EXAM2

=Examination 2=

This exam assesses the mastery of basic differentiation (using the product, quotient, and chain rules, along with all other function-specific types of rules). As before, Part 1 involves display of memorized rules (numbers 1-18 on the inside cover of the book), while Part 2 involves application and concept problems that test deeper understanding of the material. A few concepts/skills were left off the exam (for the sake of time): logarithmic differentiation, second derivatives using implicit differentiation, related rates of angles.

During the examination period for part 2 (parts were given on separate days), I stopped students after 2/3 of the class time was over and instructed them to move to the back with no papers or pencils. Once there, I put up a slide on the projected screen: As for potential aspects to change or focus on in the future, the graph in question #4 is not the best choice for the questions asked (e.g., what is the derivative at //x// = //e//?), and the simplification for #7 is tedious. Also, question #5 can be quite confusing if you haven't practiced one like that before (e.g., on the exam review).
 * [[image:Exam_Discussion.png width="224" height="187"]] ||  || I wanted to give them a chance to communicate about the mathematics, but not in a written format. I was surprised at how many chose to remain silent or ask no questions. Only one student wondered aloud if they could ask me questions (within the rules, as stated, to be sure), and I did answer one simple question. Overall, I'm not sure how much this helped their thinking, but it did seem to provide a few with ideas to pursue. [//I'll ask them later to find out what they say about the usefulness of this exercise.//] Another interesting note -- after moving to the back, their positions and orientation nearly mirrored the positions in which they sit (and few people spoke to others with whom they normally do //not// confer during class). They used their five minutes and returned to their seats to finish. ||

Overall, this is tricky material for those who have never seen calculus before (all but two of these folks). I was quite surprised by the number of students who either forgot or improperly implemented the limit definition of the derivative in #1; and I was also surprised at how difficult #9 proved to be for nearly all students. The connection to the limit concept is one of the first topics covered in this chapter, while related rates are one of the last topics. Thus, temporality may explain why these proved so difficult -- in the first case it was too long ago, while in the second not enough time was allowed to practice. I think the students' performances were fair indicators of their understanding of the material (e.g., the ability to explain concepts and apply them to novel contexts), but generally better indicators of their fluency with the algebra/calculus skills (e.g., some were unable to solve a simple quadratic equation while some still had great difficulties with even the most basic differentiation rules). Much research in mathematics education would suggest that this is a typical pattern for most learners.


 * PDFs of Exam Review and Key:**


 * PDFs of Exam and Key:**


 * Descriptive statistics regarding student performance on Examination 2:**

(n = 11)


 * __//Exam 2, Part 1//__: (20 points)**


 * Mean: 16
 * Median: 16
 * Mode: 16
 * Standard Deviation: 3.38


 * __//Exam 2, Part 2//__: (80 points)**
 * Mean: 52.7
 * Median: 53
 * Mode: 53
 * Standard Deviation: 7.5


 * __//Exam 2, Overall//__: (100 points)**
 * **Mean: 68**
 * **Median: 72**
 * **Mode: 72**
 * **Standard Deviation: 10.1**



= The following day after Exam 2, before the examination was returned to students, I requested the following from them on a sheet of paper: =
 * 1) Your name.
 * 2) Predict your score on Exam 2, Part 2.
 * 3) State and explain in detail three things that you could do to improve your performance and understanding before Exam 3.
 * 4) Please provide and explain two or three areas of need or requests or questions that you still have regarding the course or the content. This will help me as I plan and structure things from this point forward.

**Here is the data on their score predictions from question 2:**
Interestingly, the predictions ranged from massive over-estimates (e.g., 25 points) to massive under-estimates (e.g., 31) of individual's scores. Not quite sure how to interpret this, but it was an interesting exercise, I think.

**Here is a listing of the responses for question 3 (the bold phrases are me pointing out patterns and themes):**
So, it appears that "asking questions" and "doing homework" are, far and away, some of the most recognized areas of improvement. As I told folks that day, much of what can be done to improve is known only by the student. These are helpful for me know about (e.g., I know that I can make more of the homework in class to encourage students to complete it), but ultimately //most// useful to them.
 * **Ask questions**; Spend more time on homework and answer all HW questions; do extra problems in addition to homework
 * Not just **do the homework**, but "understand" it; Look over answers, go back, redo (change) if necessary; Applying what I've learned into bigger and harder questions. Comprehension.
 * **Do my homework** more efficiently and effectively; Study more often before the exam; Use the wiki and book more when I am stuck on something; Apply my knowledge to be able to do bigger, harder problems.
 * Take more notes; **Practice** more problems; be aware in class.
 * **Ask questions** much more frequently; Study every night, even if I have no homework; Read all directions/questions on exam; Time management.
 * Focus on application problems!!; **Stop not asking any questions**; Don't get too worked up on exam day.
 * Study more things I have trouble with; Do more problems in areas that I need help with; **Do the homework** section by section, not all at once.
 * Put more time into homework and green sheets to make sure that things click before being tested on it; I need to be able to **ask questions** and not fear other classmates cutting me down for it; Review longer for the exam, not just by **doing homework** but by going and making sure I know the material.
 * **Do all my homework**; Better understand all overall topics; Take longer.
 * If I don't fully understand a problems I will **ask for help**; I will make time for the exam review; I need to do **practice problems** with others to make sure we are doing them correctly.
 * Applying the things I know to problems; Do more **practice problems**; Know more about how f' relates to f.

**Here is a listing of the responses for question 4:**

 * **I don't know how to memorize all of the rules and equations without either forgetting them completely or confusing them with each other**.
 * All the rules and properties are really important parts of mathematics, and as a result, this is topic of great concern for mathematics educators -- how to help students better learn and understand many of the algorithms and rules required to solve problems. In short, there are two particular pieces of advice that I would give:
 * First, you'll learn the rules better by doing more practice with them. Therefore, if you are still feeling shaky on particular materials even after doing the homework from the book and the worksheet from class, you can look for more practice. Specifically, the even numbered problems in the book, as well as other sources of calculus problems online. There are MANY places to try -- just do a google search on your topic of interest plus something like "practice problems" and you'll be amazed.
 * Second, while I do require fluency with some things, I almost never require memorization and I never base your entire grade on memorization (e.g., I required you to memorize rules 1-18 for part 1 of Exam 2, yet it was all simple, one-step problems and only 20% of the entire exam grade). My general belief about memorization stems from my general inability to memorize all the rules myself! I would prefer you have your materials organized so that you can find what you need fast, and that you know how to use those rules. This gets back to my first point.
 * **More time in class to ask questions in class (email is inconvenient)**
 * Time is almost always the limiting factor in instruction and student/teacher interaction. We meet 4 days per week, during which (after we factor in typical distractions and the like) we have a little over an hour to go over new material, work out example problems, discuss larger concepts, practice doing all of this, and perform required in-class assessments (i.e., Exams). Finding more time for one particular activity will require shortening the time spent on others, which is why I tend to suggest outside-of-class modes of communication. Email can be difficult for those with limited access, but keep in mind other options. For example, working in groups with others, arranging for more time before or after class, looking for other outside assistance (e.g., other teachers, parents, friends, etc.). These are far from guaranteed, but still worth exploring if you continue to require more assistance. Lastly, contact me and we can always brainstorm ideas together!
 * **A wider variety of problems done in class.**
 * Agreed - variety is quite helpful for learning. However, when individual problems begin taking more and more time to complete, our ability to explore that variety becomes severely reduced (see previous response). So, part of my thinking in organizing the course materials -- notes, worksheets, online resources, etc. -- is to provide as many kinds of problems/situations as possible, while also helping you to see how one kind of problem might be very similar to another (and require the same types of solutions).
 * **Story (application) problems. Basically understanding what to use when and where when solving and setting up the problem.**
 * Story/application problems were a bit of a theme here, and I'll offer two responses based on some thoughts.
 * First, application problems are, by their nature, the type of problems that require you to apply other, perhaps more general, approaches in order to solve them. For example, almost all of the 3.7 problems required application of 3.5 techniques (implicit differentiation) and rules from 3.3 and 3.4 (e.g., product rule, chain rule). Therefore, trying to do 3.7 problems when one has a tenuous grasp on the concepts from 3.5 and 3.4 makes the application problems all the more difficult. That is, the design of the course content (in fact, the design of most mathematics courses) is very hierarchical in nature -- i.e., learn this, then use it to learn that, then use them both to learn this other thing. So, it is little wonder that story/application problems are some of the hardest within any particular theme. This fact doesn't necessarily help you, unless you take it at face value. Specifically, you need to really practice the earlier sections in order for the later sections to make sense.
 * This leads me to my second point. When did we do the application problems? At the end of the chapter -- the last stuff before the exam, in fact. Therefore, while it might seem like this would imply the content is most fresh for you, it also implies that this material is, by definition, //the least practiced of all the content//. That means we (you and me) need to be very sure we're giving sufficient practice time, and that the implicit practice is getting done. I have plenty of power over the first part of that sentence, but very little with respect to the last part. So, we need to communicate well with each other on this front in order to make sure you get what you need (and use what you get).
 * **Do more examples of problems what we might face on the homework or exam.**
 * I'm actually quite particular about how I choose problems for exams and worksheets, so I would wager that if you were to go back and examine all previous chapter 3 materials you will find Exam 2 to be highly representative of the types of problems we do in class and in the materials. In fact, I often use exact, or near-exact copies of these problems as the basis for exam questions (e.g., the same graph in part 2, question 5, was on the exam review).
 * As for doing more examples, see the above responses.
 * **Allow us a little more time to ask questions about homework or certain examples.**
 * An excellent suggestion, and one I hope to follow up on. There were several folks who suggested, in one way or another, that having more time for questions would be helpful. Therefore, trying to find a way to make that happen, while difficult (see above responses), seems well worth the time.
 * **Related rate problems.**
 * I would be more than happy to provide additional resources for you to use in reinforcing this concept. Be sure to make use of what's on the wiki, what's in the book (including trying those even problems), and what you might find on the web.
 * **Implicit differentiation.**
 * See previous response.
 * **More examples gone over in class, or videos on wiki.**
 * I've discussed the "more in-class examples" point earlier, but you also raise the idea of additional videos on the wiki. I received positive reaction from two students when I asked about the videos in class. If these really are helpful to you (and to others), please be sure to let me know. They require time to produce and post, but I am more than willing to pay that price if they prove useful for you. My hunch is that, if others were to use them, they would find them equally interesting. Even so, there are TONS of great videos out there on all kinds of topics. For example, I typed "**implicit differentiation examples**" into a YouTube search and found 72 links -- granted, not all will be worthwhile, but the odds are definitely in your favor!
 * **I need to improve on all story problems, especially those which were associated with section 3.7.**
 * Agreed. I think it is safe to say that everyone needs to improve on those. Yet, as I suggest in the above responses, this is not just because you don't "get it," but also because these are difficult types of problems that occur at the end of the chapter (and nearest the exam). So, don't be too hard on yourself (unless you aren't doing homework or practicing very much/well). :-)
 * **I need to figure out how to start complex problems. I tend to just seize up when I see a massive one.**
 * This sort of anxiety-related lock-down of thought is quite common, especially when people are in high-stakes environments (e.g., exams, interviews, performances). Some of the best, most useful advice on this topic is preventative (i.e., things you can do to prevent that from happening), so what are some things you might try? Here's an interesting bunch of advice organized by a **learning center at Penn State**. Another suggestion worth trying is to take this into consideration as you organize your notes and ask questions. For example, as we work on particular rules or strategies (e.g., the Chain Rule), be sure to keep asking questions about it until you feel it is making sense -- specifically, ask about HOW it gets applied and what are the steps in doing so. I will try to increase my level of specificity on this front, but you'll need to give me feedback and let me know when you need more.
 * **I need a way to work through story problems - when given numbers, I usually do just fine. If you throw in words, I'm lost! :-(**
 * This is a very common issue in mathematics classrooms. It's almost like the content is in a foreign language -- and, in many ways, it is! The ability "translate" within and between words, equations, graphs, and numeric/tabular values is a very important skill-set for students to develop. I try to take this in consideration when I chose problems and tasks, but I can always improve! Keep me posted on whether and how what I'm doing is working for you, and I'll try to increase the amount of "translation" activities we do.
 * **When you have a function that you have to take the derivative of, and it has multiple parts/(rules you need to use), where do you start? Which rules come first?**
 * This is, on the one hand, a very specific question since it depends on the function you're considering. Yet, on the other hand, it is also a broader question about the primacy of the rules we've learned (i.e., which comes first). Let me address this second question. The Chain Rule is the big daddy of the rules we've learned. You can ALWAYS apply the Chain Rule, and in fact, we modified each of the previous rules (e.g., the power rule, the product rule), incorporating the Chain Rule in order to make them even more generalized. That's why the rules inside the front cover of the book have "u" and "u-prime" in them. I think this is what you were asking about, but to be sure, follow up with me in class. In the meantime, here's a nice collection of **Chain Rule-related questions and answers** from Dr. Math's webiste.
 * **How to set up more story problems.**
 * This is a general suggestion for in-class activity, and I will try to do as requested. Also, see the other responses for additional thoughts on class-time and story problems.
 * **Just applying the equation to different problems.**
 * See other responses related to this topic.
 * **I want more time to learn application problems, because I usually don't have a problem with the simple ones, but when it starts getting applied in to problems. I have trouble.**
 * It sounds as though you are describing my second point about story problems from above -- they take more time, and their complexity is usually the culmination of the topic/chapter being studied. Therefore, we need to be extra dilligent about how much time we need, as well as the ways in which we use that time to study and practice. Lastly, while I don't have reason to doubt your claim that simpler problems are usually not trouble for you, I would suggest that sometimes "don't have a problem" is indication that you might not have spent enough time on the topic or problem-type. To wit, each section's problem set gets gradually more difficult. Therefore, if you were to stop half-way, believing that you understood the entire section because you were not having any difficulties up to that point, you might miss out on the more difficult problems that would cause you more difficulty. Just a thought.
 * **Seeing the teacher doing the same types of problems out on the board really helps because I am a visual learner.**
 * While I tend to be sympathetic to those who learn easily from watching problems worked out (and, hopefully, following along with their own notes), I am also aware that this is primarily because I fall into this category myself. That is, I love to watch and listen, and I learn quite well from those modes of instruction associated with these student activities. However, after many years of studying such things and working with many different student populations, I can definitely say that most students are not like this. So, on the one hand, congratulations for being quite a bit like me (though you may not see that as a happy state of being); yet, on the other hand, this means that instructors trying to reach the largest number of students may not necessarily adopt teaching techniques that conform to your learning styles. So, what to do? Well, the simplest answer is to seek those watching/listening experiences outside of the normal classroom time. For example, ask for time before/after class, ask to see specific problems worked out during class (within reason and time constraints, I'll usually oblige), and make use of videos online that provide precisely the kind of experience you require. For example, here's a fellow working out implicit **differentiation problems on YouTube**. Again, I will caution that, if you learn how to do things by watching others do them, you'll need to be very, very careful about the quality of examples you watch (e.g., some videos might just stink!), as well as the fact that you can't always ask for clarification from pre-recorded sources.
 * **Derivative of inverse trig functions.**
 * Lots of folks need more practice with these, so I think I will try to find ways to work them into more worksheet and example problems! Of course, there are also lots of ways you can practice with available materials (on the wiki and in the book) that you can investigate as well...
 * **Story problems.**
 * Plainly put, but a common theme. See previous responses for more on this.
 * **Can we do more practice problems as a group? As in, we do some and not just watch you do them. Like, after you do a few examples, we do a few and check with you to see if we are doing problems correctly.**
 * Very worthwhile suggestion -- and quite a nice connection between the common request of "do more practice problems" and the more specific point about //how// we do them. I will definitely try to rearrange the instruction a bit more so as to allow for more "together time" for students, though that comes at the price of in-class time (for me) and task-focus (for you). Specifically, if we try using more individual and group work time in class, I'll want to be sure that this is time well spent. What do I mean? Well, I will want to see folks working on the tasks/problems (rather than just chatting), and I will want to make sure that you all are able to do the tasks/problems (rather than have only a few doing the solving on each problem/task). So, I'll try to be more explicit about what I want to see, and hopefully the experience will be beneficial for everyone.
 * **Can we have more practice with story problems?**
 * As you can see from the other questions (and my responses), this is a really common request. I hope I've covered it elsewhere, and I plan on trying to make this more explicit in my instruction, but let me know if there's more that I can do.
 * **When using implicit differentiation on a story problem, how do I know what to use?**
 * This kind of question would indicate that more time on the notes, within the book chapter, and generally on practice is required. That's not to say you haven't practiced, but there are plenty of tips and strategies pointed out in those documents that will help you. For example, look for words in the problem that would indicate the variable with respect to which you'll need to differentiate -- if it says something about "300 dollars per week," you can be pretty sure that "time" is the variable and that you'll need to take "//d/dt//" of some equation that involves money. That's just one strategy, but there are others to consider...
 * **Help setting up story problems in general.**
 * Agreed - the hardest step is often this "translation" part. Trying to turn things from words to equations or sketches or graphs can be quite tricky. Some useful tips can be found in the book, within the relevant section, as well as in the notes. Also, check out **@http://www.calculus.org/** and **@http://www.ehow.com/video_4755235_solving-calculus-word-problems.html** among other online resources.