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SAMPLEPRECALCULUSNARRATIVECOMMENTFORANUPPERSCHOOLSTUDENT

This quarter, students took three major assessments, and two smaller ones. The three major assessments were on

the binomial expansion and combinatorics (40/40; A+), on sequences and series (43/44; A+), and on a

smattering of review material involving functions (37/46, B-). On the smaller assessments, Stu earned 19/20 (A)

on sequences and series, and 7/10 (C-) on function basics and transformations. What these scores show is that

Stu has a very solid understanding of the material being presentedand although he didnt perform up to his

standards on the functions basic/review material, he is doing a very good job with the new material. Thematerial in this course is often challenging, and the level of understanding that students are expected to have is

substantial. Stu has risen to the challengeand I am very proud of his work. I only hope that his great work

continues as the year progresses.

Additionally, the nightly work assigned to students was checked a number of timessometimes being checked

for completion, but often collected and marked for correctness and for explanations. (For these formal write-ups

that are handed in, students are asked to explain their work and thought processes, not just provide the answers.)

Stu always had his work, but seriously struggled with the formal write-ups. He struggled in showing and

explaining his work. One common refrain in my feedback is explain! One of the points of the formal write-

ups is to help students learn to communicate mathand based on his assessments which have a lot of explainthis problems, it is clear he has this ability. IfStu needs guidance on what is expected on these formal write-

ups, I entreat him to meet with me. Otherwise, I hope he makes a goal of improving his nightly work in the

future. He earned a C+ for his nightly work.

In Stus group, there were two very active participants, and one much more passive member. Although I went

over to the group to encourage more communication among all three participants, this didnt always happen. I

think that sometimes something as simple as switching the seating arrangement so the more quiet members of

the group sit in between the more vocal participants can help (and I suggested that). However, the dynamic

never really changed. This group lost a group member who switched classes, near the end of their work

together. Thus, the group of three went to a group of two.

Stuis a student who has a number of ah ha moments, and his insights are valuable to his group. He sees his

role in the group as a problem solver, always questioning the solutions that the group generates but also

ensuring that the solution we have come up with makes both mathematical and logical sense. I have seen that

and the animated discussions he has with [another student] are wonderful to be witness to. There is a tenacity

that characterizes the group when it comes to solving problems. They dont give up and they dont ask for help

unless they truly have exhausted their options. As I mentioned, there was one group member that was more

passive. I think the group should have slowed down a bit to make sure everyone was on the same page. A group

should be working for a common understanding, rather than three individuals working for their own

understanding. Their approach to working together is to work individually, and after some time, compare theirwork. That may work well for some groups, but the end result of this approach for this group was that two

members were able to start and work through the problems, while one group member passively waited. Instead,

talking through the problems and throwing out ideas of how to attach them initially, and then having some time

to individually work through things, might have benefited everyone in the group. Stu noted that he might have

pushed the more passive member to join in the conversationknowing that she might need a little more support

to fully understand things. That would have been a good idea, as well as talking explicitly about the group

dynamic, and how it could be improved. Often, a short discussion initiated by one of the group members can


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totally change how a group works together. (Im always happy to facilitate this also.) He earned a B for

groupwork.

In a reflection I had students write near the end of the quarter, Stuwrote: I think that the class is going pretty

well. I really like the people in the class and feel that we are all at around the same level of understanding

The pace of the class, although fast, is almost perfect. It pushes me and doesnt let me slack off by being too

slow but doesnt leave me in the dust by going too fast The difficulty and pacing put together make the class

challenging but I enjoy it since it keeps me academically stimulated and never bores me. I loved reading that!He also mentioned that he wasproud of how his groups work improved through the quarter starting out

divided, but getting a bit better. Finally, he mentioned that sometimes the length of assessments can be a bit

long. I want him to know I will take this to heart when writing assessments. Thank you for that feedback.

Stus final quarter grade is 88.5%, which is a B+.

INTRODUCTORYBOILERPLATEFORTHECOURSE

Advanced Precalculus is a fast-paced course which covers a variety of math topics, most of which are essentialprerequisites to Calculus. We began the year with an investigation into Combinatorics, the way we count in

mathematics. The next unit dealt with sequences and series. Our third unit was a comprehensive review oflinear and quadratic functions and their graphs. We studied parent functions and the graphs of their

transformations and we investigated properties of functions including symmetry, maxima and minima and

behavior over intervals. During the second quarter we will be studying analytic trigonometry, polar coordinatesand vectors.

Most notably, this course has two main features. First, students work primarily in groups. Although we will

often come together as a class, most of the heavy lifting is done by students working collectively. Second,students arent given formulas but are asked to build them up themselves so they really understand where they

come from and what they mean. I have been especially proud of the fact that students have constructed all the

formulas that weve used thus far in the course.

A students grade is calculated as follows: 75% assessments, 15% nightly work, and 10% groupwork.


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SAMPLECALCULUSNARRATIVECOMMENTFORANUPPERSCHOOLSTUDENT

Talk about a grand slam at the bottom of the ninth inning. Stu has had, to put it bluntly, a frustrating and rough quarter.

However, due to her dedication and last minute efforts in the last two weeks of the quarter, Stu has shown herself more

than capable of doing calculus. She also hopefully has learned a lesson in how important it is to keep up with the course as

we go along. The strategies that got Stu through this first quarter will not work in the remaining three.

This quarter, weve learned 17 distinct skills7 algebraic skills in our limit bootcamp and 10 calculus skills on the

concepts of limits and continuity. Of these skills, Stu has shown mastery or almost complete mastery of 11 of theseskills, a firm grasp on 4, and a confused understanding of the remaining 2.

In class, I notice that Stu doesnt always have a firm grasp of whats going on. When I met with Stu at the beginning of

the quarter, she told me she felt like other people remembered a lot of the material from previous years, and she didnt.

This course wasnt easy for her. Since the analytic part of Calculus requires a working knowledge of algebra, and since

Stu hasnt retained a lot of her algebra knowledge from previous years, this course is tough for her. Im sure at times it

was absolutely maddening. Whats clear, from the way that Stu has been able to demonstrate her understanding as the

quarter has progressed, is that Stu absolutely can re-learn these algebra skills and be successful. Whats also clear is that

Stu will have to work extra-conscientiously to make sure she understands all the algebra skills as well as the calculus

skills. This course has been challenging for Stu, but it is only through challengeby pushing the boundaries of what we

think is possiblethat we learn what were truly capable of. I hope that Stu comes out realizing that she can do this, andin the process, learns something about herself and what works for her learning.

Currently, Stu has been using reassessments to demonstrate her understanding. She has reassessed most skills, some more

than once. And each time Stu reassessed, she usually can show more than previously. For Skill 10, for example, Stu

earned a 0/4, then a 2/4, and finally a 4/4. It is actually quite common for Stu to go from a 0/4 or 1/4 on a skill to a 3/4 or

4/4 on a skill when she reassesses. This tells me, and hopefully Stu, that she can get the material, but it takes her time.

When push came to shove at the end of the quarter, Stu was able to master a number of skills in the last week. The fact

that Stu could do that, in such a short amount of time, indicates that whatever strategies that Stu was using in the past

week should be strategies that Stu uses consistently for the rest of the quarter.

In a meeting, one of the things Stu and I talked about was how she approached the home enjoyment each night. She saidwhen she got stuck, she would just give up and move on to another problem. I told her that would not cut it, and there are

multiple things that she could do: call a friend, read the textbook, look online, and review that days Smartboard notes. Stu

should then ask questions about the home enjoyment in class. And finally, she should rework the following night

some of those problems that she missed. I ask: Whats the point of doing math problems nightly if you cant learn from

them?

In addition to this nightly work, there is a student-run TEACH study group that I suggest Stu sign up for.

Stu wrote an incredible reflection at the end of the quarter, and instead of putting excerpts in, I am going to type it all in.

--

1. I like the way that even though I was falling rapidly into a hole, and it felt almost impossible to get out, once you talked

to me I became proactive and tried my best to do better. I like to continue meeting with you. I also like to continue to

participate in class and asking questions. I think asking questions in class was the biggest way for me to better understand

the topics.

2. I wish I would have started from the first day of school in this attack math mentality. I was acting very passive and like

oh I dont get it now, but I will later, which honestly was the worst thing I could have done. I also wasnt used to the

class setting and the grading system. But once you emailed me and I met with you and I know that this is a class that I

have to be in it 100%, and that your method is one that helps us actually learn, it was just beneficial. I needed that scare


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and wake up class because I was in serious denial. I became more on top of things. However, I had to dig myself out of a

huge hole that I put myself in, but eventually the rhythm has become one that I used to. And Im almost in a weird way

glad that I learned the hard way because now I truly understand Math.

--

I cant say how happy I was to read this. I hope to teach my students independence and how to be proactive learners. Stu

is learning this.

Stus final grade for the quarteris an 84%. Stu, the glee on your face when I told you in the commons that the last minute

reassessments were successfulthat was amazing. You dug a hole, but you also worked hard to come out of that hole.

Now Id like to see you work hard on keeping your level of understanding consistently upso that that smile on your face

doesnt ever leave.

INTRODUCTORYBOILERPLATEFORTHECOURSE

This year in calculus, the course is being structured so that students can truly focus on and appreciate the ideas

undergirding calculus. In years past, Ive noticed that students often times lose sight of calculus for the intense amount of

algebraic workwe do. Now, before each calculus unit, I lead an algebra bootcamp which targets the algebraic skillsneeded for the upcoming unit. The goal of this structure is to separate the calculus ideas (which are simple but deep) from

the messy algebra. In addition, students are graded solely on their ability to master the content of calculuswith the

course broken up into discrete skills. The bar has been set high, where students are asked to gain a total mastery of each of

these skills. Sometimes, however, students falterand this is okay. (It happens to all of us.) I allow students to remediate

a skill they might not be happy with their performance on, and then reassess it. My goal is to give students responsibility

and freedomand hope that they learn how to learn given that freedom.

This quarter, we have focused our attention on how to graphically, numerically, and algebraically find the limit of

functions. This unit also introduced the idea of proving statements about continuity. (In high level m ath courses,

understanding how to show something is true, instead of being told something is true, is especially of great import,

although true at all stages of math.) We ended the quarter by starting our work in the realm of derivatives (in essence, the

study of how thing change) and will continue our calculus explorations on that topic all next quarter.

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