Grouping Symbols: Jail for Numbers

I've changed over from teaching PEMDAS to using GEMDAS in Algebra 1. This week I discovered a new reason why this was definitely the right move. My Algebra 2 students had their test on radicals, which included solving radical equations.

I saw a lot of made up math. One student just dropped the radical symbol from the every equation and solved as if it had never been there at all. Others started undoing addition or subtraction that was under the radical. (It always disturbs me to see students do things like this; things I've never modeled in class. Where do these ideas come from? Is it desperation?)

In going over the test with the class, it occurred to me to compare grouping symbols, like radicals, as "jail for numbers." If 2x + 5 is under the radical, "the 5 can't just walk out of jail." For that matter, 2x + 5 can't just walk out of jail! What's the Get-Out-of-Jail-Free card? For square roots, it's squaring.

I think they got the message.

Converting Units Puzzle + Corousel

Much thanks to Sarah Hagan over at Math Equals Love for inspiring this activity. After doing a few examples of converting units (both converting within the same system and converting rates), I gave each group a set of index cards. One card said 'Start,' another said 'End,' and there were 6 smaller cards representing the numerators and denominators of the 3 fractions they would need to multiply by to converting the starting units to the ending units. I had already used tape to make a "template" on one of the desks of each table, so the students would easily be able to see where to place the cards.

Here's what one group's work looked like:

This activity worked out really well. Kids who previously did not understand why we were putting certain numbers and units in certain positions had a lightbulb moment. Other group members were able to explain to them and show by touching and moving cards why a correct solution would work. Students were also able to verify with their calculators that the numbers worked out from the 'Start'  card to the 'End' card.

They didn't all get the cards in the right places on their first try. Instead of checking and correcting each group's work, I had one or two students stay with their work, and the rest of the group rotated over to the next table to check over what they had done. This way the students ended up correcting each other's work, if necessary, and everyone got more reinforcement and confidence.

INBs with CC Algebra 1

Even though I've taught 9th grade algebra many times, this year is my first year teaching Common Core Algebra 1. I've decided I'm ready for some major changes, starting with using Interactive Notebooks (INBs) and seating my classes in groups as their regular seating arrangement.

So far I am loving both of these choices. The kids seem to be loving the INBs.

Pros: 

• Everyone can be successful in my algebra class. (Who can't master cutting and pasting?)
• Students are more organized than ever before. We're integrating adding to the Table of Contents (TOC) into our daily routine. (Will it continue? Will students be able to keep this up all year?)
• The physical demands of the INBs force me to slow down my teaching. Many of my students benefit from a slower pace in class. It also allows me extra time to do extra circulating and helping. 
• The seating arrangement silently implies that collaboration is encouraged. (And it is!)

Cons: 

• Glue stick abuse! Some students feel the need to cover every square inch with glue. I'm afraid I'll run out of glue by the end of October. 
• Perfectionist trap. Some students feel the need to perfectly trim every handout or meticulously decorate and color. At the expense of doing math. 
• The seating arrangement seems to be a distraction for some of my Algebra 2 students. Or maybe they would be talking all the time no matter how the desks were arranged. 

More updates to come soon....

Algebra 2 Mansplaining

Today in Algebra 2 and Trig, I was surprised to witness some mansplaining. A group consisting of 3 girls and 1 boy called me over to settle which student factored a quadratic expression correctly. Of course I wasn't going to say, 'You're right, and you're wrong.' Instead, I asked each person (1 of the girls, and the boy) to explain the work in the hopes that speaking aloud would allow the group to decide who was right and who was wrong.

What followed was a long explanation from the boy in a tone that made it sound like he was sure he was right. In fact, the longer he spoke, the more sure of himself he sounded. When he stopped speaking, the girl said nothing. She made no attempt to present or explain her own work. The group simply seemed ready to accept the boy's solution, though no one looked particularly excited by it or made any comment truly agreeing with it.

Did I mention that the boy was completely wrong, and the girl who shrank back and said nothing was the one who had it right?

I was really hoping that by letting the students speak, I wouldn't have to jump in and save them. Instead, I pointed out that the girls just let the boy speak and went along with his answer without questioning it at all. I reminded the girls that mathematical arguing is an important skill in this class!

It's only the beginning of the school year. But I can see already that I'm going to have to really encourage these girls to not let the boys steamroll them. And I need to encourage ALL of my students to have confidence in their own work!

School Choice? What Happened to NCLB?

I don't want to hear another person endorse the concept of "school choice" for parents without acknowledging the hypocrisy of also being in favor of No Child Left Behind. 

The idea of school choice implies that parents will choose the "better" schools, leaving the "bad" schools with little to no enrollment so that they will eventually disappear. Not only would it not work out that way, but that's the same as giving up on the students who still attend those supposedly terrible schools. 

Teachers don't get to choose their students. We don't say, 'I don't want that one; he's never going to be good enough.' Schools are not allowed to give up on their students. How is it acceptable for society to give up on our schools?

Furthermore, some people need a reality check: this is not Lake Wobegon, "where all the women are strong, all the men are good-looking, and all the children are above average." It is impossible for all children to be above average. It is impossible for all teachers to be above average. It is impossible for all schools to be above average. What we should strive for is giving every child the tools and opportunities for success. That's not only attainable, but meaningful. 

3D Printing, Here I Come!

Last night was my first of 3 sessions learning about 3D printing at Math for America with George Hart. I've been looking forward to this for so long!

We got to watch a 3D printer in action, and after a Mathematica refresher, we started playing around with attempting different challenges. Now I can't stop trying to design something fun. This is addictive. Any suggestions?

I wish I could make ALL OF THESE!

Of course, if I get good at it I'll probably want to have my own 3D printer. I'd love to hear about your experiences with different brands, opinions about cost and/or materials.

Prevention Instead of Intervention

A few years into my teaching career, the school year started on a Wednesday. On Friday, the 3rd day of school, two of my 9th graders cut my class.

Red flag. BIG red flag. I know that teenagers have been known to cut class, but 9th graders? On the 3rd day of school? That's not normal. That's a sign of bad things to come.

I saw this as an opportunity to prevent what I saw as a clear sign of the future. I went to my department chair and told her what happened and that my prediction was if we didn't step in and do something, these children would fail Algebra and maybe not even graduate on time (if ever). Her response was, "Let's wait to see how they do on the first report card."

I couldn't believe it! She was telling me that we had to wait for them to fail before we could take any action to fix the problem.

1. It should be clear that there IS a problem.
2. Once a problem has been identified, action should be taken to prevent it from getting worse.
3. As educators and professionals, we should not allow students to dig themselves into a hole that we know they are not equipped to climb out of.

Back then, I was shocked that this was the school's response. By now I'm much less surprised; this is how our system works.

What I want to know is, aside from alerting parents and guidance counselors and administrators to our concerns, what can schools do to prevent students from skipping down the path of failure instead of waiting until it's probably too late and attempting what is often an unsuccessful intervention?

(By the way, I was right about the two students who cut on the 3rd day of school. This became a pattern of behavior, among a host of other issues, and they failed tests, marking periods, courses, and so on. Once the ball was in motion, there was no stopping it.)

What Makes a "Good" School?

I am so sick of hearing people talk about "good schools" and "bad schools."

So what makes a "good" school? Let's examine some of the characteristics of school districts that are generally accepted as being the good ones.

• Students generally attend the same schools from kindergarten through 12th grade. This means that a school district can create a vision (or philosophy) for education that is developed over time. It's very difficult to take advantage of this kind of planning (or for a vision to become successful) in a place where students frequently move or when students are coming to a school from many different schools (and therefore have received many different foundations). 
• Parents are involved and/or aware of "how school works" and what's going on in their children's school. It is a lot harder for immigrant parents (even the ones who care) to understand American schools and American school culture. 
• Students score well on Regents exams and SATs, and get into prestigious universities. This is essentially saying that the students' families (generally) have money. 

My conclusion so far is that New York City schools (and other urban public schools) are assumed to be weak simply because the students who attend are by and large poor, underprivileged, immigrant, and/or non-white. All of this has nothing to do with the content, the level of rigor, the attention to students' needs, or the care provided. In other words, all of the things we SAY are what's really important in education.