# For your students’ sake: Don’t stop being a learner

Yesterday, we designed an Algebra II lesson using 3D modeling to derive the factored formula for difference of cubes. As we began to finish up, Sheila (@mrssheilaorr), the math teacher sitting beside me made a passing reference to being frustrated trying to prove the sum of cubes formula. Me, being a geometry teacher by trade decided to give it a try perhaps hoping to offer a fresh perspective. I mean, I was curious. It looked like this:

On the surface, it didn’t seem unapproachable. I quickly became frustrated as well. Most frustrating was the mutual feeling that we were so stinkin’ close to cracking the missing piece. Finally, Luann, a math teaching veteran sat down beside us, commented on her consistently getting stuck in the same spot we were stuck and then, as the three of us talked about it, the final piece fell in and it all made sense (it’s always how you group the terms, isn’t it?)

Then this morning, it happened again. Writing a quiz for Calculus, I needed a related rates problem. Getting irritated with the lousy selection of choices online, I decided that I needed to try to create my own. And I wanted to go #3Act and after some preliminary brain storming with John Golden (@mathhombre) (Dan Meyer’s Taco Cart? Nah… rates of walkers not really related…) we found some potential in Ferris Wheel (also by Dan Meyer)! Between my curiosity and my morning got mathy in a hurry.

First I tried to design and solve the problem relating the rotational speed in Act 1 to the height of the red car. That process looked like this:

Meanwhile Dr. Golden found a video of a double Ferris wheel, which was pretty awesome. Seemed a little it out of my league, so kept plucking away at my original goal.

It clearly wasn’t out of Dr. Golden’s league, as he took to Geogebra and did things I didn’t know Geogebra was capable of. (You’re going to want to check that out.)

So, what is the product of all of this curiosity and random math problem-solving? As I see it, these past 48 hours have done two things: Reminded me of what makes me curious and reminded me what it’s like to be a learner.

I have a feeling my students will be the beneficiaries of both of those products. There’s a certain amount of refreshment that comes from never being too far removed from the stuff that drew us to math in the first place. The problem you want to solve just because you want to see what the answer looks like.

And this curiosity, the pursuit, it feeds itself. In the process of exploring that which you set out to explore, you get a taste of something else that you didn’t know you would be curious about until it fell down in front of you. (For example, Geogebra… have no idea what that program is capable of, which is a shame because it is loaded on all my students’ school-issued laptops…)

And this process breeds enthusiasm. Enthusiasm that comes with us into our classrooms and it spreads. I’m not trying to be cheesy, but much has been said about math functionality in the modern economy, how essential it is in college-readiness and the like with few tangible results. Let’s remember that there are kids who are moved by enthusiasm, who will respond to joy, who will pay attention better simply because the teacher is excited about what they are teaching. It won’t get them all, but neither will trying to convince them of any of the stuff on this poster.

Now, who’s going to teach me how to use Geogebra?

# High School Calculus – Update after teaching my very first semester…

Well, I am officially through one semester of my first calculus class as a teacher. Before I get into any theories or needed revisions (and there definitely are some…) I want to simply make some observations and thoughts after some reflection:

#1. There’s no substitute for having an answer key made showing all work (multiple processes, if possible) for every single handout you give. The students seem to embrace the idea that I was relearning the calculus, but they got edgy if they got a sense that I didn’t know what I was doing or I was unprepared.

#2. Borrowed and stolen resources are helpful as resources, but you need to make your curriculum your own. I found that even making small edits to the handouts I got from Sam (@samjshah) or James was enough to invest my mind creatively in them more, which made me so much better able to embrace the holistic value of each and every activity.

#3. Advanced math students are never more than one frustratingly-difficult assignment away from behaving just like their struggling counterparts. All of the avoidance, procrastination, off-task, distracting behavior we are accustomed to from the strugglers will show up in any teenager if the math makes them feel overwhelmed and intimidated. (I will admit, “When will we ever use this?” is a question I never expected to get from a student who signed up for Calc voluntarily…)

#4. Advanced Placement might not be all it’s cracked up to be. My class is not AP. Surprisingly, the students spoke rather decisively that they were, at times, deterred from AP classes (for a variety of reasons). AP ties your hands a little bit in the schedule and topics. At the very beginning, I offered an AP prep schedule for anyone who thought they might want to take the AP exam and not a single student took me up on it. And when you consider that AP classes come with questions of grade-weighting, and exams, and GPA and blah… blah… blah (the effects of which are probably overstated anyway…) It seems like making this an honest, in-depth, investigation of calculus for the sake of investigating calculus seems to create the most favorable environment for student risk-taking and teacher responsiveness.

#5. Logarithms and radicals: these two things never quite seem to settle in for students.

Now, I am keenly aware that each group is different, and should I be blessed with this opportunity next year, many of these observations could need some updating.

… but maybe not.

# Means Extremes – Balancing the Means and the Ends of Math Class

I have been seeing this play out in my geometry classes each of these past six school years. It’s been a tricky problem for me to figure out and once I started to see what was going on, it became even harder for me to communicate it. I think I am ready to try.

Each year, a fresh cohort of young people come in straight from Algebra I. In theory, I can assume that they are fully loaded with algebra skills and thought-processes that will support them through their study of Geometry. There is one glaring hole in their understanding which I attributed to the overall youthfulness of many of my geometry students. They are 13-15 years old and, for the most part, their math experiences lack a significant diversity. So, I am able to give them a pass on some of the ways they are still developing as math learners.

But then I saw the same deficiency in my calculus students that I am only seeing now because this is my first year teaching calculus. Suppose that I give a geometry student this problem and ask them to find the angle measures of each angle.

Taken from Holt Geometry – Pg 181

Or suppose I gave my calculus students this problem.

Let’s add in as evidence that the Geometry students are used to application problems that look like this:

Taken from Holt Algebra I – pg 478

And that the calculus students have spent a lot of time looking at pages like this:

Taken from Holt Algebra II – Pg 580

The primary difference between the work I’m asking my Geometry and Calculus students to complete and the work that they are used to in the Algebra I and Algebra II is in the latter the equations are provided and in the former, the students are required to write the equation.

This is no small point. No side conversation. I am not splitting hairs. I am convinced this is a big difference.

Let’s go back to our Geometry problem.
First and foremost, recognize that there is very little natural or intuitive about this set-up as a whole. There is very little reason why angle measures are represented with algebraic expressions. The variable “x” doesn’t represent any actual value and so, the students are left to their abstract understanding of how equations are built in order to solve this problem. Their previous experience hasn’t really prepared them for this. Overwhelmingly, their mathematical experience leading up to this point has trained them to know how equations are solved.

Let’s expand this to our calculus team (of 23) of whom I noticed only about 8 or 9 who seemed comfortable modeling volume and surface area with equations and then engaging the formulas. So, even among our most talented high school students, there is a problem with the use of equations as modeling tools. Once they have them, they can operate with them wonderfully, but they struggle when it comes to writing them to specifically to match a specific situation. And beyond that, checking the accuracy of the model and then making sense of the product once they are done.

Herein lies the major issue: the paragraph above highlights a variety of skills that students (at all levels, from what I can tell) struggle with. They seem to struggle with them because they aren’t practicing them. But those are the skills that actually make mathematics worth doing to EVERYONE. The ability to do complex arithmetic on a rational or logarithmic expressions is something that is going to come in handy to people for whom formal mathematics is going to extend into their post-secondary lives. This isn’t a high percentage or our students, but these skills constitute a high percentage of the problems in our textbooks.

On the other hand, being able to recognize a situation as linear, quadratic, logarithmic or rational and have a sense of how to model that in order to make some predictions? That is something that could be valuable to a higher percentage of people outside of school.

I think that we need to recognize that the specific skills that we are teaching our math students are a means, not an end. They are the tools, not the final product.

The real goal is for the students to explore a situation, recognize the mathematically significant parts and use their math tools to model the situation strategically to help them achieve their goal. In addition to our student being better, more confident, flexible and patient problem-solvers, it seems like we’d also hear “when am I ever going to use this?” a whole lot less.

# The Fear of Trying

I teach a calculus class. It’s the first time that I’ve done that. I have a story about them.

Yesterday, I introduced the idea of limits, which, according to them, was the first time any of them had ever had limits discussed in any way in a math class. After a short lecture-style introduction about the basic idea of limits, the notation, and some anecdotal comments, I unleashed them on this handout. I was not going to collect handout. I wasn’t going to fasten any points to this handout. They knew both of those things. I gave them 12-15 minutes to try the handout. This is where life gets interesting.

About 6 minutes into the time, I began walking around to look at some work. There are 23 students in the class. I estimate that 17 of them had, for as many as 10 minutes had done zero math, but had done a fantastic job of transferring the table from the handout into their notes. I was flabbergasted. They had spent 10 minutes drawing a table. These calculus students (who, by most traditional metrics represent the most confident and talented math students in our student population) wouldn’t try the math.

Their years of math class “success” had thought them two things that created this scenario: A. Stay busy, or at least look busy. Teachers get  mad at students who sit around. And B. The only answers worth writing are correct answers. If you don’t know the correct answer, don’t write anything.

These students didn’t want to write a wrong answer. They didn’t know the right answers, so they opted to create incredibly high quality tables in their notes. Their explanation for this usually included the statement, “Well, I didn’t know how to do it.”

I responded with, “Of course you don’t know how to do all of this, I introduced it 10 minutes ago. I’m just looking for you to give it your best try to see where your thoughts are taking you right now.”

After about a 5 minute pep talk, they tried, and much to the surprise of most of them, most of their attempts where actually correct answers.

And make no mistake, this isn’t just calculus. This isn’t just the traditionally-successful math students. Anyone who has taught a math class has seen students who know that they don’t know how to solve a problem and would much rather leave the problem blank than put something down that is wrong. The wrong answer seems worthless.

It seems that we have conditioned our students into thinking that an answer on a page is an opportunity for judgement. If they write something on the page, I’m going to have the final say whether it is right or wrong. Wrong answers are bad. Grades go down because of wrong answers. This mentality would prefer to leave an answer blank. At least if it is blank, you can pretend that you simply need more time.

Instead, we should be showing our students that an answer on a page is the beginning of a conversation that ends with them learning something new. It could be that the answer on the page confirms that they have already learned (which makes for a short conversation). It could also be that the answer on the page demonstrates that more learning is needed, and the answer on the page is the window into the confusion, clues to the misconceptions or the missing understanding. When learning is incomplete, perhaps the BEST thing a student can do is show us his or her very best wrong answer.

But first we have to teach our students that anytime they put what is in their heads on paper, there is value. There is value in a correct answer and there is value in an incorrect answer. It’s true that they are valuable for different reasons, but they push toward the same end. The authentic learning of mathematics.

# The Essentials of Project-Based Learning

On Thursday, I will be facilitating a collaborative session for teachers on Project-Based Learning, which, to be fair, is a topic that I have never considered comprehensively. I’ve never been asked to define it. I’ve never been asked to explain it and, quite frankly, until the organizer of EdCamp Mid-Michigan approached me to facilitate the session, I never considered myself a practitioner of it. I just tried to design lessons that were engaging and built authentic, lasting learning.

But, this EdCamp facilitator, who’s name is Tara, has known me for quite a while. We did our undergrad together. We did our Masters work together. She’s been on an interview committee that almost hired me. She’s aware of my shtick. Maybe she wants me to facilitate this through the lens of how I work. If that’s the case, then this takes on an extra degree of difficulty because I still feel like I have a lot of development left to do before people start emulating my approach.

Besides that, my approach isn’t really that complicated. For any activity I consider offering to the students, I simply try to ask a couple of questions.

1. What can I do to maximize engagement? (I’ve seen plenty of ideas that I have had flop simply because the students aren’t drawn in.)

2. What can I do to facilitate collaboration? (The most effective use of class time that I ever see is when a group of students are effectively working together to solve a problem.)

3. What can I do to create a problem that will provide a variety of ways to solve it? (If there is only one real method to solve it, then as soon as one student gets it, the “collaboration” will become that student communicating “the way to solve it” to everyone else. My favorite evidence of this is when I hear a few “I don’t get how they did it, but this is how we did it and the answer came out pretty close to the same.”)

4. What can I do to make sure that the solution(s) are approaching effective learning outcomes? (While it can be interesting to occasionally have a group find a way to solve (or estimate a solution to) a problem effectively in ways that circumvent my desired learning outcomes, if I am continually putting together problems that don’t push the curriculum the community is trusting me to teach, then my students will be missing something.)

My favorite examples of this process working well are The Lake Superior Problem, The Pencil Sharpener Problem, The Wedding Cake Problem, and the Speedometer Problem. In each case, engagement and collaboration were high. Multiple solutions were present and most of the solution techniques required the students to make sense of mathematical procedures that could be correct or could be incorrect. “Why this formula is better than that other one.” Or “neither of our answers are perfect, this this one is better because this process has less of a margin for error.” Stuff like that. These conversations provide great opportunities to move beyond the memorization of a formula and to push toward sense-making of how that formula is used effectively.

While I’m not sure if this is “by-the-book” project-based learning, I have seen improvements in my students’ learning through the lesson design process I described above. Does anyone have any advice for me? Questions for me to think about? Corrections? Obvious holes in my logic? I appreciate this community because of your willingness to share, so feel free to do so.