Math Notebooks

I have thought of trying to incorporate writing and notebooks for a long time in math. I thought this was a way to make math more fun. I also thought this was a way to engage students in math using another type of intelligence. I hoped to engage the eloquent writers to express their understanding of math using the medium they are most comfortable with.

This was a major focus during my work towards my masters. I have been able to include more activities in my math class where students write down responses on their individual whiteboards. I still want to include more writing assignments though, but have trouble balancing computation problems with writing problems. I still want to find a way to work more writing assignments into my curriculum, but I have focused on keeping good class notes in the students notebooks.

Although after writing this blog post a thought occurred to me about how to incorporate more writing (ah the power of writing/blogging). When I start a new chapter with the students I use a directed reading thinking activity (DRTA). I ask the students what they think the title means. I ask them what they think they will learn based on the title. Then I have them look at the picture at the beginning of the chapter and the example problem that relates to it. I ask them how the two are related. I ask them to think about how you would solve the example problem. Some students get really carried away and want to know how to solve the problem that second. I then ask them how the example problem relates to the title and what we will learn in this chapter. I then have them read the title of the first section of the chapter and ask them how it relates to the title of the chapter. Then I ask them what they think they will learn in the chapter. I have them read and write their answers on their personal whiteboards. Sometimes I do a think, pair, share to answer these questions. However I ask students to show their whiteboards to the class and call on students to share with the class. I keep a list of ideas after each round on the chalkboard at the front of the classroom. As a class we compare answers and as we get more information we can see if our predictions were right. Often to wrap up this activity I read the summary of the chapter to the class which confirms and denies the predictions made. I often will read it and say this is just what student X said. This activity helps them learn to make predictions by looking at titles, pictures, and other text. I encourage students to use the strategy in other classes. It also helps build background for the students and helps me understand how much they have studied. It also helps make connections from past material to the new material. I think to build even more background I think I will do a writing activity after the DRTA. I will have the students answer a question on their whiteboard. The question will be the big idea of the chapter. I think I will then have students respond to the question again on the weekly quiz. This way they have several drafts to use before the final and they will have another chance at answering once they have learned some of the content. I will provide feedback on the written part. They should have two quizzes before the end of the chapter so they will have two chances to practice their answer. Also if for some reason a lesson runs short and I have extra time then I can have students write and/or discuss that chapters prompt. Then once we have finished the chapter they will write their final response on the page right after that chapters notes in their notebooks. I will have to instruct them carefully to save a page or two for that chapters journal entry. Although as long as they include it somewhere in their notebook and list the journal entry in their table of contents then I guess it won't matter. It just would be nice to have all the notes on that chapter followed by the journal entry over that material.

I hope that this helps the students express themselves better on the free response questions of the AP test. They are putting more and more emphasis on explaining the answers with words. The questions are really looking for calculus with words. This has proven difficult for my students since English is their second language. I also hope if they can explain a process or concept in general using the words they can solve any problem.

I will need the AP scores to back this up. The company and the Chinese administration put a lot of pressure on getting high scores. This is doubly true in math since all of the students excel in math, so they except a 5 out of 5 for all the students.

Here are the prompts I am thinking of using. I got most of these from the assessment resource book that accompanies the calculus book by Finney, Demana, Waits, and Kennedy (3rd edition) (citing my source). Some are the writing to learn question in the books exercises. Some I have tweaked or added to.

Chapter 1 (Pre-Calculus and Review of all Types of Functions): Explain why a function that is not one-to-one does not have an inverse function.
Chapter 2 (Limits and Continuity): Explain the importance of continuity in discussing limits. Also discuss the value of determining limits as x approaches infinity. For example in economic problems, the limit of a function can give important information. If a company has an increasing revenue or profit function, what does the limit as x approaches infinity tell us?
Chapter 3 (Derivatives and Differentiability): What does the derivative represent? Use the concept of the derivative to define what it might mean for two parabolas to be parallel. Construct equations for two such parallel parabolas and graph them. Are the parabolas "everywhere equidistant", and if so, in what sense?
Chapter 4 (More complicated Derivatives): How does the chain rule help us find the derivatives of almost any function? Why is implicit differentiation useful in examining the derivatives of curves that are not functions?
Chapter 5 (Applications of Derivatives): Think about the importance of the second derivative test. In economic applications, what does the second derivative test reveal about cost functions and revenue functions? And why is that information important?
Chapter 6 (Integration): Let f be a positive continuous function that is concave up. If the trapezoidal rule is used to estimate an area between f and the x-axis, will the result be an overestimate or an underestimate? What if the midpoint rule is used instead? Explain how you know if it is an overestimate or underestimate. (often a question on the AP test)
Chapter 7 (Integration Techniques and Differential Equations): Compare and contrast non-separable differential equations with separable differential equations. Analyze the following: equations, solution method/process, how to use initial conditions, and slope fields. Use the following vocab words: general solution, particular solution, initial condition, non-separable differential equation, separable differential equation.
Chapter 8 (Applications of Integration): Describe in what situations should you use the disk method, washer method, and shell method? Think about rotating the same curve about the y-axis and the x-axis. What similarities could be seen about the volumes of these solids? Are they the same? Would you use the same method to find the volume?
Review: The students will write about the mathematicians we have studied going to a dinner party and talking about the topics we have studied so far. The assignment will let them be creative and help them review concepts. I will give them guidance on what to write about, but still the ability to choose and be creative. I am really looking forward to reading these.

I came up with the review entry after taking a ESOL class online from Willamette. The example came up in the textbook we were using for the class and was tailored to an English class. The example was student writing about author's from a certain time period at a dinner party. I read that and thought why can't it be about mathematicians. The textbook I use for calculus has some really interesting comments about the different mathematicians for example it talks about how Michel Rolle spent a long time making fun of Calculus, but then ended up having contributing a theorem (Calculus, Finney, Demana, Waits, and Kennedy, 3rd edition). So I flipped through the book to see who else was mentioned and for other paragraphs that mentioned the mathematicians contributions. I had fun actually reading my Calculus textbook and finding little gems in it that I had previously skimmed over.

The one for Chapter 7 I came up with by myself. I noticed last year that my students could not put the equations with the vocab. They could solve a separable differential equation no problem if you put it in front of them, but if you asked them to describe what they did using the above vocab then they couldn't. The differential equations questions on the AP test are loaded with the vocab I listed. I put more emphasis on the vocab this year and had students write "a comparison essay" as homework comparing non-separable differential equations and separable differential equations. I got some really creative pieces. I think it helped solidify the vocab so when they saw it on the test they knew what the question was talking about. Most of the students could describe how to solve a separable differential equation at the end of this year, but there was still a few that did not know. They were the few that struggle normally though and was not shocking to see that they did not know, but it was disappointing  Although hopefully now after reviewing what the words mean before the AP test they will do well on the free response question that is over separable differential equations. It just shows that some students need constant reminder and review for the words to sink in.

I suggest that if you want to do something similar look through the assessment resources that come with your textbook. Hopefully I can continue this if I go on to teach other subjects and out of other textbooks. I would need some ideas for writing prompts at least as a start. So hopefully other books come with alternate assessment ideas in the resource books that accompany it.

I think the dinner party review assignment can be done in any math class at any level. Most math textbooks include short biographies on mathematicians. You could also turn it into a sort of mini research project. You just have to suggest topics or people that relate or connect to your content. So I think the hard part would be to come up with suggestions for students to write about because that would take some time on your part to research the background of the math you are teaching.

I plan on reading these and checking these journal entries when I collect the notebooks. I will just have to figure out how to grade them. I think I can use part of the rubric I use for mathematical presentations and then I studied rubrics to use with ESL students in some of the ESOL classes I took online. I will have to look back through the textbook I have to find a rubric that will work. I want it to be something quick and easy to use. Any suggestions or links to rubrics would also be greatly appreciated.

Let me talk about what I have done with the notebooks so far and what I plan to do.

During my student teaching I had students keep a table of contents for their notebooks and number the pages in their notebooks. I thought this kept things organized and if their was something specific they needed to study they could easily find their notes on that subject. It also makes grading them easy because then you know which page to turn to when you are grading something specific. I continued that work with my Chinese students. Some students are good about doing this and others have not. I just have the students label what part of the textbook we are working on and use the title of the section from the textbook. I always have this displayed on the first slide of that day's powerpoint. Some students leave an empty line between each entry and others leave extra space when they start a new chapter. I also have a student that put in tiny stick notes on the pages where each new chapter starts in her notes.

I started working with my students on using an organized system for their notes during my student teaching and I tried a couple of things. This year I focused on using a Cornell notes like set up. The students put the main idea of the content or summarize what the notes on the right are about. So they could put the name of the theorem in a small column on the left of notes and in the large section on the right they could have the theorem written out. I gave students this main ideas at the beginning and then wanted students to come up with them on their own. I wanted them to either come up with them in class so they are processing information during class. The other option being they review their notes and add the main ideas in the left. I noticed once I was not giving them specific main ideas students were either not including them or giving really general main ideas. Some students would just write example, but not what it was about or why it was different. Some would just write equations and I really wanted them to use words to summarize the notes. Last year I found when I handed out study guide sheets that listed the objectives to be tested on the exam the students did not know the English words connected to the math that they knew. I could put a problem in front of them and they could solve, but if I asked them to describe what type of problem it was using words they couldn't. I hoped to solve this problem by having students include some of those words in the main idea column.

I don't want to feed them the main ideas in class either on the whiteboard or on the poweroint because I think they gain a lot more learning coming up with them on their own. I have one student who always asks how to classify what we are learning so she can label it in her main idea column, but only a few students listen to my answer. I am not sure how to help them develop more specific ideas. I wonder if I could stop and ask them what we are learning on this slide and how it relates to what we have been doing? It could be a way to check in with students and the class about their understanding and help them come up with main ideas.

Then they use the bottom five lines of the page to make notes about tips and strategies for the AP test. I often remind them or give suggestions of things to do on the AP test. I began writing these tips on the board and then expected them to listen for them. I noticed a lot of people stopped writing them down. I need to get the kids to recognize that when I say AP test that should be a cue to them to use that bottom part of the page.

I have a part on my rubric that talks about underlining, highlighting/color coding, starring, boxing, or using other techniques to make information stand out. However still were not make the big key concepts like formulas,derivative rules, vocab, or even titles stand out. So I think I need to mention that and go over that more. Although when I first started in Chapter 1 and 2 I talked about it a lot and modeled it some. However maybe I need to give more reminders. The students can do this during class or after class when they are reviewing their notes. Some of my students have really found a way to make color really effective and organize their notes.

I also wanted students to relate or connect ideas in their notes. For example how derivative equations of different trig functions relate. They could also relate to things they learned in physics. They can connect definitions to examples. I don't think I gave them enough examples of this. I think next year I will bring some of the best notebooks from this year to illustrate to the next years class what I am talking about. I think I will have to make photo copies so I will always have examples to go along with my rubric.

I just recently finished grading the notebooks and was sick of finding handouts shoved willy nilly in them. One student had English handouts in their notebook. I made a comment about that, but no one listened. I think next year I will take away one point from the format category for having loose papers or handout in their notebook. I have had some students who taped or glued my study guide handouts in their notebooks after that chapters notes, but they listed it in their table of contents and it was placed in a logical place.

Rubric

I took most of the rubric from the AVID programs rubric, but I added some of my own ideas that tailored to math and what I wanted. I also got rid of categories that I thought didn't really lend itself to math. I told the students the AVID program uses a similar rubric and those are some of the best students who are working hard to go to college. I told them they start doing this in 7th grade and they were shocked. It showed them that other people in America felt you needed these tools in college. They seem shocked and impressed at first at least.

Now let me explain how I use the rubric. I circle each numeral in each category. So this means that students could get a 5 for using plenty of space between their notes, but a 3 because they are not using abbreviations. This way the students can see how well they are doing on each skill. Then what I do is I average the score in each category. I always round up to the nearest tenth. Then I added up the averaged score in each category and then find an average for the overall score. So it is only worth 5 points in the end. This way they get a wholistic score on the notes. I have debated about averaging in each category then making the notes 20 points (5 points for each category), but I am not sure if that is too much. Next year I hope to grade the notebooks every three weeks until I have looked at notes for each chapter. So that means that in total that will be 40 points for the notebook. Although next year I am putting more weight on the scores so I should probably keep the points low. I have also thought about finding the average score between all four categories then multiplying by ten so there are no decimals. Then each chapter would be worth 50 points, but that seems too much. I will have to think it over.

The scores are often low using the rubrics since I am holding them to high expectations. I often will curve the grades or give back points in the end. I often find to when I start grading them I start mean and become more lenient. So if I go back and give more points it evens things out. However some students rise to the high expectations and take amazing notes. I recently let those students have extra credit because they took such good notes. I am not giving too many extra credit opportunities this year so I thought this would be a good chance to get some extra credit.

After I graded their notebooks the first time I gave them some written feedback to improve which described some examples. I will have to give this type of feedback more often next year. I will probably give this to the students every three weeks or every month. We finish a chapter a month usually so that will help them refine their notes for the next chapter.

My Calculus Class Next Year

I have decided I am sick of grading homework and want to give more meaningful homework next year. So I am restructuring my class. I thought of this over vacation and wanted to get it down in front of me. I will work on creating and adjusting resources for next year after the AP test is over.

Grade break down:

Final: 20%

Tests/Midterm: 30%

Quizzes: 15%

Projects: 15%

Notebooks: 10%

In-class: 10%

This year I tried doing a project that helped students prepare for a free response question as well as develop their ability to explain their reasoning using different methods. The AP test emphasizes understanding the material using a graph, algebra, tables of values, and words. I started off with a project at the beginning of the year to help students work on justifying their answers using those methods. The project covered concepts that we had recently covered, but it also included concepts they need to review because we were going to explore them deeper. I enjoyed the sort of review and preview of material this year. So I will do the project again this year. However the students procrastinated on it. So I need to break it down some more. So each week they will have a small piece of the project to work on. So they will have to find the domain, range, x-intercepts, and y-intercepts of their given function. Then support their answer using a graph, table of values, algebra, or words. The problem I had this year was I couldn't grade the homework as well as the projects. Now the projects will be my main assignment to grade. I think this may be harder to grade than the homework out of the textbook I have been giving, but I think it will help them prepare for the AP test better. They will be preparing for the free response questions that are what most of my Chinese students do poorly on. They will also be developing analytic skills that will help them in other disciplines as well as their future math skills. Some of my students really wanted to attack the problem using each and every method. So they were developing the ability to view the problem in different ways and using different information. I will collect these each week and try to get them handed back in a week. Then at the end of the project I will take time in class to have students peer review the projects. Each student in one section of the class will have a different equation to analyze. The more advanced students will be given more complex equations and the weaker students given a less complex equation. So then I will take the projects of the students who studied the same equation in the different classes and have them grade each others paper. I hope the students will learn a lot from grading the other students paper and give good feedback. I think I will then take a quick glance at the projects and the scores. However I think I will just average the scores that were assigned by the students. I will include in the in-class category points for giving good feedback. So if a student did not provide enough comments, reasons, or the scored was not at all accurate then they will loose points. I will have to outline what good feedback is and have the students take it seriously. 

Project 1: General Properties of a Function

I plan at the beginning of the year reviewing basic properties of functions: domain, range, intercepts. Then I will have them apply the knowledge they just learned: continuity and using limits to find asymptotes. Then I will have the preview/review material we will learn about in more depth later: increasing, decreasing, concave up, and concave down. The resources include handout describing the task, detailed rubrics outlining how to get the points, and a rubric grading sheet for each small piece of the project as well as the final project. 

So the students will work on this project in September and October. It may even extend into November. I will see how small I can chunck up the project. 

Project 2: Connecting a Function to its Derivatives
Sometime in November I should wrap up teaching about derivatives. Once that is over I am going to have the students start the next projcet. The students will have to find the derivative and second derivative of the function then apply them. They will have to find the local extreme values using both the first derivative test and the second derivative test. They will have to find the absolute extreme values. They will have to find the point of inflection and justify it. They will have to find where the function is concave up and concave down. I will require them to use the words to support their algebra. I will make sure they show all their algebraic work as well. These sort of questions are always in the free response section of the exam. I will give a lot of feedback about if they have justified their answer enough for the AP test on this project. I know that my students can determine where the function has these propeties, but they do not know how to support their answer using words or showing enough work. The resources include handout describing the task, detailed rubrics outlining how to get the points, and a rubric grading sheet for each small piece of the project as well as the final project.

Project 3: Fundamental Theorem of Calculus

The following semester I will have the students work on a similar project but the equation that they will be given will be the equation from the previous semester, but with an integral. They will have to use the fundamental theorem of calculus as well as what they learned in previous units. The resources include handout describing the task, detailed rubrics outlining how to get the points, and a rubric grading sheet for each small piece of the project as well as the final project.

The equations I plan to use are listed in the assignment table chart that you will find with each project. I decided to have students focus on intervals when doing the second project so that they would have to check the endpoints when looking for extreme values. The classifications column is where I labeled the equation as easy(E) or hard(H). The other numbers and letters refer to groupings. The numbers all refer to equations of the same family. The letters that accompany them match together equations that are similar and often differ by a sign change. Each student in the class will get a different equation but this chart will help keep track of the students in the other classes that have the same function. When I assign functions I will type in their names.  

The other project that will go into the project grade will be the review project I started doing last year. I started it to help students review for the test and to help students work on their English presentation skills. I think I will give them a copy of this article and discuss it before presentations begin so they can focus on how to improve their presentation skills. I will pass out a packet of old free response questions from college boards website. I give this to them a month or so in advance. I remind them to keep working on them. I also give them some time in class to work on them. Then around the end of march or beginning of April I have students pick which problem they want to present. This allows them to choose a problem they are comfortable explaining to the class. Then they have to give a ten to fifteen minute explanation of the problem to the class. This means they have to solve the problem using multiple methods, explain every step, explain how to solve on the calculator, or explain key things the college board is looking for when grading to fill the time limit. I grade them both on content as well as presentation skills. I want them to review the material, but I also want them to work on their presentation skills. I had to give presentations in several math classes and as a part of my major. They will have to present to future colleagues as well. I made a rubric to grade this by combining a lot of different rubrics I found online. Feel free to use in your classroom or adjust what I have made to use in your classroom. I will be trying out a new rubric next year. It is based on the one I used before however I added more emphasis on the math content area. So I increased the point value in the content area. I ran into a problem this year where students solved the problem using a second method, but that method was not really appropriate for the AP test. I didn't mind them including those solution methods, but they should have included information about which method to use on the test. I added that to compare methods as well. I also added a category about making sure the students explain why they can use a certain theorem. The students need to know that they can only use theorems when the function is continuous. The AP test often has questions where you can't use a certain theorem and must explain that you can't because the function is not continuous. So I hope by including this category it will prepare them for those type of questions. 

General Rubric
Detailed Rubric

This year I am going to get the audience more involved. Three students will grade the students using this rubric. Then three other students will grade the solution of the problem using the AP rubric. I will also require students to ask one question during the course of the presentations. I wanted to make it more than one, but was not sure if I made it more that there would be time for all the students. I was going to check this off and put it in the grade book. 

I will look at their feedback and check that they are doing it to give them a feedback grade that will go in the in-class category. I will check off who has asked a question and give them points for that in the in-class category. 

I will average the scores given by the students using the AP rubric and put that score in the homework category. 

I have also considered averaging the students scores on the presentations with mine. 

So next year I will be teaching 5 periods of class a week instead of 6. So two days a week I will see a class for 90 minutes and one day a week I will see the class for 45 minutes. So during the 90 minutes I will teach one section of the textbook. So I will teach two sections of the textbook a week. Then on the day I have the students for 45 mins I will have the students take a short quiz. Then once they are done with the quiz they can work on their project. Depending on the schedule I will probably have them turn that weeks part of the project in at the end of the period. 

So instead of homework I will have a weekly quiz. I am think the quiz will have five multiple choice questions on it and the students will have 15 minutes to answer the questions. That is the amount of time they will get on the AP test. The remaining class time I will check in with students on their projects and give them time to work on it. If a student is done I can let them get started on the next part. I will have to have the handouts ready far in advance though. I am planning to use AP questions on the quiz. 

I made a rubric to grade students notes and taught the students how to take good notes this year. I wanted to grade and collect the notebooks a lot, but did not have time since I was spending it grading everything else. I have also learned that I can only grade one class set of notebooks in one week. So I think each week I will collect the notebooks of one class. That way I am grading and giving feedback on their notes every three weeks. I will talk about how I grade notebooks and what I have discovered so far this year in another post.

For the in-class portion of the grade I will grade any activities that are done in class. However next year I am planning on using an app called class dojo to monitor student behavior and participation. One of my colleagues here in China uses it with his classes. He gives points for things like critical thinking and takes away points for speaking Chinese. At the end of the six weeks he looks at the summary points and gives them the grade according to that. I have to decide what criteria I will be looking for. I will also have to make sure I can use the app in the classroom since their is no wifi and the internet. I know it can be done, but I will have to test it out before I use it next year.

Here is my criteria:
Positive:
Different solution method
Critical thinking
Questions
Helps others

Negative:
Unprepared
Speaking in Chinese
Off task

I want to make these categories meaningful. I am try to think about what behavior in class makes a good math student. I would love more suggestions of categories to add.

I will try to keep you updated on how using class dojo impacts my classroom.