theta_2\big\}\) with \(0 \leq R_1 < R_2\) and \(0 \leq \theta_1 < \theta_2 \leq 2\pi\). Show that \[\iint_D f(x,y)\,dA = [G(R_2) - G(R_1)] \space [H(\theta_2) - H(\theta_1)], \nonumber \] where \(G\) and \(H\) are antiderivatives of \(g\) and \(h\), respectively.
65) Evaluate \(\displaystyle \iint_D \arctan \left(\frac{y}{x}\right) \sqrt{x^2 + y^2}\,dA,\) where \(D = \big\{(r,\theta)\,|\, 2 \leq r \leq 3, \space \frac{\pi}{4} \leq \theta \leq \frac{\pi}{3}\big\}\).
- Answer
- \(\frac{133}{864}\pi^2\)
66) A spherical cap is the region of a sphere that lies above or below a given plane.
a. Show that the volume of the spherical cap in the figure below is \(\frac{1}{6} \pi h (3a^2 + h^2)\).
b. A spherical segment is the solid defined by intersecting a sphere with two parallel planes. If the distance between the planes is \(h\) show that the volume of the spherical segment in the figure below is \(\frac{1}{6}\pi h (3a^2 + 3b^2 + h^2)\).
67) In statistics, the joint density for two independent, normally distributed events with a mean \(\mu = 0\) and a standard distribution \(\sigma\) is defined by \(p(x,y) = \frac{1}{2\pi\sigma^2} e^{-\frac{x^2+y^2}{2\sigma^2}}\). Consider \((X,Y)\), the Cartesian coordinates of a ball in the resting position after it was released from a position on the z -axis toward the \(xy\)-plane. Assume that the coordinates of the ball are independently normally distributed with a mean \(\mu = 0\) and a standard deviation of \(\sigma\) (in feet). The probability that the ball will stop no more than \(a\) feet from the origin is given by \[P[X^2 + Y^2 \leq a^2] = \iint_D p(x,y) dy \space dx, \nonumber \] where \(D\) is the disk of radius \(a\) centered at the origin. Show that \[P[X^2 + Y^2 \leq a^2] = 1 - e^{-a^2/2\sigma^2}. \nonumber \]
68) The double improper integral \[\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-x^2+y^2/2}\,dy \, dx \nonumber \] may be defined as the limit value of the double integrals \(\displaystyle \iint_D e^{-x^2+y^2/2}\,dA\) over disks \(D_a\) of radii \(a\) centered at the origin, as \(a\) increases without bound; that is,
\[\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-x^2+y^2/2}dy \space dx = \lim_{a\rightarrow\infty} \iint_{D_a} e^{-x^2+y^2/2}\,dA. \nonumber \]
Use polar coordinates to show that \(\displaystyle \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-x^2+y^2/2}\,dy \, dx = 2\pi.\)
69) Show that \(\displaystyle \int_{-\infty}^{\infty} e^{-x^2/2}\,dx = \sqrt{2\pi}\) by using the relation
\[ \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-x^2+y^2/2}\,dy \,dx = \left(\int_{-\infty}^{\infty} e^{-x^2/2}dx \right) \left( \int_{-\infty}^{\infty} e^{-y^2/2}dy \right). \nonumber \]
Contributors
-
Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org .
- Problems 1, 2, 34 - 37 and 50 - 51 are from Apex Calculus, Chapter 13.3
- Edited by Paul Seeburger (Monroe Community College)


Chapter 13: Special Relativity
Modern relativity is divided into two parts. Special relativity deals with observers who are moving at constant velocity. General relativity deals with observers who are undergoing acceleration. Einstein is famous because his theories of relativity made revolutionary predictions. Most importantly, his theories have been verified to great precision in a vast range of experiments, altering forever our concept of space and time.
-
- 13.1: Prelude to Special Relativity
- It is important to note that although classical mechanics, in general, and classical relativity, in particular, are limited, they are extremely good approximations for large, slow-moving objects. Otherwise, we could not use classical physics to launch satellites or build bridges. In the classical limit (objects larger than submicroscopic and moving slower than about 1% of the speed of light), relativistic mechanics becomes the same as classical mechanics.
-
- 13.2: Einstein’s Postulates
- Relativity is the study of how different observers measure the same event. Modern relativity is correct in all circumstances and, in the limit of low velocity and weak gravitation, gives the same predictions as classical relativity. An inertial frame of reference is a reference frame in which a body at rest remains at rest and a body in motion moves at a constant speed in a straight line unless acted on by an outside force. Modern relativity is based on Einstein’s two postulates.
-
- 13.3: Simultaneity and Time Dilation
- Two simultaneous events are not necessarily simultaneous to all observers—simultaneity is not absolute. Time dilation is the phenomenon of time passing slower for an observer who is moving relative to another observer. Observers moving at a relative velocity do not measure the same elapsed time for an event. Proper time is measured by an observer at rest relative to the event being observed and implies that relative velocity cannot exceed the speed of light.
-
- 13.4: Length Contraction
- All observers agree upon relative speed. Distance depends on an observer’s motion. Proper length is the distance between two points measured by an observer who is at rest relative to both of the points. Earth-bound observers measure proper length when measuring the distance between two points that are stationary relative to the Earth. Length contraction is the shortening of the measured length of an object moving relative to the observer’s frame.
-
- 13.5: Relativistic Addition of Velocities
- With classical velocity addition, velocities add vectorially. Relativistic velocity addition describes the velocities of an object moving at a relativistic speed. An observer of electromagnetic radiation sees relativistic Doppler effects if the source of the radiation is moving relative to the observer. The wavelength of the radiation is longer than that emitted by the source when the source moves away from the observer and shorter when the source moves toward the observer.
-
- 13.6: Relativistic Momentum
- The law of conservation of momentum is valid whenever the net external force is zero and for relativistic momentum. Relativistic momentum is classical momentum multiplied by the relativistic factor. At low velocities, relativistic momentum is equivalent to classical momentum. Relativistic momentum approaches infinity as uu approaches cc . This implies that an object with mass cannot reach the speed of light. Relativistic momentum is conserved, just as classical momentum is conserved.
-
- 13.7: Relativistic Energy
- Conservation of energy is one of the most important laws in physics. Not only does energy have many important forms, but each form can be converted to any other. We know that classically the total amount of energy in a system remains constant. Relativistically, energy is still conserved, provided its definition is altered to include the possibility of mass changing to energy, as in the reactions that occur within a nuclear reactor.
Thumbnail: A diagrammatic representation of spacetime. Image use with permission (CC-BY-SA 3.0; Stib).
Contributors and Attributions
-
Paul Peter Urone (Professor Emeritus at California State University, Sacramento) and Roger Hinrichs (State University of New York, College at Oswego) with Contributing Authors: Kim Dirks (University of Auckland) and Manjula Sharma (University of Sydney). This work is licensed by OpenStax University Physics under a Creative Commons Attribution License (by 4.0) .


3.5: Sources of Law- Administrative Law, Common Law, Case Law and Court Rules
Administrative Law—Agency-Made Law
State and federal legislatures cannot keep up with the task of enacting legislation on all the myriad subjects that must be regulated by law. In each branch of government, various administrative agencies exist with authority to create administrative law. At the federal level, for example, the Environmental Protection Agency enacts regulations against environmental crimes. At the state level, the Department of Motor Vehicles enacts laws concerning drivers’ license suspension. Administrative regulations are enforceable by the courts provided that the agency has acted within the scope of its delegated authority from the legislature.
Common Law
[1] LaFave describes the process by which common law was derived in England.
“ . . . Although there were some early criminal statutes [in England], in the main the criminal law was originally common law. Thus by the 1600s the judges, not the legislature, had created and defined the felonies of murder, suicide, manslaughter, burglary, arson, robbery, larceny rape, sodomy and mayhem; and such misdemeanors as assault, battery, false imprisonment, libel, perjury, and intimidation of jurors. During the period from 1660 . . . to 1860 the process continued with the judges creating new crimes when the need arose and punishing those who committed them: blasphemy (1676), conspiracy (1664), sedition (18th century), forgery (1727), attempt (1784), solicitation (1801). From time to time the judges, when creating new misdemeanors, spoke of the court’s power to declare criminal any conduct tending to “outrage decency” or “corrupt public morals.” or to punish conduct contra bonos mores: thus they found running naked in the streets, publishing an obscene book, and grave-snatching to be common law crimes.
[3]
Stare decisis , discussed below, plays no persuasive or binding role in the civil law tradition. Under the common law tradition though, new substantive law generally adds to, rather than replaces, old substantive law.
“Though the description of the Anglo-American system as a “common law legal system” notes an important distinction between it and the civil law system, that description should not lead one to ignore the fact that legislation also constitutes an important source of law in the Anglo-American system. That system is actually a mixed system of common law rules and statutory rules. The common law rules established by American and English courts have always been subject to displacement by legislative enactments. Indeed, the courts have the authority to develop common law standards only where the legislatures have not sought to provide legislative solutions. … In our country, the common law also is subject to the legal limitations imposed by federal and state constitutions. The supremacy of the constitutions extends over all forms of law, including the common law. Just as legislation cannot violate a constitutional limitation, neither can a common law rule.” [4]
Judge-Made Law: Case Law
case law refers to legal rules announced in opinions written by appellate judges when deciding appellate cases before them. Judicial decisions reflect the court’s interpretation of constitutions, statutes, common law, or administrative regulations. When the court interprets a statute, the statute, as well as its interpretation, control how the law will be enforced and applied in the future. The same is true when a court interprets federal and state constitutions. When deciding cases and interpreting the law, judges are bound by precedent.
Callstack:
at (Bookshelves/Corrections/Introduction_to_the_American_Criminal_Justice_System_(Burke_et_al.)/03:_Criminal_Law/3.05:_Sources_of_Law-_Administrative_Law_Common_Law_Case_Law_and_Court_Rules), /content/body/div[1]/p[5]/@if, line 1, column 3
Callstack:
at (Bookshelves/Corrections/Introduction_to_the_American_Criminal_Justice_System_(Burke_et_al.)/03:_Criminal_Law/3.05:_Sources_of_Law-_Administrative_Law_Common_Law_Case_Law_and_Court_Rules), /content/body/div[2]/p[2]/@if, line 1, column 3
Court Rules of Procedure
The U.S. Supreme Court and state supreme courts make a law that regulates the procedures followed in the lower courts- both appellate and trial courts- in that jurisdiction. These court rules, adopted by the courts to facilitate the administration and processing of cases, are generally limited in scope, but they may nevertheless provide significant rights for the defendant. For example, the rules governing speedy trials may be governed generally by the Constitution, but very specifically by court rules in a particular jurisdiction.
Local courts may also pass local court rules that govern the day-to-day practice of law in these lower courts. For example, a local court rule may dictate when and how cases are to be filed in that jurisdiction. Generally, the local bar (all the attorneys in the jurisdiction) are consulted, and a workforce consisting of judges, trial court administrators, and representatives from district attorney’s office, the public defender’s office, assigned counsel consortiums, and private attorneys will meet every few years to decide on the local rules.
Okay, so where do I look to see if my behavior is prohibited?
stare decisis , one red flag that your behavior may be unlawful is that, in the past, the courts have found behavior similar to yours to be unlawful.
-
Kerper, H. B. (1979).
Introduction to the criminal justice system
(2nd ed., pp. 27). West Publishing Company.
↵
-
LaFave, W. R. (2000).
Criminal law
(3rd ed., pp. 70). St. Paul, Minn: West Group.
↵
-
Kerper, H. B. (1979).
Introduction to the criminal justice system
(2nd ed., pp. 27). West Publishing Company.
↵
-
Kerper, H. B. (1979).
Introduction to the criminal justice system
(2nd ed., pp. 28-29). West Publishing Company.
↵
-
Vasquez v. Hillery
, 474 U.S. 254 (1986)
↵
-
Kerper, H. B. (1979).
Introduction to the criminal justice system
(2nd ed., pp. 47-49). West Publishing Company.
↵
-
Kerper, H. B. (1979).
Introduction to the criminal justice system
(2nd ed., pp. 49). West Publishing Company.
↵
-
Kerper, H. B. (1979).
Introduction to the criminal justice system
(2nd ed., pp. 49). West Publishing Company.
↵
-
Cardozo, B. N. (1924).
The Growth of the Law.
New Haven, CT: Yale University Press.
↵
-
Kerper, H. B. (1979).
Introduction to the criminal justice system
(2nd ed., pp. 51-52). West Publishing Company.
↵
-
LaFave, W. R. (2000).
Criminal law
(3rd ed., pp. 86-89). St. Paul, Minn: West Group.
↵
-
LaFave, W. R. (2000).
Criminal law
(3rd ed., pp. 89). St. Paul, Minn: West Group.
↵
-
LaFave, W. R. (2000).
Criminal law
(3rd ed., pp. 96-97). St. Paul, Minn: West Group.
↵


8.4: Context- School and Extracurricular Activities in Middle Childhood
-
- Last updated
- Save as PDF
Learning Objectives
By the end of this section, you will be able to:
- Identify features of the school climate that foster development
- Describe types of teacher-student relationship s
- Discuss the goals of the positive youth development model
Yvonne wants to go to school and learn. They love the thrill of studying something new, memorizing information for a test, and pushing their mind in new directions. The adrenaline rush of writing a math answer on the board or raising a hand to provide an answer is fulfilling and fun.
But despite this, school is not a place where Yvonne feels comfortable. The other students in class are not enthusiastic about learning and seldom share Yvonne’s eagerness for reading long books or perfecting long-division skills. The teacher seems scattered, sometimes losing his place while teaching. Yvonne feels understimulated and underappreciated.
In this section, you’ll consider the demands of the class on students, and the way the school environment can influence child development outcomes through the teacher-student relationship.
You’ll also learn about the social aspects of school and other contexts of youth development.
Academic Demands
One of the biggest changes in middle childhood is navigating formal education , or elementary school. School is a social place, in which peer interactions are expected and demanded for most of the day, and in which children encounter new social and emotional demands. As children enter middle childhood, their formal schooling becomes more structured, and assessment becomes increasingly important. Elementary schools have become more focused on teaching skills that are assessed on standardized tests (Russell, 2011). These academic demands require attention and self-control from students (Degol & Bachman, 2023; Pener–Tessler et al., 2022). Children may also face learning obstacles and new relationships with peers and teachers.
Students are expected to remain at their desks for most of the day and follow along with group activities, complete tasks on time, and keep their bodies still. In fact, children spend about 64 percent of their school day sedentary (Egan et al., 2019). Sedentary behavior is linked to more inattention and increased hyperactivity (Suchert et al., 2017). It can be particularly demanding for children who feel the need to wiggle, dance, and walk to concentrate, and especially difficult for children with attention-deficit/hyperactivity disorder (Evans et al., 2020). Further, research has shown that being physically active has important academic benefits (Norris et al., 2020).
Finally, the classroom is a place of learning, particularly about literacy, math, and other academic subjects. By the end of middle childhood, children are beginning to take tests and quizzes and complete individual projects. A focus on academic performance can put emotional strain on children as they learn to measure up and perform at the expected levels. Academic stress is associated with lower academic performance (Luo et al., 2016). This stress and pressure can come from a range of sources including the child’s personal academic goals as well as from school staff or family members (Sotardi, 2016).
Parenting style is a particularly important influence on the child’s academic achievement. When parents show high levels of warmth , it can help a child build confidence in their academic abilities, while lower warmth from parents may increase a child’s experience of academic pressure (Luo et al., 2020). Parents’ and teachers’ expectations also matter. Having higher educational expectations for children can lead to positive academic performance, but very high expectations can exert pressure and generate academic burnout (Luo et al., 2020). In some cases, failure to meet parents’ academic expectations can lead to child maltreatment. For example, one study found that incidents of child abuse increase when report cards go home (Bright et al., 2019). Parents who place a lot of focus on measures of how their child is doing in school (grades and test scores) may also unintentionally teach children to focus on grades rather than on the process of learning (Pinquart & Ebeling, 2020).
Parental expectations about academic performance vary across cultures. For example, parents in the United States tend to focus on academic success and praising their child, while many parents in China are more involved and tend to focus more on academic failures and mistakes (Pomerantz et al., 2014). Although parental expectations for academic performance tend to be lower among children in families with lower socioeconomic status (SES) in Western societies, parents of children in East Asian societies more consistently have high expectations across families with lower and higher SES (Li & Xie, 2020). 6 Multiple factors including parent education, culture, and financial resources are likely to influence how parents express their academic expectation s to children. Generally, research indicates that a hybrid approach, emphasizing learning from mistakes while praising successes, may be beneficial to most children (Pomerantz et al., 2014).
Classroom Climate
How well children navigate and meet the demands of the classroom largely depends on classroom climate , the emotional culture of the learning environment. For instance, a positive classroom climate is characterized by encouragement from peers, well-organized learning activities, respect for the asking and answering of questions, kindness, warmth, and happiness (Wang et al., 2020). It also includes positive disciplinary practices and a focus on developing positive interpersonal relationships between teachers, students, and their peers (Wang & Degol, 2016). Children who experience positive classroom climates are more likely to participate, volunteer answers, and enjoy being in school (Wang et al., 2020).
In contrast, a negative classroom climate is characterized by chaotic or disorganized activities, teachers and peers who are critical and judgmental, bullying behavior, or an overly strict and disciplinary atmosphere. Children in this climate are more likely to disengage, withdraw, and form a negative opinion about school and learning (Wang et al., 2020).
Given the importance of classroom climate for academic success, many countries, including Canada, China, Germany, England, France, Israel, Singapore, and the United States, have initiatives designed to improve it (Wang et al., 2020). For example, some schools have focused on improving school quality through improving the relationships between teachers, parents, and students, while many other schools have moved toward a focus on promoting positive behaviors in classrooms (Horner et al., 2015; Wang & Degol, 2016).
Link to Learning
Education for elementary school children is often approached quite differently in other countries. Some elementary school rules in Japan are presented in this video.
Teacher-Group Relationships
Teachers have a major impact on classroom climate. A teacher who is aloof, distant, and disorganized may influence students to be off-task, overly active, and talkative (Wang et al., 2020). Teachers who are intrusive, overdemanding, and authoritarian, however, may foster a fearful learning environment in which students experience anxiety and stress over their contributions and performance. A teacher who provides a high level of warmth, encouragement, and support for autonomy can create a positive learning environment in which students feel comfortable participating and can rise to their full potential (Quin, 2017).
Teachers’ classroom organization and teaching strategies also influence the climate, including how responsive teachers are to student questions, their instructional formats and learning activities, and their sensitivity to student needs (Miller & Wang, 2019). Effective teaching strategies that allow children to work independently and develop competence, practice in small groups and with challenging learning problems, while promoting a warm and helpful learning environment can all help children stay engaged in the learning process (Miller & Wang, 2019).
Teacher-Child Relationships
In addition to fostering classroom climate, teachers also form one-on-one relationships with their students that can influence the way each child perceives school and learning. Teachers who promote communication and are open to student feedback are more likely to foster healthy teacher-student relationship s, which in turn increases student motivation (Liu, 2021). Teachers and students who are excited to see each other, look forward to exchanging ideas, and are comfortable approaching one another are considered to have a close teacher-child relationship. Close relationships are beneficial for learning both because they allow children to be happy to attend school and listen to their teacher and because they motivate teachers to excel in their profession. Children who form close relationships with their teachers are more likely to have a high level of prosocial and cooperative behaviors (Portilla et al., 2014).
Some children may form dependent teacher-child relationships in which they are particularly clingy, overly attached, or possessive of their teacher. This relationship may occur with children who are nervous, shy, or new to formal education (Bosman et al., 2021). They often desire more one-on-one attention from their teacher than is possible, they may require extra assistance during learning activities, they may become upset when their teacher is absent or unavailable, and they may use their teacher as an emotional support system. However, when teachers are provided with training and support to improve teacher-student relationships, children and teachers are more likely to form healthier relationships and learning environments (Bosman et al., 2021).
Finally, children and teachers may form conflicted relationships, in which their exchanges are characterized by arguments, suspicion, power struggles, and conflict (Mason et al., 2017). Some children who form conflicted relationships with their teachers may struggle with motivation to do well and instead may dislike and harbor hostile feelings toward school. For example, one study found higher rates of teacher-perceived behavioral problems among Norwegian school-aged children who had conflicted relationships with their teachers (Drugli, 2013). Conflicted relationships can also have a negative impact on the teacher. For example, teachers in such relationships are at greater risk of burnout and struggle more with class organization and quality (Hoglund et al., 2015). Promoting greater classroom support for teachers and positive behavior in the classrooms can improve teacher self-efficacy and teacher perceptions of student behaviors and increase teacher-student relationship quality.
Cultural expectations may also influence the development of conflicted relationships. Teachers may enter the classroom with expectations based on their own culture or influenced by stereotypes they hold. For example, children of color as well as children from low-SES backgrounds are more likely to be perceived by their teachers to have more socioemotional problems (Ginsberg, 2023). White teachers often see their relationships with White children more positively than their relationships with Black children, particularly leaving Black boys at a higher risk of poor teacher-child relationship (Rudasill et al., 2023). However, when close teacher-student relationships are present for children of color, including Black and Latino boys, it can serve as a protective factor and promote learning gains such as in language skills (Goldberg & Iruka, 2023). 7
Over the course of middle childhood, this racial gap in teacher-child conflict widens (Spilt & Hughes, 2015). Unintentional biases among school officials can interact with systemic racism and students’ adverse childhood experiences to foster conditions that increase academic failure and dropout risk, particularly for historically marginalized racial and ethnic children. However, these trajectories can be shifted by greater classroom diversity, an increase in teacher diversity, and provision of educational interventions at the school level that promote positive behavioral support and cultural competence (La Salle et al., 2020; Rasheed et al., 2020).
Deficit versus Asset-Based Models in the Classroom
Valencia (1997, 2012) pointed out that for many decades, the education system has blamed the poor school performance of children of color on genetic, familial, and cultural deficits. This deficit model assumes that little can be done to assist the students who are at risk of failing in class and overlooks the wealth of cultural communication styles, intellectual skills, and creativity children bring to the classroom. A strength- or asset-based model, in contrast (Flint & Jaggers, 2021), assumes that all children have internal, familial, and cultural assets or strengths that they use in the classroom, learning and thriving despite facing odds like poverty and systemic racism.
In fact, the asset-based model values the insights students bring to the classroom from their cultural experiences, while at the same time striving to reduce injustices based on oppression and other social factors (Muhammad, 2020). For example, educators have documented educational projects that highlight the diversity of identities in the classroom, such as a “Who Am I?” project (Zhang–Yu et al., 2020) that allows students to connect the learning of new information to their prior knowledge and personal experience.
Interventions can take the form of training and teaching strategies that recognize and value students’ cultural backgrounds in all aspects of learning. In line with a strength-based model , Muhammad (2020) suggests focusing on identity and building on the creativity and talent children possess. Hammond (2014) notes that culturally inclusive teaching environments are not only about celebrating Black History month or Cinco de Mayo but rather about truly understanding the diverse ways in which families communicate in different cultures, celebrating and valuing those diverse communication patterns, and leveraging them in the classroom.
Numerous programs have been developed in schools using an antibias framework to improve outcomes for marginalized children. For example, summer programs that support children in economically disadvantaged households or neighborhoods can reduce achievement gaps in academic skills. Programs are more likely to be effective if they include interactive learning, highly trained teachers, and small class sizes (Lynch et al., 2023; Trends, 2009). Improving inclusion in the classroom can have a positive influence on students’ growth, learning, and feelings of safety in the classroom. One study of Canadian teachers found that when teachers were more culturally sensitive and aware, they were more likely to focus on and promote Indigenous students’ overall skills and knowledge, which in turn improved students’ academic achievement (Perso, 2020).
A U.S. study found that nationwide almost 75 percent of K–12 teachers surveyed said they provide antibias education in different forms in their classroom (Woo et al., 2022). The study recommended developing and adopting clear and shared standards and evidence-based practices for antibias education and investing in curriculum development and selection and hiring practices (Woo et al., 2022). Teachers who used antibias curriculum materials said their students were more respectful of diversity and developed more positive social identities (Woo et al., 2022). For example, New Jersey’s department of education and division on civil rights have developed several resources and a plan to prevent, interrupt, and deal with instances of bias in schools (New Jersey DCR & DOE, 2023). However, there is still a long way to go to ensure equitable practices for all children’s social and educational development to benefit.
Link to Learning
This article describes a study by Nelson and Johnson (2023) that examined White teachers’ responses to working in racially diverse schools where they were either in the majority or the minority.
Positive Youth Development
Outside formal education, middle childhood is a time of expanding our network to include new extracurricular activities and connections. Children in this developmental phase often engage in music, sports, art, and civic activities that allow them to practice teamwork, leadership, and personal growth skills. Members of scouting groups, for example, often learn many interpersonal and self-regulatory skills that help them to better communicate, understand, and interact with the world around them. Engaging in fundraisers for charitable organizations, organizing community events, and learning about their local, regional, and global issues helps children to better understand their part in the world.
Despite clear benefits, however, extracurricular activities also highlight disparities in SES . Though they have the most to gain from after-school programs, children from lower SES backgrounds have the least access to them due to barriers such as expense and transportation needs (Heath et al., 2022; Smink, 2013). As communities recognize this, many are taking steps to make extracurricular activities more accessible to families with lower SES. For example, many foundations provide support for after-school programs designed to reach children from families with lower SES and immigrant families (Heath et al., 2022).
Whether extracurriculars consist of a coding club at the library, arts at school, sports activities, or environmental activism, the goal is often to facilitate the development of resilience (Figure 8.11). As you’ve learned, resilience is the emotional ability to respond to stressful events with determination, perseverance, and adaptability. For instance, a child who is studying dance may need to commit long hours to performing a routine (determination), continue performing over many weeks and months (perseverance), and learn to accept feedback and become better at certain parts of the routine (adaptability).
Actively helping children by understanding and promoting predictors of well-being, thriving, and prosocial development in childhood and beyond is the focus of positive youth development (PYD) programs (Johnson & Ettekal, 2023). There are a variety of PYD models and theories, and they may use a range of techniques to engage youth, but at their core they all focus on promoting thriving in healthy youth development (Johnson & Ettekal, 2023). The PYD programs use a strengths-based model to focus on helping children develop by using their existing resources. For example, participation in sports is one way to engage youth and has been linked to positive socioemotional and mental health outcomes in children (Whitley et al., 2019) as well as to the learning of life skills and feelings of competence and confidence (Bruner et al., 2023).
References
Bosman, R. J., Zee, M., de Jong, P. F., Koomen, H. M. Y. (2021). Using relationship–focused reflection to improve teacher–child relationships and teachers' student–specific self-efficacy. Journal of School Psychology, 87 , 28–47. https://doi.org/10.1016/j.jsp.2021.06.001
Bright, M. A., Lynne, S. D., Masyn, K. E., Waldman, M. R., Graber, J., & Alexander, R. (2019). Association of Friday school report card release with Saturday incidence rates of agency–verified physical child abuse. JAMA Pediatrics , 173 (2), 176–182. doi.org/10.1001%2Fjamapediatrics.2018.4346
Bruner, M. W., McLaren, C. D., Sutcliffe, J. T., Gardner, L. A., Lubans, D. R., Smith, J. J., & Vella, S. A. (2023). The effect of sport–based interventions on positive youth development: A systematic review and meta–analysis. International Review of Sport and Exercise Psychology , 16 (1), 368–395. doi.org/10.1080/1750984X.2021.1875496
Degol, J. L., & Bachman, H. J. (2023). Early self–control and sustained attention problems: Associations with youth achievement, motivation, and engagement. Cognitive Development , 65 , Article 101290. https://doi.org/10.1016/j.cogdev.2022.101290
Drugli, M. B. (2013). How are closeness and conflict in student–teacher relationships associated with demographic factors, school functioning and mental health in Norwegian schoolchildren aged 6–13?. Scandinavian Journal of Educational Research, 57 (2), 217–225. doi.org/10.1080/00313831.2012.656276
Egan, C. A., Webster, C. A., Beets, M. W., Weaver, R. G., Russ, L., Michael, D., Nesbitt, D., & Orendorff, K. L. (2019). Sedentary time and behavior during school: a systematic review and meta-analysis. American Journal of Health Education, 50 (5), 283–90. doi.org/10.1080/19325037.2019.1642814
Evans, S. C., Cooley, J. L., Blossom, J. B., Pederson, C. A., Tampke, E. C., & Fite, P. J. (2020). Examining ODD/ADHD symptom dimensions as predictors of social, emotional, and academic trajectories in middle childhood. Journal of Clinical Child and Adolescent Psychology: The Official Journal for the Society of Clinical Child and Adolescent Psychology, American Psychological Association, Division 53 , 49 (6), 912–929. doi.org/10.1080/15374416.2019.1644645
Flint, A. S., & Jaggers, W. (2021). You matter here: The impact of asset-based pedagogies on learning. Theory Into Practice , 60 (3), 254–264. doi.org/10.1080/00405841.2021.1911483
Ginsberg, Y. C. (2023). Socialization discontinuity as an explanation for disparities in teacher–child relationships. Early Childhood Research Quarterly , 65 , 170–178. https://doi.org/10.1016/j.ecresq.2023.06.006
Goldberg, M. J., & Iruka, I. U. (2023). The role of teacher–child relationship quality in Black and Latino boys’ positive development. Early Childhood Education Journal , 51 (2), 301–315. https://doi.org/10.1007/s10643-021-01300-3
Hammond, Z. (2014) Culturally responsive teaching and the brain: Promoting authentic engagement and rigor among culturally and linguistically diverse students. Corwin Publishers. https://us.sagepub.com/sites/default...tem_140839.pdf
Heath, R. D., Anderson, C., Turner, A. C., & Payne, C. M. (2022). Extracurricular activities and disadvantaged youth: A complicated—but promising—story. Urban Education , 57 (8), 1415–1449. doi.org/10.1177/0042085918805797
Hoglund, W. L., Klingle, K. E., & Hosan, N. E. (2015). Classroom risks and resources: Teacher burnout, classroom quality and children's adjustment in high needs elementary schools. Journal of School Psychology , 53 (5), 337–357. https://doi.org/10.1016/j.jsp.2015.06.002
Horner, R. H., Sugai, G., & Lewis, T. (2015). Is school-wide positive behavior support an evidence-based practice?. https://aztap.org/wp-content/uploads...ak-Handout.pdf
Johnson, S. K., & Ettekal, A. V. (2023). The five Cs of positive youth development: configurations of thriving in four U.S. adolescent samples. Journal of Research on Adolescence , 33 (2), 656–679. doi.org/10.1111/jora.12806
La Salle, T. P., Wang, C., Wu, C., & Rocha Neves, J. (2020). Racial mismatch among minoritized students and white teachers: Implications and recommendations for moving forward. Journal of Educational and Psychological Consultation , 30 (3), 314–343. doi.org/10.1080/10474412.2019.1673759
Li, W., & Xie, Y. (2020). The influence of family background on educational expectations: a comparative study. Chinese Sociological Review , 52 (3), 269–294. doi.org/10.1080/21620555.2020.1738917
Liu, W. (2021). Does teacher immediacy affect students? A systematic review of the association between teacher verbal and non-verbal immediacy and student motivation. Frontiers in Psychology , 12 , Article 713978. https://doi.org/10.3389/fpsyg.2021.713978
Luo, Y., Deng, Y.T., & Zhang, H. (2020). The influences of parental emotional warmth on the association between perceived teacher-student relationship and academic stress among middle school students in China. Children and Youth Services Review, 114, Article 105014. https://doi.org/10.1016/j.childyouth.2020.105014
Luo, Y., Wang, Z., Zhang, H. & Chen, A. (2016). The influence of family socio–economic status on academic burnout in adolescents: Mediating and moderating effects. Journal of Child and Family Studies, 25, 2111–2119. https://doi.org/10.1007/s10826-016-0400-2
Lynch, K., An, L., & Mancenido, Z. (2023). The impact of summer programs on student mathematics achievement: A meta–analysis. Review of Educational Research , 93 (2), 275–315. doi.org/10.3102/00346543221105543
Mason, B. A., Hajovsky, D. B., McCune, L. A., & Turek, J. J. (2017). Conflict, closeness, and academic skills: A longitudinal examination of the teacher–student relationship. School Psychology Review , 46 (2), 177–189. doi.org/10.17105/SPR-2017-0020.V46-2
Muhammad, G. (2020) Cultivating genius: An equity framework for culturally and historically responsive literacy. Scholastic Teaching Resources.
Nelson, J. L., & Johnson, T. D. (2023). How white workers navigate racial difference in the workplace: Social-emotional processes and the role of workplace racial composition. Work and Occupations , 51 (3). doi.org/10.1177/07308884231176833
New Jersey Division on Civil Righta & Department of Education. (2023). Addressing Bias in K12 Schools. https://www.njoag.gov/wp-content/upl...12-Schools.pdf
Norris, E., van Steen, T., Direito, A., & Stamatakis, E. (2020). Physically active lessons in schools and their impact on physical activity, educational, health and cognition outcomes: a systematic review and meta-analysis. British Journal of Sports Medicine , 54 (14), 826–838. doi.org/10.1136/bjsports-2018-100502
Pener–Tessler, R., Markovitch, N., & Knafo–Noam, A. (2022). The special role of middle childhood in self‐control development: Longitudinal and genetic evidence. Developmental Science , 25 (5), Article e13270. doi.org/10.1111/desc.13270
Perso, T. (2020). Teaching Indigenous students: Cultural awareness and classroom strategies for improving learning outcomes . Routledge. doi.org/10.4324/9781003117728
Pinquart, M. & Ebeling, M. (2020). Parental education expectations and academic achievements in children and adolescents – A meta–analysis. Educational Psychology Review, 32, 463–480. https://doi.org/10.1007/s10648-019-09506-z
Pomerantz, E. M., Ng, F. F. Y., Cheung, C. S. S., & Qu, Y. (2014). Raising happy children who succeed in school: Lessons from China and the United States. Child Development Perspectives , 8 (2), 71–76. doi.org/10.1111/cdep.12063
Portilla, X. A., Ballard, P. J., Adler, N. E., Boyce, W. T., & Obradovic, J. (2014). An integrative view of school functioning: Transactions between self–regulation,school engagement, and teacher–child relationship quality. Child Development, 85 , 1915–1931. doi.org/10.1111/cdev.12259
Quin, D. (2017). Longitudinal and contextual associations between teacher–student relationships and student engagement: A systematic review. Review of Educational Research , 87 (2), 345–387. doi.org/10.3102/0034654316669434
Rasheed, D. S., Brown, J. L., Doyle, S. L., & Jennings, P. A. (2020). The effect of teacher–child race/ethnicity matching and classroom diversity on children's socioemotional and academic skills. Society for Research in Child Development , 91 (3), e597–e618. doi.org/10.1111/cdev.13275
Rudasill, K. M., McGinnis, C. P., Cheng, S. L., Cormier, D. R., & Koziol, N. (2023). White privilege and teacher perceptions of teacher–child relationship quality. Journal of School Psychology, 98, 224–239. https://doi.org/10.1016/j.jsp.2023.04.002
Russell, J. L. (2011). From child’s garden to academic press: The role of shifting institutional logics in redefining kindergarten education. American Educational Research Journal 48(2), 236–67. doi.org/10.3102/0002831210372135
Smink, J. (2013). A proven solution for dropout prevention: Expanded learning opportunities. In Terry K. P. (Ed.), Expanding minds and opportunities: Leveraging the power of afterschool and summer learning for student success (pp. 50–57). Washington, DC: Collaborative Communications Group. Retrieved from http://www.expandinglearning.org/sit..._0.pdf#page=52 .
Sotardi, V. A. (2016). Exploring school stress in middle childhood: interpretations, experiences, and coping. Pastoral Care in Education , 35 (1), 13–27. doi.org/10.1080/02643944.2016.1269360
Spilt, J. L., & Hughes, J. N. (2015). African American children at risk of increasingly conflicted teacher–student relationships in elementary school. School psychology review , 44 (3), 306-314. dx.doi.org/10.17105/spr-14-0033.1
Suchert, V., Pedersen, A., Hanewinkel, R., & Isensee, B. (2017). Relationship between attention–deficit/hyperactivity disorder and sedentary behavior in adolescence: a cross–sectional study. Attention Deficit and Hyperactivity Disorders , 9 (4), 213–218. https://doi.org/10.1007/s12402-017-0229-6
Trends, C. (2009). Effective and promising summer learning programs and approaches for economically–disadvantaged children and youth.
Valencia, R. R. (2012). The evolution of deficit thinking: Educational thought and practice . Routledge.
Valencia, R. R. (Ed.). (1997). The evolution of deficit thinking: Educational thought and practice. The Falmer Press/Taylor & Francis.
Wang, M. T., & Degol, J. L. (2016). School climate: A review of the construct, measurement, and impact on student outcomes. Educational Psychology Review , 28 (2), 315–352. psycnet.apa.org/doi/10.1007/s10648-015-9319-1
Wang, M. T., Degol, J. L., Amemiya, J., Parr, A., & Guo, J. (2020). Classroom climate and children’s academic and psychological wellbeing: A systematic review and meta–analysis. Developmental Review , 57 , Article 100912. https://doi.org/10.1016/j.dr.2020.100912
Whitley, M. A., Massey, W. V., Camiré, M., Boutet, M., & Borbee, A. (2019). Sport–based youth development interventions in the United States: A systematic review. BMC Public Health, 19 (1), Article 89. https://doi.org/10.1186/s12889-019-6387-z
Woo, A., Lawrence, R. A., Doan, S., & Kaufman, J. H. (2022). A Snapshot of Anti–Bias Education in U.S. K–12 Schools: Findings from the 2021 American Instructional Resources Surveys. Santa Monica, CA: RAND Corporation. www.rand.org/pubs/research_r...RRA134-12.html.
Zhang–Yu, C., García-Díaz, S., García-Romero, D., & Lalueza, J. L. (2020). Funds of identity and self-exploration through artistic creation: Addressing the voices of youth. Mind, Culture, and Activity 28 (2), 1–14. doi.org/10.1080/10749039.2020.1760300


3.6: Concluding Comments
selected template will load here
This action is not available.
To this point in the course, you have learned important concepts related to the demand and supply side of the market. The elasticities covered in this chapter reinforce these concepts. Moreover, they allow questions such as those posed in the introductory section to the chapter to be answered with much more precision. Problem sets 2 and 3 represent plausible business decisions that can be answered with elasticities. In problem set 2, you are asked whether you should increase price by a given amount. In problem set 3, you will need to use an advertising elasticity to determine whether to increase advertising expenditures. Later, you will learn that provided you have a precise estimates of demand elasticities, you can answer these types of question much more accurately. In fact, if you have a good estimate of the elasticity of demand facing the firm, you can calculate the actual price that maximizes profits. This will be addressed further in Chapter 7.
In the next chapter, you will put elasticities to work in earnest. Elasticities will be used in models of equilibrium to understand and estimate how markets respond to shocks to demand or supply.  It is time to put them together to understand markets. This is the topic of Chapter 4.