Sunday, March 22, 2015

Solving Exponential Equations (No Logs)

This is my second year teaching a "not advanced" Algebra 2 class. One of my goals has been to avoid simplifying or watering down the curriculum, even though my students are lacking in pre-requisite skills.

Instead, I've been taking more time for almost everything.

I've learned that the extra time is often better spent on the front side of a new concept, rather than after the fact like "Oh no, they didn't understand that so let's just repeat it for a couple more days".

I try to start with what they know, then gently stair-step them into the new thing.

Case and point, our recent venture into solving exponential equations (without logs) like these:


On the first day, I equipped students with white boards and said we were going to work on some puzzles. (Don't you just love the word "puzzle"?). Try to write an expression that equals 8, using a base of 2 with an exponent.


Step 1: We did a whole bunch of those, starting with positive integer exponents and working up to the use of negative integers and even a few fraction exponents. Students needed some reminders here and there, but we kept practicing until they were answering them all with accuracy. Then . . .


Step 2: We continued with more, but now students needed to choose what base number to use. There were some great moments here to capitalize on, like when students tried to use a base of 9 to get 243 but realized it wouldn't work and had to use a 3 instead. Or, when two students each used a different base and both answers were correct. I made a huge deal about these when they happened. Next . . .


Step 3: I gave students TWO goal numbers. They must write an expression for each, using the same base number for both.

And now, without realizing it, they've really learned how to do the hardest part of solving the equations for tomorrow's lesson.

Step 4: (Same as step 3, really) To finish the class period, I grabbed a copy of the practice set I wanted them to do the next day and pulled sets of goal numbers directly off of it.

On day 2, students easily transitioned into solving these.

On day 3, extra practice with Add Em Up.

BONUS! From here I introduced logarithms, and found that students were completely prepared to answer the question "What exponent goes with this base number to make this value?" The extra day spent on the front side of solving exponential equations ended up having a double payoff.

Friday, February 20, 2015

Giving Immediate Feedback Without Breaking a Sweat

Our entire math department is transitioning towards SBG. Recently, we were having a conversation about scoring of assessments and giving feedback and such, and I thought about THIS. I remembered conversation about Frank's orange pen strategy a few years ago, and I can't imagine why I only just decided to give it a try.

For the setup, I wrote out a key for the assessment. Since students would be looking at this, I tried to include more detail than I would for my own personal key. I made copies and set up these little stations around the room. My pens were red, pink, orange, purple . . . any color I could find that wasn't green. Green is my signature color.


Students did their assessment-taking in the usual fashion and then, before turning it in, they made a stop at one of these stations to mark what was wrong and make corrections.

And Voila! That's it.

The good:

Any given day, when I take the time to write feedback on a student's paper, I have no idea if the student is actually reading that feedback and taking it to heart. I don't know if they understand what I wrote, either. When a student writes their own feedback, I can clearly see their level of understanding of their mistakes.

I used this in my lower-level Algebra 2 class first. These students who are not usually super interested in their assessment results had more buy-in than usual. A few of them immediately stopped by my desk to explain their mistakes to me . . . just because they wanted to share!

Another student left a perfect note to herself after making a classic mistake while squaring a binomial. She was able to find exactly what she had done wrong and identify what she needed to do differently next time! I am sure this note IN HER OWN HANDWRITING is way more meaningful than anything I could have written on her paper.

And, at the risk of sounding lazy, I am also going to mention how much faster I am able to finish grading a set of assessments! Of course I still look at each paper carefully, but in most cases the student feedback is adequate. I just determine the level of understanding and BAM! Done. No additional comments needed.

The not-so-good:

Not every student left stellar feedback to themselves. A few just put a slash through the problem number if their answer was wrong without identifying their mistakes. I sent a few of them back to write more. One of them gave me a heavy sigh. I can live with this, but I will continue to coach students regarding my expectations here. I also updated my instruction sheet. Much better.


I really love #3. I have found that, if a student can convince me they understand their mistakes, it can actually influence the level of mastery I select for that skill. The writing of feedback almost becomes part of the assessment itself. An unexpected result that I am happy with.

Other thoughts:

If I want students to assess and do feedback in a single class period, I have to make sure that my assessments are not too long.

One colleague is concerned about students early in the day sharing information with those who take the same assessment later in the day. I have four different preps, so I am never giving the same assessment more than twice in a day. And even if an answer were shared . . . a student can't demonstrate understanding by simply writing the correct answer. I just haven't felt that this is a problem in my classes.

The first time through I underestimated the number of stations I would need. To avoid students waiting very long for a station, I found I need about 1 station for every three students.

Tuesday, December 2, 2014

On Review and Remembering

Recently, our high school math department met up with the middle school math department to discuss all kinds of things. Part way through the discussion, someone mentioned the need for a magic pill to help students remember what they've learned.

We all agreed. We have all experienced the frustration of believing that our students have mastered a concept, only to discover they can't remember what they've "learned" just days, weeks, and certainly months later. By the next year, we're hearing that they have "never" seen whatever thing we are expecting them to remember.

In the middle of this conversation, it occurred to me that maybe we are teaching students to forget. A traditional math classroom (including mine until I started SBG) looks like this:

1.  Teach a unit
2.  Review the unit
3.  Test over the unit
4.  Move on to next unit
5.  Start forgetting previous unit

The end of each of my units always had a review day, where I would have students practice a bunch of problems that were really similar to the ones they'd see on the test the next day. Only the numbers were changed. The next day, my students would (generally) do pretty well on the test. I would pat myself on the back for my good teaching ability, and away we'd go to the next unit.

Now I'm thinking . . . Do those big-review-days-that-look-just-like-the-test-right-before-the-test-day just train students to stuff in the information, hold it in for 24 hours, and regurgitate it the next day? If our students do very well on such a test immediately following such a review day, does it give us a false representation of what they've actually learned? What if the "forgetting" we see is really just "never learning"?

What should review look like? When do you review? What do you review? WHY do you review?

Sunday, November 30, 2014

Fighting Assessment Freak-Out

Our school was asked to present a "high school" perspective on preparing for state assessments. I don't know that we are doing anything all that unusual, but here are some of the things we shared.

1. Teach the standards. I don't want to over-simplify this, but at a time when so much is unknown, it makes sense to focus on what we do know. In the CCSS, we have a document that states what our students need to know. Make sure you're teaching that stuff.

2. Focus on the essentials. While it is our goal to teach every standard, we know that might not be possible. We printed out the standards for each of our classes, one to a page. One at a time, we took the standards for each class and sorted them into "most", "somewhat", and "less" important. Then we took the "most important" stack and narrowed it down to the 8-10 most essential standards for each class. If we don't do anything else, we'll make sure our students don't leave our classes without mastering those standards.

3.  Assess what they've learned. All of our departments are working on regular formative assessment. Data is brought back to weekly PLC meetings where we discuss the teacher side (how can the teacher approach this topic more effectively) and the student side (what interventions can we provide for students who didn't learn).

I am piloting standards-based grading, soon to be joined by the rest of our math department and more. Each skill is based on 1-2 standards from the CCSS.  The difference is the data. Instead of identifying that a student scored a "74% on chapter 3", I can identify exactly which skills each student has mastered, and which ones need more work. If the class doesn't learn a particular skill, I can devote more class time and/or spiral back to that skill as we move forward in the curriculum. If individual students don't do well on a skill, I can provide opportunities for them to continue to work on that skill.

4.  Provide interventions. We have all kinds of interventions.

If a student is deficient in a lot of skills: We provide an extra hour of instruction in addition to their regular math class, called "math lab" or "opportunity room". Here they spend a portion of the time working on skills like math facts and solving equations. The remaining time is spent reinforcing what they are doing in their core math class. Right now this option is only available for math. English will be next.

If a student needs help on a particular concept: They may work with the teacher before or after school, or be assigned to our tutoring center "irish hub" to work with a national honor society student. We could not do this without our NHS students. They are amazing, and they often explain something in a different way that clicks with a student. This intervention is available to all students for all classes.

If a student just refuses to do something: They go through the "non-compliant" side of our intervention plan, which involves a lot of follow-up and administrative assistance to help the student be successful. This is also a school-wide intervention. Here's the flow chart:



I was surprised that standards-based grading was the portion of our talk that received the most questions/responses when we were done. There are so many in the world of blogging and twitter who use SBG that I forget it is actually not that common (in our area, at least, SBG is fairly rare). The room was full of mostly administrators who were very supportive of the idea, but they were meeting resistance. The idea (specifically the unlimited re-assessment) is apparently a tough philosophy for many to embrace. 

Tuesday, September 23, 2014

Changing Everything, Using Blueprints

My Algebra 2 classes have been textbook-free since 2007. It started when my school purchased a textbook that I eventually came to hate. I started changing the order that things were taught, customizing lessons, re-doing units, adding activities, and so on. Over time I developed an "Amy Gruen" version of Algebra 2.

All was well and good until Common Core. I read the standards and I tweaked things regularly and added a few new units here and there, but I couldn't shake that "re-arranging chairs on a sinking ship" kind of feeling.

The problem is that I was planning for Common Core in the way that I think a lot of teachers are planning.  I started with the hodgepodge that was Amy Gruen's Algebra 2 curriculum on the left, and over there on the right was the pile of CCSSM. I tried to file those in where they fit, but I had some left over. And there were things in my curriculum on the left with no matches in the CCSSM.


So here I am. I am ready to pitch everything in my Algebra 2 classes and start fresh. After much thought and reading about lots of approaches, I keep coming back to a great session I attended at TMC14 . . . Blueprints*.

Blueprints are a joint project by Kate Nowak, Mathalicious, Illustrative Mathematics, and others. The Blueprints include a sequence of CCSSM that makes sense, justification for decisions, and links to activities that coincide with each standard. What draws me here above other approaches is that they have STARTED with the common core standards. None of this taking a thing that's already done and stamping it up with the words "common core".

Now I can reverse the planning process . . . starting with the CCSSM on the left and a pile of resources on the right and filing those in where they fit.


I've decided to go all in. Start from scratch. Possibly planning just a few days ahead. I am the kind of person who usually shows up August 11 with the year (somewhat) planned out, so this is a bit of a stretch for me. But it feels good and right to start each lesson plan with a standard rather than matching the standard to an already-existing lesson.

I am excited for this new adventure.

Update since I started writing this post: So far I've used the Blueprints to plan a few weeks and I am already seeing a lot of positives. I will write about those soon I hope.

*I got a sneak peak at the "almost final" version at TMC14, and I'm using that to get started. The final version is to be published this fall.

Wednesday, September 3, 2014

A Desmos Challenge for Everything?

Last year, I used the animation feature on Desmos a to assess students' understanding on parent functions and their transformations. This year I expanded the activity by giving students a series of animation challenges (with increasing difficulty), like so.


Then I ended the unit with a Desmos question on the assessment. I color-coded the questions from easiest to most challenging. I was so happy that most of my students selected the most difficult level. Here are the cards I used:



I still feel like there is so much more that I could do with this. I am thinking of creating a set of Desmos challenges to coincide with every unit.

For example, one of my classes is working on systems right now. I could have them create systems in Desmos with various constraints such as a given solution, a solution in quadrant 1, no solution, and so on.

I am inspired.

Desmos challenges for everything! (Or at least lots of things.)

Tuesday, August 26, 2014

Scenes From My Classroom

Here are a few this and thats from around my room. I have a lot more to write about, but for now this is all I can muster. I am full-on beginning-of-school exhausted.

Inspired by Glenn Waddell at TMC14, I thought I would use this progression of (h, k) forms. That might be changing (more on that later), but I still love my yellow brick road with ruby slippers to show where we are. I still want an emerald city, but at some point I needed to stop decorating and plan lessons. So that didn't happen.


(I printed a brick pattern on yellow scrapbook paper, then printed the equations on top of that).


My weekly schedule board. I felt really clever when I used a 2 for "two's day".


I downsized my supply baskets this year. Then I noticed that the red/green/yellow cups fit right on the handle. That's handy.

Red/green/yellow posters.


Pretty syllabus inspired by Sarah's post.


On the first day of school, I combined "telling them how awesome I am" with a plicker quiz. I asked the students "Which pet does Mrs. Gruen own?" and such. It turned out to be a yawner. There is nothing worse than boring the children to tears on the first day of school. I managed to make it fun by following up my quiz questions with little stories, but next time I will put the questions in a slide show and follow them up with a picture answer. When it comes to perfectly executing the first day of school, I am not there. YET.


I use an apple TV to project iPad and/or laptop in my classroom. As a security feature, I have to enter a 4-digit code each time I connect wirelessly. A different code is randomly generated by the apple TV every time, and last year I had a few students start to try and guess the code before it popped up. I thought I would start the year by having every student guess a 4-digit number for a fun on-going game. There could be a prize when someone's number comes up. I just set up a google form to collect guesses and students immediately started talking about probability! How many prizes will I need?


Finally, my buddies and I rocking our TMC shirts. Bring on the school year!