Unknown Quantity

Another opener I lucked into

2018-08-09T20:27:00.000-04:00

I found another great graph that I used as an opener in class today. My only complaint is that it was a bit hard to read, but it had so much to talk about.

My classes talked about the red line in the graph. They talked about how the car line and the bike line were very similar and what that meant. They talked about the axes being reversed from what they usually see for position-time graphs and why the author of the chart may have chosen to do that. They talked about what a steep line means in this context. They talked about why the lines don't all start at 0.

They had some great thoughts. I have several students who have become "frequent responders" to my questions. I appreciate their willingness to answer, but I'm going to have to find some ways to bring other voices into the class while still respecting the introverts enough not to shove them into the limelight. That will be my challenge for myself next week.

When to share a cool idea?

2018-08-08T19:44:00.001-04:00

Today most of my classes wrapped up the problem that we have spent a few days on (happy dance). We spent several days on the problem and they were ready to be done, but they found the answer.

Some students found the answer by adding 1+2+3+...+101, while others found an interesting area approach built off of creating a square and then adding to that (because of the zig-zags). It was really fascinating. One group surprised me by trying to solve it by "peeling the onion," looking at the outside layer, then working in... it was really a cool idea that I want to explore more soon.

Then, since we had some time in each of the classes, I thought I would share an elegant solution to the triangle number problem. Specifically, I showed them a diagram like this one (source: solvemymaths):

In retrospect, I'm not sure that I should have done this. I prefaced this by trying to congratulate them on solving a difficult problem and spent some time commending the different methods I saw. Then I said I had a particularly interesting solution I wanted to show them. I showed them this idea and explained it to them. Some kids got big smiles on their faces as they saw what a cool way to handle the problem this was. But others got disappointed in themselves for not finding this (it "seems" obvious once you see it).

So what I'm wondering about right now is whether I showed them the idea prematurely or if I should have handled it differently.

...

Maybe I should have drawn the picture and just let the class study it and see if they could develop the idea themselves. That would have let them own it more. Or maybe I should have just held off and let them revel in their success. I could have saved it for a day or two.

It's all fun and games until...

2018-08-07T20:44:00.000-04:00

In my algebra 2 classes I began the year by showing a pattern from visualpatterns.org. Specifically, the third pattern, which looks like this:

We spent a few sessions trying to figure out what the perimeter of the 100th figure would be. We are also still trying to figure out how many squares the shape has.

I have been letting the students work in groups, and I am using Visibly Random Grouping. My favorite thing about randomly grouping is that when we didn't finish the problem one day, some of the ideas spread around the room the second day. I loved watching the ideas spread. In addition, as they worked at the boards, some of the students would intentionally or unintentionally overhear on of the other groups and get an idea from them. I find this "cross-pollination" of ideas to be a great thing. This activity was more about group work, sharing ideas, and communicating effectively.

I don't want good ideas to be greedily hoarded in the classroom, I want them shared. We can build on good ideas as a class.

Anyway, all of it has gone great and students were very engaged in the process. We had to take a break from the problem for a couple of days, and when we came back, most of the classes were able to solve the problem.

Then came the hard part: the written solution. After we had talked about the solutions they had and I pressed the class with several questions, I set them free to write their own, individual solutions to the problem. I figure this individually written solution will be a good practice of writing skills, plus it will help me see who understood it and who was just faking it.

I'm still really liking the year, and it has been fun watching them argue about representing numbers and how to accurately count the sides along the diagonal. I sense great things out of this group.

"Day 4: Data Analysis and the NFL"

2018-08-06T20:09:00.000-04:00

Class today started with this question that Ben Jones @DataRemixed posted:

Stats Quiz: The average NFL player is about 25 years old, just over 6'2" in height, weighs a little more than 244lbs and makes slightly less than $1.5M in salary per year. Can you tell which distribution goes with which trait? See why the shape of the distribution matters a lot? pic.twitter.com/pnfQ8odurZ
— Ben Jones (@DataRemixed) July 26, 2018

So, I asked my classes which trait goes with which distribution. They argued with their table-mates and we eventually talked about what the different distributions meant.

We tackled this in Algebra 2, Pre-Calculus, and Calculus. It doesn't technically fit the content standards for all of them, but who cares? Data analysis and making connections between various representations of data is important across the board.

Plus, it was interesting. And that comes from someone who doesn't like football at all. Data is fun.

Day 3: "What Mr. Shores is pondering"

2018-08-05T08:39:00.001-04:00

Several years ago the teachers in our school were asked to put up signs on their doors that said "What I am reading..." and then we would post pictures of the books we were reading. Think of it as hallways full of physical manifestations of profiles from Goodreads. It was a great idea and it was a way that we communicated the importance of reading to all of our students.

This year, I decided to up that just a bit. I decided to post the things that I am pondering. I have posted this picture on my door, as well as a couple of the books I have been reading recently (mostly Harry Potter, because that was the theme of my summer reading).

More will be added as the year goes on. I will add other books that I have been reading, quotes that have resonated with me recently, tweets that have gotten me thinking, problems that I have been working on, and anything else I can post on a doorway.

Day 2: Area of a circle

2018-08-02T20:10:00.005-04:00

Good things were happening in my classes today. Algebra 2 and Pre-Calculus continued to work on a problem from the Visual Patterns site. More on that soon, but I'm waiting until they solve the problem. There have been some good ideas and I am so happy with all the students. They have been working hard.

Today in Calculus I started class by having them give me the area of a circle. They had no problem giving me the correct formula. Then I asked why that was the formula. Where does the formula come from? Nervous laughter. These kids have had me for two years already (super small school, and this bunch I had for Geometry, Pre-Calculus, and now Calculus), so they pretty quickly said, "You know what? I have no idea where the formula comes from." Smiles.

So then I reminded them that pi is the ratio of the circumference of a circle to its diameter. We discussed some of the consequences of that idea. Then I directed them back to the original question: why is the area formula for a circle pi*r*r?

I split them into groups (visible random grouping for the win), and let them wrestle with the ideas. One quick pause in the middle to discuss some ideas they had. One group thought about inscribing a square. I suggested they build off of that and maybe think about circumscribing a square about the circle as well.

No answers from them yet, but a lot of great ideas. I'm pretty confident that the results of this conversation can tie in nicely to limits (especially limits at infinity) and definite integrals eventually. It also has the nice side-effect of giving them a chance to review some geometry and algebra. I even heard a bit of trig bandied about today.

Locally linear (Day 1) #teach180

2018-08-01T21:29:00.002-04:00

I have come to love first days. I love the excitement and possibility.

My favorite thing today was in my calculus class. I used an activity I got from an AP training several years ago. The short version is that I display this graph and ask the students to find the equations of the lines.

Of course, they find this pretty easy. They get this pretty quickly. We spend 30 seconds talking about slope and finding the equation of the line. Simple, quick review.

Then I pull the rug out from under their feet. Nobody asked about the scale of the graph. It turns out that the scale is small. Each grid line represents 0.001 units. I have them rethink their answers based on that. This gives us the chance to talk about fractions involving very small numbers and review some basic arithmetic ideas.

After this, I pull the rug out one more time. I point out that if you look carefully, the blue "line" isn't really a line.

We zoom out to find various familiar curves.

I close with a line about "close ups of curvy things often look like straight lines."

What a nice quick introduction to local linearity while reviewing some basic algebra ideas.

Preparations

2018-07-31T21:52:00.000-04:00

Year Sixteen: Day 0

I spent most of today finalizing my plans for today and putting the finishing touches on my classroom. Of course, no classroom is complete until students sit in the seats.

Tomorrow I begin my 16th year of teaching. It surprises me that it has been that long. Like most teachers I talk to, I still have the nervous/excited feeling as I think about tomorrow. But perhaps I'm finally easing into this career because it is less nervous and more excited.

Anyway, here are the required pictures of the fruits of my labors. I have left a large part of the wall blank (not pictured) so that I have space for student work. The yellow, green, and blue papers that are impossible to read are the standards of mathematical practice, Glenn Waddell's "Three essential rules of algebra," and the things mathematicians do (taken from the chapter titles to TracyJohnston Zager's Becoming the Math Teacher You Wish You'd Had).

I'm looking forward to using Visibly Random Grouping (#VRG) and making better use of my extensive whiteboard space. I have enough that I think I can claim two. One will be there for things like quotes, notes to myself, and making my after school "office hours" more clear to my students. The other whiteboard I'm claiming I am really excited about. I'm making it the "What Mr. Shores is thinking about" board. I will try to use it to model my own interest in mathematics and problem solving to my students. Currently I have some beginning thoughts on a variant of this Exploding Dots problem posed by Kent Haines.

That's it. The adventure continues tomorrow!

Introducing Transformations

2017-01-09T22:36:00.003-05:00

Today in class we started to study transformations. To level the playing field some and to ease us into the unit we started with a (pretty weak) definition of a transformation and then I gave them a worksheet and asked them to describe the transformation of one segment to another. The goal of all this was to create some needs. I wanted students to need vocabulary to talk about the ideas, I wanted them to need a better definition of transformation, and I wanted them to get a general idea about what this unit was about.

Since this was an introductory activity, I gave my students minimal information and minimal coaching. If students asked me "What is it called when you flip the picture over the line?" I would say, "Hmmm... good question. I think flipping sounds good for now. Go ahead and put that and let's see what your classmates say in a few minutes." If they had correct vocab, great. If not, no biggie. I figure I was either activating their prior knowledge or I was creating a need for some words. We'll dive into those words in the next few days.

My favorite part of this activity is that most of the problems have multiple transformations that work (and that are reasonably obvious). This meant lots of discussion and debating. Some classes debated more than others. The problem that drew the most interest was this one.

While most of my students have an eye for reflections across one of the two axes, they are not yet trained on y = x or y = -x reflections. This didn't even cross their mind. So instead of the reflection I had originally had in mind, most students went through a convoluted series of transformations... something like "reflect over the x-axis, then rotate 90 degrees clockwise around A, then reflect over y-axis."

They were flabbergasted when I told them that there was a single step that can do all that at once. I set them back to work looking for the "one transformation to rule them all" (a movie/book reference almost nobody in my class caught... sad). Now they were trying some sort of rotation. Since that was the majority opinion I opened up Geogebra and started messing with rotations. I plotted a center point at any point they wanted and we saw all sorts of ridiculously wrong answers, but it was fun and they enjoyed guessing and seeing what happened. When they finally guided me to the correct center point of (-1, 1) they were kind of amazed it all worked.

I was tempted to bring in some more serious math at this point and talk about what it all means, but time was running short and I didn't want to trade the small amount of capital I had built up for a rushed explanation.

All in all, I really enjoyed today. The students were thinking and trying things out. They were exposed to the idea that multiple routes were possible. Some of them even were on the verge of "playing" with the mathematics. I'll call this a success.

Planning: Transformations (part 1)

2016-12-30T23:50:00.000-05:00

Subtitle: "Finding the perfect (working) definition"

Background

When school starts up my geometry students will begin exploring transformations. I have been unhappy with what I have done in the past and so I am scrapping it all and starting from scratch. I've been thinking a lot about the ideas I want my students to bump into; which ideas are important enough that I will point them out if my students miss them and which ones are minor enough to just lurk below the surface of what we do?

As I brainstormed and messed around with ideas it became clear that I want my students to do as much of the work as possible. I prefer it when my students come up with as many of the definitions as possible. However, I think I will need to define "transformation" to get us started.

Other people's ideas

Most websites give one of two types of definition: definition by example ("A transformation is a translation, reflection, rotation, or dilation of an object") or rigorous definition from advanced mathematics ("A transformation is a bijection from a set of points to ...").

I am not wild about either type of definition.

The second type of definition is fantastic at more advanced levels, but I don't plan on explicitly connecting transformations to functions right away. That may be one of those things that I try to nudge my students toward, but it doesn't feel crucial as we begin to poke around. In addition, that kind of definition seems to be 100% opaque to my students. I need a definition that my students can latch onto and say, "Yes, I know what that means. Here is an example..."

The first type of definition seems worse to me. On the one hand, it doesn't feel rigorous enough. Defining by example doesn't feel right. But the more pressing issue is that at the beginning of our explorations I want my student to be using language that is more natural to them. I want to be able to give my students various graphs showing transformations and ask them to describe possible transformations. I want them to be able to use their language first and then bring them around to the more accepted words like "reflection" when the time feels right.

So... I guess what I'm saying is that the first kind of "definition" is too helpful while the second kind is not helpful enough.

A starting point

To recap, I'm looking for a "definition" of transformation that lets my students start to explore the landscape of transformations without giving them too much.

Here are a few that I'm batting around.

A transformation is a rule explaining a change in the location, orientation, or size of a geometric object.
A transformation explains how to move a point (or set of points) to another point (or set of points).
A transformation can be thought of as a rule about how to move a point (or set of points) to another point (or set of points).
A transformation is a rule explaining the relationship between one geometric object and another.

None of them are perfect and none of them are rigorously correct. I'm willing to circle back to the definition later in our explorations and poke it until it is more technically correct.

I would love feedback from other geometry teachers on what they think. Are any of these good starting points? Am I making a mountain range out of a mole hill?

Summer Reflection & Planning #13: WODB

2016-06-09T23:02:00.001-04:00

This is the summer of me getting caught up on some of the cool things that have been working their way through the #MTBoS over the last few years. I decided to try my hand at a Which One Doesn't Belong prompt. It was much harder to do than I expected. It took several days to flesh this one out. I thought I would share my journey with you.

I wanted to create something for a calculus class to look at early in the year that would nudge them in the direction of talking about tangent and secant lines. While my students have a passing familiarity with those words in the context of circles, they are not used to thinking about them in the context of more general curves. I wanted to prompt conversation that would give kids the need for the words secant and tangent and at the same time begin the process of developing some intuition about them.

My first attempt was just a rough draft sketched out.

There are things I like about this, but in the middle of this I started also thinking about the Mean Value Theorem and my various lines had too much to do with matching the slope of the secant line connecting endpoints of the interval. I felt like I was overloading the prompt.

After some feedback on twitter I have settled on this one.

I feel like it will root out some of the common misconceptions about tangent lines and secant lines before they have a chance to settle in. For example, the top right graph has a tangent line that also clearly intersects the graph somewhere else. The bottom right graph has a tangent line that "crosses through" the graph, which is different from the behavior of tangent lines in circles.

I'm pretty happy with this one. I will probably field test it with my calc class on the first couple days of school (Early August). Let me know what you think in the comments.

Summer Reflection & Planning #12: First Calculus Acitvity

2016-06-08T22:03:00.000-04:00

I worked today on tweaking a first day activity that I recall reading about in an old AP document about teaching Calculus. (I can't remember the details, but will try to update when I do.)

The activity centers around this graph and is built off the core idea of Local Linearity. The main goal of the activity is to introduce the idea that, "If we zoom in on curvy stuff, it starts to look more straight,” (quote from Jason Wilkes' Burn Math Class). There are also some sneaky ideas embedded in this activity.

Labels matter (the axes are labelled in this diagram and that plays a role in the activity).
You should be able to write equations of lines in Calculus (AKA, the stuff you learned in previous courses still matters).
Surprise and wonder are a part of this class.
You should recognize other functions besides lines and be able to make a rough guess about what their equations might be.
Thinking about math and then talking about it will be important.

Here is a link to my "Teacher's Guide" to the activity. I would love your input on it.

Summer Reflection & Planning #11: More Organizing

2016-06-07T22:15:00.000-04:00

I went in to school again today. The math office area was a bit of a mess. Now it is a bit better.

Before

After

Summer Reflection and Planning #10: Starting Class

2016-06-06T23:54:00.000-04:00

So I'm back to the first few minutes of class again. I still haven't really settled on a name, but I am getting more sure of what I want that time to look like. I created a slide that I will use at the beginning of class to get things rolling. The specifics for each day can be filled in later... right now I just have a template... feel free to steal/borrow/change it if you like it. Besides a place for opening instructions and supplies students will need to gather I have also included a slide that has space for a description of the day. The teachers in my building all post an agenda for the day somewhere in class. Each teacher puts their own spin on it, but we all include what things the class is doing and what the goal of the class is, so I have included space for those things.

Agenda (Google Slides)

Summer Reflection & Planning #9: Going over homework

2016-06-05T22:25:00.001-04:00

This past year one of my students was a foreign exchange student from Spain. Near the end of the year I had a conversation with her about how her experience here was different than her schooling in Spain. She had many great comments, but the one the struck me the most was the difference between how I handled homework checking and how she had done it in Spain. Most days in class I would circle through the room checking off homework while they worked on some sort of opening activity. After I had checked the assignments for completion we would discuss the answers, either by projecting the correct answers and work, by reading out the correct answers, or by randomly calling on students to share their answer. Occasionally, to spice it up, I would simply collect the work without going over it.

In contrast, my student remembered that most of the time in her previous school students would be chosen to write up the solution to a problem on the board and then the class would discuss. Do any of you do this? How do you manage the time issue and keep it from dragging on throughout the whole class period? Do you have students explain their work or do you just ask students to survey the work on the board and ask questions? Do you point out errors or ask for students to do that?

I was wondering how this practice might fit in my class and whether it would work to have one student write up the solution and another one try to interpret the work to the class. Thoughts?

Summer Reflection & Planning Day 8: The first few minutes of class

2016-06-04T22:18:00.001-04:00

(As an aside, why can't I shut my brain off for a few days?)

I really dislike the typical names that teachers have for the first few minutes of class:
Warm-up: It implies that whatever work the student has been doing up until this point in the day was not adequate.
Do now: Nothing like a command to set the tone of class long session of dialogue and conversation about the abstract ideas of geometry. I prefer not to set off the "fight or flight" instinct as they walk in.
Bell ringer: Who is ringing the bell? Why? My school's "bell" is actually just a tone played over the intercom. Do we have a bell ringer we time to conclude right at the dismissal bell as well? Actually, I don't have a good reason for not liking this one except it just sounds weird.
Opener: This is the least disagreeable one of the standard set of names. If I can't find something better soon, this is probably what I'll settle on for another year.

Anyway, I sent out a twitter appeal the other day and got some good ideas for anyone else tired of using the same old names.

Looking for a new phrase for that beginning of class activity. Dislike "do now," "warm-up," and "opener." Thoughts? #MTBoS
— Andy Shores (@abshores) June 3, 2016

Here are some of the great responses I got.

@abshores countdown. Mind stretch. Problem pique. if tech driven excite byte. Robot themed-thought-bot a-la-carte if they have a choice
— pam j wilson (@pamjwilson) June 3, 2016

@stoodle @abshores @classroomchef uses appetizer in their book. Trying to decide if kids would get the proper connotation by using that term
— Lisa Henry (@lmhenry9) June 3, 2016

@abshores "Last time in math class . . ." or recap
— Rachel (@Seestur) June 3, 2016

@abshores @lmhenry9 aperitif?
— Ashli (@Mythagon) June 3, 2016

@abshores How about "launch"?
— David Wees (@davidwees) June 3, 2016

And perhaps my favorite string:

@abshores @lmhenry9 “Hey look kids, a thing to do, do the thing!” is something I say often, the exercise itself is untitled
— Jonathan (@rawrdimus) June 3, 2016

@abshores @lmhenry9 maybe make up an acronym, Don't tell them what it means, let them wonder. Kind of like #wcydwt.
— pam j wilson (@pamjwilson) June 3, 2016

@abshores @pamjwilson Could be based on what @rawrdimus says - hlkattddtt!
— Lisa Henry (@lmhenry9) June 3, 2016

I still haven't settled on anything yet, but I'm sure I'll figure it out (or perhaps go back to the boring "opener"). In the meantime I saw this tweet and I fell down the rabbit hole of great warm-up resources.

Planning weekly routines for next year: mental math monday (thx @Veganmathbeagle), daily estimation (thx @mr_stadel). What else? #mathchat
— Mark Kingsbury (@mhkingsbury) June 3, 2016

The replies included the following links:
@MaryBourassa's semester long warm up sets
@lisabej_manitou's warm-up filing cabinet

If you haven't already seen those resources, you can thank me now for losing an hour of your life perusing the awesome ideas.

Summer Reflection & Planning Day 7: Summer Reading

2016-06-03T18:57:00.001-04:00

This next year I will get to teach a class that is new to the school. It is an elective students can take in addition to their math class that is "Mathematical Problem Solving." The plan is to present a bunch of problems from several different areas of math and then get the students creating their own conjectures and questions about mathematical situations. A significant portion of the class will be spent allowing students to pursue their own mathematical interests, wherever those interests may land on the Pure-Applied math spectrum.

In preparation, I'm trying to skim/read as many books with as many interesting problems as possible. I think at the beginning of the year the students will need a bit of guidance about creating conjectures and asking questions. My experience is that high school students often have been playing the "school" game for too long to feel comfortable throwing out ideas about what they notice. It will take a bit of coaxing.

I'm really looking forward to the class, but as I'm perusing these books I'm also finding many great resources for my other classes. I hope to share the activities I create from these problems over the next year.

If you have any book suggestions or problem suggestions, please add a comment!

Summer Reflection & Planning Day 6: Organizing

2016-06-02T20:51:00.003-04:00

I spent a few hours at school today. It was time to clean out the math office bookshelves.

We are getting rid of textbooks that were 12 years old. I made some good progress, but there is more to do.

I also finally got rid of this nasty two year old mousetrap.

Summer Reflection & Planning Day 5: Standards

2016-06-01T21:54:00.001-04:00

I decided today that I needed a list of the standards for the classes that I teach. Of course there are documents that have the standards for my state, but I want all the standards for all my classes in one easy to edit document. I also have two sets of objectives for the Calculus class I teach. Students can opt to take the AP test (and so I need that list of objectives) or they can get dual credit through Ivy Tech (the statewide community college system.... with another set of mostly overlapping objectives but in a different format). Unfortunately those documents are in pdf form (not easy to edit) and more importantly they are not laid out in a fashion that is helpful for planning purposes. There is usually a bunch of pages explaining what the standards are, where they came from, etc. I just wanted a list in word format or a google doc. But no.

Feeling Good.

So I spent some time today and yesterday putting all my standards in one place. If anyone teaches Geometry and/or Calculus in Indiana, feel free to use my document.

[Update: June 6, 2016] I realized that my problem solving class would use standards from a variety of different courses. I was going to do 8th grade on up to all the different sets of standards in the high school, but I decided if I was going to do that many, I may as well add all the standards. So here they are, all the math standards for the state of Indiana. I kept Geometry and Calc at the top since that is what I'm teaching, but other than that, they are in grade level order and then in the order that seemed most logical to me for the high school classes.

Indiana Math Standards plus AP Calc and Ivy Tech Objectives

Summer Reflection & Planning Day 4: Homework Policy

2016-05-31T21:54:00.000-04:00

Yesterday I took a long look at my philosophy of homework and how it has evolved over the 13 years I have been teaching. Today I wanted to talk about something new I plan on putting into place with my Geometry class this upcoming school year.

I will not allow a student to take a test until they have completed all assignments (classwork/homework) for that unit.

I'm doing this for a variety of reasons, but the most important one is that I think the work we do in class and the work I assign for outside of class is meaningful and that a student is not ready to be successful on a test until they have completed the work.

A secondary reason is that I allow my Geometry students to retake each test one time. By the end of the year I had a lot of students using the first test as a trial run and then when they realized there was a lot of content they hadn't prepared for they would decide to do a retake. They needed the trial run because they hadn't gotten the practice they needed on the homework. My homework counts as about 10% of the grade and so students were not seeing the value since it wasn't making much of a grade impact. They had trouble connecting it to the test.

I'm hoping the new policy will help matters, but I also will work on trying to connect homework problems to assessments. I need to work on giving the students more transparency on the assessments so they aren't surprised and so they see the homework as the practice I mean it to be.

Anyone else have a similar policy? Any landmines I should be prepared for?

Summer Reflection & Planning Day 3: Homework Philosophy

2016-05-30T20:30:00.001-04:00

Over the last few days my twitter feed has seen a lot of conversation regarding homework and how so many teachers have stopped assigning it for a variety of reasons. I have been thinking about homework a lot and trying to flesh out my philosophy during these past few days. During my time teaching I have held just about every position that is possible to hold regarding homework. When I first started teaching I assigned the "1-41 odd" type assignments, indiscriminately throwing problems at students with very little thought. As I grew in my practice I became a bit more careful in the types of problems I assigned.

Eventually I got to the point where I thought homework was unnecessary and perhaps even wrong for me to assign. I spent most of a school year in this position, but I began to realize that my students were not gaining the proficiency with certain skills the way I had hoped. This was despite the fact that I had built substantial practice into the classroom routine. For some reason the practice they did in class did not seem to translate into proficiency for more summative assessments like end of unit tests. I had a large number of students saying that while they enjoyed having a light homework load they did not feel like they were getting enough practice.

So I began assigning short homework assignments (designed to take around 5 minutes). These still didn't seem to do the trick, so I have slowly gone back to assignments that should take my Geometry classes about 15 minutes tops, while I expect my Pre-Calculus and Calculus classes to invest about 25-30 minutes of practice on assignments. On average I assign about 3 homework sets per week. Depending on the size of the homework set, we sometimes spend no time in class on it and other times I give them most of a class period to start it and then finish it off for homework.

I think the ideal assignment should have a few basic practice problems and a few "novel" problems that cause a student to tweak the work we have done in class in order to find a solution. I feel like the only way for students to be comfortable with any type of math problem they encounter is to encounter a lot of "non-standard" math problems.

So here is my current homework philosophy:

I will continue to assign homework. I will be careful to not make it burdensome. I will be careful to assign meaningful work. I will continue to hold high expectations regarding completion of homework.

I would love to hear your ideas about how much time is appropriate for high school students to spend on homework in the comments.

Summer Reflection and Planning Day 2: Geometry Problems

2016-05-29T22:17:00.001-04:00

I've been re-reading Paul Lockhart's Measurement. I forgot how much I love this book. One of the ideas I love is how the book asks questions without providing the answers. For example, Lockhart provides a diagram of the medians of a triangle (although I don't think he names them as such) and then leaves it to the reader to figure out why they will always cross. He gives some guiding ideas, but he doesn't spell it out for you or give away the answer. I've been wondering if this idea would work in my Geometry class.

In addition, I've wondered recently about the "goal" of Geometry. I don't mean the purpose of the class. Depending on who you ask, you might get several different answers ranging from "learning to think logically" to "figuring out how shapes play a role in the world around us" to "using shapes to solve problems in Engineering, Science, Architecture, and other arts and sciences." When I talk about the "goal" of Geometry, I'm talking about the ideas it builds towards. Since one of the purposes of Geometry is to prove various geometric propositions, I have been thinking about what all those propositions are headed towards. For example, at least in a loose sense, the 13 books of Euclid's Elements is building toward a description of the Platonic Solids. The material in the first 12 books sets the stage for the 13th book. Likewise, one of the main goals of a Real Analysis course is to prove the Fundamental Theorem of Calculus. In that vein, I wonder what my high school Geometry course could build toward. Where are all these conjectures, propositions, postulates, and theorems headed?

I want to separate this from the idea of the "big ideas" for Geometry. I personally see the big ideas of Geometry being the deductive system and the set of tools we use to prove things (I'm thinking of similarity/congruence, parallelism, angle sums in polygons, the Pythagorean Theorem, and Trigonometry). However, what I'm talking about is more like this:

What questions or problems are interesting enough to drive a class to develop the big ideas of Geometry as a side effect?

I have been thinking for a few days and this is a partial list (some stolen from Lockhart and others from some of my favorites). Actually, most of them are just a picture that have an associated question or set of questions.

Regular Polygon Tiling (stolen from Lockhart): which regular polygons (or groups of regular polygons) tile the plane? This forces the issue of the measure of polygon angles (at least for regular shapes).
Why do the medians always meet at a point? This is a fascinating question and so simple to explain to a group of students. I feel like leading off with this was a good idea on Lockhart's part and may be a good way to start the class off. Set the problem and let it simmer there for as long as it takes. I feel like once students create a few triangles and measure the sides and find midpoints by hand to create the medians they will be pulled in. It will only increase the mental anguish to see it on a computer. This problem will likely need to be set aside and revisited every once in awhile to see if any progress can be made.
How much of a rectangle does a triangle take up? I wonder how many students would reach for the formula for the area of a triangle when the question is presented in this form. The solution is a marvelous way to explain the area formula for a triangle.
This set of pictures is enough to "prove" the Pythagorean theorem, at least at a level most HS Geometry students are ready for. More rigorous proofs can be given as the year progresses. (Side note, I wonder if trying to force too much rigor early in the course can kill its effectiveness for students.)
Is the midsegment parallel to the base? Why? If similarity and congruence haven't shown up yet, then surely this question has a way to force the issue.
This idea blew my mind a few years ago. It is still pretty amazing to me that connecting midpoints for any quadrilateral will form a parallelogram. I totally get the proof, but I still feel like this is one of those ideas that shows that once we create some math it gains a life of its own and will do whatever it will do without us being able to stop it.
Thales' Theorem and the more general version about an inscribed angle compared to a central angle. These are just cool. A nice way to show how rad circles can be.
Tangent theorems. They are nice applications for the Pythagorean Theorem and open up a whole new world of circle problems that can be pondered.
I think classification of shapes helps students really get their minds around what they mean when they say certain words. Once we get that all sorted out, it is another place where we have created mathematical monsters that have properties that we didn't initially spell out (for example the fact that the diagonals of a rhombus are perpendicular.... that isn't part of our original meaning for a rhombus, but it is brought into existence by the the definition).
Finally, some "application problems" :) Also, this is a nice way to introduce the ideas of Trigonometry and to revisit the key ideas of similarity.
More application problems, plus there are plenty of cool ideas about "how do these two areas compare" or "which of these two figures is larger?" Ditto for volumes.
This was added after I had thought of the other 11 problems. Clearly it would be brought up earlier, but parallelism and the angle sum of a triangle are deeply connected. I'm not sure what I would ask here, but there are lots of good possibilities.

What have I missed? What good questions could move a Geometry class forward in their thinking? They need to be simple enough to grasp the meaning of the problem quickly, but that may take some time to develop a solution.

Summer Reflection & Planning Day 1: Relationships

2016-05-28T22:18:00.000-04:00

Breakfast at the farmer's market. pic.twitter.com/2HB9WKzFnZ
— Andy Shores (@abshores) May 28, 2016

Today was the first day of break and it has been pretty mellow. I started the day off by taking the family to the local farmer's market and having a delicious breakfast. After that I became pensive. I think it is because I'm so introverted that the first several days of summer break are usually just spent in a mellow zone where I don't speak much. I slip into reflective mode once the frenetic pace of finals is over. Today was no different. I spent many of the quiet moments of the day thinking about what went right, what went wrong, and what needs to change.

I had many successes this year. On the surface level, the chess tournament I organized was a success and the academic team that I coached went to State. I had a lot of fun with the chess kids and the academic team kids. Even more importantly, several students connected with me and felt like they belonged in my class. A few were convinced that math was more interesting than they had previously believed. I even had a couple who were inspired enough to begin asking their own mathematical questions.

I had many failures this year. Too many students struggled to figure out how my class connected to their lives. Several left liking math even less than they did when they first entered my class. A few never connected well with me or their peers in class. To some degree that is the nature of working with young adults (and people in general). I can't be everything to everyone. I work with an amazing staff and most of the kids in our building connect really well with at least one teacher. But I'm always bummed when a student doesn't seem to have a good relationship with anyone in the building and doesn't see success in any class. I think about all the little interactions I had with them and wonder if there was something different I could have done.

Anyway, that got me thinking about my classroom culture and my relationships with the students. I would like to believe that most students find my class to be intellectually challenging and emotionally rewarding. However, that is certainly not true for every kid that walks through the door and I have been thinking today about what I can do about that.

Inverse Function Ideas

2015-11-12T21:54:00.000-05:00

A story in three acts (no, not those three acts).

Setting: A pre-caclulus classroom. On the previous day, students had learned about the definition of inverse functions (technically review from Algebra 2, but I take nothing for granted) and how to find the inverse function of linear functions and some slightly tougher functions.

Act 1:
I asked students to find the inverse of another linear function. I don't remember which one for sure, but let's pretend it was f(x) = 2x + 4. I told them it is to help me check their retention of the idea from the day before. Students found the inverse easily.

Now I told them that the goal for the day is to investigate the graphical connection between functions & their inverses, especially when the functions are lines. So naturally, I have them graph the two functions. Since this is pre-calc, the graphing went off (mostly) without a hitch and only took a couple seconds.

I then asked them to talk to their neighbor about what they noticed. I asked for their observations and I wrote them down on the board. I wrote down correct and incorrect statements. I gave no indication of what I was "looking for" or even if I was looking for anything at all. If a statement was incorrect, I let it stand unless a student fought against it.

I repeated this several times with several different linear functions. Sometimes I purposely picked a function that would break one of the observations that had seemed true based on the other graphs. We tweaked the incorrect conjectures that can be saved and we deleted the ones that are hopelessly wrong.

This went on for most of the class. They came up with some interesting ideas. The lousy picture below is what one class generated. The starred ones are the ones that they felt were most interesting/intriguing at the end. It is worth noting that in another pre-calc class students were pretty sure that y = x was a line of symmetry (although they worded it as the line that is at a 45 degree angle through the origin... I accepted this language initially, but after we referenced it that way several times, I pushed them for the equation of the line).

Thus ended our first day of thinking about the graphs.

Act 2:

Today I walked into class and I told them that we would try to prove a couple of the things they had noticed. I didn't tell them up front, but I wanted to prove the idea that the coordinates "switch" for inverse functions (an idea we had talked about but wasn't listed in our conjectures... it had almost been taken for granted by the students) and that inverses were reflections across y = x. Along the way we checked off some of their other conjectures.

I thought it would be best to work with specific points first and then replace the specifics with variables later. I reminded them of two key facts:

If a point (a,b) is on the graph of f(x), then f(a) = b.
If f & g are inverses, then f(g(x))=g(f(x))=x.

I told them that we would prove everything using the above two facts, some algebra, and some geometry. I used the following line of questioning to guide the discussion.

1. If (3,1) is on f(x), what is on the f-inverse? Why?

This was tough for them because I forced them to use the definition. They wanted to just use the "switch x & y" thing, but I wanted to show this was proving that thing using our definition.

2. If (5,5) is also on f(x), what else do we know?

This actually gives us a lot. The thing I most wanted out of this is that (5,5) is the intersection of f and f-inverse.

3. Why are the slopes of f and f-inverse reciprocals?

This is pretty trivial after the preceding work, but I wanted to check off one of the things they had noticed.

4. How do we know that (1,3)—the point on f-inverse— is the reflection across y = x of the point (3,1) on f.

This first required a side-quest into the definition of reflecting across a line. (Side journey of our own, I couldn't find a good definition online. I looked at >10 websites and found nothing useful. See if you can do better. Can you find a definition that would help us answer the question above?) After a bit of them suggesting stuff and me poking holes in it, they eventually settled on P and P' are reflections across a line if the line connecting P and P' is perpendicular to the line of reflection and P and P' are equidistant from the line.

After we had a definition, we had a good time with coordinate geometry (showing that we had perpendicular lines) and more traditional geometry (showing we had equidistance either with congruent triangles or Pythagorean Theorem). It was a good time for me.

Act 3:

We had just a few minutes left, and I made a mistake here. I quickly went back and rushed through replacing each particular number we had used with a variable and then checking to see if the explanation still made sense. I didn't spend nearly enough time on this part. In retrospect, I should have saved this for the next class session so that we could wrap it up. Part of me also wonders whether using variables is even necessary, but then the Geometry Teacher in my mind says, "YES! It isn't a proof if it is for particular numbers." Here is another lousy picture of the result with all the variables.

In the end, this matched up well with many of my curricular goals. I like to move slowly and deliberately through the content, giving students enough time to get comfortable with the objects we are talking about before stating facts about those objects. That is why I found it important to do a days worth of conjecturing. I also find it really important for students to see (and often to reproduce at some level) an explanation of where things come from.

Without these explanations students start to see math as an arbitrary collection of strange facts.

I want students to see math as a rich subject that makes sense and where ideas from one area (functions) have connections to another area (geometry).

The conflict I always have is whether I should be spending so much time doing stuff like this. In the end, we will spend 3 days talking about the graphs of inverse functions, which is a topic more "traditional" pre-calc classes will spend 10 minutes on. Is it worth it?

Setting ourselves up to be suprised

2015-11-04T22:56:00.001-05:00

Here is a short list of things I have been thinking about recently:

1. The role of multiple choice questions in assessing certain things.

For years I have been a "free response" kind of guy. I don't do multiple choice, true/false, matching, etc. My logic has been that I want to see what the students are thinking, and often times that gets hidden in a multiple choice setting.

However, I have a young coworker who pointed me toward Gradecam as a possible way to simplify my life. We spoke for awhile about the merits of multiple choice questions and somewhere in the conversation, he reminded me that formative assessments are important and that multiple choice questions may be perfect to test certain types of things.

I still stand by the idea that most calculation based problems should be free response to encourage the type of answer that I can analyze on the individual level. However, multiple choice is a good avenue to get at some conceptual ideas and it makes it easier to spot class-wide trends. One perfect example from Calculus would be giving a graph with a few points labeled and then asking which point has the largest value for the derivative. This would require students to understand the meaning of the derivative in a graphical context and then make a judgement based on that knowledge.

I can see this type of multiple choice question becoming more important to me this year. I already have a few in mind for Calculus and the beginnings of ideas for Geometry. It may take a bit longer for me to find something for Pre-Calc. (On an unrelated note, even after 11 years of teaching Pre-Calculus, I still have trouble finding the overarching conceptual truths I am hoping to teach. It still feels like a bag of unrelated tricks that get used to scare students from higher level math.)

2. The value of really listening for student thinking during class time.

I know that I should be listening more than talking most days. Of course, I find this much more difficult to pull off than I would hope, but it is something I am working on. However, I need to remember to listen to student conversation or read student work with an eye toward formative assessment. Too often I think about what students have done wrong when I should be looking for how they are attempting to make sense of the problem. This was partly inspired by this tweet.

Think about students as sense makers compared to mistake makers by listening to their thinking thanks to @davidwees #bsd100math
— Anna Van Asselt (@annavanasselt) November 4, 2015

This line of thinking was also inspired by a great talk by Michael Pershan. He spoke to the Global Math Department yesterday about how to improve as a teacher and the importance of reflecting on our teaching. I walked away from that talk with the idea that it is important for me to make a prediction about how I think students will perform, or the way in which students will solve a problem, or the types of mistakes they will make. The reason for this prediction is that it makes it more likely that I will be surprised, which will make it more likely that I will learn from the experience and grow as a teacher.

3. The role of personal relationships in this messy business we call education.

The following happened just a few days ago. Student didn't turn in his assignment when it was due (he was one of many, as usual). Later that day, for reasons that had nothing to do with his missing assignment, I spoke to him about life and what he was up to and other non-Geometry things. A short while after that conversation, he asked me about making up his missing assignment. While I can't be sure, I find it likely that the interest I showed in him was the main catalyst in him showing interest in my assignment.