# Category Archives: Big Ideas

## New Blogger Initiation 4

I didn’t get to do as much with this one as I would’ve liked, but I chose:

1. Read another blogger’s post for the Math Blogging Initiation. Write a comment on their post.

Back in round 3, I almost responded to:

5. Statement: “Algebra 2 and Precalculus are a hodgepodge of ideas.” If you agree, what are some unifying and fundamental themes/ideas/concepts that can frame these courses so they can designed to be less of a mess and be something more coherent.

I’ve read a few of the responses to this one in my reader (but I’m getting hopelessly behind so I’m sure I’ve missed many more) and particularly like the responses by Bowditch’s Apprentice and Compact Spaces. I feel like the prompt is a reference to A Mathematician’s Lament by Paul Lockhart. I mostly agree with Lockhart’s main ideas and plan to share the opening analogy, a nightmarish method to teach music and painting, with pretty much all of my math classes in the future despite the fairly harsh criticism of the average/typical math education.

Here’s the table of contents for an Algebra 2 book from a major publisher (you won’t be able to tell which one; they’re all about the same):

1. Expressions, Equations, and Inequalities
2. Functions, Equations, and Graphs
3. Linear Systems (includes a little bit of matrices despite the chapter title not mentioning it)
5. Polynomials and Polynomial Functions
6. Radical Functions and Rational Exponents (also includes some advanced function concepts: composition and inverses)
7. Exponential and Logarithmic Functions
8. Rational Functions
9. Sequences and Series
10. Quadratic Relations and Conic Sections
11. Probability and Statistics
12. Matrices (why isn’t the lone section of matrices from Ch. 3 included in here?)
13. Periodic Functions and Trigonometry
14. Trigonometric Identities and Equations

After chapter six, you may prefer some other shuffling of the chapters (I’d probably go something like 1-7, 9, 13, 11 as ‘musts’ for Common Core Algebra 2 … then use any remaining time on 10 {even though I don’t care much for conics for some reason}, 8, 12, 14 (harder trigonometry is definitely more ‘Pre-Calculus’ than ‘Algebra 2’ at that point in my opinion) in that order, but I would love to have this book as I’m currently book-less. I basically refuse to use our Algebra 2 book; it’s a ‘classic edition’ that was adopted like 7 years ago and was already old then – a teacher who was new to the district got to pick and chose what they knew they liked and then left two years later. He wanted ‘no tree frogs’ (his version of dogs in bandannas) but left us with a book that was completely visually unappealing (no color in the book except grayscale and lesson objectives in blue text) to use with students who are digital natives.

Every chapter title is a reference to the underlying theme of sets, structures, relationships, and functions in my opinion. Maybe I’m cheating and just being too broad with my unifying or fundamental ideas though. Obvious and not-as-obvious explanations (by the way, obvious is an extremely dangerous word in mathematics – I personally detest it almost as much as variations upon “the proof is left as an exercise for the reader” – thanks scumbag mathematics PhD):

• Chapter 1: the “algebra 1” they’ve probably forgotten – emphasize solution sets
• Chapter 2-8, 10, 13: basically have graph or function in the title … I’m only worrying about a lot of the conic section stuff because I’m theoretically legally obligated to include it in Algebra 2 – I really don’t feel like I ever really learned about ellipses and hyperbolas as the high school teachers never got to it and the college professors assumed I knew it (sad as it might be to admit that). I can complete the square but don’t know all the forms and focus business by heart.
• Chapter 9: A sequence is a function from the natural numbers to the terms of the sequence (in case you’ve forgotten)
• Chapter 11: Probability is a function of a random variable (the notation P(event) made so much more sense after I started teaching this stuff), and statistics is concerned with sets of data.
• Chapter 12: Matrices and vectors are probably the easiest algebraic structure for students to consider besides the real numbers (or subsets of the real numbers)
• Chapter 13: Trigonometric relationships abound … this is the chapter I personally would always run out of time before ever getting to in Algebra 2; too easy to put off until Pre-Calculus.

By the way, now that I’m basically finished with the post – I can merge my themes …. sets are a type of structure, and I was only including relationships to get at non-function conics. Thus, Algebra is the study of structures and relationships. Now I just need to check my work against a few more of the other newbie blog posts that I haven’t gotten to read yet …

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Filed under Big Ideas

## New Blogger Initiation 3

or “Why Two and Two Makes Fish”

Almost out of time yet again (so glad this was timed to coincide with the start of school), but I suppose part of the point is to see if you can handle reflecting as you go during the school year. My choice for week 3:

1. Introduce and show the solution to a math problem that you particularly like.

When I read this prompt, I thought of one of my favorite random teaching tangents (finally worked that in!); I have a few things I try to work in to each class whenever I need to kill some time a student asks a question that I can address with this idea/concept/problem. Unfortunately, these favorite mini-lessons are often not exactly tied to any particular required content, but I’ll have students a year or two later remember these and not much else.

So here’s a somewhat paraphrased, somewhat fictional account of one of my favorite problems and its solution:

Student: I liked math when it was easier; 2 + 2 = 4 is always true … why can’t algebra always work the same way?

Teacher: Well actually, you really need to remember math is completely made up by humans. I mean 2 + 2 = FISH is honestly just as valid if you know what you’re doing …

Student: Stop joking around …

Teacher: Let me show you – but you have to be willing to let me bend the rules and change the meaning of a few things. Math is a game; when you know the rules well enough, you know how to bend, break, or even make up your own rules.

I’m going to almost use normal addition, but you’re limited to combining four symbols: 1, 2, 3, and FISH (Greek alpha). First, I need to tell you that FISH is sort of like zero but not exactly. Also, I’m going to use ‘circle-plus’ since this isn’t quite normal addition …

<scribbling on board>

Student: … um, isn’t that the exact same thing as usual???

Teacher: so far yes, but how can we complete the table without using any new symbols (only FISH, 1, 2, and/or 3), and the table be consistent – it has to make sense

<student suggestions, teacher prompting/questioning, more scribbling>

Student: are you just making this up?

Teacher: I already said I’m just making it up, but it has to make sense! Okay, let’s try another one … with less numbers and multiplication instead of addition … yeah, this should do it … we’re going to use the symbol i; there’s a rule that $i^2=-1$ by the way {yes, I can sort of use LaTex}

<scribbling, questions, more scribbling>

Student: Okay, you can play games and move around squiggles on a piece of paper just so … what’s the big deal?

Teacher: You happen to skateboard, right?

Student: Yeah, so?

Teacher: Come here … face the class; this is position zero. Show me a 180° … good, reset then show me a 90° … okay, same thing but 270° … fine, 360° … wait that’s the same as the starting position?

Student: duh

Teacher: Let’s make it interesting then … let’s start build tricks or turns on top of one another … show me a 90° followed by another 90° without a reset.

Student: 180° of course

Teacher: Reset, then show me a 180° followed by a 270° … you might want to sit there and actually work through the turns.

<student attempts, teacher helps, asks for a few other examples if necessary>

Teacher: So tell me what we just figured out …

Student: Well, you can kinda sorta add angles together but if you get to 360° you start over – it’s the same as 0° in a circle.

Teacher: Great, now consider a square in the coordinate plane … it’s basically like the skateboarding stuff we just discussed?

<scribbling>

Teacher: We’re just adding angles of rotation together, so let’s make a table … you should be getting the hang of this by now … work with a partner

<work, work, work>

Student:

Teacher: Great, notice anything yet?

Student: Not sure … this table starts over like the FISH stuff?

Teacher: Yeah! You’re getting there – can’t we really just think of these angles as multiples of 90, though?

Student: a 180 is two 90s … 270 is three … 360 resets to zero …

Teacher: You’ve just about got it … look at all three tables … color code them if you have to …

<student looks it over>

Student: The pattern is the same on each table isn’t it !?!!?!?!!?

Teacher: Awesome! The relationships are the same even though the ideas seem completely unrelated. On the first one, I used FISH instead of zero, because I wanted you focusing on relationships to get started. This problem demonstrates the basic concept of something called a group. You were just working on college-level math by the way – the kind for math majors even. We’re going genius-level in here …

Student: You’re kidding me – that seems easier than what we’re working on now!

Teacher: It turns out once you make it past calculus – math is mostly things you already know how to do but more abstract and even more interested in the ‘why’ than the ‘how’ … too bad so many people get turned off by the tedious calculation bits along the way

Student: This was the best thing we’ve done all year – it makes more sense than a lot of the other stuff …

Teacher: Remind me to show you the one about the empty trashcan, empty bag, and empty bottle not being empty anymore sometime … we should probably get back to factoring quadratics before class is over

Student: Yuck

Teacher: I know, I know … but this is a topic guaranteed to be on the state-sponsored high-stakes exit exam.

[Partially due to internet issues this is now over an hour ‘late’]

Filed under Big Ideas

## New Blogger Initiation 2

This is going to be a quick one (until I start rambling). I’m tired, this is “due” in about 2.5 hours, I’ve spent most of the last 10 days getting back into school mode and fighting with technology in my spare time, and I don’t even care that this was an entire paragraph’s worth of run-on sentence.

Here’s the prompt I’m choosing:

1) Find one worksheet or activity or test or unit or question or powerpoint slide or syllabus or anything that you are proud of. Share it.

Here’s my pride and joy …

(and the guided notes to go along with it … ya’know, if you’re into that sort of thing)

I can’t tell you how much time and effort have gone into creating that one 40ish minute video. I’m proud to say I’m following through on my promise to myself to try different things this year; we’ve been in school 10 days, and I’m yet to really teach anything from the Common Core. We’ve done some team-building, some activities, some pre-tests (paper-based Algebra 1 and ALEKS), some almost interactive notebooks, a really awesome trick you into agreeing with the idea of ‘flipped classroom’ (turns out most students really connect with the idea of “why are we doing the easy stuff in class and then sending you home to work on the hard parts by yourself?”), some not quite interactive notebooks (with bonus not quite foldables paper-folding!), talked about how grades will work once we actually have some, and spent TWO-and-a-HALF DAYS watching that video together last week with frequent pausing. By the way, there is something quite bizarre about watching yourself teach your class in asynchronous real-time (not a paradox apparently).

Oh, and here’s the pre-test since I don’t think I’ve shared it on here yet …

(thus proving I kinda sorta know how to embed documents)

I survive the day; kids are coming by later  to pick up a DVD (I’m going to have to take a screenshot of the disc menu; I was impressed with how good free software is becoming) in a sandwich baggie since the first batch went quick. Kids are mostly flipping through the PowerPoint and trying the crossword with a decent level of seriousness. At some point, I figure out I can burn DVDs on school desktops amazingly fast when I have about two more left. I bump into him in the front office during lunch – “how’d you enjoy the experiment this morning?” I venture … <insert follow-questions and comments from the boss here>

Much later, I’m settling down to type this and notice a school email …

I cannot wait to share your flipped classroom and ALEKS hybrid with the staff.

I will probably be presenting what I’m trying and how it’s working at one of the first few faculty meetings. Maybe he’s just doing his job and following up on a class visit with a short email, but I’ll actually go with optimism for the moment – I deserve it in exchange for all the sleep I’ve been giving up.

PS – I’m sure I’ll realize this was quite the hot mess when I re-read it later.

Filed under Big Ideas, Getting Started, Technology

## Time for the Obligatory Dan Meyer Post (and more!)

I was at a conference two weeks ago now (thought about this then but just getting started and such), and we watched Dan Meyer‘s TEDtalk during a breakout session on STEM education. The presenter prefaced the video by saying she didn’t feel like she could make the case any better than Dan and asking those of us in the audience to raise our hand if we were familiar with him and his work. I was surprised when I was the only person in the room who raised their hand – in fact, I kind of blurted something like “C’mon, I can’t be the only person!” to the other fifteen or so. Later, I gathered that most of the other teacher were actually science teachers, but I was still a little surprised. At the time of writing, the video is creeping up on a million views on TED.com (not sure if that includes the views from their official YouTube channel or not). Personally, I consider this essential viewing for any other math teachers and have shared it with several others. In case, you’ve somehow managed to find this post without being aware of Mr. Meyer’s blog:

I’ve now watched this at least five times and still find it interesting; now if I could just perfectly internalize his approach!

I want to paraphrase something that a science teacher said to the group that I wish I had been able to capture the exact quote…

I feel sorry for the math teachers; they are downtrodden. … extra meetings to create plans to meet goals this time and stress from supervisors constantly looking over their shoulders … math test scores are almost always the lowest. … Some kids just aren’t prepared, and you work really hard to get them as far as you can even if it won’t be enough to ‘pass the test.’ … You don’t hear it enough, but you’re making a difference.

Not an exact quote, but it gives you the flavor of her sentiment. I appreciate her comments, but I’d rather they weren’t necessary. The work being done by the best blogotwittosphere is pretty amazing. I’ll try hard during this next school year to be more like some of the teachers that have chosen to make themselves online leaders (i.e. they are putting their content and thoughts out there to be ‘followed’ or ‘subscribed’ to …). It won’t be easy, but I want to be more effective at helping to answer the question “Is Algebra Necessary?” with a resounding ‘yes, and let’s make it a less miserable experience while we’re at it.’

One more fairly keen insight from that particular breakout session before I forget about it …

The presenter passed out strips of paper with ‘habits of mind’ printed on them. We were supposed to sort them out groups “Mathematics,” “Science,” and “English/Language Arts.” Here is the list; try to sort them for yourself:

• Determine the meaning of symbols and domain specific words
• Integrate information expressed in words with visual representations
• Ask questions and define problems
• Make sense of problems and persevere in solving them
• Construct explanations and design solutions
• Reason abstractly and quantitatively
• Construct viable arguments and critique reasoning of others
• Engage in argument from evidence
• Distinguish among facts, reasoned judgment in research, and speculation
• Plan and carry out investigations
• Analyze and interpret data
• Develop and use models
• Model with mathematics
• Obtain, evaluate, and communicate information
• Compare/contrast experimental results with information text
• Cite textual evidence to support analysis
• Use appropriate tools strategically
• Attend to precision
• Follow precisely a multi-step procedure
• Look for and make use of structure
• Analyze the structure used for organization and to enhance understanding
• Look for and express regularity in repeated reasoning
• Use math, information/computer technology, and computational thinking

I discovered pretty quickly I was fairly terrible at this activity. I have read the Common Core Standards for Mathematical Practice several times, but I do not have them memorized. I wasn’t accidentally grabbing just science statements either; I wanted to include a few from English as well. For any readers who might want an answer key:

Common Core Standards for Mathematical Practice

1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.

The science standards were slightly modified (verb tenses):

Next Generation Science Standards Key Practices

1. Asking questions and defining problems.
2. Developing and Using Models
3. Planning and Carrying out Investigations
4. Obtaining, Evaluating and Communicating Information
5. Constructing Explanations and Designing Solutions
6. Engaging in Argument from Evidence

She modified some part of the Common Core English Language Arts to create the other red herring statements, but I’m not as sure about the specifics of what and how on these. [EDIT: I think this is the source.]

Of course, the point is that academic domains are artificial creations by humans; the core skills (some refer to them with buzzwords like Habits of Mind or 21st Century Skills) that we really want our students to develop long-term are not mutually exclusive clumps that can be easily labeled.