# Tag Archives: learning

## On Curriculum: What Doesn’t Work

For some reason I feel compelled to write about teaching even though I just quit teaching. Quit? Yes, I resigned from my job this year and am taking a break from teaching high school math. Why did I do it? Honestly, I don’t think I can adequately articulate it, and I don’t owe anyone an explanation, but quite simply, I needed a break. I have a sneaking suspicion that I shall return to teaching again some day (probably sooner than I realize), but in the meantime, I’ve been pursuing some of my other passions and working on acquiring some new skills.

I’m still very interested in being part of the conversation on teaching high school math, and I still use Twitter every day to keep up with it. In fact, I have so much to say on the topic that I figured I might as well blog about it. I hope that blogging will be cathartic for me, helpful to other teachers out there, and helpful for me if/when I return to teaching.

So that was quite an introduction to a post in which I wanted to talk about curriculum.

I’m inspired to write about curriculum because the school where I taught had such a horrible, out-dated curriculum, and it was a huge burden for me. I’m pretty sure the curriculum pre-dated my own high school years, so I was shocked that I was required to teach it to my students. It was the most rote, procedural, and repetitive mathematics that I have ever come across. It made me think of the Cold War era, which I actually don’t really know anything about as I was born after that time, but if I could imagine it, I imagine different countries putting their young people in little school factories to see who could solve equations by hand the fastest.

Such was the imagery in my head because the entire curriculum at my school was built around solving equations algebraically. Here is the procedure for solving quadratic equations. Here is the procedure for solving exponential equations. Here is the procedure for solving trigonometric equations. And so on. Naturally, this led to an incredibly teacher-centered classroom. For each lesson, there were pages of notes that the teacher talked about. Then the teacher did some examples. Then the students were supposed to mimic the teacher exactly on a worksheet of 25 identical problems. It was brutal. I felt so sorry for… everyone involved.

Now, I don’t mean to say that we shouldn’t teach solving equations. The concept of what it means to solve an equation is a fundamental part of mathematics. During my first year of teaching I quickly realized the lack of conceptual understanding my students had as a result of our pathetic curriculum. Our assessments would be filled with equations to solve, but not a single student could answer the questions: What does it mean to solve an equation? What does it mean if a number is a solution to an equation?

When I discovered this discrepancy, I just felt terrible. Why were we making students do something that they didn’t understand? Hey kids, memorize exactly what the teacher did, regurgitate it on an exam, and then do it again. There’s no need to understand it. Heck, you can get an A+ grade without actually understanding anything. I quickly realized that no genuine learning was happening. It was sad.

My last two years of teaching I incorporated the two italicized questions from above into the first non-review unit almost every day. (Yikes, don’t get me started on how our curriculum wasted the first unit of every year on “review”.) Last year, I finally had more students answer the first one with something along the lines of “find the values that make the equation true” than students who said “IDK” or “get the answer”.

Besides a lack of understanding, our curriculum lacked efficiency and modern technology. Before becoming a teacher, I was first and foremost a mathematician, and I assure you that no mathematician was solving by hand some of the equations we made our students solve by hand. Mathematicians use technology. If I were to come across an equation that I knew I could solve by hand, but that would take me more than 60 seconds, I would turn to my computer or pick up my Iphone and use Wolfram or Desmos to find the solutions and then carry on from there. I don’t waste my precious time doing a rote procedure when a computer can do it so much faster. I spend my time on bigger and better, more important and more relevant mathematical ideas. Our students should be doing likewise.

Again, I’m not saying that students shouldn’t know how to solve equations or that mathematics is purely conceptual. However, telling students to memorize a specific procedure isn’t that important or that useful. Rather, let students explore equations, find methods that work for them, and develop fluency. As they do this, they will acquire and practice important mathematical skills such as: manipulate equations, model with equations, create different representations of equations, and solve equations efficiently.

Finally, this type of curriculum is also very boring. Be silent. Watch teacher. Work in isolation. No creativity. No thinking. No discussion. There is a complete lack of genuine student engagement. Those poor kids. It’s no surprise that so many dislike math/school. And poor teachers! Our curricula can set us up for failure or, at least, prevent us from seeing the successes our classrooms are capable of.

Wow, am I still complaining about my old curriculum? It appears so. I guess I had to get something off my chest. I should probably stop complaining and maybe write a post called On Curriculum, Part Two: Making it Better. Although I could probably write a whole book on that topic. Now there’s an idea…

Filed under curriculum

## End of 3rd Quarter

Hi everyone.

It’s the end of 3rd quarter, and we’ve got a grading day. Actually half-day. So I thought I should blog since I got the time! No kids! It’s some sort of miracle. I’m fortunate at my school to have a prep period and a department planning period… but during my prep, I can expect to supervise 8 to 10 up-to-no-good-but-so-lovable seniors. Up-to-no-good is definitely putting a positive spin on it. During my plan period, I can expect to supervise two or three accelerated freshmen for whom school comes easy and are mostly bored with it, and two or three sophomores who care, but need me to give them 1-1 tutoring in Geometry.

In short, having some time to myself in my classroom is some sort of miracle. It never happens. Now if only I had something interesting and substantial to blog about.

The kids are the best part of the job though. Forget grading, planning, and prepping. I do what I do because I believe in those little punks. They’re beautiful, lovely, funny, and smart. They deserve the best.

OK, here’s something worth blogging about. I just taught right triangle trigonometry to my Geometry kiddos. I love introducing trig. This year it conveniently followed a similarity unit, so I introduced it with a quick lab measuring sides of triangles and computing SOHCAHTOA ratios. Huh, weird, for any 30 degree angle in a right triangle, the ratio of the opposite side and the hypotenuse is the same. Huh, weird. (Similar triangles, anyone?)

Then we do some boring, but straightforward practice. Then the next class we go on a field trip. I love to advertise this next bit as a field trip, even though we only go down two floors to the Commons.

I start by having them estimate the height of the ceiling in the Commons (we regularly do Estimation180 in Geometry). Then I have them take out their telly-phones and download a free clinometer app. The only issue is the kids who say, “but I don’t have any room on my phone”. Maybe if you deleted some of those dang selfies, kid.

I crappily, but enthusiastically, model what they’re supposed to do. (My teaching career is a work in progress, OK? Year two is better than year one, at least.) I pass out the awesome, giant tape measures that the math department owns. We disperse down to the Commons. Chaos ensues, naturally, but we’re on a field trip in math class, so it’s a good thing. Tape measures are being stretched out, kids are pointing their phones at the ceiling, and most kids are sketching a triangle and writing down some sort of trigonometric equation. It’s my favorite day of the year.

Eventually we return to the classroom. My least-focused kid (one of those with an ADHD star next to his name in Infinite Campus) happily sits down and gets to work solving trig equations. How could he not? I just let him run around the Commons for 10 minutes.

They’re beautiful creatures, ya feel?

8.4 trig lab

8.5 trig invest how high is ceiling (I think this was adapted from something from Tina Cardone @ drawingonmath ??? Not sure. But I definitely stand on the shoulders of giants. Thank you all.)

Filed under fun, Geometry, grading, trig

## The teacher learns

I’m getting better at making my expectations clear. Giving quick, short directions right away and repeating them until all students are with me sounds obvious, but it’s easy to move on without some kids and then you never really get them back.

Always be one step ahead of the kids. Pass out and explain the next task to the kids before they start their quizlet so that kids who finish early have something to do.

I freaking love warm ups. Haven’t figured out a system for them yet though. Should I preprint the questions on a half sheet? Should I grade it? I think the answer is probably yes to both of those questions, but I don’t love the idea of using more paper or having more things to grade.

Graphic organizers are great. A few phrases in a few boxes is more writing than we usually do in math class. They work on it individually, then in groups, then I solicit answers and go over it as a class.

That reminds me: cold-calling = awesome. I have cards with student names on them that I use. Open ended questions or questions with more than one right answer (give me one of the transformations we’ve talked about) are best.

I’m almost half-way done with my first year! The lows have been low, but the highs have been high, and I keep reminding myself just to be better than I was yesterday. Always learning.

Filed under classroom management, productive struggle

## Making Corrections is Valuable

I love, love, love having the kids make corrections on quizlets (formative assessments) and tests (summative assessments). It requires them to actually look at my feedback and to maybe even learn from their mistakes.

For corrections on the last test in FST, I also had the kids write down one thing they’re proud of or one thing they thought they really learned. I got some great responses back, and I hope it helped remind them of what they did well rather than just focusing on mistakes, so I’m glad I had them do that.

Sample responses:

“I did really well on my factoring. I was worried about it and it went better than I thought. Happy about it.”

“I did well on the quadratic formula.”

“I liked Part 1 because I didn’t get any points off, and I did good with my negatives.”

“Last year I feel like I didn’t get a single quadratic problem correct. I feel like I understand them a lot better this year.”

“Factoring went well and I really have cemented the material in my brain.”

“I learned how to graph equations and find the x-intercepts.”

“I think I really mastered the factoring aspect of this unit.”

“I showed my work.”

“I slowed down this test!”

“Overall, I did OK.”

“I did well at taking my time and going through my work.”

“Test was easy but J. did better than me, so I’m salty. I’m over it. Otherwise, test went pretty well. What really helped was coming in and reviewing with you. Thanks, Ms. Cummins.”

Sometimes these kids drive me crazy, but sometimes they can be thoughtful and serious and make me proud.

Filed under formative assessment, FST / Algebra 2, grading

## A Flop

Had a pretty big lesson flop today. I was frustrated at the end of the particular class, but I’ve thought about some things I can do to improve. Fortunately, I’m on an A-B schedule, so I get to re-do the lesson on Monday with another class. (My A-day kids always get the flops.)

I’m almost too embarrassed to write about the lesson because so much was wrong with it. I am tweaking every part of it for Monday. I thought about scraping the main task altogether, but it’s a good task that was ruined by poor implementation.

First, I am not going to assume the students remember how to do something even though I know they studied it last year. Flying through one example and saying, “This is familiar, right?” isn’t going to cut it. It turns out what they learned last year was a “trick” anyway, so I definitely need to do a better job explaining explicitly what is happening conceptually.

Second, I am not going to throw a handout at them and expect them to get to work. I am going to do a better job explaining the task and modeling how they should get started. I just read the phrase “model curiosity” while surfing some blogs, and I think it perfectly describes what I need to do at the beginning of a task.

Third, I’ve got to follow through on my behavior expectations. I had too many non-participating, off-task students. Worst of all, I let them behave that way. I let them get out of their assigned seats. This is my problem. I don’t like telling people what to do. I just want them to do the right thing. But I have to remember that high schoolers are still kids, and they still need guidance. Basically, I’ve got to toughen up. I’ve got to enforce my expectations.

Fourth, I want to do a better job structuring group work. I think this will also help me with my classroom management issues. I think I need to bring the groups back for a whole-class check-in more often. If there are four parts to the task, then I think I should bring everyone back together to go over each one before we move on to the next. In contrast, today I just said “do it” and consequently lost a lot of people, who never came back when I tried to go over everything at the end. So, on Monday, as students make progress on part 1, I’m going to bring us back together and have groups share. Then I’m going to explain part 2 and let them go. Then I’m going to bring them back again for a whole-class discussion on part 2. Then I’m going to explain part 3, and so on.

The tricky bit will be bringing everyone back. They’ll want to keep talking to their friends, but I need them to pay attention to me or whoever is sharing. I really need something to get everyone’s attention back. Maybe a timer, but students might work more slowly or more quickly than I anticipate. Another new teacher, who is in the English department, shared with me her method for bring everyone back. She simply says, “I need everyone back up here in 3.. 2… 1.” That sounds magical to me.

I can probably pull it off. I can do anything, right? I think what will work for me and for my students is to explain to them at the beginning what it’s going to look like. I will explain that I will let them work on part 1 for a bit, but that when I say “I need everyone back up here in 3, 2, 1” they need to stop where they are, turn to the front, and listen because we are going to share ideas at that point.

Overall, I think I need to be a better communicator. Specifically, I need to be more explicit with my directions and my expectations. More explicit with some of my explanations of content would also be good. Again, these are kids, not adults. They are learners, not experienced mathematicians. They are relying on me to communicate well.

1 Comment

Filed under classroom management, group work, planning

## My Automathography

I’m taking Justin Lanier’s smOOC called Math is Personal, and one of our first assignments is to write our “automathography”. So here’s mine. Enjoy!

Mary’s Automathography

I love math, but I didn’t fall in love with it until college. I was good at math in high school, but I was good at all my classes, so nothing stood out about math in particular. I definitely had a fear of getting the wrong answer in math class, and I was happy to just follow the procedures given to me by my teachers. At this point in my life, I don’t think I understood what mathematics actually was. I won the conference quiz bowl in math my senior year, and it was great to get that recognition, but I graduated high school thinking I would study chemistry in college.

I soon discovered that I did not enjoy working in the lab, but that I did enjoy my math courses, so I ended up majoring in math. I went to a huge university (40,000+ undergraduates), so my first two years of math classes consisted of lectures with 300 students. Despite this, I found myself completely inspired by the professors. I was enamored with how passionate and genuine they seemed. In other subjects, I felt like the professors and TAs were egotistical or arrogant. In contrast, everyone in the math department seemed friendly and easy going. I’ll always remember when one of my calculus professors introduced Euler’s identity. His voice wavered, and I thought he might even cry when he described how this one equation related the most important numbers in mathematics.

Even those first few years of college, I was still focused on answer-getting. This quickly changed when I started taking courses like Real Analysis and Modern Algebra. In these classes, I was finally challenged to think for myself. There were no recipes to follow, and it was completely up to me to decide how to prove or demonstrate something. It was both terrifying and liberating. Math became a creative endeavor for me, and I loved it. I truly came to understand and appreciate Georg Cantor’s quote: “The essence of mathematics is its freedom.”

Besides the creative aspect of math, I also thrived on its collaborative aspect. Getting to know the other students in my classes was so much fun, and struggling with them on math problems late into the night will always be one of my favorite college memories. I also always appreciated how there wasn’t a competitive atmosphere in math, compared with most of the science classes I took. Simply put, I learned so much from doing and talking math with my peers. I became more confident and began to embody the mathematical habits of mind.

In particular, I will never forget the group I worked with in Real Analysis. The professor assigned problems every class which were due the following class (this course required more of my time than any other), so the five of us would get together almost every day, sometimes for several hours, to struggle through them. We would meet in the student union in the evenings, staying later than everyone else and having conversations about math or maybe not about math. Before class, we would meet in the math library to share any last minute insights, often getting looks from others for being too loud. Naturally, a strong bond formed between the five of us. On weekends (or Thursdays, or whenever we could no longer stand to stare at our papers) we would go out and get drinks together.

The experiences I had in classes like Real Analysis really transformed my idea of math. I learned the value of productive struggle and collaboration. I learned how to be creative in math and make it my own. I really felt mathematically strong at the end of it all.

Fast forward to the present- five years after that Real Analysis class. I am now about to start my first-year teaching high school math. I hope I don’t suck.