These lessons, as well as the others on the Mathalicious site, are designed to add an authentic and real-world approach to learning mathematics and, from what I can tell, it's working. The geometry kids did VERY well with theirs. It was interesting to watch which kids struggle with which parts of the assignment and how I was able to redirect their thinking using pointed questions, many of which were suggested by the Mathalicious lesson planning guide.
It was clear that the students had seen the material before but that it had been a while. Some of them needed a bit a of refresher, which I told them to get from their partner or small group. Several students gave up early, claiming they had no idea what to do. After some directed questions, it became obvious that they DID know what to do, but fell into one of two categories: 1) They didn't want to work or 2) They were having trouble with directions that were not explicit. They wanted step-by-step Ikea-style directions to follow and when they weren't given those, they were uncomfortable and shut down. Some poking and prodding got them back up again and back to work.
Overall, I was pleased and am looking forward to extending some of these in the next few days. I hope to do this same lesson with the Math 8 kids near the end of the year, once they've talked about linear equations.
The Math 8 lesson did not go as well, but I don't think it had to do with the lesson design. I think I vastly underestimated the skill level of my students as well as the way they think about math problems. I need to find a class set of calculators, but more importantly, I need to train my students to answer a certain question before they complete a task:
Why am I doing this operation?
All too often, they say "We just multiply, right?" but can't tell me why they want to. They don't know how to analyze a mathematical situation to determine the best course of action, or even a sensible course of action.
I don't blame them for this in any way. We have trained them from elementary school that math comes in small, discrete packets. We will provide them with one question and expect one answer. "What do we do next?" is important, but so is "How do we set this up?" and we don't spend enough time on it. On top of that, as soon as they see decimals or fractions, they shut down. I don't want to modify the lesson to make the numbers "pretty" because then it loses some of the authenticity. I even hate to pick an easier lesson because this one is designed for 6th graders.
Perhaps I simply need to have a more open discussion about how to find what is being asked for instead of relying on them to simply know. I have plenty of time with double periods. I can take the time to explain these things, but I REALLY don't want the class to turn into lecture.
I have faith that the more of these activities and projects I do, the more students will begin to understand what is expected of them.
In response to my query about good math practice and function over form, the brilliant and forward thinking Christopher Danielson suggested that I test my students' understand of the equal sign by having them answer the following question:
8 + 4 = [] + 5 What number goes in the []?
The idea being that students who have a less accurate understand of the equal sign would put an answer of 12 because the sum of 8 and 4 is 12. These were the results:
It's not nearly as bad as expected, but clearly, we do need to have a discussion of the equal sign and how it does not mean "Put the answer here."
The second class did a bit closer to my expectations, which is unfortunate. There are some interesting answers here, including ones that I think are clearly calculation errors instead of conceptual.
On the personal front, I had a student walk into my room today clearly in a funk. She immediately put her head down and when I asked her repeatedly to pick it up, she got an attitude with me. I set the class to a task and pulled her outside.
Me: "What's up with you today? You did great work for me last week and now you're giving me attitude."
S: "I don't want to be in this class."
Me: "I can understand that. Have I done something to upset you?"
S: "No. I just don't want to be in this class."
Me: "Well, unfortunately, there's nothing I can do about that and nothing you can do about it, right?"
S: "I guess."
Me: "So if there's nothing that either of us can do about it, then you giving me attitude is just making things tougher for you and for me, right?"
S: "I guess."
Me: "Alright then, I know everyone has bad days. Hang out here for a few minutes, take a couple deep breaths and then come on back in ready to work."
S: "Ok."
AND SHE DID!!! She did great work today! I made sure to find her at the end of the day and thank her again for her great work.
P.S. Since attending TMC13, I've been thinking about every day things in terms of how I could use them in my math class. When I saw my kids swinging, I immediately thought about Max Ray, the Drexel Math Forum and "I notice, I wonder."
I think there's a great lesson here about periodic motion and pendulums!
Hi Justin!
ReplyDeleteSometimes I think our lessons make teachers uncomfortable, not because they're not scaffolded enough, or too hard (complaints we hear sometimes), but because they do a great job of illuminating students' (really quite alarming) weaknesses. Applying procedures haphazardly without knowing why is a weakness. Not checking the reasonableness of their outcomes based on their intuition is a weakness.
And you're completely right that it's not their fault. It's the result of many years of schooling where they're completing exercises instead of solving problems. And they're not making decisions as a matter of course.
Fundamentally changing instruction is, of course, not an easy road. But you seem to be willing to do it, because you know it's necessary if your kids are going to learn something. I don't know how to communicate how much I respect you as a professional.
So! Specifically from your experience with New-Tritional, it sounds like they could use some visceral engagement with unit rates, and they need to be asked over and over to make decisions about when to divide and when to multiply, and to justify their decisions. Luckily, the MTBoS is chockablock with unit rate resources. Mathalicious for example has Civic Hybrid and Big Foot Conspiracy (or, Been Caught Stealing is a PBL lesson, if you want to go the d=rt route), the WCYDWT tag on Dan's blog is a unit rate playground, and really, thank goodness it's this really useful thing you might feel driven to remediate, and not subtracting polynomials or something.
I don't know what to say besides, don't give up! They need you. And, thank you for being awesome. And, thank you for reporting back. Reports from actual humans doing the messy business of teaching and learning helps us make our stuff better.
It's always difficult to discover that your students are lacking in skills, but I think it may be more difficult in math, where so much of what we do is cumulative. "How am I supposed to teach probability when my students don't know how to work with fractions?" or "How can I teach them to analyze equations when they have no idea what they mean?"
DeleteIn reality, we end up teaching those skills in tandem with the new ones. It's especially frustrating because all of the education classes I've taken glossed over the idea that I may have to reinforce (read: teach) basic concepts like subtraction. I certainly was never prepared for that.
It's frustrating and confusing. I don't know how to teach those concepts at the level that the students need them, but I'm working on it.
I worry about walking the line, trying to figure out how much information to give them and how much to let them figure out. I want them to struggle to have success, but if they struggle too hard, they'll give up.
I think I need to spend some more time on dimensional analysis, identifying units and estimation. I think it might help to clarify which operations to perform and what answers might be reasonable.
Have no fear, dear lady! Even though this didn't go exactly the way I wanted, it was FUN! And it was the first time doing it. The second class did much better with working through their struggles than the first and I attribute it more to my familiarity with the content and the ideas of the lesson itself.
They aren't used to this type of lesson and neither am I, so as we do more of them, they better they will get. Plus, now I know I need to do more unit rate problems! We went over the concepts after they worked, so I'm excited to try out the projects tomorrow! ALL THE PROJECTS!!!
I just spent an hour watching your kids swinging.
ReplyDeleteJust kidding. That would be creepy! I didn't mean it like that; I meant, you know, pendulums are ... hypnotic ... whatever.
Thanks for reminding me about Mathalicious. I have some other stuff planned for tomorrow--Lots of Geometry fun with Constructions!--but maybe I could dive into their site some time this evening and find something apt for Algebra. My Algebra 1 classes are going pretty well but my 1A class is pretty miserable. I'd tune in to the MS math chat right now if I weren't running off to a meeting. I'll catch up with more of your writings there later.
In the meantime, thank God for your prolific output! You are a f#@king champ!
My kids are stupid cute. I would not blame you for spending you time watching them NOT screaming at me for refusing them ice pops.
DeleteI am excited to get into constructions but I haven't had a chance to look at the curriculum enough to know when the high school teachers want me to do it. I know how you feel about the 1A's too. That's a rough class. Hit me up sometime and we'll see if we can talk strategy.
The #MSMathChat was pretty epic tonight. I'll try to Storify it in the next day or two so you can get the gist.
Prolific, maybe. Productive and useful? I doubt it. :-)
I hope that I'm helping someone other than me.
You might have fun, too, with working on guessing and predicting before jumping into problem solving. Some of the number sense that you harvested today could be leveraged to help kids (sloooowly) get better at having a gut feeling about how operations should come out.
ReplyDeleteA 6th grade I get to work with occasionally blew my mind one day because we were having this crazy, freewheeling conversation about this made-up-math-problem kid (Jasmine) and how many minutes she spent on the computer each week and month if she spends 30 minutes a day on the computer. The kiddos could not make good predictions, struggled to convert minutes to hours, etc. They could talk all day long about how long they spent on the computer, and immediately knew that a kid who was on 5 hours a day was on longer than a kid who was on 20 hours a week when it was real kids they were arguing about. But whenever Jasmine the fictional kid came up, they started telling me operations and rules, nothing about actual time or computer use! They could compare their computer use to their classmates', but not to Jasmine's. It was eerie, like Jasmine lived in a math-free twilight zone!
So... their teacher and I tried to help them bridge the gap between numbers in the world that have to make sense, and numbers in math class that they don't believe could ever make sense, using Estimation180 for fun and then having the kids estimate an answer in every contextual problem before doing ANY calculation. And we banished the whole notion of calculate first and then round, by using Estimation180-style visual prompts, withholding key pieces of information from stories, or having them relate the problem to their own life and estimate about themselves or people they knew first.
Anyway, if you're into games and stuff, bringing in betting (including indicating confidence in one's bet by wagering, offering double-or-nothing bets, etc.) and just plain old "who can guess closest" for all kinds of problems, from Estimation180 to word problems to "naked number" problems can be fun and help tie math to thinking (crazy right?)
I think that as math teachers, we often shy away from estimation because math is supposed to be exact, but there is tremendous value in estimation. Even when I'm doing math for my own purposes, I use estimation to get close to my answer and then tweak it.
DeleteIf I'm using this skill, why am I not teaching it to my students? I'm very much into games and need to include them more. Also, I'm thinking I need to include questions on my tests that use phrasing like "What is a reasonable answer for..."
Getting kids comfortable with uncertainty is just as important as getting teachers comfortable with it.
I would love for you to come and observe what I'm doing and give me notes. You have an open invitation.
Thanks! I love visiting classrooms, especially when I can get there on SEPTA. I can probably come towards the end of September... maybe once the weather is cooler? (Ha! That's mean 'cause you're stuck there no matter what. Sorry!)
DeleteSadly, SEPTA doesn't make a run to Pittsburgh... Megabus and Amtrak do! I'd be happy to put you up for as long as you want to visit! :-)
DeleteOh! For some reason I thought you were coming to TMC by way of SEPTA. Hmm... does that mean you brought us muffins all the way from Pittsburgh? Or were you staying with friends? I love Pittsburgh, and owe some relatives there a visit anyway. It may take me a little longer to make it happen with work, but it will! And maybe I'll go visit Lisa Henry, too!
DeleteI was staying with my mom in the suburbs and came in via SEPTA. The muffins were baked in and transported from Pittsburgh though.
DeleteYou could also visit Jami Packer and Steph Reilly while you're out here too!