Sunday, April 30, 2017

My Experience Writing a Conference Proposal for NCTM

This year at NCTM in San Antonio I realized that I have reached my 25th anniversary of NCTM membership. For twenty-five years I've attended inspirational talks, read well-informed ideas, and engaged in important conversations. And each year I think to myself... "maybe I could contribute something." 

Well, this is it. I've officially submitted my first conference proposal for NCTM 2018 in Washington DC. The process of writing the proposal was grueling but undeniably useful. A couple things I've learned? It is SO difficult to:
  1. fully develop an idea now for a potential workshop one year away... so that it feels fresh and current both now and then.
  2. write 'explicit' and 'specific' descriptions that are engaging and thorough, and within the character limits defined by NCTM.
  3. appreciate just how important it is to focus on struggling and under-represented learners.
Thank you to Robert Kaplinsky and Dan Meyer for your professional leadership and to my friends, family and colleagues for your motivation, support, editorial advice, and encouragement. We'll see what happens, but the proposal has been submitted and here it is:

"From Number Lines to Logarithms: How Forgotten Instincts Can Spark Deeper Understanding"
Research suggests that logarithmic thinking is innate. We are born with a number line in our heads, but the numbers are spaced in equal RATIOS instead of equal distances. We will explore how reigniting this instinct can pave an enlightened pathway from number properties and fractions, through ratios and means, to rational exponents and logarithms.

Write the participant learning outcomes of your presentation, including an explicit description of what participants will learn. Please also provide an overview describing how time will be allocated during this presentation.
In four parts, this session seeks to answer these guiding questions:
  1. Introduction - Given a number line from 1 to one billion, where is 1 million? Where would your students place it? What research-based implications can we make? How can we turn our confusion into flexibility and sense making around additive and multiplicative mindsets?
  2. Foundations - How does this flexibility illustrate basic number properties? How does it change our understanding of fractions? How does your vision of later connections impact the way you teach foundations?
  3. Connections - How do ratios compare to fractions? How does our new flexibility change the way we think about "the middle" and "equal sharing"? How do factors and terms help us recognize what kind of thinking is required?
  4. Breaking Barriers - Can we use our new flexibility to break down the confusion surrounding exponential models, logarithms, rational exponents, inverse functions, geometric vs arithmetic sequences and means... and more?
What is the key mathematics content that is a focus of this presentation?
This presentation is about how exponential models/patterns have many connections and similarities to linear models/patterns. Noticing and celebrating these similarities helps make sense of: 
  • Additive and Multiplicative Number Properties
  • Fractions and Ratios
  • Linear vs Exponential Models
  • Logarithms as Inverses and Related Facts
  • Integer and Rational Exponents
  • Arithmetic and Geometric Sequences
  • Arithmetic and Geometric Means

How does your presentation align with NCTM’s dedication to equity and access?
I attribute part of my success in mathematics to an occasional willingness to accept that “it will make sense later.” This kind of persistence through adversity in mathematics is admirable, but it also pushes the limits of student engagement and promotes boredom and withdrawal. My presentation is rooted in the idea that logarithmic thinking is innate. Embracing this instinct makes sense-making and flexibility with multiplicative and additive mindsets accessible to all... now, not later.

Thursday, September 8, 2016

Magic Doesn't Always Happen in my Classroom, but When it Does...

I used an activity this week where I introduce the method of using Lowest Common Denominators to eliminate fractions and decimals in 'tricky' equations. On the activity page, I refer to this trick as a 'good idea,' but in class I said, "It's like a magic trick: I wave my LCD wand and presto, no more fractions!" They laughed, so I kept it up.

"Ack. This one looks yucky. I need some magic." Snicker.
"See, here's where the magic happened." Chuckle.

At one point, a student raises his hand and says "You know Mrs. Allman, it's not really magic, it's just logic."

Nice. But aloud I said, "You know what you are? A muggle."

Jaws dropped. Did she really just call him a muggle? Putty in my hands now.

The next equation had some decimal coefficients: 
1.2x + 0.4 = 7.6

I asked, "What's the LCD?"

Someone said, "There aren't even any fractions." (I swear, he wasn't a plant.)

I spun around and looked again, "There AREN'T?"
So I waved my hands and said, "What's this number?"
"One point two."
"Didn't your 5th grade teacher tell you never to say the word 'point' in math class? How are you supposed to say it?"
"One and two tenths."

There was an audible gasp from the class. I cannot make this up.

So now they're fighting over whether 5 or 10 is the LCD of this group (cool) and whether it even matters, and which is more magical, and I could NOT have planned this better. At one point, a student shows me his work "Mrs. Allman, I used a different number to multiply both sides and I still got the right answer. It doesn't matter what you use."

"Ah," I said, "And that's a beautiful thing. You've confirmed one of the properties of equality: that you can always multiply both sides of an equation by ANYTHING, and it will not change the solution. The magic comes from knowing which numbers will make the decimals (or fractions) disappear, and (hopefully) make your life a little easier."

There might still be pixie dust on the floor.

Wednesday, November 4, 2015

Powers, Roots, and Logs are Related Facts

This year, in my Algebra 2 class, I prefaced our individual function units with a overview of functions in general. One thing that happened is that before we studied exponential functions, we had a decent understanding of inverses and how several functions are related in this way. Several weeks before I ever needed to hint at the existence of logarithms, the students saw the need for an inverse to an exponential function and also were stymied by the relationships that are already comfortable to them: namely, the existing inverse relationships between powers and roots.

So this month, when we got to the middle of our exponential function unit, I decided to present logarithms as a group of THREE related facts in a fact family.

We listened to a fabulous Radiolab program that presents numbers and logarithmic thinking as a human interest story. I am so grateful for programs like this that do my hard work for me!

Then we talked about how powers, roots, and logarithms are all different ways to say equivalent things, while each highlighting a different feature.

Fact families are something that students are familiar with. No one batted an eye.
When I asked them what is meant by logarithmic thinking, I was happy with how their explanations centered around exponents and thinking about 'how many times a number is doubled or tripled,' etc.

We concluded with some fact practice by using fact triangles and naming the three related facts. I made these awesome octahedral dice with fact triangles in base 2, base 3, base 4, and base 5. Their job? Roll a die and record the three facts that can be written from the trio of values.

The students whipped through it, never complained, and had 100% accuracy on our quiz the following day. I'll count it as a worthy addition to my filing cabinet.

Wednesday, October 7, 2015

Algebra 2 is All About Exponents

I don't know if this makes function families easier or more complicated, but I realized this week that everything we cover in Secondary Algebra (1 and 2) can be reduced to two basic function families: f(x) = n^x and f(x) = x^n (trig functions excluded, but we don't cover those until precalc at my school). I don't remember this ever getting pointed out to me when I studied functions, and seriously, how many years have I been teaching this? I just find it very interesting.

We made these organizers in class today (and yesterday). We've already got our brains wrapped around transformations and compositions, although we have thus far stealthily avoided talking about operations on functions (adding/ subtracting and multiplying/dividing)... other than to say, "Ugh, two x's, that looks messy." Which of course, it IS, right? And isn't that the point?  At least, I think that's my point this time through. Basic functions that involve these families, simple transformations, and compositions where one step follows another in a specified order... this kind of function is not hard to work with. They are logical and orderly. It's only when we start multiplying and dividing or adding/subtracting functions that stuff gets tricky and we need to start pulling out new tricks like factoring and zero product rule, and complex numbers, and extraneous solutions, and reducing rational expressions, and limits, oh my.

Tuesday, October 6, 2015

Why aren't exponents and roots always inverses?

We've been studying functions in my Algebra 2 class. I'm taking an entirely new approach this year. It's going well, but the jury's still out. So far, one thing I'm really happy about is their excellent grasp of inverse functions as a process of 'undoing' whatever the original function did. Today in class we were organizing our thoughts about the different function families. We'd drawn out some nice examples of expoential growth and decay and I asked if we could figure out the inverse functions. I EXPECTED them to see that they could not write a function (we haven't done logs yet), but that they could use the tables and graphs that they just made to draw an inverse graph. Our function was a simple "2 to the power of x."

But they surprised me. "The inverse would be the xth root of 2," someone said.

Several agreed. I was dumbfounded. Why not?

After all, isn't it TRUE that:

for all real values of n?

That is, I know there are some issues with this as a blanket statement. For example, even values of n only work if we restrict the function domain. But for the most part, this is entirely true and logical.

So why is it FALSE that:

Maybe you, like me in class today, are scratching your head now and wondering... could that be right? It isn't. We checked. Choosing an input for the original function, applying the function and then applying the inverse function will not return us to our original value. But WHY?

Help me out here. Can anyone provide a purely sensible argument for why this will not work? Not just a demonstration of HOW it doesn't work. I can supply several of these now. I want to know why.

Monday, September 28, 2015

Number Line Movement as a Function Intro... Continued

I've been using a new approach to introduce functions to my Algebra 2 students (who presumably aren't completely new to functions). As much as I can, I'm trying to let need dictate the math. At this point our definition is somewhat incomplete, but working well so far. Today we see a need for more explicit notation, because it's not always clear to us what exactly the function rule is:

I was amazed how quickly they got comfortable with function notation, given this set-up, but I am willing to accept that it may be partly due to previous teachers and previous foundations.

From here we jumped right into the idea of function compositions (but not composition notation, because we don't need it yet):

It was entirely logical to follow these with a discussion of inverses that are a bit more complex:

Here are two samples of their student work at this point. There still seems to be some ambiguity around the word opposite (a great entry point for an upcoming lesson). Beyond that, however, their understanding appears to be rock solid. Everybody did a nice job of explaining the function. Only one got mixed up on defining the inverse. A teacher's dream.