Reflective Teaching

It's summertime which, for me, means a lot of planning a new course (is there ever enough time to do it right the first time?), reflecting, and … teaching summer school.  I teach an accelerated Pre-Calculus course for 12 students looking to take AP Calculus their senior year and every time I teach this class, I’m confronted with the concept of reflective teaching.  I think reflection is one of the most understated component of not good but great teachers.  This summer school curriculum (and most of the materials) was assembled over the years by my mentor teacher, Steve, and is a historical snapshot of what great reflective teaching can be and should be.  

In Steve’s 1,826 files for summer school he has titled each document specifically.  You might see “Concept of a function”, “Concept of a function v09”, “Concept of a function v10”, “Concept of a function v11”, and finally “Concept of a function v13”.  Most of his instructional activities have been rewritten multiple times and then passed onto a group of his mentees.  Each iteration has a sometime small but profound change that enhanced his students learning.  Sometimes he swapped some problems out for other, more meaningful, ones.  Other times he might add a section and other times, take a section out that led to misunderstandings.  

Steve had documents in each folder that were titled “0-tn-functions” which means “Teacher Notes”.  This document historically outlines these iterations and reflections to then create meaningful changes each subsequent year.  He listened and observed student learning then reworked each instructional activity to better bridge his curriculum to his assessments which were vertically aligned to the AP Calculus course (another blog entry).  Steve had phenomenal results with the largest group of students taking AP Calculus in Chicago as well as one of the highest pass rates.  I would contribute a certain portion of this to his reflective practice.

A Snapshot of the Quadratics Folder

I find that we have so many things on our plates that we don’t take time to reflect.  I encourage everyone this summer to start somewhere - reflect on each unit in a course: what worked? what didn’t?  Also, set yourself up for success - add a “Reflections” box to your lesson plans or unit plans or put a note in Google calendar to reflect once a month.  I’m working toward these goals but am no where near where I need to be.  I’m open for suggestions on what you do to make this a part of our daily practice.

Why we can't find Malaysian Flight 370

This entry is with full respect to those individuals and their families who have been missing since the flight's disappearance.

Our textbooks like to give “Real World” problems. Too many times these are at the end of the section or under a section called “Modeling” and they are so canned, they are unrealistic - thus defeating their purpose. This pedagogical paradigm that our textbooks frame assumes that we should teach skills before content but this is like teaching vocabulary without context and in the world of vocabulary learning, context is everything (think spelling bee... Can you use this in a sentence?) We need to be always teaching context for math just like any other language always uses context and the Malaysian Flight 370 disappearance is a perfect example. Many people are wondering – why haven't we found the floating stuff that we see with the satellite?

What a perfect related rates problem.

If I wasn't on spring break right now, I'd ask my calculus class this exact question. Here's the information:

-7 days ago the Australian government spotted a big object floating in a location 1,400 miles south of Perth

-The current in the South Indian Ocean flows at 1m/s.

-We can imagine that the search area becomes a sector area (assuming its still floating) – we can make up some reasonable angle such as it can move 15 degrees in either direction from the center line of the current.

Now ask the question again given this information and see what happens in a calculus class. A follow up question might be: what is the rate at which the search area is changing every day/hour/minute/second. I would love to see students problem solving (making diagrams, asking questions, etc) and seeing why its so hard to find debris in the ocean.

I found this information on CNN.com so why not integrate literacy into the lesson by having students read for information (your English/Social Studies/Science teachers will love you because they always assume that students are learning math the way they did 25 years ago). As of yesterday, CNN also states “ the current search area is 2.97 million (Thanks to Dan Meyer for checking me on the details) square miles” - that's roughly the size of the continental US. Can you imagine 15 planes and 30 ships trying to find something 100 feet long somewhere in the continental US?

A great follow-up discussion would be how to maximize the efficiency of searching, hoping that they find the objects soon would be even more remarkable.

(Want to make this an Trig/Precalc lesson? Ask students about how the sector area and how height changes the distance which a person can see on the ocean – how many ships are necessary and of what height to cover the search area?)

For more problems like this check out Stu Swartz's "Ripped From the Headlines" at mastermathmentor.com

Horizontal Asymptotes & End Behavior

It's been a while since we've updated this blog, but we're back!

This is a lesson that we did way back in October (better late than never!) in my honors precalculus class. In this short activity students investigate what happens to rational functions toward infinity and negative infinity. The activity is fairly simple, but it helps kids improve number sense a whole lot more than memorizing "if the degree of the numerator=the degree of the denominator"....

Horizontal Asymptote Exploration

 


The "Quick Solve" was fairly straightforward for the students. It was easy for them to see that as x gets extremely large the "plus 4x" starts to matter less and less. As x approaches infinity the "minus 3" barely even matters at all. Once they move on to the rational functions they can start thinking about how the numerator and denominator grow in relation to one another, with just looking at the terms that has the most impact on the fraction.

I liked that this activity got the students away from trying to memorize rules to find horizontal asymptotes of rational functions. Once the students completed the table on page 1 we discussed as a class what these functions look like in the coordinate plane as you approach positive and negative infinity, ignoring what's happening between the extremes. This got us into the whole discussion of end behavior (they have already been working with limit notation) and the kids started to think about the "ends" of these functions when there was no horizontal asymptote.

All in all, I think this activity did its job - there's no real practical applications embedded here, but it makes the students just think about numbers in general; something I've been really trying to push with my students this year.

Google Day #1

So I introduced the Google Days project in my Precalc class last Friday.  I tried my best to get the students excited without telling them what we were doing on Friday.  I posted signs up around the entire school that said "I know what Mr. Doudican did last summer!!" I even teased them and asked them if they had asked him and figured out what the big deal was all about. Of course he didn't give anything away because on Friday during class he presented to the class what he did this summer.  He worked with a friend of his from Google on a project that is to be released soon.  (I can't actually say what he did until it goes live in a couple weeks.) Let me just say that I think we're off to a great start with the project.  The students shared their ideas for a project, or a topic they were interested in, and I think that there is definite potential here.  I think after Friday the students had a better idea of the type of things that would be appropriate for this project. I also gave them this after we introduced what we are going to try to do in this class.

I'm still really excited about this project.  Next we need to have the students officially submit their topic/idea and we will assign project advisors after that.

As always, your comments are welcomed!

Math Humor

The math geeks were trying to make some fun signs to help get our points across during the strike.  Just thought I would post this.  #Studentsfirst!

I believe in public education

I first want to thank all of the math blogger initiation folks (Julie, Fawn, Anne, Megan, Bowman, Sam, Lisa, John, @druinok, Tina, Kate, Sue) that started this whole thing.  You've gotten so many more math teachers to start blogging and that's only going to make us better math teachers.  It's exactly why I'm in this, and I wanted to say thank you!

I haven't posted in about two weeks, (another math teacher at my school and I are splitting the duties on this blog) and I'm not going to write about one of the suggested topics for this fourth and final week of the blogger initiation.  Instead I'm going to talk about this hot button issue that I'm currently in, the Chicago Teachers Union (CTU) strike.  There are some rules for this post though that I want to make clear.  First, I'm not going to talk about unions because either people hate them, or support them and I don't want start a never ending argument.  I'm also not going to get into whether or not teachers deserve better or if they are being made out to be the bad guys.  Look, we all chose our respective professions and we all have our opinions.  However, we don't really know how someone else's job is unless we've actually worked it.  I definitely think that we don't do enough research and reading on our own to get all the facts whether it is with politics, business, or education to issue some of the statements that we do.  What I am going to talk about is something I believe in very strongly: public education.

(Note: I've linked most of the facts here with articles.  Check them out for more background knowledge.  I didn't want to make this post extremely long.)

The CTU has been on strike for two days now.  We as teachers have been picketing at our respective schools in the morning, and rallying downtown Chicago in the afternoons.  Let me say, it's been a very strange and interesting experience.  The media has concentrated their reports on teacher compensation, which is currently 1st or 2nd to NYC teachers, depending how you do the math, and the city says they are offering a 3% raise the first year and 2% for the next three years, which the city is saying equates to a 16% raise.  Again, not sure who did the math on that.  We've already started teaching a longer school day, and Rahm Emanuel promised compensation but took it away despite an independent arbitrator deciding that we should get paid 35.7% over four years.  BUT!  Let me say that this is not about the money.  The compensation is such a small portion of this.  Also, the entire country is experiencing tough economic times and there are so many people out there looking for work, or working extremely hard just to get by and I definitely feel guilty in some ways having that issue on the table.

Let's get to the real issues.  Our recently elected mayor is busy imposing his own agendas throughout the city. (Oh and BTW he was running a super-pac up until yesterday while also running the third largest city in the country. PRIORITIES!!)  He's politicized the changes that he's wanted to impose on Chicago's schools, and he says that it's for the students.  The truth is that he doesn't believe in public education.  He wants larger class sizes so he doesn't have to hire more teachers.  He wants to create a privatized school system in the way of charter schools.  He says that charter schools are the way to go, but the charter schools in Chicago are actually doing worse than Chicago Public Schools.  The real reason why he likes charter schools so much is because they are cheaper.  Teacher turnover in charter schools is extremely high, thus giving them lower overhead costs for teachers.  He's not trying to create better schools for Chicago's children, he wants a smaller budget line in his budget.  Can you imagine a public school system where the teachers have one, two, MAYBE three years of experience?  I'm just starting my fourth year of teaching and I feel that I have so much more to learn and it's still a struggle despite receiving high ratings from my principal last year.  Plus, if Rahm really believed in public education then why does he send his children to private schools?  The way I see it, if we don't invest in public education and provide a quality learning experience for ALL children, then our country will suffer.  There are so many benefits and positive outcomes the more education and learning people pursue.  I really think that serious problems we have in this country like poverty, crime and health care costs would improve if we invested in public education.  However, if Rahm has his way and failing charter schools are the norm, parents will be forced to send their kids to private schools.  They'll pay twice, once on their taxes for the failing charter schools and then again for tuition to private schools.  I don't want to have to send my future children to private schools.

I'm a product of public schools.  Public education works.  I'm fighting for public education.  Why not join me and wear red if you support public education!

Get to the point.

All teachers have heard "why do we have to learn this?" from confused students waiting for us to give them answers.  To many students math is a distant, far-off land for math teachers and math teachers only.  To them our job is to "dump" our knowledge into their heads and as teachers we hope that somehow the information will stick.  Their understanding of mathematics is that it is a means to an end: the answer in the back of the book, the 28 ACT score, the letter on their transcript.  The students simply want us to get to the point already.

So why do students really have to learn this stuff?  The simple answer is that everyone needs to learn math to make informed decisions.  Let's imagine we've assigned a word problem on linear systems.  What do we want kids to do?  In an ideal world students would...

1. Read the problem and pull out necessary information.
2. Make a plan of attack.
3. Execute the plan.  If something goes wrong, try a different approach!
4. Verify the answer.

This crazy foreign language of word problems that math teachers speak is really just our way of helping students learn how to analyze a situation and make an informed, educated decision.  Unfortunately, as teachers, our intentions often get lost in translation.  We need to view our classrooms as places where we can help shift a student's thinking.  We need to challenge them, and we need to make the practice of mathematics something that the students connect with.

One way we can work towards this is by helping students learn about social justice through their math courses.  In this, we are creating knowledgeable citizens who know how to look at situations and perform an educated analysis - little mathematicians!  Something pretty cool that I'll be using this year in my AP Stats course comes directly from the Bock, Velleman, and De Veaux book Stats Modeling the World and analyzes the relationship between race and the death penalty.  Students are given a data set and are asked whether race has an influence on the verdict in death penalty cases.  Types of activities like this show students that math is used to study the world around them, not just the world in their Statistics text book.