tag:blogger.com,1999:blog-3071719252136968205.post6666180249381099080..comments2017-04-17T06:49:25.243-07:00Comments on Following Learning: Mandating the materialsSimon Gregghttp://www.blogger.com/profile/07751362728185120933noreply@blogger.comBlogger6125tag:blogger.com,1999:blog-3071719252136968205.post-52907354062146242122016-11-27T20:10:37.526-08:002016-11-27T20:10:37.526-08:00Thought provoking and thoughtful comments. When I...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.<br /><br />I 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.<br /><br />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. <br /><br />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.Nina Sudnicknoreply@blogger.comtag:blogger.com,1999:blog-3071719252136968205.post-47451098516967680802016-11-27T17:05:03.701-08:002016-11-27T17:05:03.701-08:00I've been following this discussion the last f...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.<br />http://wp.me/p75zqr-8bAnn Elisehttp://www.blogger.com/profile/04157564126954310744noreply@blogger.comtag:blogger.com,1999:blog-3071719252136968205.post-55361048413237270752016-11-27T16:31:57.228-08:002016-11-27T16:31:57.228-08:00I’ve been mulling over the “mandating” question. I...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. <br /><br />I 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.<br /><br />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. <br /><br />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. <br /><br />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?) <br /><br />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. <br />Marilyn Burnshttp://www.blogger.com/profile/16127025673987723020noreply@blogger.comtag:blogger.com,1999:blog-3071719252136968205.post-64737375096987723252016-11-27T13:14:55.500-08:002016-11-27T13:14:55.500-08:00I'd be against mandating use. Mandating exposu...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. <br /><br />It'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. John Goldenhttp://www.blogger.com/profile/18212162438307044259noreply@blogger.comtag:blogger.com,1999:blog-3071719252136968205.post-58005811181064511862016-11-27T11:15:49.199-08:002016-11-27T11:15:49.199-08:00Mandate. What an ugly word. Defined as "an of...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.<br /><br />So say you want all students to learn a Cuisenaire rods or a tape diagram. How do you go about it?<br /><br />1) Curiosity, invoke the students' - What you described with Graham's 3-Act tasks is but one nice way to do this.<br /><br />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.<br /><br />Create a need ---> see value in ---> explore / learn / understand / apply the tool<br /><br />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<br /><br />regarding this passage about two students discovering, through exploration, the algorithm for equivalent fractions:<br />"...but they completely understood it because they had built it, tested it, and found its strengths and limitations."<br /><br />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.<br />Here's an article about the California aftermath. It is worth a read but the abstract alone will give you the gist:<br />http://ci512-summer2011.wikispaces.com/file/view/Cohen+%281990%29+Mrs.+Oublier.pdf<br />Mark Pettyjohnhttp://www.blogger.com/profile/02894064823822296340noreply@blogger.comtag:blogger.com,1999:blog-3071719252136968205.post-8083261131552230812016-11-27T10:18:41.970-08:002016-11-27T10:18:41.970-08:00So much great reading between this blog and Kim...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/<br /><br />Now, 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! <br /><br />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! kgrayhttp://www.blogger.com/profile/00385187692877248447noreply@blogger.com