Kim was kind enough to comment on my last post. Here's part of what she said
It's been a delight to work with Graham Fletcher's 3-Act tasks over the last few years. Is there anyone who doesn't know these yet? Just in casre, let me tell you, it's an excellent approach to problem-based learning, where, in "Act 1" children watch a scenario, and then are asked to notice things and ask questions about it. Hopefully, and usually, there's a "how many?" question that comes up naturally, just the right one to pursue. Children can then estimate an answer to the question, and think about what additional information they need. Then, in Act 2, they're given some supplementary information that should help them to get started on calculating. While they're doing this, they can select the resources, the materials that will help them best, whether it be pencil and paper, number line, hundred square, snap cubes or whatever.When they've had plenty of time to struggle with the question and come up with answers, there's Act 3, which shows them the answer being revealed.
So, to repeat Kim's question, is there another type of lesson, where students simply get to know and explore what a material or a representation can do?
I would say a definite yes to this. Especially with younger learners, I want them to spend time getting to know number lines, ten frames, counting, snap cubes, counters and of course Cuisenaire rods, just to see what they can do. I still want there to be a degree of openness in the task; I never want students to just follow instructions, but the challenge can be to achieve certain things, or explore possibilities with the manipulative.
Take a recent lesson with number lines. We'd had a really interesting discussion about where numbers should go on a line, and I then had a great lead in to an idea I'd seen on Kristin's blog, children themselves placing numbers onto an empty strip of tape (a literal one this time).
And we're regularly doing this with the Cuisenaire rods. Just on Thursday, the task was to find different ways of making a "train" of two rods that is the same length as the orange rod.
I've gone into more detail about what we've been doing in our Cuisenaire lessons here; one things's for sure - I've definitely mandated their use. I want my students to be really familiar with them and use them for exploring how numbers work, for asking questions and investigating. And hopefully to have a real "feel" for numbers because they've navigated the model in lots of ways.
A couple of examples. During the Thursday lesson, M, who knows I like his questions and observations, called me over. He had something to say. He said making the same-length train with the blue and the white rod was "fussy". I'm puzzled, and, after a bit of questioning, he asks a neighbour about it, in Spanish. No, it's not "fussy", it's "easy"! I ask if I can video him talking about it:
I've been working with tape diagrams for several years now, so I do feel like I know them (though I can always learn more), but it's true that they were foisted upon me by EngageNY and so it wasn't a voluntary process. I guess one question I have about that as a teacher, though, is: is there ever a time when we should make a particular tool or model mandatory because we are trying to help kids become familiar with it, so that later they can have a choice about whether to use it or not? My leaning has always been toward not mandating any tool or model ... toward always leaving it open to the child to explore and choose the representation that makes sense to him/her. But when I started working with younger kids, it seemed like it might be necessary to "mandate" certain models (like the tape diagram, or the number bond, or the "quick tens" drawings) for at least a couple of days as a way to lay a foundation. I wonder deeply about this, because it goes against my instincts to mandate, but then I think that maybe as a 4th grade teacher I was just benefitting from the groundwork that my colleagues had already laid through some of their "today we're all going to try this model" work. Would love your thoughts on this.It's a really interesting comment, and I've been pondering it through the weekend.
It's been a delight to work with Graham Fletcher's 3-Act tasks over the last few years. Is there anyone who doesn't know these yet? Just in casre, let me tell you, it's an excellent approach to problem-based learning, where, in "Act 1" children watch a scenario, and then are asked to notice things and ask questions about it. Hopefully, and usually, there's a "how many?" question that comes up naturally, just the right one to pursue. Children can then estimate an answer to the question, and think about what additional information they need. Then, in Act 2, they're given some supplementary information that should help them to get started on calculating. While they're doing this, they can select the resources, the materials that will help them best, whether it be pencil and paper, number line, hundred square, snap cubes or whatever.When they've had plenty of time to struggle with the question and come up with answers, there's Act 3, which shows them the answer being revealed.
So, to repeat Kim's question, is there another type of lesson, where students simply get to know and explore what a material or a representation can do?
I would say a definite yes to this. Especially with younger learners, I want them to spend time getting to know number lines, ten frames, counting, snap cubes, counters and of course Cuisenaire rods, just to see what they can do. I still want there to be a degree of openness in the task; I never want students to just follow instructions, but the challenge can be to achieve certain things, or explore possibilities with the manipulative.
Take a recent lesson with number lines. We'd had a really interesting discussion about where numbers should go on a line, and I then had a great lead in to an idea I'd seen on Kristin's blog, children themselves placing numbers onto an empty strip of tape (a literal one this time).
A couple of examples. During the Thursday lesson, M, who knows I like his questions and observations, called me over. He had something to say. He said making the same-length train with the blue and the white rod was "fussy". I'm puzzled, and, after a bit of questioning, he asks a neighbour about it, in Spanish. No, it's not "fussy", it's "easy"! I ask if I can video him talking about it:
I love it when students bring up something we can explore further, in this case how easy making a set of equivalent trains is when the white rod is part of one of them.
Earlier on, in free play with the rods, T had made a German flag.
Something must have really intrigued T about this, because he then started trying to make same-colour trains in other colours, adding a white rod at the end if necessary.
I'm looking forward to sharing this with the class again and exploring this idea together.
Now, this isn't problem-solving in the way we most often talk about it in maths lessons, but this sort of inquiry is, to me, worth sharing with the class and pursuing together.
As we build up an understanding of the way numbers work like this, I'm expecting that the children will be flexible thinkers with numbers, because they've seen how they work. I'm hoping that, eventually they won't need to pull the Cuisenaire rods out every time, because of the number sense they've developed through them. There will be new things we explore where we'll need them again, so they will always be a place for them. But there will also be the "mastery of structure" as Goutard calls it, that means students carry all sorts of acquired understandings with them mentally.
But I'm interested in Kim's instinct about mandating. How would it respond to the kind of work I'm talking about?
So much great reading between this blog and Kim's! I struggle with exactly the point Kim was making when we began multiplying fractions on bar models in 5th grade: https://mathmindsblog.wordpress.com/2013/11/22/modeling-mathematics-developing-the-need/ & https://mathmindsblog.wordpress.com/2013/11/15/fractions-of-fractions/
ReplyDeleteNow, I see the same thing happening in younger grades...the 100s chart, number line, even down to the 10 frame in Kindergarten. We begin with that frame to show students there is a way we can organize our counts, see structure, and build on those understandings of 10ness. With all of these tools or representations, I had to take a step back many times to think about the goal of its use. I think, as with the tape diagram, students are exposed to these tools to begin to see structure and eventually move to abstraction, but like Kim says, what if students are ready for it or what if they don't have the need? I have no good answer!
I love Simon's exploration of the tools and have tried to make an effort to do more of that in the classrooms I work (speaking of which, I really need to blog soon:). I think the more we start there and ask students to make connections between the tools and representations, the more flexibility and understanding of the structure of number students will have. Speaking of which, I wonder how students would respond to putting up the same problem, solved on a number line, tape diagram, 100s chart and numerically (all by different students) and asking them to find the similarities and differences. Then, ask which they prefer and why. I am always curious about what they prefer!
Mandate. What an ugly word. Defined as "an official order to do something," that nags at me. It seems antithetical to the classroom cultures you, Kim, and many others are creating. I would rid that word from the discussion. It's the wrong angle to approach from.
ReplyDeleteSo say you want all students to learn a Cuisenaire rods or a tape diagram. How do you go about it?
1) Curiosity, invoke the students' - What you described with Graham's 3-Act tasks is but one nice way to do this.
2) Create a need - Engineer a problem / inquiry / exploration where students will see a need for something beyond the tools they know, then introduce it.
Create a need ---> see value in ---> explore / learn / understand / apply the tool
Way back on my blog about the Math Celebration you commented "This is a great paragraph. It’s a really robust kind of learning!" - Simon Gregg
regarding this passage about two students discovering, through exploration, the algorithm for equivalent fractions:
"...but they completely understood it because they had built it, tested it, and found its strengths and limitations."
That's what I see you putting your students in positions to do with Cuisenaire rods. It's not what it sounds like is happening in classrooms where the tape diagram is mandated. The tape diagram mandate reminds me of what happened in California in the 1980's when they tried to move from mechanical mathematics to mathematics understanding. I believe Marilyn Burns was out there during that time so if she chimes in she may be able to provide some additional context on this point.
Here's an article about the California aftermath. It is worth a read but the abstract alone will give you the gist:
http://ci512-summer2011.wikispaces.com/file/view/Cohen+%281990%29+Mrs.+Oublier.pdf
I'd be against mandating use. Mandating exposure... I'd have to think about. The problems should be front and center. Sometimes a problem is model specific, like your use of the cuisenaire rods. But you encourage the use of other models (drawing, for example) if it's helping them think about it.
ReplyDeleteIt's hard for me to think of a reason to demand students be able to do something -this way-. Students need to be able to add and decompose, not show what the addition looks like in unifix cubes. I like Mark's point about finding situations that support use of the model you're sharing or investigating. Having students using different models is going to lead to some fantastic discussions.
I’ve been mulling over the “mandating” question. I’m committed to giving students opportunities to deal with concrete models that have the potential to help them understand mathematical structures. And I’ve been committed to this throughout my more than 50 years of thinking about math teaching. (That’s 54 years, to be precise.) And over the years, my teaching has changed. There was a time when I was committed to using Dienes’ multi-base blocks. Then I changed my commitment to using only Base Ten blocks (a subset of the Dienes materials). My commitment was based on my belief at the time that the blocks could reveal to students the kind of understanding of our place value system that was essential for making sense of standard algorithms. I even made a video tape (in the late 1908s) about using Base Ten blocks to teach long division. I believed deeply that I was uncovering, not covering knowledge. But I’ve changed my mind, not about the potential of Base Ten blocks to help reveal the structure of our place value system, but about how I was using them with students, thinking I was bringing meaning to an algorithmic procedure so that students had a shot at thinking relationally not only procedurally. Hah! What I really had done was increase my own understanding about the sense behind doing long division, and then I made the error (ouch!) of trying to impart that understanding to students, which I did by having them manipulate the blocks as I prescribed, thinking that they were understanding as they did so. Is this beginning to sound like a rant or a confession? I’m really not terribly prone to either, I don’t think. And I did receive a fair amount of feedback from teachers who felt that the videotape helped them see the sense behind a division procedure that they had learned mechanically. The horrid truth is that it’s possible to use the Base Ten blocks mechanically and be fooled into thinking that necessarily results in understanding.
ReplyDeleteI think that shifting from the Dienes multi-base blocks to only Base Ten blocks was an unfortunate choice. In a way, only if you can understand how numbers in any base make sense can you really understand how our base ten system works. There are many reasons why I dropped pushing the benefit of the multi-base blocks, at the core of which it required teachers to interact with mathematics that can be difficult to understand. Yes, I know we can’t teach what we don’t understand, so maybe if I focused just on base ten, I’d be helping teachers more.
So, from first mandating (yes, I’ll try and get back to the issue at hand) multi-base blocks, to then mandating base ten blocks, I now mandate neither when I work with teachers.
I’ve learned how important it is to hold my beliefs as opinions. And how important it is to keep my eye on what is the mathematical purpose of anything I offer to students. And then how important it is to observe and listen to students so that I can try and understand what they are perceiving and learning. It all comes back to the mathematics.
I’ll probably be sorry I posted this response as I typically edit, rethink, and re-edit my own words to avoid being misunderstood. But I’m still feeling the glow of Thanksgiving and what it means, and I so appreciate your opening yourself and sharing your thinking, your photos, your commitment to Cuisenaire rods. (Hey, aren’t the Cuisenaire cubes, squares, and rods the same as Dienes’ multi-base blocks without all those annoying scoring marks on the materials?)
If asked by a teacher for what would make a good starter set of manipulative materials, I have different ideas for different grade levels. And my starter set suggestions change from time to time. And they are suggestions, not mandates.
I've been following this discussion the last few days because it is something I'm experiencing in my work as Math Specialist in a 3-5th grade school. I wrote a response here, but the characters were too long so I put it on my blog instead. Here you go! I would love to know your thoughts.
ReplyDeletehttp://wp.me/p75zqr-8b
Thought provoking and thoughtful comments. When I was using a textbook it offered particular models like the tape diagram. I introduced the models and sometime the student could replicate them...at least for a short period of time. In my heart I knew this was not the end result I desired. Now that I teach in a more student-centered manner, with lesson ideas created by Marilyn Burns, John Van de Walle and others, I find that the students create their own models that help them make sense of any problem we pursue. They often share their models during our whole group discussions. If one student's model makes sense to another student they will often adopt and adapt the model for themselves. This is an end result I desire.
ReplyDeleteI do believe that concrete materials truly deepen understanding. The concrete materials we use are Cuisenaire rods, linking cubes, Place Value blocks, tiles, patterns blocks and the like, I will recommend certain materials for certain problems but, after a while, the students will choose the materials that works best for them for the problem. Sometimes the materials I offer are only used once, probably because the materials only made sense to me (as Marilyn elaborate above.) Other materials are used time and again by the students. I am not sure why that is the case and it can vary from year to year. I think this is why it is important to introduce them all and have them available for the students. I so appreciate the work of Simon and others as they introduce the concrete materials and allow the students to explore mathematics with those materials. The more often students are exposed to these materials the more they will view them and use them in ways that help them navigate through the concrete stages of learning new content.
I also think it is valuable to use the same materials for multiple content standards, too, so they students can see them as tools for understanding, not just something we use to learn about fractions or area.
Time wrap up my thoughts...I am not inclined to use prescribed models but I do think the students find deeper understanding with concrete materials.