Pedagogical Predicaments

What They Don't Teach In Teacher Training School

2011-11-02T20:52:00.000-07:00

Two years ago, I would rush from school as soon as I could. I would get into my car and drive 30 miles until I reached Maple Crest, the nursing home where my grandmother was living out her last moments.

I would stay there until I was almost too tired to drive and I would almost always end up in tears before I reached the big hill out of the town, driving 45 minutes until I was exhausted and would fall into bed. I would wake in the morning and repeat the whole thing.

She was dying of cancer and I watched her change from a feisty, stubborn woman that still had much to do in this world to someone who wouldn't accept the watermelon or strawberry ice cream that I tried to feed her, the only foods that she would still eat in the last weeks. I watched her become listless, moaning in pain but still refusing the drugs that would calm her. And now I'm watching one of my students do the same for his best friend, a 15 year old, who lies dying of leukemia.

I asked him to stay after class earlier this week. We talked a little, but were continuously interrupted. Our conversation ended with no resolution, other than he knew that I understood where he was, I knew the pull that he was dealing with, and that I would give him whatever I could to make this easier.

It's been two days since we talked. I still don't know how either of them is doing. I sit alone at my desk and stare despondently at the wall. I was never taught how to deal with this.

Finding and Accepting My Place

2011-04-14T16:37:00.000-07:00

This first day of the White Privilege Conference has shown me that I what I need to do is to act. I can sit and contemplate my own actions and privileges, and although that brings me closer to being a caring, compassionate adult, it does nothing towards fixing the inequalities that exist right now.

There are many atrocities in today’s world that I have watched happened and not spoken up for. When friends have come to me with troubles that they have had while living in one of their labels, I have stated my solidarity with their fight and then I’ve encouraged them to hide in their own skin. I have never gone to a protest to stand up for gay rights, an MLK rally, or signed a petition for equal access. How can I say that I am an ally, that I am not prejudice, if “silence shows acceptance”. I can say that I am not silent but my actions have not shown it.

I spent this morning in a workshop entitled “The Critical Liberation of White Woman” where I finally verbalized my trepidation of creating close relationships with white (straight) women. I allowed myself to own my stereotype against this group, to claim with pride my belonging to this group, and to pledge to work with other white woman to investigate our place in society and then to use the power we do have to affect greater change. I am learning to love my label, and to take pride in who I am. Please help me in this plight, whether to give me time with my white female friends, to ask me about my progress, or to spread this acceptance of self within your own social groups. Let me also offer to help you in conversations and acceptance of who you are and what entitlements or disadvantages your labels give to you.

I also went to a very moving keynote speech from Michelle Alexander entitled The New Jim Crow: Mass Incarceration in the Age of Colorblindness. First off, I must insist that you do more research about this matter. Her basic thesis, and I state this no where near as elequently as she does, is that our ‘war on drugs’ has really been a race war, where minorities have been targeted and incarcerated for barely offenseable crimes.

I remember being in college and finding out that if someone was caught in possession of marijuana, in any amount, they would lose all of their federal financial aid and would never again be allowed to receive aid. At the time, that punishment did not seem to fit the crime. I imagined my (white) friends and would happen to them if this happened. It seemed crazy and I could not understand why anyone would put such a hard punishment against a person.

Note too that marijuana possession is a felony in many states, meaning at least a year of imprisonment. Not to mention that on any job application after being released, one would have to mark that they are a felon, probably passing up any possibility of getting hired again. That hardship seems strong enough. Unfortunately, if you are a felon, you also lose your rights to public housing and to food stamps. So you may end up with a year of your life lost, no home, no job, and no food. What are these people to do? (And as an aside, you also lose your right to vote. For possessing as a little as half a joint.)

Now if that doesn’t seem harsh enough, think again of who is the target. As a privileged white female, I have never been searched for drugs in some cases where my shiftiness should have warranted such a thing to happen. But I have privilege, I have followed a good path (primarily), and any cop looking at me knows that it would be a tragedy to ruin my life in such a way. But what about for the young black man, walking around in “the bad part of the city”? Why is it not a tragedy to do the same to him? And he is only the end user and not the one leading the spread of the drug.

So what do I do with this information? Should I fight for the legalization of marijuana, so it can’t be used for a race war anymore? That might be a start but there is something much deeper at work here. If marijuana becomes legalized, does that stop the harassment? Definitely not.

I’m not sure where to go with that one yet. I will stay in search of the answer. I will find a way to act.

Volumes in Calculus- Part 3: Rotation

2010-04-04T08:10:00.000-07:00

This is part three in a series about teaching volumes in calculus. The first post can be found here.

My students return for day three of volumes. They have spend the last night trying to figure out how high up the glass the Pepsi will go. Some of my students have random guesses, some have looked up volume equations for frustums, others have tried to use volumes by cross sections and proportions to make their guesses.

Then I put up my guess, with three decimal places. They know I've done some crazy math, but they're not quite sure what I did. I hold their interest while I pour, and believe it or not, I'm the closest.

I show them my work. I have set up an equation for one the side of the glass, I explain how I revolved the line, created disks, and then put the integral and the volume I want into my calculator and found the intersection of the two.

I expand this discussion of rotations and we pull our our Play-Doh again. Before rotating the line y equals x squared around the x-axis, the students practice visualizing it:

I've found the Play-Doh to be the best thing that's happened to my teaching of volumes. Not only does it force all of the students to solidify their mental picture (pun intended), it also gives me a great way to quickly evaluate where my students are in their understanding.

Next up: The final project that ties everything together.

Volumes in Calculus- Part 2: The Cup

2010-03-29T16:51:00.000-07:00

This is part two of a four part series about teaching volumes in calculus. It starts here.

After reviewing the cross-section problems from the homework from last class, I pull out a can of diet Pepsi and a glass. I open the can and the whole class is watching me. I try to pretend like nothing special is going on, but they can't help but oogle at the can- I don't drink soda. I'm a health nut. And it's what my department head drinks- one per class period, as the class informs me. I tell them that I had to wrestle her for it and after hearing them go off about how bad for me it is (diet soda apparently means double calories for anything else you eat for the day) I tell them that I didn't really want to drink it anyway. All I want to know is how high up the glass it will go.

My glass is pretty much a cylinder, so I pass it around with a ruler in inches (just to really throw off all my international students). We have a discussion about measurement in inches, that's it's not 4.6 inches but 4.5 and one eighth inches. Someone finally decides to estimate the volume of the soda and the class realizes that the volume is on the can- no estimation is needed. Of course, the volume is in milliliters and then we have to figure out how to change it to cubic inches. After some texts to Google, finding converters on our cell phones, and using unit conversions, we have determined the volume to be 21.663 cubic inches and we can focus on finding the height the Pepsi will go in inches.

I pour. And our guess ends up being off by a little. Not much, but a little. So we discuss why. The Pepsi doesn't actually start at the table; there is glass in between. Still, that doesn't account for all the error. I pick up the glass and we look at it's sides. The bottom is slightly curved. They blame the error on that. If the sides were straight, they said, they could have guessed it perfectly.

So I pull out another glass, one with completely straight sides, and ask again- how high will the Pepsi go?

My students pass around the glass and all take measurements. (Note for doing this in the future- have multiple glasses of the same shape so that multiple students can be taking measurements at once.) By the time we agree on the measurements, the class is over. Their homework- figuring out how high the Pepsi will go and the next class will start off by answering this question.

Volumes in Calculus- Part 1: Cross-Sections

2010-03-28T17:07:00.000-07:00

This is part one of a four part series about teaching volumes in calculus.

I showed my students a prototype paperweight made of Play-Doh. I told them that I was hoping to mass produce them and sell them to other math teachers. But first, I would have to know how much clay I would need to make one and since the nearest art supply store is over 30 miles away, I needed to know exactly how much clay I needed.

I asked them what shape they thought the base was. At first, they said a half circle and after asking them to think what it would look like if I put two of these together, someone quickly said, "It's the first half of the sine function." Amazingly brilliant, my students.

Then I told them that I made this to be a very mathematical paperweight- if it was cut down the middle at any point, there would be a semi-circle:

I then threw all the book vocabulary at them: "Let R be the base of a solid that has semi-circles as cross-sections perpendicular to the x-axis..." We remembered how the integral of a length gave us area and conjectured that the integral of area would give volume. Now it was time to figure out what area we would want to integrate and after some discussion, we determined that adding up the area of all of the semi-circles would be the easiest, so we finally came up with this integral for our volume:

We then had a great discussion of how to evaluate this by hand and eventually came up with our answer. I thanked them very much and said I had another shape for them to calculate the volume of (this time, using only my big mathematical jargon): Let R be the region bounded between y equals x squared and y equals x cubed in the first quadrant. Let R be the base of a solid that has squares as cross-sections perpendicular to the x-axis. Find the volume of the solid.

As they all scratched their heads, I went behind my desk and pulled out a big box of Play-Doh and gave each student a tub of their own. I told them to make the solid first, and then we could go about finding the volume. I brought around dental floss and chopped up their shapes to see if they were accurate. Then we created the integral, focusing on the area of the cross section. They went home with Play-Doh and more problems to practice making solids and integrals for.

caN YOu Decode this?

2010-03-24T18:42:00.000-07:00

I'm happy when my students learn good math in my classroom. I love solving real problems with my students and having them go out into the world with a better understanding of why things work. Even more rewarding is when they tell me how they solved a problem that occurred in their life with mathematical reasoning.

But what I get the biggest kick out of is students using math with each other for no good reason other than they think it's cool. Monday night in my cryptology class, we talked about Bacon's bilateral cipher and all the students got charged with the idea that they could hide a message in seemingly innoculous text just by converting letters to numbers in binary and bolding the appropriate characters in the regular text.

After a long day at work yesterday, I got home and checked my Facebook. I saw one of my students trying to bold his status and failing.* So I left him a little message, letting him know in Bacon's code that I was sorry he couldn't bold. (I just used uppercase and lowercase letters instead. More obvious but usable.) Next thing I know, other students are leaving messages in Bacon's cipher and a student that isn't even in the class figured out the cipher just because he didn't want to be left out!

When I finally logged out of Facebook, I was nearly radiating from all the geekiness.

*Yes I currently friend students. You can scold me on that later. I go back and forth on the reasonableness of that decision daily.

Notes Again?

2010-03-21T15:11:00.000-07:00

I'm having a hard time with Algebra IB students. They are sophomores in their second year of Algebra I. My school teaches traditionally, although the students all used the Connected Math program through middle school. To not mess up the students too much, I try to follow our department basic lesson plan: review homework, lecture, allow students to do homework.

I always add in warm-up problems at the beginning of each lesson. This week, as we're working on factoring polynomials and the students are asking me why they need to know how to factor, so to sneak in an extra tidbit, I decide to throw up some basic solving for x problems. The expressions are factored into binomials and set equal to zero, so the students only have to realize that to multiply to make zero, one of the two binomials needs to be equal to zero. After that connection, it would be a very simple solving a linear equation problem.

It was pure mutiny. The students yelled, screamed, banged their heads on the desk. They accused me of being a fraud, for not lecturing to them about how to do something as complicated as this and just expecting them to be able to find the answer on their own, even after I walked them through one of the problems. They wanted notes, a step by step guide to follow.

I guess I should have seen this coming. These are the same students that don't allow me to start a new lesson by trying a problem or two and having us work together to come up with the method to solve all problems like this one. They have trained me that I need to first write out the steps and then we can try to attack the problem as a class, which takes all of the fun out learning math in my opinion.

Once the tears had dried, they asked for a project about factoring. Any ideas?

Big Week Ahead!

2010-03-21T11:34:00.000-07:00

Here is a teaser for upcoming posts:

This week in my calculus classes we are going to start our discussion of finding volumes. I've already picked up a supersized box of Play-Doh and dental floss so that we can easily explore area by cross-sections. I'll make my famous mathematical paperweight out of a sine curve and semicircles and we'll try to estimate the volume.

After the Play-Doh, I'm modifying Dan Myer's WCYDWT volumes of rotation with glasses project. I have some glasses to bring in and we'll test out our ideas. I'm thinking a good follow-up would be to have students create their own Coke bottle that holds 20 oz of fluid. They will make their own curve and give me the integral. I wonder if there is a good software program or applet out there where students can easily enter their own curve and have the program rotate the area for them? My mind is quickly filling with ideas about this project. I can't wait to see the student results!

And one other big thing- I'm doing my first live, in person conference presentation this Friday at the ATOMIM spring conference in Bangor. It's not too late to register online! If you're there and reading this now, you probably don't need to be attending my presentation about Edublogging, but stop by and introduce yourself!

Grading Scales and Regression Lines

2010-03-16T06:43:00.000-07:00

I make hard tests and quizzes. Because of this, I usually incorporate a scale of some form. The simplest one to use is a square root scale: take the square root of the number of points that the student earned and divide by the square root of the total points. While definitely helping scores, this gives a much more generous scale to students who do very poorly while giving almost no scale to those who do well. My tests are hard, but I don't like the amount of increase it gives to students who fair poorly on my tests.

Because of that, I started establishing cutoffs for the different letter grades. I would enter these into the lists in my calculator and create a cubic or quadratic regression line through the points. Although this helped normalize what was happening for those who were passing the exam, it again gave very large scales for those who were only getting a couple of questions right.

Last test, I used a logarthmic scale with my cutoffs and I am much happier with the results. This places the high scoring students closer together in final grades while lower scoring students are not getting as much of a cushion. I'll have to test this out on a few more assessments to see if I like this scale more, but right now I think it'll give my students who have been coasting the incentive to work a little harder.

Free Response Quiz

2010-03-14T08:52:00.000-07:00

As I start preparing my students for the AP exam, I like using several class periods to go back to common free response questions and have students work on solving them. One of my favorite ways to get students engaged is to do a group Free Response Quiz.

At the beginning of class, students are broken up into groups of three and are given fifty minutes to work on the problems together. I recommend that one student solves each problem and then they share their solutions. At the end of the time, I break the groups up and each student of the group randomly gets one of the three questions in a regular quiz format. Grades are calculated by adding up the scores from each person in the group. This way students are motivated not only to learn the other questions but to make sure that their partners fully understand their question.

I did this Friday with my BC Calculus students looking strictly at questions about series. I love watching the students work with each and explaining the material. I believe the best learning happens when I remove myself from the process. It was great looking over the work my students turned in and seeing how much they had gained from working together. I love this time of year.

Best Work Day of the Year

2010-03-11T18:30:00.000-08:00

Yesterday, a student walks into my class carrying his uncovered book. Writing warm-up problems on the board, I just point and say, "Naked," referring to the lack of cover as I usually do. He looks down at the book and then deadpans, "I like it more that way."

Maybe too much stress had built up from all of the exams last week but I could barely contain myself.

Other great news of the day: Four of our students were in the top five ranked juniors and seniors in our regional math team league for the year. Our top team also won first place in division B. Overall, it was a very good day!

Cheating?

2010-03-03T18:29:00.000-08:00

The latest lunch table teacher drama has been about students taking a long February break. Like, an extra two weeks for a one week break. (It was Lunar New Year on Valentine's Day, for those of you that aren't teaching Asian students.) So many of them went home and took a few extra days to celebrate with their families and really, who can blame them?

My last students arrived back yesterday, a good week and half after break had ended. They knew before they left that they would be having a test over the Fundamental Theorem of Calculus when they returned. In fact, I was a kind enough to push it back until this week and not have it the week directly after break, since so many students were still missing. So today, I had a two students, X and Y, walk into my classroom that I hadn't seen for close to a month and being the strict teacher that I am, I made them take the test. Although student Y pleaded to take the test after school, I refused, knowing that I had another meeting to attend and couldn't stay to proctor the test.

But this story isn't about them, really. It's about my sweet, quiet student Z who rarely speaks, even when called on and struggles with his English. He is a sweetheart, stubborn in his ways until the end, and if you tell him to do one thing, by gosh, he'll work on that one thing until he perfects it and then he has the biggest grin as he shows you what he has accomplished.

Today we were done with part two of the multiple choice section of the test and I asked the students to hand their papers forward. My darling Z looks surprised that the end had come. As others are passing their work forward, I notice him open his test back up and write feverishly on his hand. I go to him and ask to see his hand, which he refuses to show me, while smiling bashfully. He tells me another student asked him to copy the answers but he swears he wouldn't give them to someone to cheat. As what is required of me, I send him to the office to talk to the Dean about it. I told him that the sad part was that the students taking the test after him were taking a completely different version of the test with different questions and different answers.

Later, I talk to the Dean. He says that Z told him that student Y (who remember, ended up taking the test at same time as Z did) had asked him to write down the answers to give him after the test. I think that maybe Y had hoped that Z would take the test, give Y the answers, and then Y could use those answers when he took it after school. Unfortunately I foiled this plan and Z, being the ever diligent person that he is, ended up still writing down the answers to give to Y, not really comprehending the full reason for his copying.

So, I'm going to let Z finish taking the test tomorrow and he's going to get full marks on his test. Y... Well, his grade is going to stand on it's own, although I have a feeling that without any changing from me, it's already not going to look pretty.

Hurrah for Math Competitions!

2010-02-24T17:17:00.000-08:00

The week before February break was amazing. My students took the American Math Competition. I was a little afraid of turnout because it is completely voluntary but the day of the exam, 38 students showed up. I was excited to see the turn out but a little disappointed that two of my previous high scorers didn't attend. I guess I can't have everything.

Last year, one student of mine qualified for the USA Math Olympiad. This year, he beat his AMC score by a few points. I have a new student who I thought might be near the same level. He ended up outscoring my student from last year. I couldn't believe it. I am thrilled to have two students who are exceptionally gifted in mathematics. We are looking ahead to the AIME's with anticipation!

To top my week off, I was going through my mail and I had some information about Moody's Mega Math Challenge. On a whim, I brought it to the math team meeting to see if some of the boys might interested. Boy, was I wrong. They were ecstatic to be able to spend fourteen hours on one problem! They started working out who would be on the team and immediately got online to start researching the competition. Since then, they have been studying statistics in their spare time and trying to do anything else they can think of to prepare. I did nothing but show them a couple of pieces of paper. How did I manage to create such monsters? I must be doing something right.

The FTC: From Scratch

2010-02-10T18:03:00.001-08:00

I like making my students discover ideas on their own. I like to lead them down a path and then make them put the pieces together to solve the puzzle. For all of my teaching years, it had seemed like when we finally discussed the Fundamental Theorem of Calculus (the supposed most wondrous moment of their mathematical careers up until that point), my students would just look at me and say, "And?"

The books I used made students practice taking antiderivatives by evaluating indefinite integrals way before the discussion of area ever appeared and I blindly followed along. The students were already used to the idea that integrals and antiderivatives were related, so when we found that derivatives and integrals were really inverse concepts, it was obvious because we had started off by defining them that way. Sharing that the trick to evaluating definite integrals was to use antiderivatives made them think I wasted their time making them calculate area in any other way. Wasn't that already coming? This year, I wanted them to feel the awe of the FTC.

After midterms, I had them work on taking antiderivatives by giving them differential equations and asking for the original function. I also told them to hide their books and promise not to try to read them. (Those over zealous students of mine always outdo themselves!) We worked on finding area using Reimann sums and then explored how the limit of the Reimann sum evolves into a new symbol, the integral. We practiced graphing equations to find the geometric area between the curve and the x-axis to evaluate integrals. After covering some of the basic properties of how integrals work, we worked on the first worksheet from the FTC curriculum module made available by the College Board.

This was the perfect spring board. We were able to review finding area beneath the x-axis, what happens when endpoints are "backwards", and finding area of trapezoids. Although the students understood that the integral symbol meant finding the area from a to b, this worksheet really helped them solidify that connection. Then we were able to look at the general formulas for the integral functions given and the derivatives of those equations.

When we reviewed the results from the worksheet, students began seeing "+ Cs" which happened when I began changing the lower endpoint of the integral and asking what happens to the general formula. This made them think of antiderivatives. Then they saw the relationship between the integrands and the "shortcut" formulas we were getting for evaluating the integral. They told me that to evaluate definite integrals, you must take the antiderivative. They were the ones that put together that you substitute in the upper endpoint (which in the worksheet, was the variable x) and then to find the value of the "+ C" they had told me about earlier, you take the opposite of the antiderivative evaluated at the lower endpoint. From there, they ended up working out how to take the derivative of integrals with any kind of endpoints you could imagine. They figured all of this out on their own in less than seventy minutes. It was beautiful to watch to their ideas develop and I think that they were impressed with both parts of the FTC because they saw the pattern and created it themselves.

I can't wait to do this again next year.

Pyramid Testing

2010-01-30T08:00:00.000-08:00

While in college, my geology teacher used pyramid testing for our assessments. Basically, he gave us a very difficult exam, expecting us not to be able to do all of the problems. After we turned in our individual attempts, we worked in groups of two or three to redo the test together and talk about questions. He used a weighted average to get our grade. Although a little more time consuming than a regular test (both in the classroom and for grading), the test then became a tool for learning and not just assessment.

I enjoyed this (although I was always upset that I had problems with the questions the first time around) but I hadn't incorporated it into my teaching. I graded midterms this past weekend and was dreading the two days of standing in front of the classroom, going over the exam, question by question. (A good teacher should get out of the way and not be the one talking all period.) My exams are hard and no one had a perfect score on the multiple choice questions the first time through. I decided to modify the pyramid test and on the first day of the next quarter, I gave the students the question packets back (I kept the answer sheets with the right and wrong corrections) and had the students work in groups of two or three to redo the problems for an extra point on their 2nd quarter average.

What I saw was amazing. My students that seem so shy to speak up in class, often so distracted while working on problems in class, were arguing with each other about discontinuities. Hands were waving in the air, drawing imaginary graphs to explain ideas. Pencils were scratching at paper, calculators were being passed around, and even the students with the least confidence in their ability were describing their reasoning to others. Having the motivation of a test in front them totally changed how they attacked the problems.

Then I got their new set of answers back. The number of correct answers more than doubled. The problems I needed to share because there was class-wide misunderstandings were few.

I think I might need to do this for every test, maybe even every quiz.

***
I just did some googling and found this more in depth explanation of pyramid testing, a few variations of how to do it, and why it works so well in reformed calculus classroom.

LaTeX

2010-01-20T08:44:00.001-08:00

In part of my re-creation as the best teacher I can be, my goal is to post about one great thing that I've done during the week. (Hence, I must do one great thing worth blogging about each week!) This being midterm week, I thought that I was going to have to wait to commence my record of cool things but as I was getting ready to start typing up my mammoth two calculus midterms, I realized it was time for me to start using LaTeX (pronounced as Lah-Tek) to make my exams instead of fighting with Word.

For those of you have haven't yet crossed over, it is easy and elegant. The amount of joy I experienced in making my first exam almost brought me to tears. LaTeX is not a word processing program, but a *free* document processor. You type everything into a basic text file, run a compiler, and your commands make a beautiful PDF document with professional looking equations. If you have any experience in computer programming, this is a piece of cake. You have complete control and don't have to deal with the enforced numbering system and automatic formatting of Word, the drop-down menus of Equation Editor or the sheer amount of memory resources it takes to make WYSIWYG equations. (Just the first of four sections of my calc midterm had over 100 embedded equations!)

To all of those that have never used LaTeX before, it is well worth your time to become fluent. In less than an hour, I was typing up my midterm in record time, popping back and forth from websites with instruction commands back to my exam. The information is all readily accessible online and you can do the basics with no problem. I already started to go beyond the basic page template and have been researching special page formats and I now hope to organize all of the questions I type up so that I can easily reuse questions. It will now be no problem holding thirty plus multiple choice questions in one document to easily copy and paste into a current quiz or test.

Here's a sample of what you can produce in a little over ten minutes with LaTeX. (This is my first homework assignment for my Number Theory Through Cryptology course.)

If you want more information about LaTeX, check out their website for downloads, explanations, how-tos, and more resources. My favorite resources for learning the equation commands have been the Art of Problem Solving website and the wikibook LaTeX but there is tons of information out there. Google can supply you with whatever information you need.

Finding My Motivations

2010-01-17T07:17:00.000-08:00

As I read Dan Myer's latest post titled "This Blog is Counterproductive," I realized that in some way I empathized with him.

I mentioned before that my teaching had become an island. My comparisons for how I was doing were only made with others in my department, let alone in my whole school or even my whole state. My department head regularly told me that I worked too hard and put too much into my teaching and that I needed to back down. My dean of academics told me that he thought I was best calculus teacher in the state, which I fully knew was not true. Yet, those statements made me become complacent. Then I noticed the status quo kept slipping lower and lower.

It is in the reading of blogs like Dan Myer's, Kate Nowak, and Sam Shah that I have been reminded that I had once been inspired to be the best teacher I could be. Not the best in my school or the best in my state or the best against anyone else. I wanted to give everything I could to my students academically and emotionally. I may borrow the ideas of the great three but I am now working on making my own again. I hope Dan realizes that his blog has been anything but counterproductive.

New Year, New Courses

2010-01-01T14:52:00.000-08:00

I just spent a snowy afternoon by myself, thinking about my Cryptology/Number Theory class that is going to start at the end of January. There are a few things that I love at my school, such as my ability to choose whatever semester long, college level class I want to teach. (The downside- I do this on top of my regular teaching schedule and the class meets for two and half hours one night of the week.) But, they give me complete autonomy to do whatever I would like with the students.

This semester, I'm teaching Linear Algebra, which has been quite the experience for both the students and me. It's definitely a harder course that what I had in my undergrad, but I don't think that's too hard to do. We're spending quite a bit of time focusing on proofs which had been totally ignored in my undergrad class but was done in detail in my masters Lin Alg course so I'm trying to find the middle road.

Anyway, I like to do one "traditional" course a year and one fun course. I taught the Cryptology course two years ago and the students loved it and were begging for me to have it again, although it would have been the same kids in the same course. (I don't know what they were expecting I would do the second time around.) I'm trying to set these up so that they will rotate on a two year schedule. In the past, course offerings have been Multivariable Calculus (with college credit from one of the local state universities), Proof and Problem Solving, and math history.

I'm wondering where to go next. Multivariable is hard to offer because there are really only two or three students a year that have the background to take it and I hate to limit the students that can be in it. Many high schools offer differential equations but whenever I've talked to my students about it, they seem to cringe at the idea. They too are looking for something fun and engaging. At the same point, I would love to have some mathematics included that would help my students for math team, the AMCs and beyond. (One of my current students took the USA Math Olympiad last year and I think I have another student that may have a shot at it.)

Here are some ideas for other courses that I've come up with: Probability, Non-Euclidean Geometries, Combinatorics, Fractals, and/or Game Theory. The first two courses I would have no problems in offering because I already have taken the courses and so, I know the basics. The second three are pretty much completely new to me and I would have to teach myself the material before I could teach it to them. Does anyone else have any comments or ideas that could help me figure out my possible courses for next year?

Growing: Reflections on the Last Four Months

2009-12-22T12:10:00.000-08:00

This year has been amazing for me as a teacher. I am now into my sixth year of teaching; I have passed that invisible line over my fifth year, proving my choice in the field was correct. (I have read many sources saying that over half of new teachers don't make it past their fifth year.) Once that line is crossed, it seems that one has become a teacher for life.

I realized at the beginning of this year, the newness was definitely gone for me. I wasn't anxious in the two days leading up to the first day of school and I had started to feel comfortable. This told me that something was wrong. My teaching was missing the excitement, I was going through the motions that at one point had been challenging and now were comfortable. I knew I wasn't happy with my teaching even though in my department, I'm considered the radical one.

In September, I made a little pledge to myself. I had just started reading dy/dan and some other blogs and I was getting excited. I told myself that my goal should be to add to my blog twice a month, instead of the once per month posts that I had started with. Since my writing became more important to me, so did my reading and the next thing I knew, I was listening to other teachers and having conversations about teaching that resembled the type of connections that I had made during my time studying education.

All of this has been so meaningful to me and the group involved in this seems so small that I am ready to expand. I'm going to be submitting my application to present about professional blogging at the ATOMIM conference this spring. I'm only on the beginning of my journey but thanks to everyone who has inspired me and encouraged me to become a better teacher. I hope I can now help others to do the same.

K-W-L: Reflection

2009-12-22T11:43:00.000-08:00

I finally completed my K-W-L's with my calculus students. We spent one class day where I threw up a random graph and asked them to come up with some ideas about derivatives and how the graph looks. It started off simple, "The derivative tells you the slope of the function," to having a girl say, "The function goes up when the derivative is increasing." I put up her statement as said and then we had to explore some slopes on our graph and one student ended up coming up with the more correct statement "If the slope is decreasing, then the graph will have a maximum."

Wow! In only thirty minutes, we had moved through critical points, the first derivative test, and were talking about concavity and the students were creating the rules for the second derivative test all based on an initial incorrect statement (she knew what she meant, she just used the wrong words and it was a wonderful teachable moment).

This was in my first period of calculus. It was one of those magical classes that you always dream of, when the students see why things are working as they are and you are so proud of them because of how they are creating the most amazing ideas on their own. Then I had two more periods of calculus.

The second two times, the K-W-L was still much better than what I could have done lecturing by far, but we stayed with discussions of the first derivative. I just had the students take a quiz over the relationship between graphs and derivatives and hopefully the success of using a K-W-L for introducing this material will be evident.

Oh and that random graph from the first day? It was taken from the 1989 free response question highlighted in the movie Stand and Deliver. At the end of the unit when we were preparing for the quiz, I pulled out the question again and we walked through all of the parts. It was a great way to conclude the unit and reflect on what we had learned!

K-W-L's

2009-11-14T12:10:00.000-08:00

Anyone who has gone through a teaching program has probably heard rave reviews about the knowledge activation process known as a K-W-L. You start a unit by folding a paper into thirds and writing across the top of each third, "What do I know/What do I want to know/What have I learned." I agreed in my college education classes that it sounded great, especially when your students have some basic ideas about the topic you are about to delve into.

Then I began teaching high school math. I never used it. The acronym came across my mental field of vision occasionally and I scoffed it off, thinking how it might be used in a middle school or elementary setting, or even high school science or history, but not high school math. Come on now.

Flash forward: I have been at a conference for the last two days where I heard a presentation about creating a professional culture at your school and ways to encourage faculty to expand their teaching repertoire pedagogically. The presenter was pulling out examples to highlight his main points and of course pulls out the old standby, a K-W-L. And suddenly there it was: maximums and minimums in calculus!

How had I never came up with this idea before? The students know about maxs and mins, we've graphed sine and cosine functions and I've asked the kids if there are any points where they know exact value of the slope and of course, they point out the maxs and mins. There must be thoughts bouncing around their brains about these points and zero slope; if I could just access those and get them excited to find these points easily with their knowledge of derivatives...

Hurrah for K-W-L's! I'm starting this unit after Thanksgiving and I'll let you know if the K-W-L transforms my unit and is worth all the hoopla it has always received.

More Than Just Worksheets

2009-11-08T03:49:00.000-08:00

I usually teach seniors. At that point, they are so close to finishing high school, they understand that to graduate, they must do their work. Occasionally, they skip a homework assignment or two, but if I give them time in class to do work, they work. I can give worksheets or book work; students are fine working individually and if they need help, they ask a friend and learn how to solve the problem.

This year, I have two periods of sophomores. My usual hands-off approach doesn't work. These students still need extrinsic motivation to solve their problems. I'm working hard to try to come up with new ideas and projects that will not only force students to work together but force them to actually solve the problems.

Force is a strong word. I'm trying to be more creative in the method that the problems are presented to get them motivated to solve them. Whether it's review Bingo, speed math, group problem solving*, or my one problem per person class quiz**, I've got to get them motivated to solve. There is no motivation to solve on their own.

Which brings me back to my seniors. Yes, the vast majority of the students are doing the work and learning the material. But would they learn more and do more if they problems were presented in a more intriguing way? This may change how I teach all of my classes.

*: Like group story writing. This works well in problems that have a lot of steps. Each student writes only one line of the solution. The next student has to use the previous line to create their line. This can be done as a full class or with students broken up into smaller groups. I've used this before to analytically create a graph in calculus or to solving a system of equations in algebra.

**: A regular quiz is given but each student is assigned to one problem. Students may elicit help from each other, but once a solution has been passed in to me, it can't be changed. The whole class gets the cumulative grade of quiz. Works as a great pop quiz.

Like the First Snow, Falling

2009-10-08T12:25:00.000-07:00

I'm not sure if you know about Taylor Mali, but you need to. He is an outstanding poet whose work usually deals with teaching. As I was sitting here, grading my calculus homework and thinking about today's lesson, I thought I wanna teach like that, like the first snow, falling.

There are moments now when I think the students are really intrigued and on the edge of their seats, but there are also so many days when I'm just working to push through what needs to be covered next in the book. And that's where my problem is. I follow the book too closely. I am a slave to the next section, to the order they present things in. If we start a conversation about graphing derivatives and it's not talked about for two more sessions, I follow the conversation for a few minutes until their interest is semi-satisfied, then I say, "More on this in two days." Why not follow the topics they want to talk about and go back to today's lesson some other time?

I think that simply making that change, to actually teaching on what's relevant at the moment and not what comes next, is the change that my teaching is looking for this year. Something needs to change. I'm flat lining. Modesty aside, I'm good teacher but I'm no where near a great teacher.

I'm afraid. What if we don't have enough time to cover everything that we need? What if I skip over a concept or a theorem that they were supposed to know, but I just missed it somehow by not following the book? I think I've been teaching my calculus material long enough now that I'm familiar enough with it to know about skipping things. And maybe I will have to take days where I cram a bunch of "book facts" at them instead of concepts. I know I can find a balance and it won't destroy my students.

My New Year Resolutions

2009-09-07T04:43:00.000-07:00

Getting caught up on my teaching blogs this weekend, I found many people making their "New Year Resolutions." Following the herd (because heading into my sixth year of teaching, I know that I still have so much to improve on!):

1) Keep my room tidy and organized. So selfish of me! But I have started off fairly well. (I still have some books to put away in my new bookshelves and I have some posters to find a home for.) This resolution has gone further than my own personal sanity and reputation though. I have created the missing day folders where students can go to pick up handouts they have missed. I have the homework turn-in boxes all setup (but still needing labels). It is starting to come together. Now I just need to keep it up and file things the hour they are no longer needed out.

2) Beef up my teaching of my non-calculus class. Each year, I am thrown one or two random courses that are not my calculus course and it always changes each year. Because of that, I never take the time to really think out a good unit or great projects because I probably won't teach it again. Nor do I organize the materials I create. That then affects how I teach the course. Without taking more time in the planning, I am basically saying that I don't care about those students and I've realized that in my conserving of my own resources, I have not been giving them the best education that I could. There is no creativity in the course and I am not happy, nor are they.

I've started out well with this so far. I have my first unit outlined and I started with an amazing first day to test their previous knowledge. I had each student tell me their name and something that they like to do. At the end, I introduced myself and told them I liked to bake and to share my baking with my classes. So I put up three different recipes and had them help me double one, halve another, and take a third of the last one. We worked on fractions and conversions. (I brought in my measuring cups and spoons. So when they told me: 1/12 of cup, I showed them I couldn't measure that so they needed to change their unit of measurement.) I just need to keep finding more of those things to do!

3) Not let my students fall through the cracks. With almost 100 students, that is hard to do. But I need to pay better attention to each student. I've gone back to collecting homework every night to keep and read, which will help somewhat. But I need to catch when students aren't doing well before it's too late and I've lost them for the year. It's watching the quality of their work, talking to them when I see a change, and letting them know that I'm paying attention to them. Some days, it's hard to spend all of that time thinking about the kids and evaluating where they're at, but I know that it's worth it.

Back to Functions

2009-09-05T07:19:00.000-07:00

It is almost a year later and I think another piece of the puzzle about abstraction and functions has finally fit into place. I was speaking with the co-coach of my school's math team and the resident physics teacher, when he brought up the fact that students may learn things in math and then can't apply them in physics. He related it to the math team topic Arithmetic with * Operations to explain when things are tweaked a little, students no longer understand. As he spoke, I got a faraway look in my eye as things started connecting.

In calculus, it's easy to use function notation to manipulate functions. You can even use as a function to have students differentiate and then interpret at different points. In precalculus, you can give them a graph of an abstract, piecewise function and ask them to transform it. But how do you make functions more abstract in Algebra I and Algebra II? One way would be to give them lots of abstract notation to interpret.

In teaching my SAT Math Prep class, I have noticed students who are in precalculus that can't solve a problem like . In fact, they don't even know where to begin. They have never seen an asterisk in a problem before but to them it looks a lot like a multiplication symbol. Once I finally get them to see it is just a symbol that signifies a rule that explains how to manipulate a pair of numbers to associate them with a third number, they become confused if I try to use an asterisk again in a different problem and assign it to a different set of points.

Just this week in my calculus class, we reviewed functions. I gave them a set of three points and asked them if it was a function. They looked at me quizzically and then as if I had a second head. "No," a student said. When pushed to explain his answer, he said, "That is only three points. That can't be a function." I brought them back to our definition of a function: A set of points where every x-value is associated with at most one y-value. Our three points made a set. Each value of x that was defined had a unique y-value. All other x-values had no associated y-value, therefore, it fit our definition. My students were blown away. Never had they seen a function with such a limited domain and range.

As I move on through my work on my masters and I think more deeply about the material I teach, I wonder about the other math teachers in my department. What would be their answer if I gave them three points and asked if it was a function? How often do they expose their students to abstract ideas and different notation? I think I am ready to put together a small talk for my department about functions and abstraction. I'll be sure to post my results and further thoughts.

Different Methods of Solution

2009-06-13T14:14:00.000-07:00

In my third year of teaching, one of my students said to me, "In Vietnam, our teachers ask us to solve each problem three times; each time, finding a different way to come to the same solution."

This statement, so small at the time, has become a way to define my philosophy of teaching. I hate to show my students algorithms and often try to pretend that their books don't have anything of use for them. Many students learn a way to solve a problem and that is all they have- a method with no understanding of what is happening or why.

This has always been my problem with elementary mathematics education. Instead of having students find their own ways to solve problems and allow them their own methods, students are told that they must add or multiply using one given algorithm. I grew up on this system and was very afraid when I first started teaching because my arithmetic skills were so poor, I knew that the students would have a hard time trusting I knew anything in math.

But as I continued teaching, I watched the different methods that my students used and started to pick up on their tricks to solve problems. I started making my students do their homework problems on the board, not just to get me out of the limelight, but to show methods of solution and ideas that I would have never thought of. My students learn from the varied solutions of others and can look at a problem a peer solves and ask, "Why did you do it like that? Here is what I saw..."

I don't regularly ask my students to solve problems multiple ways but I know that they can. If a student is done early and has a particularly messy solution, I'll ask them to go back and try to solve the problem again. Sometimes, I go around the room and find as many different ways to solve the same problem as possible and have all the students go up and share their different solutions and then we evaluate them as a group.

What my Vietnamese student gave me was not just a teaching strategy but a reminder of the creativity involved in mathematics and beauty of each solution.

Do Our Students Need Algebra?

2009-05-17T07:39:00.000-07:00

I am a calculus teacher but I must say, I almost never use traditional algebra in my daily life. It feels like a sin for me to utter those words out loud.

I really should quantify that statement. I do not write equations and solve for x. Nor do I create graphs of linear equations to understand a problem better. I do not multiply polynomials nor do I factor. The only time I use the quadratic formula is to solve a problem given in class.

This is not to say that I don't use math on a daily basis. I do. I am constantly calculating rates: how fast did I bike fifteen miles? How long will it take me to do the sixty mile ride at that speed? Is my speed increasing over time? I can't read a newspaper or magazine without being inundated with ideas about statistics and how to interpret them. I routinely make directed graphs to show the flow through a system and easily explain paths to others.

Yet, where is all of this covered in a traditional high school mathematics course? Calculating rates are quickly covered early on but I still find many of my seniors in calculus don't know how to calculate the average miles per gallon for their car nor can they figure out how to start by analyzing the units. They do not know how to interpret the rates that are given to them in terms of a problem. How does a 5% interest rate effect the $1000 they put away for retirement at 20 years old versus a 3% interest rate?

Then there is statistics. I don't believe that our school is out of the norm to only teach students about averages and not about measures of variation. What is taught is only how to do it and not how can those numbers be manipulated to give a support to whatever conclusion is wanted. Nowhere do we start talking about percentiles, hypothesis testing, or confidence intervals. These are ideas that will come up again and again through their adult life but we harp on about how they must learn to determine the degree of polynomial!

Maybe I'm progressive but I believe that our first focus should be on teaching mathematics that the students will use in their daily adult lives. They should know about more than the process but also how to use and interpret the results in different situations. Our focus should be on making our students mathematically literate enough so that they can calculate the tip they need to leave at a restaurant and know when they are being ripped off in a business transaction.

That is not to say that I believe that our current program is unneeded. As a calculus teacher, I want students passing through the traditional curriculum so that they can get to my class with sufficiently strong symbolic manipulation skills so that we can start attacking the concepts without worrying about the rules involved. I just do not believe that this traditional background is necessary for all students.

A new goal arises then: creating a course called "Math For Life". What topics, concepts, and understandings do you believe need to be included in this course for being successful in daily life?

Deeper Arithmetic

2009-04-22T06:54:00.000-07:00

I have always believed the mathematical truths that my teachers told me growing up: multiplication is nothing more than repeated addition and exponents are shorthand for repeated multiplication. Of course, I knew there were flaws in this- how does really mean taking the square root of a number? Eventually, I could make sense of this idea and I reasoned through it by thinking of multiplying a number by half of itself would mean that if you multiplied two halves together you should get back the original number, which followed logically.

A bigger question I had was about . Somewhere in high school, I decided that this was a function that I needed to graph and explore. To the right of the y-axis, everything made sense; the graph was a smooth continuous line. To the left, I knew there were places where the function was defined and yet, I saw nothing. It got me thinking: if x was a simplified rational number and the denominator was even, then there would be a hole. If the denominator was odd, the function would be defined. But what about the irrational numbers? Where was their place in this? If all of the defined points were graphed and looked at from afar, would there still be a "smooth" pattern?

Somewhere, this hit a disconnect for me. A process as simple as repeated multiplication seemed to tell me that the graphed solutions should be continuous. Something was wrong with exponentiation that was further illuminated when I later studied Taylor series and was able to produce Euler's Formula: . (Sorry, I can't seem to make the exponent look correct. The exponent should be: .) While this formula blew me away mathematically, what really gave me pause was rewriting it to read: . This meant that e, a positive number, when multiplied by itself some mathematical amount of times would yield a negative number. True, there was an i in the exponent, but I could not reconcile this equation with the idea that exponents were just repeated multiplication.

Keith Devlin explained the idea of how multiplication and exponentiation cannot be simplified forms of other operations in his articles "It Ain't No Repeated Addition" and "It's Still Not Repeated Addition." It is amazing to me that although I had evidence that exponents were not multiplication, I still could not help but teach my students that same fallacy. (Although after my own experiences in high school, I have always had my BC Calculus students explore Euler's Formula to help them start to ponder what exponents really mean.)

Devlin leaves us K-12 math educators to determine how to rectify this situation. Part of me wants to say that working with whole number exponents gives students a chance to explore the basic rules and make sense of them before applying them in abstract situations. Begin by talking about multiplication as repeated addition and addition as counting items in a set. But I know the type of student that arrives in my Algebra I class that still does not know how to add and subtract signed numbers. They have to work to memorize the rules for multiplying and often fail. How are these students being served correctly by being told that math is simple and based on concrete representations when it is not? Not everyone can naturally abstract on their own and reconcile untruths.

I'm not sure I have a good solution to Devlin's challenge yet, although I will try to expand the minds of those students who come to me already believing an oversimplification of an abstract system.

Passing Notes

2009-02-16T03:41:00.000-08:00

I was a student that was just hitting the beginning of the technological wave. We had a computer lab in my high school and several other computers for use in the library, but they were used only for word processing and (if you could score a computer in the alcove where the librarians didn't often walk) sending email.

At that time, some information was available online but it was not a resource tool. Teachers did not bring their classes to their computer lab (that space was reserved for keyboarding class) nor were there enough computers in the library to have a whole class go to use them. But, even more importantly, what could they be used for besides writing papers and email? It would be more hassle to make sure that students stayed on task writing their papers and not checking their email than it would be to just give them a piece of paper.

But think what we would do with pieces of paper! If asked to write an assignment by hand, we could draw a doodle in the upper corner, embellishing our name with flowers and swirls instead of creating an essay. Paper airplanes could be made, which would distract the whole class. And worst of all, we could communicate with each other- writing notes secretly back and forth, where the language might not be completely pure. We would shorthand names and events so that if the note was intercepted, it might not be understood. As we left the classroom, we would scrunch the paper into a ball and drop it in the trash can.

Today, I read an article from the New York Times about cellphones being embraced (but more likely not) for usage in the classroom. Four schools in North Carolina gave students a cellphone with 900 minutes of talk time and 300 text messages per month. The teachers are required to go through the students' text records to see if they said anything naughty and cutting them off if they did.

It makes me think about about my history teacher in high school. He must have seen the paper fly across the isle between me and best friend. He said nothing, although we did try to be slightly discrete about it. I can't imagine that after we left the classroom, he went to the trash to dig out our paper and check for the appropriateness of our words. And if our messages happened to not be appropriate, I can't see the need for a school wide ban for us to not use paper because we used it once for communicating about our private lives with each other.

In the long run, isn't that what school is about? Between the snippets of learning, you get a chance to communicate and socialize with others your age. How is checking email and sending texts any worse than passing notes back and forth?

The Abstraction of Number

2009-01-14T09:08:00.000-08:00

As my department was looking over the new state learning results, I came across an interesting sentence:

"It is expected that when working with measurements students... understand that a number without a unit is not a measurement, and that an appropriate unit must always be attached to a number to provide a measurement." (From Maine Learning Results 2007 Mathematics Grades 9-Diploma, Standard B)

I immediately wondered why is this so important that it needs to be listed in concepts that high schoolers should grasp? Yes, many students forget to write units at the end of answers, but how does the proper usage of units show more sophisticated thinking?

These questions brought me back to why man first started using numbers. Our first evidence of counting comes from the Ishango Bone, found in the Congo dating back to 20,000 B.C.E. When man first started using numbers, they represented a one-to-one correspondence to elements in a set of physical objects. In fact, even as we began to use negative numbers and fractions, they still had meaning in the physical world, such as the area of land owned or how many bags of grain were owed in a business transaction.

Eventually, the study of numbers and patterns became a pursuit of its own that lost its connection to the real world. A number, such as 3, could be thought of without associating an image of three goats or a length of three cubits. The number existed on its own, holding its own special properties such as being odd and prime and that adding 1 to 3 made 4. The change from having numbers exist only with units to an abstract thought of its own took mankind thousands of years to develop.

Considering how long it took our species to make this leap, it is a major leap for a child to make and realize in their understanding of mathematics. How many high school students still use their fingers to add and subtract? They do not have an abstract idea of number and still see numbers as representing how many objects belong to a set. Their understanding of mathematics is prohibited by their inability to abstract numbers without units. It is obvious why these same students flounder in algebra, an abstraction of the abstract idea of number.

My question now is not why this is important for high schoolers, but why is this standard not included earlier? A further question is how do we foster this deeper understanding of numbers with students that have gotten this far without the abstraction?

Function Composition

2008-12-18T07:43:00.000-08:00

I was in the process of creating a list of things that my students of calculus need to know but don't seem to and one of the ideas that came to my mind was function composition. It's tricky to say exactly what it is that they don't know. They have no problem composing two functions and they can usually correctly identify the functions that lead to an already composed function. But when they start using the chain rule to take a derivative, the whole idea of the outer function and inner function are totally lost.

Then, I read a posting from Keith Devlin's article, "How Do We Learn Math?" and he had this to say about functions:

Functions, as defined and used all the time in mathematics, don't do anything to anything. They are not processes. They relate things. The "doubling function" relates the number 14 to the number 7, but it doesn't do anything to 7. Functions are not processes but objects in the mathematical realm.

Suddenly, I knew where my students' problems were: they see functions as processes not objects. This is a common misconception, even up through undergraduate mathematics courses. It isn't until Abstract Algebra that students are taught to think of functions like they are objects and find properties about them such as their additive inverses. Before, talking about properties only happened with numbers and how functions look when graphed: even, odd, or neither. Many of my teaching peers looked at further abstraction and ran away in tears or forgot these ideas entirely.

To encourage students from the beginning to think about functions as objects, using the idea of "function machines" may help. Nowhere in our school's curriculum do we explain functions through this lens, although I have often contemplated using this model in my classroom. I have yet to do this because I was unsure of how beneficial it would be and I didn't want to waste any of my precious time with students. I am now rethinking that view.

I know that function machines changed my personal view of functions in high school. I did start to recognize functions as machine (object) that related one number to another, although there still was a process of "doing" happening inside of the machine. This idea did help me to see that to solve a trigonometric equation, I didn't have to divide both sides by sine (as many of my students seem to believe) but instead I had to use the "opposite" function machine (the inverse function) to undo the sine machine.

Another way to have students see this is to work with functions in a much more generic way, giving more examples of functions without giving a specific rule that is associated with it. This happens occasionally in precalculus in terms of looking at piecewise graph, but even at that point, the forward thinking student could create the equation for the function if they so wanted. In calculus, that option happens less frequently because we create functions such as f(x) = x g(x) and ask them to take the derivative. Here they have no option of creating an equation for g(x) but must accept it as an unknown object.

I have even gotten to the point with my calculus students that before we start talking about inverses of functions through function composition, we start by looking at group properties of real numbers under addition and multiplication. Then I lead them to figure out what the identity element of function composition is. From this building, they start to think of functions as objects that work together that have properties as greater structure; they see them as a group of objects and aren't trying to dissect the individual equations. It is a struggle for many students but some start to see the larger picture.

As I think down to the heart of why my students can't see functions as objects and why they struggle when I try to make them think this way, I start to question how my colleagues see functions. If I asked my colleagues to define the word function, what would I get for an answer?