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.