mathlovergrowsup

My geometry students have actually started asking to find out about what we are learning in Calculus class. It all started when I realized(thank you Randi Engle) that showing my students how geometry would be useful in real world Calculus problems might increases their ability to transfer knowledge to new situations. I’d already been showing kids where geometric formulas appeared in the long Calculus problems left over on the board from the class before but now I had research based impetus to really spend some time trying to explain to kids why Geometry was invented and what it has to do with physics, velocity, motion, science, etc… I taught them that the integral symbol meant area under a curve and had them do some simple integrals of linear functions first. Then I got a bit more ambitious and had them solve a problem which gave a velocity versus time graph and asked a number of questions about position and acceleration. They did surprisingly well, especially one particular 9th grader who makes fantastic logical leaps. He tends to solve challenge problems first and get really excited about pushing his brain. I hope that he’s thinking that this is one of the first classes where he’s truly been able to use all of his reasoning abilities. There are a few students who know that Calculus is not going to be on a test any time soon and sit back and relax while other kids dig into the hard problem. I know that my methods of show and tell with the Calculus problems are not ideal and that I should have more carefully designed a learning environment where I’d be able to check for understanding a bit better. The enthusiasm and willingness to ask questions by the vast majority of the class seems to outweigh the cost of not having a carefully structured worksheet that allowed me to see if kids were getting things. I think kids like lessons that they see developing on the fly. it’s as if they have gotten the teacher off track and are learning something outside of the curriculum and somehow that is more exciting.

Today I tried to teach the students that there are infinitely many prime numbers. I think I did a fairly good job at dispelling some notions about prime factorization and divisibility but I’m pretty sure that only 15% of the class understood the mechanics of the proof by contradiction. I didn’t even try a repeat performance the next period and am actually considering finding an SAT problem involving number theory so the lost and confused kids don’t think I wasted 45 minutes of their time confusing them. I suppose it makes sense that people struggling with basic two column proof would find proof by contradiction subtle and confusing but I suppose it is worth doing something really hard once in awhile just to push the limits of what is possible.
Eighth period wanted to do logic puzzles and I gave them one my good math major friend taught me over Pitas late Friday night after a thrilling econ lecture. (You can tell how cool I am right?)
Suppose you have eight bottles and one of them is poisoned. Your job is to figure out which one using three mice. If you give the mice the poison they will die in exactly 24 hours but not have any symptoms until then. You have 24 hours to find the poison meaning you are not able to give the mouse some poison and then wait and see what happens before giving it a different bottle.
How can you find the poison? It’s fine to mix bottles, put the mice in cages, or do anything else that seems logical in the situation. I promise the answer is quite deductive, involves binary numbers and is not going to make anyone groan in frustration.

Here are the two logic problems my kids brought to class.
A man was found dead in a room with a rock. How did he die?
You meet two men on a fork in the road. One always tells the truth and the other always lies. One road leads to death. What one question can you ask them to figure out where to go?

The answer to the first is that the man is superman and the rock is kryptonite. The answer to the second is something you’d be able to figure out without asking questions or introducing additional stories or circumstances.

The take away question for me is: What are my kids learning from all of this? They do know how to count to 31 in binary on their hand now and how that relates to finding poison and computers. Is knowing this better or worse than knowing a theorem about circles and chords? None of it is on the SAT directly so does it matter that I pick what seems to make them happiest? Joy=investment= learning?

I read an interesting article today that sums up so many of my feelings about mathematics. It was written by a mathematician who quit teaching at the university to teach “real” math to students. Math that involves conjectures, ideas thinking, justification, proof, beauty.
“Why aren’t we giving our students a chance to even hear about these things, let alone giving them an opportunity to actually do some mathematics, and to come up with their own ideas, opinions, and reactions? What other subject is routinely taught without any mention of its history, philosophy, thematic development, aesthetic criteria, and current status? What other subject shuns its primary sources— beautiful works of art by some of the most creative minds in history— in favor of third-rate textbook bastardizations?

The main problem with school mathematics is that there are no problems. Oh, I know what passes for problems in math classes, these insipid “exercises.” “Here is a type of problem. Hereis how to solve it. Yes it will be on the test. Do exercises 1-35 odd for homework.” What a sad way to learn mathematics: to be a trained chimpanzee.

But a problem, a genuine honest-to-goodness natural human question— that’s another thing. How long is the diagonal of a cube? Do prime numbers keep going on forever? Is infinity a number? How many ways can I symmetrically tile a surface? The history of mathematics is the history of mankind’s engagement with questions like these, not the mindless regurgitation of formulas and algorithms (together with contrived exercises designed to make use of them). A good problem is something you don’t know how to solve. That’s what makes it a good
puzzle, and a good opportunity. A good problem does not just sit there in isolation, but serves as a springboard to other interesting questions.”

I’m not sure how I feel about the part where he says schools of education are complete crocks but I do like his belief that mathematics is beautiful and interesting and can arise naturally from the world. I’m glad that I’m going to enter a school of education and spread my profound appreciation of mathematical beauty to the world. His opinions about geometric proofs made me feel better about the time I spend doing logic problems instead of proofs. I do need to come up with some more problems that are actually related to Geometry however. And at least take a peek at an SAT because those problems tend to be decent. (Or at least more likely to assess real understanding than what I find in most textbooks.)
He has a vision for math classes rooms where kids solve problems and learn skills so that they can solve more problems. He worries about the lack of mathematical ability of most teachers. But I suppose I’ll be able to fix all this. Or at least in a few situations. And I have a new problem for tomorrow.
Check out the complete article at this website.

As I was riding home from school today the middle school principal at my school commented that I always have such great ideas at faculty meetings. I told her “I love coming up with ideas, and I’d be happy to share more.” We had an interesting discussion about formative assessment and teachers hanging on to traditions.
Our school is involved in the process of responding to national trends surrounding formative assessment, tracking based on standards and backwards planning. Although the private school isn’t tied to any particular system when it comes to how we grade and what we teach we are influenced by national trends in education. Of course we have the flexibility to adopt only those trends that we are interested in and hopefully escape the perpetual cycle of new panceas that new administrators introduce to old teachers who already feel like they’ve heard it all before. There are definitely veteran teachers at my school who don’t enjoy all of the renovations in professional development and compensation models but everyone is willing to think critically and discuss pedagogy enthusiastically at staff meetings.

The most recent meeting revolved around grading practices. The principal wants to follow a national trend which suggests that giving students grades for completion of tasks encourages them to learn how to play school instead of learning content. He wants to adopt the goal of having the primary determining factor in a grade be a summative assessment of content knowledge specific to the discipline. This does not just mean tests and actually necessitates that teachers think carefully about alternative assessments so that kids who do poorly on paper and pencil tests still have a shot of demonstrating mastery of objectives. Greater emphasis on assessments in grades will hopefully force teachers to improve assessments to more fully capture student mastery of topics. Additionally students must receive formative feedback before taking an examination so that they know what is expected of them and have a chance to improve before their work counts for a grade.

A whole host of arguments are raised by teachers who are rightfully cautious of revising their grading methods. Some wonder if grades will go down when students no longer have the additional pad of getting 100% on assignments for completion. They will not longer be able to work hard on homework and have that balance out weak test scores. It certainly gives an advantage to students who master concepts quickly and disadvantages students who struggle with traditional assessments. It’s doubtful that the school as a whole will think of a way to make our assessment favor all learning types equally. It is so much easier to test verbal linguistic and reasoning mathematical intelligences that I doubt we will ever be able to appropriately assess kinesthetic, musical, social, or other types of skills on a balanced percentage of assessments. Some teachers worry that students will be discouraged from working hard. I think that we just need to be creative with our grades. If we want to reward hard work then there are ways to incorporate that into the grading. Presumably we value hard work because it helps the students achieve. We might realize that some kids will never be quite as mathematically fluent or as gifted as writers no matter how hard they try but I think we all believe that there will be progress. We hope that they will learn that intelligence is malleable and that by working hard they can improve so that they will transfer that skill set into other areas of their lives. I don’t think the teachers would argue that hard work for the sake of hard work is really the goal of the class. No one thinks I should give extra credit to the student who comes and digs holes in my yard and fills them back in just because that took a lot of hard work.
So perhaps in the new system where we no longer have the homework padding for hard working students who struggle with demonstrating mastery we can pad them by giving them additional opportunities to show what they have learned. Small amounts of oral tests, retakes, alternative assessments and more could be used to give kids a second chance on a summative assessment. I think if there is student buy in(and at my school there usually is) kids will still want to do well on tests the first time because they won’t want to come in for a second round of testing. I think it makes sense if the ultimate goal is to learn and teach that hard work on specific weaknesses pays off that it’s not a bad idea to give second chance assessments. This definitely allows hard work to impact a grade in the way it used to with homework but hopefully it is more focused than simply giving points for completion. It will also force kids to think carefully about what they are learning and why and how they will correct errors on future tests.

There is also debate about what are appropriate learning goals. Many teachers have ideas about instilling hard work, respect, organization, and other non content related skills into their instruction. Organization is important and so is respect and the debate is whether or not that should be tied to a grade. One teacher surprised me by saying that kids are respectful because they have 6 or 7 classes where they receive grades partially based on respect. THey thought that if we take the respect grade away kids will know we value it less. This shocked me because I’ve never considered giving a grade for respect. Sure, my first year teaching the disrespectful kids got sent to the dean, missed class and had lower grades as a result but this wasn’t about a direct respect grade. I do have to admit I give bonus points on tests for kids who make me laugh while grading. But the effect on the final grade is so small that it’s basically a token thank you for improving the grading process. I’ve definitely become much less strict about when homework is handed in for absent students. I’ve decided that I’ll trust them if they tell me they couldn’t do it for a particular reason and if they are really actually slacking on it I’ll find out on the test.

We spent some time discussing being tied to tradition and what a hard time teachers have of changing their ways. I’m definitely guilty of that myself. I’m completely excited about constructivist pedagogy and am not very willing to make my class revolve around direct instruction. I know some of my kids yearn for direct instruction because they like the comfort of notes, steps, procedures and formulas but I’m just not so sure where those take them.
I felt really respected by the administrator and really enjoyed discussing the changes in grading at the school and possibilities for implementation. I kind of wish she’d been my administrator because(and I know this sounds like bragging) I think I had a lot of ideas and energy that I could have contributed to the school if my strengths had been better assessed by my administration. I felt like we got so bogged down in the details of my relational skills that we overlooked all of the ideas about curriculum and teaching and organization I could have contributed to the school. Or maybe I’m just on a bit of a high from getting accepted to some absolutely fantastic grad schools and getting positive reinforcement from some brilliant professors.

There was definitely a clearly thought out pedagogical reason for having all of my students walking around the classroom marked with post its indicating whether or not they would “die” of an infectious disease. It was either that or geometric proofs with algebraic properties for an entire 65 minutes. I know it’s traditional but I have my suspicions that two column proofs were a bastardized way to present proof to high schoolers when they decided to move the conceptually difficult topic from the college curriculum. And appeals to tradition bother me more than any other logical fallacy.

I also know that being able to prove cool things in math requires a lot of time spent on the details. I once endeavored to prove that odd + odd = even and other seemingly trivial facts. At least the proofs of the Mean Value Theorem(If you go from below sea level to above sea level on a walking trip you must have been at sea level at some point) was pretty cool from what I remember of it when I learned it the final year of college in Advanced Calculus.

I’m trying really hard to figure out why we do two column proof, how I feel about talking about axiomatic systems explicitly and just how wrong a proof is if the kids forget the angle addition postulate.
This brings us to the point of proof. To convince others? To practice logical argument? Why do we learn proof in high school? I love it. I know it’s important but which aspects of it are important and why? What kind of nonsense can I gloss over and what is really important. Mathematics is too beautiful of a discipline to only show them the tedium of really rigorous arguments. To what extent must they experience this tedium to be able to prove the awesome stuff I did later? Or perhaps they don’t see it as tedium. I find myself a bit flummoxed when trying to explain why we need the angle addition postulate. (If you have two angles that share a ray the sum of the angles equals the big angle formed when you put the two together.) If we are teaching them to argue this point is so boring as to be unnecessary. And I’d guess a professional mathematician doesn’t put steps like this in. Of course he could if he wanted to and maybe that is the rub with this argument. Perhaps I do need to have my kids focusing on these details. I know that they will never encounter a question like this on a standardized test and that while logic will be important in all fields, the use of the annoying postulates that seem super obvious will only be helpful if they take a class on proof in college. (Of course I’m sure the all will since through the wonder of Wikipedia I’ve shown them how much cool math is out there.)

What does this have to do with monks? A boy I had a huge crush on once sent me a text message with the following problem:

“There is a colony of monks who can’t communicate in any way and who cannot look in any sort of reflective surface. No exceptions. Someone comes to their monastery one day and says to them at the meeting that one or more of them are infected with a disease. The only symptom they will ever have is a red dot on their forehead that they cannot see or feel or touch. He tells them that if they have the disease they must commit suicide so that they do not infect the rest of the monks. The infection will spread in a month or so but for the moment the number of monks with the disease is fixed.

The next day all of the monks return to morning meeting and no one has committed suicide. The stranger repeats the message and the monks listen and look around.

The next day it happens again and still no one has committed suicide.

On the morning of the fourth day the monks with the disease have killed themselves and everyone who was well is at the meeting.

Given that every monk follows perfect deductive logic all the time, how many monks are in the monastery?”

There is a perfectly satisfying and logical answer to this question(I promise the answer is not, they used sign language, or communicated telepathically, or used God to help them or anything silly of that sort.)

By getting the kids to put their heads down and marking some of them with a disease I was able to get some of the kids to come to the answer. They all wandered around looking at everyone’s post it and got excited about conjecturing and trying to figure out how to decide. Everyone was really good at not revealing who was going to die as well which was surprising given how fun it might be for high school students to laugh about their friends being the marked monks. Even once those kids had explained and demonstrated with the post it note infection notices some of the kids were still extremely confused about the answer. It was interesting to see that the ones who were capable of geometric proof were similar to the ones capable of deductive logic in this problem as well.
The of course was evidence for my claim of this being a meaningful mathematical task for a logic and proof class even if I did ignore some stuff on Geometric proof while I was at it.

After we finished I said “Do you guys want to do some easier proofs now?” and most of the class happily worked on seemingly “obvious” geometric proofs. A few kids said “Can we please just keep working on harder problems instead?” So adorable. At the end of class kids talked about how much they liked wandering around class and being active problem solvers. I told them that if they helped me lesson plan we could do more things like that but that it was hard to find problems that interesting on a daily basis. Hearing them this happy was sure good motivation to try. And thank god I don’t have a standardized test I’m trying to get them to pass. And thank god no one is really going to know just how many times they practiced two column proofs. And maybe this will even help them.

In other news: My study hall turned into a dance party. A Rocky Horror Picture Show inspired dance party. And a lot of cartwheels happened. I got some tests graded then wondered if I could transfer infections though coughing on the tests when I laughed really hard at my super smart computer nerd kid demonstrating the “pelvic thrust.” So I had him look up the length of time bacteria stays on paper. Then sprayed the test with disinfectant. I promise I’m still taking my job seriously. Because my kids are legitimately great at music and dance and involved in school theater I’m trying to get them to write and perform a musical number detailing just what does happen in study hall at my school. Good thing I’m leaving for graduate school and won’t have to answer to the consequences of days like today!

Today in Geometry Class I started it off with a quick round of “do you want to see what we did in Calculus today?” Before I erase the board I try to point out to them any instance where we are using algebra or geometric formulas to solve something. Today we used the area of triangle formulas to find integrals of linear functions.

After that we looked at a pattern in blocks that I learned at UNLV in a math methods course that I’d used in Calculus. The pattern starts with one block, then 1 +3 in a pyramid shape. You keep adding rows so you get 1 + 3 + 5, then1 + 3 + 5 +7. If you actually draw this out you’ll see that you can always rearange the blocks into a perfect square. To find the sum of the first n odd numbers you just square n! One of my students who tends to struggle with algebra was the first one to figure this out. I’m always surprised what happens when I put non-traditional problems up on the board. She had a smug and satisfied look on her face when she solved the Calculus problem.

(Nerdy side note:The reason we were doing it is Calculus is that you can relate this series to left Riemmann sums to approximate the area under the curve 2x+1. Then we see that the area has to do with x squared and the anti-derivative.)

One kid exclaimed, “we are doing real math now” because he liked the problem so much. I said, well, I try not to teach fake math too often. Then of course they wanted to know what fake math was and I told them fake math was “memorizing formulas you don’t understand because your teacher told you to.” Someone said “every single test I ever take in history is fake history” and I was pleased that they saw the connection between memorizing random historical facts and memorizing random formulas. Of course, some students thrive on memorization because it is lower level thinking that they know they can do but I’m trying to show them just how much fun math can be.

Next a student asked “do you ever have Calculus students do Geometry problems?” I thought for a moment and pulled out “The World’s Hardest Easy Geometry Problem” The kids were able to understand the problem immediately. It was a triangle with some angles listed and an x in one part of the triangle. It seems like it is going to be possible because they’ve been able to solve every other problem like this. This problem is incredibly hard however to the point that unless I sat down with it for at least 2 or 3 hours I don’t think I’d even approach a solution. I know one student solved it last semester but it took him and his family all weekend. The solution requires drawing a bunch of additional lines and use of SAS congruence postulate. The students were hooked. They barely wanted to correct answers on their homework. I decided not to even try to do what I’d planned for the day (easier geometric proofs) because I figured they were excited about math and practicing the same type of reasoning that was on the homework anyways. One student who has forgotten most of Geometry A and never seemed interested in class being hard asked “Can we do problems like this for homework?” Another said “I’d rather solve one really hard problem than a bunch of easy ones.” Problems are not the solution to all of problems in math education but kids respond to them. Even if they are as pointless as finding x in a triangle that has no real world connections. It seems that the quality of the thinking required to solve the problem matters just as much as the potential utility of whatever is being practiced. I’m excited to get into the book Thinking Mathematically at the conclusion of our unit on geometric proof because it has lots more problems like this.
At the end of the class one student asked “what types of proofs will be on the test?” and I literally had to say, flip over the triangle problem because I want you to listen to the answer to this question. I’ve never had to ask kids to flip over a math problem before! Then a few kids were late to the next class because they said “we are so close…” and agreed that as a group they would continue to work on the problem in the next class.

I thought a lot about lesson design today. Our school enrolled in Key Curriculum Press’s service which provides pre-made Geometer’s Sketchpad (GSP) lessons with worksheets. The one we did today allowed the students to change the limits of integration graphically by changing the amount of area under a curve that is calculated. As they drag a button to increase the area under a curve the area is plotted as a function of the upper limit of integration. it’s easy to see that the area under a constant line is linear and the area under a linear function is quadratic. I’d say 30 % of kids saw the connection to derivatives when they saw the area graphed in the same place. it was pretty exciting. I told them that they were as smart as Newton. Which was probably a lie. One who figured out the relationship particularly quickly said it was because they were studying area under the curve in physics. Another pointed out that the timing was perfect because while we learned methods to find area under the curve in Calculus they were finding physical interpretations in physics class.

I did get annoyed a bit at the worksheet which seemed to me resulted in kids getting bogged down in unimportant details. The problem with technology is that they end up figuring out relationships that don’t end up being something that gets study in first year Calculus. It’s really hard to design a program and a list of questions that go with it that focus kids on the key points and not the fun of moving random graphs around. I definitely decided that there was not enough white space of the GSP worksheet and that all of the text made it too many things to really concentrate on in the time given. So I told my kids to use the worksheet as a guide, to skip problems if they felt like it and to focus on finding a formula for the function of the area plot.
We summed up the day with me explaining why it was so crazy that the area under the curve followed formulas we’d developed for slope. Like, change the world amazing. Best thing that happened in the 17th century amazing! As a result of visiting Berkeley this weekend I decided to start asking kids more often what was going on in their heads. I asked them if the relationship between area and slope seemed random or impressive. One said that it seemed obvious. I took them back to the limit definition of the derivative and the integral and asked again if it seemed obvious those things were connected and maybe so they could go to lunch they agreed that it was special.
I remember thinking the Fundamental theorem of Calculus was anti-climatic because it was obvious that integration was the opposite of differentiation because that is how our teacher explained it when we first saw it. I hope my kids see it differently. Tomorrow we need to take all of our conjectures and use some summation notation to look at how we write all these things rigorously. My kids have been involved in a few days of semi-guided exploration and it will be interesting to see how that might change their ability to interact with formal notation. I hope that they actually get the point of it instead of seeing it as a bunch of crazy symbols. I hope by having them problem solve about area using their own reasoning they see that these were conclusions arrived at logically.
One kid asked about the proof of the Fundamental Theorem of Calculus and I had to admit I’d forgotten it. I better look that up for next year!
So Research Questions for the day:
How does having kids figure out their own methods of finding area affect their conceptual understanding of integration?
How does teaching anti-differentiation as a guess and check process affect self-efficacy when faced with an integral problem?
What is the optimal design for GSP activities surrounding the derivative?
How can the design of applets help focus kids on key points?

Today we had the second in a round of community conversations designed to help our school have discussions about race and privilege. In the first rounds students led conversations about identity that centered around family traditions, food, culture and holidays. Today we split into affinity groups based on race and discussed the benefits and challenges of looking a certain way.

I was in the European American faculty group which was so large it had to be divided into two rooms. The other groups for faculty were Asian, African American and interracial and those were much smaller. We started the conversation with a discussion about our heritage. I said “I’m Irish, but I don’t think it affects who I am. If I was German I don’t think that my life would be any different.” Most teachers felt similarly to me and the discussion felt a little pointless. Someone asked if anyone identified with their ancestry and a few mentioned having certain holidays or food or traditions from their European country of origin. The point of the conversation was to think of one myth about being white and it was proposed that the myth was that we should all be lumped together because of our skin color and that we came from lots of countries. People seemed to have trouble with the fact that we were put into the white box. I pushed back. Do you really think about your ancestor’s country all that much? Would you life be very different if you didn’t eat certain food on holidays? Do you think about your race every day when you interact with the world? They all admitted that we didn’t and we decided that a benefit and challenge of being white was being able to not think about our skin color.

We had the inevitable discussion about our student’s lack of awareness of their privilege and our professional responsibility to do something about it. We talked about a school exchange where students trade schools for a day with a struggling school and one teacher says it’s like “yeah, field trip! We are going slumming.” Another equated it to a field trip to the zoo. We also wondered how it would make the kids feel from the disadvantaged school. What are we supposed to do with our privilege? It’s so easy to get lost in academia. They read Kozol’s Savage Inequalities in class and then forget about it as soon as they are done with their essay. Life is stressful and full and it’s hard to be bombarded with inequity. It’s hard to be white and even talk about race without worrying that you’ll say something incredibly offensive. I remember my first year teaching when a group of students told the administration I was racist because I gave them some statistics about the achievement gap. I was also called into the principal’s office for the comment because he wanted to make sure that I didn’t think that I was at “one of those schools” you hear about in articles about the achievement gap. I wanted to ask “why were you so desperate as to hire a 22 year old with no experience then?”

After the conversation we went back to class and I asked the students what they had talked about. The big theme was that sometimes the school prevents potentially offensive messages in the school plays because they offend a few people. They thought that it was better to offend and have conversations. it’s a tough call as an administrator. I made the point that reading Huck Finn is not condoning the use of the n-word or the racism it describes. I told my students about TFA. About crying on the floor after class. About how I used to fail 40 % of my students and now I fail none and my grading is much harder now.
We talked about how 100% of my school now goes to college and only 50% of my old school graduated. I told them how hard it was. Some admitted to having no idea what other schools were like. One offered that the best we could do was try to make a difference and to give back. Couldn’t we be doing more of that now? We spend so much time on sports(which are great), school activities and fill our days too full to actually interact with less privileged students in the community. And I’m not sure what exactly to do about it. At some point hanging out in our privileged white world becomes immoral but also at some point spending our whole life in misery becomes counter productive. They asked me if I had a culture shock when I arrived at my new school. I told them I was in shock that my printer worked and couldn’t believe kids would complain about the food. 20 minutes later we were working on Calculus. Lost in integration and far away from thinking about what is happening at other schools.

I’m teaching Geometric Proof. And it’s hard. Some kids understand what I’m talking about when I mention making a list of reasons that prove something. They totally understand when I say, don’t use what you are trying to prove as one of your reasons. Don’t jump to conclusions about something that would be true if the conclusion was true even though you know the conclusion is true and that that fact will also be true. Today we were proving that if you draw an altitude in an isosceles triangle it divides it into two congruent triangles. If you can imagine that you’ll see that the altitude is the leg of two right triangles and that to use the Hypotenuse Leg Theorem you need to technically say that the two triangles have congruent legs because they overlap. Technically, this is called the reflexive property. rnMy students want to know why they need to include this step because it is so obvious. I know that “real mathematicians” wouldn’t always include this step because it is too obvious. Calculus books wouldn’t include that step if they were trying to write a proof of a theorem involving shapes. It’s obvious. It’s boring. I understand that it’s there but how in the heck do I explain why it’s important to the students. I know that details of proofs should only be skipped when you understand them but when they barely understand the structure of a proof, crap like the reflexive property is so much harder than actually thinking logically through the problem. rnrnIn another proof that vertical angles are congruent first you notice that the angles form two linear pairs, then you notice that linear pairs are supplementary and then you notice that supplementary angles by definition are 180 degrees. A bit of substitution and subtraction property of equality later you’ve got the theorem. Vertical angles are congruent. YUCK! This is not proof. This is some sick bastardization of proof. Must proofs be dry and boring and obvious to be easy enough for a high school student? Of course the answer is no, but must we continue doing these banal geometric proofs with the stupid reflexive property. Does it get us at increased clarity of thinking? Where does pickyness about reflexive properties get us. rnOne of my students who has a visual impairment has a geometry tutor and returned to class with a precisely written proof, reflexive property and all. I was a little taken aback. How did you already know how to do this I asked? We haven’t even learned this formal structure yet. Her tutor had helped her was the obvious answer though I didn’t want to admit to her that I entirely doubted that she could produce such work in a weekend. She said “My tutor told me to ask you if we need all these details. She said some teachers require them and some don’t.” Later in class a student, kind of annoyed by the reflexive property asked me if I was going to require such detail on obvious facts. rn”It’s up to me. Real mathematicians don’t include details like this in proofs because they are obvious. They know they are there and could include them but don’t. The number of details that is included depends on the audience.” She wasn’t satisfied with that. “So will you require them on tests in this class?” And once again I’m faced with this question of details. I know that how I respond to this general line of questioning is going to influence how people think about proof. If I say we must always include the reflexive property kids are going to kind of resent the redundancy. I don’t like writing things like that either. If I say that we don’t need it I do lose some mathematical rigor. So I tried to go back to first principals. What am i really trying to do. Logical thinking. Deductive reasoning. Writing clear arguments. I hedged the question “If you know it’s supposed to be there, please put it but I’m not going to attach huge points to it.” While checking proofs that day I told kids who’d put “same line” instead of reflexive property that their thinking was logical and good. I think I’m grading these proofs on logic more than conventional details but I’m struggling to assess what really matters when they ask. rnrnFrankly, I want to just get to thinking mathematically and ditch all these boring Geometric proofs as soon as I can. Boring is such the wrong word too. There are some beautiful and wonderful geometric proofs that seem way to hard for my kids at the moment but which might be great later. I have no problem with geometry as a field, but I don’t want to be marking down points for did you include “definition of supplementary” as the reason for “angles add to 180″ as opposed to skipping straight to the needed result. Though, honestly, if I look at my math career I was perfectly content to prove that two evens added are always even and that there are infinitely many primes in 10 different ways. The actual utility of the item proved does not always matter to me. I don’t always care if I’m already convinced something is true with inductive reasoning. There is something satisfying about understanding infinite decent in Fermat’s Last Theorem or why the harmonic series diverges and the sum of the square number is connected to pi. rnrnAfter being a little on the spot confused about what I really wanted from the kids in terms of proof I said “Do you want to do a proof that isn’t obvious and boring?” I wonder if these are actually more efficient at helping them understand the point of proof because they are not struggling to identify what they know for sure and what they also know because they already know the answer and everything is so obvious but can’t technically use in the proof. rnrnAlso. Two Columns. I read somewhere that these were invented as a scaffold for high schoolers who were not developmentally ready for deductive logic. How do I feel about them? In some ways I like the idea of a scaffold. There must be some way to help my kids who have a blank page staring at them get something accomplished with the proof. One of my students(one who also said she loves notes and structure) said to me today that she doesn’t understand where proofs come from. All of these ideas come out of no where and I don’t know which ones to pick. I started talking about music and how I have no idea how people write songs. How do they decide which note comes next? How can they pick from the millions of combinations. There is no formula for that. And I admit that I feel defeated when sitting with a guitar and asked to write a song. I can however make some random guesses, see how they sound and honestly as much as I might resist I probably could learn some general rules of song writing and be able to write a passable song. One that my mom might enjoy at least. rnWe returned to the triangle and altitude problem. She started with the given. I said “well, they use the word altitude, so even if I’m not sure it’s going to be helpful I look up the definition and write that down.” Then I notice that there are triangles and we are trying to prove they are congruent so I read through all the congruence triangle theorems until I find one that matches. I wonder if this is actually what I do and if my thinking applies. I know when faced with a ridiculous proof on a take home final I read through the book until I found something similar and started with that. It felt almost like cheating because I realized that I was throwing excessive time at the proof as opposed to excessive cleverness. What do I actually think about in Geometric proofs? I can see all the steps from beginning to end on these so reflecting on the minute snapshots of my thinking and laying them out for others to see is hard. And I’m not even sure if my thinking is the model.rnI have not done many proofs on the board for them because I don’t want them to think that proofs are something to copy or to memorize. I wonder if that is a mistake and if more direct instruction would help. I learned proof in college with lots of direct instruction. rnrnTomorrow I think I might try the “even + even = even proof” or perhaps the harmonic series diverging. Of course divergence is a really hard topic because kids think it must diverge because you are adding up infinitely many terms. At least these proofs seem more interesting. My co-teacher was going to look for accessible proof for the kids but I might need to do some looking myself to struggle to find something interesting and possible. rnrnOne final note on proof before I wrap up this longish post. After we went over a proof in class I asked the kids “honestly, and I promise that I don’t care what you say, what is your opinion of these proofs?” I’d already expressed a bit of negativity about the reflexive property and was curious what they thought(though of course they’d been influenced.) One said “I can see how this will help us eventually in life.” Another pointed out that it seems like it would make more sense to prove things that we not obvious. I’m sure someone mentioned the picky little detail steps being annoying. And I know some are still wondering where all of this comes from. I now understand why some people say “some kids get proof and some don’t.” With equally small amounts of instruction half the class comes back with proofs and the other half comes back confused with white spaces on their paper. I’m already planning differentiation.

What do explosions, Calculus, cookies, Tetris, raps songs, ballet, and 16 excited teenagers have in common? My classroom. All in one day. I came home thinking that it was the most exciting day of teaching yet. Mainly because of the dry ice and water explosion. As far as I can remember no one has even worn safty goggles in my classroom before (though years ago my dad did offer to send down a box of goggles with a paintball gun to help me control my freshmen boys.) Wow. I’ve come a long ways from that.

Now I’m left with the curriculum design question: How do I make every day project day? How do I let them see how much math has to do with the world while still covering important procedures? The students projects were filled with negative exponents, tricky constants, ugly fractions, long algebraic manipulation, real world interpretations, unit analysis and more. It was all of the things you teach them in little chunks coming together into one awesome problem. But it took a week of class time. One obvious solution is to have them write up problems and not make posters and presentations and cookies to go with them. Less fun but certainly more efficient. Part of what drove the creation of these tough problems was the drive to find out how dancing or their cars worked.
One of the more epic events was the rap written about Related Rates. I’ll need to post it here because it really accurately described what the problem was asking. What happens to the forward motion of a car when a wheel is inflated at a certain rate while the axel is rotating at a fixed rate. The theory is that the radius of the “torus” will expand and then the car will go faster. The end of it concluded with a line about using Calculus to break the speed limit “with the help of Ms. B” It was adorable.

A group of girls brought in a lazy susan type device borrowed from the physics teacher and demonstrated the effects of moving your hands closer to your body while spinning. The physical demonstration was incredibly helpful and great fun to watch. E*** and R*** are two adorable, sweet girls who are not always extremely confidant in their mathematical abilities. R**** seemed disappointed with earning a B on a tough final and E*** writes about needing lots of extra help. Their project required them to read ahead in the physics book and use equations that I had never even studied. After learning something new and challenging they did an incredible job illustrating the situation by drawing dancers and labeling the appropriate parts with dv/dt and dr/dt to indicate the spin of the dancer and changes in the radius of her arms. The entire poster was so clear and interesting that I’m sure they could make a good career designing instructional materials. Of course, I’m probably the only nerd in the room who is trying to do that! I hope(and based on talking to them I think it worked) that they took away just how hard of a problem they could solve with their own abilities. I think that they were quite impressed with themselves. Of course two days later I was trying to get students to figure out the area under curves using Reimman Sum approximations and R*** seemed to think that she couldn’t possibly make sense of a table of values from a speedometer. With a hint I got her going again but she still seems resistant to the idea of figuring out math without my prior example.

A**** and D***** presented their results about relating the surface area of a ball to it’s volume while inflating it. While the equations got rather hairy in the middle of the problem most everything canceled out and they were left with a simple formula relating the rate of change of surface area to volume. They were pleased at the elegance and seemed so excited to have actually conceived of a problem, done a bunch of messy algebra and had the result come out neatly. They already are on the path to the satisfaction of mathematics. Of course it helps that A***’s dad is a professor of mathematics and a prestigious university. He teaches his son Calculus in Polish. I know A**** thinks he’s smarter than me and I know that I should care but I definitely like to be the smart one in the room. I’m going to have to give that up if I get a MA in math from Berkeley! The best part of their project(aside from the elegant result) was that they wrote the entire problem on a basketball. It immediately captures everyone’s attention and shows them that the math problem is connected to the real world. In fact it’s connected to something you can pick up, throw around and inflate at a constant rate. You can touch, see and comprehend what is going on and then the awestruck middle schoolers who were playing with it today at least have the sense that some very complicated looking math is describing something quite simple.

While visiting Berkeley I sat in on Alan Schoenfeld’s research group “Functions” and listed to a presentation of a pre-service teacher who was trying to understand why students didn’t like word problems.

Despite her inexperience with teacher she thought up the great warm up to linear equations and was trying to form research questions around the data. She gave students the linear equation 3x+8=17 and asked them to write a story problem related to that equation. The results seemed rather like a mad-libs game and revealed a few important things.
If Sally buys a greeting card for 8 dollars and buys x pens for 3 dollars each and pays 17 dollars in total how many pens did she buy. Alan pointed out that 8 dollars was an absurd amount for a greeting card and that kids were playing the game of story problems. Their books are filled with nonsensical word problems where reason about the real world is abandoned. The game is to translate between a certain class of word problems and they symbols that they represent. If students use real world thinking and point out that greeting cards are not actually 8 dollars and you might want to check the price tag they are asked to suspend their reason about the situation. I wondered if the new teacher had told the students not to make the problem like an example and hadn’t given them numbers if the results would have been any different. I’m not sure if the success I saw with related rates would extend to Algebra or not. I certainly saw my kids coming up with “sensible answers to sensible questions” in that class and not trying to mad lib some new words and situations into the same old ladder falling or sphere expanding problems. Of course sometimes the real world situation got so complicated that simplification to the point of absurdity was required.
The group trying to model the rate of licking of a tootsie pop decide to make it a log function thinking that people would slow down licking as they got sick of it. Of course no one licks food like a log function and only those truly committed to finding how many licks it takes to get to the center of a tootsie pop(google says 419) are going to have the willpower not to bite it towards the end.
The cookies we backed were ellipsoids and the ramps we analyzed were frictionless. Real world problems either seem impossible simple(Three friends are spitting a 20 dollar bill and want to tip 18 percent. How much does each owe? Give the combination of ones and fives they have how can they all pay evenly?) or way too hard. When you take air pressure, friction, air resistance, the vagaries of human behavior and everything else into account no one, not even me knew how to solve the problems my kids were coming up with.

I’m not convinced that problems must be related to the real world to be good. The truncated chessboard problem is one of my favorites and intrinsically interesting even though no one really cares about covering chessboards. (Actually, maybe it has to do with city planning or some other design, who knows. Need to go to school to give me the time to find out!) Real world sure seems to motivate though.

Finally teaching makes sense in my life. Teaching is so much fun now that I know I only have a few more months of it left. (Of course, I’m sure I’ll be involved in teaching, tutoring, etc) in some regards forever.

I wake up in the morning and walk into my school bubbly happy because I got into Berkeley and am already having amazing conversations about education.

I’m just warming up for this post, lots and lots has been happening in life.

Big concepts…. emails with my possible future PhD advisor who is a brilliant and incredibly entergetic math ed researcher. I wrote to him about my ideas about designing an interactive online math “book” that allowed kids to interact in ways that best suited their needs. He wrote to me that “perhaps you’ll be surprised to find out(or perhaps not) that researchers are investigating a lot of your ideas). Sentences like this make me feel like I’ve finally found my feild. Of course, questions in math education are not as complicated to pose as questions in math. What type of textbook helps kids learn the most? Quiet rows or lively discussions about everything from math to what’s being served for lunch? What type of teacher training? What order should we put things in? How should we introduce integrals? What things can teachers do to be effective? Am I effective!?

Even though the questions are easy to pose, I’m not sure that everyone thinks like this. As I read papers on math education I can envision being able to write them. It’s a good feeling to think that a PhD is possible in this subject because you have to have a lot of good ideas to write a great dissertation. And I’m just so excited about it.

I’m reading about transfer of learning for Berkeley and thinking about how I set up lessons in the class. This week I’ve told my students that integrals were useful in a wide variety of subjects and showed them webpages from physics, statistics and photography that included integral signs before teaching them what the integral sign meant. One student said that he felt like he was learning one of the fancy symbols you see on blackboards in movies about genius children. And they are, it’s true. But it’s also just area.

The things that I’m saying to students are changing subtly as well. We are finding area under curves using a variety of methods and I have not really spent any time on direct instruction of Reimman sums. I gave students a problem asking them to find the total distance traveled by a car if you measure the velocity at a number of intervals. The next two questions involved finding the distance again but from a graph and from a function of velocity.

Kids were able to come up with all sorts of ways to solve the problem. They did left and right approximations, midpoint approximations, trapeziod approximations, counted squares, estimated the error and averaged various methods. When they were trying to find area under a linear function they used the area of triangle formlas and when the car was driving backwards they were able to interpret the area under the x-axis as a negative distance.
They could relate left and right hand sums to the graph.
And when I say, I mean that at least a few people in each class were able to figure these things out. I tried to select speakers based on their ideas and have them present on the board to get a picture of all of the different methods we’d come up with. I found myself having a hard time not stepping in while they explained because they struggled to explain concepts they’d just come up with. In both classes kids came up with the idea of making smaller rectangles to find a better approximation and connected that to limits. One said “I don’t like where this is going. Can we all drink a big glass of water before seeing where this leads?”

I purposefully didn’t introduce notation about Reimman Sums the first two days on integrals because I didn’t want them to lose their own agency. I want them to understand that they can make sense out of this without the formulas that describe it and without subscripts and sigmas. I am hoping that the conceptual understanding of what is happening will motivate and help them to learn the notation when we get down to the details next week.
I found myself saying “If that is logical, then it is correct” to a lot of students who’d come up with a variety of other ways to find area. I justified my focus on left and right hand sums because I said that they would be easier to write notation for later.

In Related Rates news, two students are trying to use the ideal gas law to relate the change in temperature, number of moles of gas, and pressure together. The Calculus part of the problem is easy in the sense that the formula is not hard to differentiate. Deciding which parts of PV=nRT are constants and variables is really tricky and required the help of the Chemisty teacher. She is also buying dry ice for class tomorrow so that they can conduct their experiment.
It’s amazing how difficult it was for me to use Calculus to solve a problem that wasn’t from a book. When there was no obvious way to decide which peices were constant and I had to look to the real world to decide the problems became much more difficult. It made me realize that I have a lot of science to learn about if I ever want to be really effective in creating problems with real context and real applications.
A lot of my students are not sure if they have a correct answer because the problems they created ended up so complicated and hard to verify based on the knowledge we have. I reassured them that I was impressed and asked them to reflect on their process. the group who collected data on the rate of change of a lunch line realized that doing a regression on their data was not going to work and is going to write about how the problem they invented is way too hard to solve using the tools they know. They ended up inventing a simpler formula to model their situation.
I didn’t expect the result of this lesson to be that students realized how difficult it was to actually solve real world problems. As a teacher, I still feel like I don’t really have a concept of what real scientists actually do with data and formulas. There are so many additional variables and considerations when a problem is not from a book. I find that my solution is to keep making things fit into a format that makes sense to me. An equation. An oversimplification. A cookie is a hemisphere. A line of people is modeled by a log graph.

In Geometry I’m struggling a bit with my introduction of proof. We are reviewing words from Geometry A and learning how to set up proofs. I don’t love class because it feels really teacher driven but I don’t want to take forever learning these words through exploration again. I honestly have no clear idea about how I’m going to teach proof. Some kids get what I’m saying about not using what we are trying to prove as a reason and some don’t. For the kids who don’t understand the basic logic I try to make examples that are simpler about cat’s and dogs and how many legs they have(if 4 legs then cat is false) but I’m not sure if that is effective. After getting the class to help me create a proof on the board I had them discuss all of the steps. I’m not sure if having a class help me create a proof is better or worse than telling it to them. Things seem more linear and clear when I’m the sole arbiter of what goes on the board. However, I don’t want kids to get the impression that they can’t actually figure these things out on their own. When they lose that sense of agency, I feel like I’m not accomplishing my biggest goals.
I keep wondering after I say something like “you can’t use the conclusion as part of your proof” if kids really understand. Often the statements they use are things that they know are true because a teacher told them. For example “If a shape is a parallelogram, then opposite sides are congruent.” They know this from Geometry A and struggle to decide which peices they can use and not use. And I havn’t even tried to touch on axiomatic systems(largely because I’ve read about the levels of Geometric thought and realize that my kids need to do proof first.)How do I explain that at the beginning you need to prove this property and then later you can use it in a way that actually makes sense. Prove them in this order because I handed out the packet in this way seems too arbitary.

The other thing that I’ve noticed lately that is probably occuring for a whole host of reasons, is that I’m actually running and bouncing around the classroom. The jokes are easier to tell. I do ridiculous things. I’m like an actor who is living in the moment and channeling my impluses into my body and words. I’m more me and less of a script than I’ve ever been before. I’m not trying to please my administrators, or think about how my actions today might affect my students behavior when I’m observed tomorrow. We went outside two times this week in 8th period because it was sunny and I wanted to go outside and the kids were fantastic. We look up random cool things(only momentarily) because I feel like it. I’m so open about my motivations mathematically and as an educator. I guess perhaps class is less planned and more planned all at once. I’m incorporating learning theories in what I say even as I come up with them on the spot. I think that my answers to some questions will help students see their own abilities in math in the way I want yet those answers are spontaneous. Perhaps the extra thought has given me the ability to ad lib easily and effectively.
The thing is, I have no real way of knowing if me feeling great about my job is actually helping my kids learn more. It might just be making my classroom crazier and less well managed. I’m certainly not being strict at all and I know that could get away from me.

I should wrap this up. To describe one day of teaching could take a book though. And I’m not going to have myself as an experiment for much longer. Who knows if we can generalize from me in any case.
I need to observe more classes, make videos of my class. I feel like I’m already starting my PhD. And I NEED to review my undergraduate math as well.
But, luckily, I have this new boost of energy and happiness and direction. It’s amazing how much positive reinforncement works for me.

I woke up to an amazing email this morning. Admission with possibility of fellowships at UC Berkeley to study math and math education.
Scary Amazing. Do they really think I can hang with Berkeley math PhD students?

Free school. What could be better? And the chance to learn the things I need to know to keep changing math education.

Then I arrived to school and found at that one of my awesome students was admitted to RISD(a super selective design school in Providence.) We threw balloons around the classroom and watched movies about three dimensional fractals to celebrate.(Google mandlebub and you will not be dissapointed.) Of course the kids had no idea I had been accepted this morning as well because I don’t want them to know I’m leaving just yet but I pretended that I was bouncing around the room excited over Roya. The goal of the balloons is to answer questions about math while keeping the balloon up in the air as a class.

Next was project day in Calculus which is so much fun I wonder how I can make project day a part of every day. The students create real world related rates problems and try to solve them. One group wants to use chemistry formulas to relate temperature and pressure. Another wants to use radial motion to relate how changing the radii of car wheels affects forward motion. Another group is baking cookies to relate the rate of the change in height of a cookie to the change in the radius. Another is measuring the rate of change of the school lunch line. I love the interesting, relevant and vivid conversations that come up about the applications of math to the real world when we talk about projects. Google queing theory I tell one group, find out how they do use Calculus in Chemistry because I would like to know I tell another. Another group wants to relate the speed of a space shuttle to its orbit height. I told them “I’m not sure I know how to solve that problem off the top of my head but go ahead and try.”
Kids are happy. I am happy. Hearing about their college admissions is so much sweeter because I get to go back too!

To top this off it is sunny(seattle, winter, rare), and before I even found out about admission I already agreed to a dinner date with a cute boy tonight.

Oh dear. I might explode. Hopefully not on my date’s shirt.
Now I see what happens when you work hard for years