**Found problems that connected directly with the content material that was already part of the course.**

There are many problems that lend themselves to the content found in traditional MS and HS classes. For example, many of the problems in the Interactive Mathematics Program, Years 1 and 2, lead to students creating rules for specific scenarios or functions, including linear, exponential, and inverse ones. The Mathematics in Context and Connected Mathematics series have some great problems that can be integrated into traditional Pre-Algebra and Algebra 1 classes. The drawback with trying to connect everything back to the traditional content is that there's lots of material for which I have not found good problems, such as factoring, operations with rational expressions, and radical functions and expressions. Back when I taught Algebra 2 and Pre-Calculus, I had similar difficulties finding rich problems for much of the content. There's also the issue of time - I'd like to ideally have at least one rich problem every week or two, which eats up a lot of my class time if done well. Finally, using only problems that have a clear connection to the traditional curriculum leaves out a lot of rich, awesome problems that I still want to include.

**Assigned problems to be completed outside of class. Some were connected to the traditional content, some were not.**

This gave me a lot more flexibility in terms of good problems to use and took up much less class time. But I never found a good way to support struggling students, develop the writing and problem-solving skills that are at the core of this type of work, and make explicit the connections between the assigned problems and the rest of the curriculum. The problems gradually petered out as both I and the students lost steam and assigning the problems became stressful and unproductive. If I do this again, I will need to spend some class time teaching students how to wrestle productively with open problems and will probably need to do some ramping, with easier problems at the start of the year.

**Provided problems to interested students outside of class. Not required, problems were usually unconnected to the content.**

This was definitely the approach that involved the least amount of work. I had a pretty straightforward system: a folder with copies of the current "Problem of the Week" stapled to the wall outside of my classroom and another folder stapled just below that where students put their completed write-ups. At the end of the week, I would read through the submitted work, write feedback, and award candy to those students who demonstrated good work on the problem. I had a spreadsheet where I kept track of students who completed these. Some positives were that I got kids who weren't even my students to participate, just because they thought it might be interesting, and because it was not required, it was very stress-free and emphasized the "fun" aspect of figuring out math problems. The cons were that there was little connection to the curriculum and the students who participated were those who already enjoyed math and the students who could stand the most to gain from this type of experience avoided it altogether.

So, my thoughts for this school year are that I would like to do all three of these options (hooray for overachievers!). A mix of #1 and #2 make the most sense for my class - doing those problems that have a clear content connection in class & spending more time on them, while reserving those awesome, random problems for the times when I can't find anything good that connects to what we're studying. Option #3 can co-exist as optional, more challenging or more "fun" type problems for students to do just because they want more. My biggest enemy right now is time: time in class for students to discuss and time outside of school for students to think and do math and write up their thinking and mathing. Oh, and did I mention that my students only have math for 45 minutes four days a week??? Clearly, I can't just add on more stuff without cutting anything, so I'm wondering how others have found time to do this - what do you cut?

I know what you mean about the difficulty in finding rich problems for different math topics. To make matters worse, it seems that these topics (eg. rational expressions)would benefit most from such problems.

ReplyDeleteGreat idea about exploring/collecting rich problems with multi-entry points in the Global Math Dept.

Thanks for sharing,

Blaise @blaisej

This is definitely a goal that I have for the next two - three years. I have the same struggles as you - finding authentic problems for the curriculum and finding the time. I keep reading blogs in hopes of finding some answers! The one thing that I have done so far and it worked well, was that I used the problems as a review for the assessment. The students were then able to work in groups and learn from each other. I set it up in the form of the Amazing Math Race, so the skills that I couldn't find tasks for became my "clues" in the form of puzzles.

ReplyDeleteI have started using some of Dan Meyer's three act problems with my classes on random occasions. These generally don't tie to the content, but they do to our goals of math (problem solving, logical thinking, etc)

Good luck this year!

Carey

Those are great suggestions, Carey - thanks! I like the idea of using problems as review activities - I haven't done that very much. This would give me more flexibility & probably take less time than looking for problems that could introduce or be a way to teach the topic. I still like the notion of students discovering or making sense of concepts through a problem context, but usig problems to review could be a great compromise when that's not possible.

DeleteAnna, the three points you've made

ReplyDelete- connect to current content or not?

- assign outside of class or do inside?

- required assignments or extra (enrichment?)

might all be addressed by shifting to an idea of having problem solving more as a process that gets a little bit of time in class but spills over into the hallways and outside-math-classroom time.

What if at the end of class some day you end what you're doing 3 minutes early and you say to the class, "I'm going to read you a story." You proceed to read a problem solving "scenario." At the Math Forum we call a problem where we've intentionally left off the question, a scenario. Depending on the length of the scenario, maybe that day all you have time to do is read it. That done, you leave it at that. "See you all tomorrow!"

The next day or maybe two days later, again at the very end of the class period (at the most 5 minutes of time) you read the scenario again but this time you ask students "What did you hear?" The students respond with a variety of things that they heard. You don't record them. You don't repeat them. This time encourages the students to have that scenario in their minds. Again, you leave it at that. "See you all tomorrow!"

Here are some reasons why I suggest this as a starting point of the process:

* it takes very little class time

* and yet because it's started in class, it is assumed to be part of the class experience/record

* if I start with a "scenario" instead of a problem, there isn't a question to answer and be over and done - instead the expectation is to think about what's happening, think about the quantities and relationships involved - maybe even construct your own question and ponder what might happen or what results you might get -- always leaving it open to not finishing because you don't know yet what the question really is.

The third time you do this (again maybe the next day or maybe a few days later) you display the scenario and ask the students to "turn and talk" (with "turn" referring both to physically turning to talk with someone but also taking "turns" talking). Have them talk about what they've noticed. Depending on how that goes, maybe there will be time to share out or maybe you just wander around and listen to their conversations. Leave it at that. "See you all tomorrow!"

The fourth time you take a few minutes at the end of class, ask the question "What do you wonder?" Maybe you'll do this whole class, maybe you'll have each student list their own wonderings, maybe you'll have pairs or groups talk about it, maybe you'll have them make written lists -- each variation has an advantage and, perhaps, depends on how you do other things in class.

Here are next steps now that you have all students familiar with the scenario and thinking about possible questions -- you might decide to have them pick one of the wonderings and that then becomes the problem ... or ... you might have different groups work on different wonderings and so you have different versions of the problem ... or ... you might just hand out a copy of the problem that generated the scenario.

You can continue the idea of just taking a few minutes at the end of class to continue the process. (I always use the end of class because if I try to use the first few minutes at the beginning -- it's much, much harder to stop and go to what I had planned!)

Continuing the problem solving process next steps might be:

* working out an answer and talking about it

* going from talking to writing a draft

* getting feedback from others on the draft (feedback both on the problem solving but also on the communication)

* revising (if the revision is of the problem solving -- adjusting an answer. If the "answer" is correct then the revising might focus on the communication)

Do any of these ideas resonate or respond to what you were pondering?

Thank you so much for the thoughtful feedback, Suzanne! That is a very helpful way to structure problem-solving in the classroom, and I can see how easy it would be to implement and ramp up over time. It's very reassuring to think about problem-solving as a progression and not something that I have to dedicate a lot of class time to from the get go.

DeleteOne way that I like to think about what I will do in class vs. as a take-home activity is to think of myself and what I would prefer if I were solving the problem. When are the times that I like to have other people to bounce ideas off of, and when are the times I would rather be in a quieter place?

ReplyDeleteI find that in working to understand a problem, I often need a friend to talk things through with so I dont feel like I'm barking up the wrong tree, to help me generate some different strategies to try, and to check my understanding against.

When I have some ideas of different ways to approach the problem, such as trying a few simple cases, making a guess, organizing data into a table, I usually like to work on my own without fear of interruption... at least until I get stuck!

When I get stuck I really need someone to share ideas with and even get them to help me do some of the strategies that feel too daunting (making an organized list of cases, systematic guess and check, making an algebraic model).

If I get to a solution, that's another time I really like to have people around. I like to compare my approach to theirs, both to check the accuracy and to see what I can learn from comparing one working (or almost working) strategy to another. Might there be a more elegant way to solve the problem? Can I find a more general solution? What do the two approaches have in common that can reveal some underlying math?

So for me, the phases tend to be:

-Understanding the Problem and Making a Plan -- good to have others around (though I might need to noodle on my own & have quiet thinking time too)

-Carrying Out a Plan -- good to be on my own, until I get stuck

-Getting Unstuck -- really need other people!

-Reflecting, Checking, Revising, Comparing -- really need other people!

When I plan how students will work on a problem, some of the considerations then are: I wonder how setting aside a few minutes at the end of class, as Suzanne described, or at other times of the day, or even online, can be leveraged for students who need to talk through their understanding or get unstuck? I wonder how the reflecting & comparing portions can be part of problems students do mostly outside of class? I wonder how I can support students to take the quiet time they need to carry out their plans during in-class work?

I also wonder: do other people have different needs when they solve problems? People who are more introverted or extroverted or learn math differently than I do?

Thanks for such a thought-provoking post, Anna. By the way, I'm really curious what the illustration in your blog's background is from!

I think these are GREAT ways to think about best structures for problem solving. I have also thought a lot about making sure that I'm not assuming that my students would benefit from the exact same structures as me. I absolutely think that different people learn better in different contexts, and it's really hard to create ones that fit everyone. Personally, I really like to give students alone thinking time immediately after we discuss what the problem is saying because if we discuss the "making a plan" step, everyone's plans look the same, and that's no fun! I also like varying the interaction structures, so students would start making a plan alone at first, then consult with a partner, then join two partnerships to discuss as a group of four, then maybe discuss as a class. Guiding or structuring this process often feels more like an art than a science to me, unfortunately.

DeleteOne other issue that I thought of since posting this is that so many of my students get help from outside sources at home (parents, siblings, tutors). If students feel nervous or focused on the grade, it can be easy to get help and let someone else take over. I'm not sure how to prevent this, other than emphasizing the process, not the answer.

Thanks so much for your insights - this is definitely something that I wish there was more professional development around.

P.S. The background is from an illustration for the Russian version of Thumbellina, as is my avatar. Combine that with the "borscht" in my blog name, and you probably get the picture that I'm of Russian descent. :)

Delete(Sorry, Anna, I thought I'd left a comment here right after Blaise's because I read it soon after you'd posted it. #losingmymind, again.) Suzanne and Max have this down, that's why I love using mathforum.org not just for the problems themselves, but for the resources! Clearly they're both generous with their feedbacks online also.

ReplyDeleteI'm surprised you only have 4 days of math for 45 minutes each! Two years ago we doubled our kids' math time to two 55-minute periods, but the "advanced" algebra 7th graders and geometry 8th graders still only had 1 period. That is until this coming year when ALL students will get 2 periods of math, but they aren't necessarily blocked together either. Selfishly I LOVE this as I can no longer say that I don't have time. A lot of teachers are very interested in making problem-solving a mainstay and are looking for resources, so thank you for contributing to this important topic, Anna. And of course, thank you so much for the mention!

No worries, Fawn! Thanks, as always, for stopping by - I appreciate your comments and feedback a lot. And I am crazy jealous of how much time you have for math. For the 1-year Algebra 1 class, it feels like a year-long sprint. We are supposed to be deciding on a new school schedule soonish, and I am hoping to convince everyone that Math needs more time. It really, really does! But for now, I have to work with what I have.

Delete