“It is generally agreed that concrete experience must be the foundation of mathematics learning. When children find it difficult to understand arithmetic it is at once suggested that this is because it is too abstract; for small children the study is then simply reduced to the counting of objects. It seems to me that there has perhaps been too great a tendency to make things concrete and that perhaps the difficulties children experience spring from the fact that they are kept too much at the concrete level and are forced to use too empirical a mode of thought.” (p2, my emphasis)What kind of abstraction is she looking for then? Exactly the kind that Connecting Arithmetic to Algebra is recommending: looking for general patterns in the way simple arithmetic works.
"I find it of limited value to ask children a large number of definite, restricted questions whose answers they obtain through manipulation of the rods. On the other hand, I find it most profitable to start with vast questions which can be seen in a number of ways and which permit a continuous analysis of the dynamics involved. This is why I shall consider here families of equivalent additions, of equivalent subtractions, and of equivalent products and quotients." (p3)And as well as having a clear idea of the kinds of areas that are fruitful to investigate, Goutard had a very strong view of the role of the teacher.
"The teacher is not the person who teaches him what he does not know. He is the one who reveals the child to himself by making him more conscious of, and more creative with his own mind. The parents of the little girl of six who was using the Cuisenaire rods at school marveled at her knowledge and asked her: 'Tell us how the teacher teaches you all this', to which the little girl replied: The teacher teaches us nothing. We find everything out for ourselves.'
It is evidently very difficult to give the child so complete an impression of non-presence, and to convince him that he alone is the artisan of his own education, but this is the way in which free and conquering minds are formed." (p184)You can see that Goutard has an abstract way of writing, quite philosophical and psychological. I find I want to ask her, "How did the teacher teach her?? Yes, I know she stood back and gave her space. But how did she set up her sessions? How do you reconcile the seeming oxymoron of having the student genuinely following their own way, and at the same time having the teacher directing them towards the vast questions? How, not just in general terms, but how does a lesson go? What do you say? What do you do?" I want to see dialogues, with just a little bit of analysis, like you find in Connecting Arithmetic to Algebra. I want to get a feel for actual lessons.
But we haven't got that.
Luckily, all is not lost. If we don't have a clear roadmap, we've got a clear destination and a bearing. And I'm getting a more and more clear idea about the details. It helps to have colleagues like Caroline Ainsworth bringing Goutard's ideas to life. And #MTBoS collagues like Tracy Johnston Zager, Kristin Gray, Mike Flynn, Elham Kazemi and Kassia Wedekind and many others who while they understand the vital importance of the agency of students, also seek ways for them to connect with the 'vast questions'.
I think the square can be circled.
Our beginnings are small:
After which we go off to write our own ones in our journals.
The next step is to take one of these, and focus on it. (For example, I find the relation between the five threes and Jinmin's triangle number way of expressing it worth following. Worth getting the rods out and investigating.) And then you're listening out for the generalisation, the claim. Up on the claims board. Do you agree with it? Then how would you explain it? Show it with pictures, words, equations, a story...
Year 4 arrange Cuisenaire rods to match written equations #istlive pic.twitter.com/T8edXvL49O— Simon Gregg (@Simon_Gregg) November 3, 2015
I hope we can do our bit to reanimate Goutard's brilliant philosophy. I hope too to capture a few of the moments while we're at it, and perhaps share them with you.
Today we got our "Mathematical Claims" board started. To get the ball rolling I showed an image and asked for remarks. After the obvious features, I asked if anyone could say something more general. "What does general mean?" someone asked. We talked about that a little and then Tibo said:
to which Jinmin adedd:
These ones came a bit too quick and easily; they'd talked about them with teachers before. But they went up on the board to get the idea going. I'm looking forward to claims that come out of a more immersive experience of trying things out, reflection, and talking things through, ideas that are a little more hard-won, (and hard-defended).
"The teacher is not the person who teaches him what he does not know. He is the one who reveals the child to himself by making him more conscious of, and more creative with his own mind."ReplyDelete
I appreciate those sentences so much because I realize how difficult our job really is. In creating these spaces, there is so much thoughtful planning, questioning and feedback by the teacher. All of this is so evidenced in your work that you do Simon!
I love all of this work around conjectures and claims so much but I am wondering how you think about these vast ideas? Is there a progression that you have structured in your mind? I find that I do this work, but I go by the order by which my curriculum flows and add this work in. I am wondering how you think about your planning because your activities pull out such great noticings and wonderings!
Kristin - I agree, it is a difficult job, and there are a so many things to balance, beyond what I've mentioned here!Delete
The curriculum is one of them, and it'll mean that certain things have to come in at certain times.
Then there's a kind of natural progression.
We know the ones that stand out. Generalisations like:
a + b = c is equivalent to b + a = c
a x b = c is equivalent to b x a = c
a - b is not usually the same as b - a
a ÷ b is not usually the same as b ÷ a
a + b is the same as (a + 1) + (b - 1)
a - b is the same as (a + 1) - (b + 1)
(a x b) + c is not usually the same as a x (b +c)
and so on.
Goutard is very good at seeing how these families of equivalent operations might be explored with rods, so that generalising can come slowly out of experience.
These are just the ones that pop into my head first. Maybe we need to work on a list, all of us together. The field is much richer. The work we did on consecutive numbers, which starts out with how odd and even numbers add, leads to a rich seam. And other work where I tend to lead the students, with exploration with manipulatives, to key ideas could lend itself to being discovered by the students and formulated as generalisations - I'm thinking for instance of our factor and prime number work.
I don't feel that I've at all done justice to your question - but then maybe it needs a whole book, rather than a quick reply.
I'd add that there should be some unpredictability in this though, and flexibility on the part of the curriculum and teacher. If a student comes up with a conjecture, it seems to me that that is often the one to go with, if we are to develop that responsive kind of classroom that's going to allow students to do the thinking.
What is powerful here to me is that you are teaching generalizing as a process. Inherently more powerful than teaching generalizations.ReplyDelete
That is an insightful contrast John.Delete
Yes, the verb seems a much safer bet.Delete
For some reason this quote popped into my head:
"If God held in his right hand all truth, and in his left the ever active impulse towards truth, even subject to the condition that I should remain perpetually in error, and were to bid me make my choice, I would fall humbly before his left hand, and would say, Father give me that; pure truth is for Thee alone."
The problem is, "pure truth" seems so much more accessible in maths; it's like the goose that lays the golden eggs. People want to get straight to the gold; "can't we dispense with that tricky laying-the-egg bit?"
Love this quote and I couldn't agree more. (Looks like it's from G E Lessing, someone new to me.)Delete
(I don't know Lessing either. I think Kierkegaard quotes him somewhere.)Delete
Also, I think a lot of Goutard's concern about the time on concrete is that people would teach the tool, as opposed to the content the tool was supposed to support. In your students' work it looks more like an understood representation in which they can think and express.ReplyDelete
"teach the tool" - that's another nice articulation. Mind if I steal that phrase?Delete
I totally agree. The critical step for me is bridging from the concrete to the abstract. C-rods continue to be a useful tool for my sixth grade mathematicians.ReplyDelete
I'd be curious about the demographics of the children involved, and specifically in regards to the family's socio-economic status.ReplyDelete
Tell me more about this, Owen...Delete
Striking that it might be our goal to create, "An impression of non-presence." Later, as an adult, the child may realize how much craft that actually took on her teacher's part. I'm not entirely convinced, however, that difficulties are caused by being, "Kept too much at the concrete level." I've always thought that many difficulties spring from rushing too quickly through the concrete level, or from not even having a concrete level to begin with.ReplyDelete
As always, lots to think about!
David Rock has some nice books about neuroscience that bring it down to layman's terms. If I had to summarize I would say that facilitating an insight is a much more powerful way to learn and then apply, than being shown or told. I see parallels to Rock's ideas in "an impression of non-presence." If the student has the insight, it's theirs, not the teachers. I'm curious, when you say striking, does that carry any negative or positive connotations?Delete
Re:concrete. Well said!
Thanks Joe. I like it when you tell me the times you're not convinced!Delete
You know I think there's a huge place for the concrete, that can continue right the way through schooling.
Like Alex Overwijk's use of cubes for quadratic equations:
Or using Lego for probability:
And I think the "impression of non-presence" is about not rushing students towards the mastery of structure that you're hoping for. It takes time.
Simon: It's great to have a safe space to work all this out. Much appreciated!Delete
Mark: I mean striking in a positive way. Creating the conditions in a classroom where students can be "artisans of their own learning" and we become less the center of the action is hard work. Too often observers and evaluators focus on the teacher, like they're watching a performance, rather than on the students and what they are doing. That needs to change.
Can students ask-and-answer vast abstract questions without being taught?ReplyDelete
My first thoughts before reading further:
1) It depends, but probably not.
2) What does being "taught" mean? Procedures, processes, or something else?
Why do I say no? Largely anecdotal, but at least in the U.S. the vast majority of mathematics is taught in the abstract and symbolic. Ask 10 Americans about math and you'll probably hear "I'm not a math person" more often than not. So it seems to me that we need to invert the idea of beginning with abstract (or only ever teaching in the abstract).
Cue Thomas Carpenter, Elizabeth Fennema, Megan Franke, Linda Levi, Susan Empson. Also, Michael Battista.
Children need concrete experiences with new concepts first. Carpenter calls this direct modeling. The concrete experiences help students to develop more abstract counting strategies, and derived fact strategies. These ideas are drawn from the book Children's Mathematics: Cognitively Guided Instruction (CGI) by Carpenter and colleagues as well as Michael Battista's Cognition Based Assessment and Teaching (CBA) series. These have been a huge help in allowing me to better understand the development of children's mathematical thinking. It made sense as I read it, and it made just as much sense when I saw it play out with students in my classroom.
Off the top of my head I see these ideas of direct modeling leading to or facilitating abstract thinking in the work of many other mathematicians that I admire such as Marilyn Burn, John Van de Walle, Cathy Fosnot, Arthur Baroody, Deborah Ball, and Magdalene Lampart.
"they are kept too much at the concrete level"
- The way I read this, the teacher is the one asking the student to use a concrete manipulative. Teachers don't have to force it. Students need concrete manipulatives to make sense of new ideas, but once they have internalized that concept, they will jettison the manipulatives and work more abstractly when they are ready. I see the teacher's role in this is just as Madeleine Goutard says "the one who reveals the child to himself by making him more conscious of, and more creative with his own mind."
Can students work on big open-ended, abstract problems? I think the answer is very much so, but first they need to learn how to. They need the invisible hand of the teacher that Goutard describes to help them develop thinking, reasoning, and problem solving strategies.
Open-ended questions are fantastic elements of a good math curriculum. I can see how valuable it would be to have the dialogue from a task presented in a Marilyn Burns book accompanied by an open-ended task.
I sketched a little diagram of how I see a path playing out during open-ended problems. I think it's a longer path to generalizing because there are more things to notice, and problems to solve with an open-ended task. This is where I think you can just jump in and as you do I am certain that you will learn what those details are.
Another note: I had never heard of Goutard before this post. I'm now excited about reading her book. It reminds me of finding out about W.W. Sawyer. He has books that came out in 1943 and 1964 (forever ago!) that espouse teaching and learning mathematics in ways that make sense. It speaks to a bigger issue, these ideas aren't new! So what can we as educators do to gain more momentum so many, many more students benefit from having instruction from a Simon Gregg?ReplyDelete
Lastly, thanks for writing this. I'm not only excited about Goutard, but I'm excited about my colleagues. Goutard's philosophy dovetails nicely with the K-6 mathematics vision that was developed by a team that my colleague Nina Sudnick led. They talked to every single math educator in the district K-6 to gather input about what was working well for them, and what challenges they faced. From that feedback they crafted a vision of where they wanted to go as a district that could guide big decisions made about mathematics. I see Goutard in the vision that was crafted, and that makes me proud of my colleagues. Here's the vision:
"Students’ positive disposition towards mathematics allow them to use their own thinking and reasoning to discuss and learn mathematics in situations engineered by teachers who are collaborative learners, who understand student thinking, and who support the development of students as problem solvers."
What a great thing to come together and come up with a statement of vision like that! Yes, that is the kind of thing that excites me professionally too - us as colleagues discussing, sharing our insights and approaches. Thank you so much for taking the time to share your thoughts in these comments Mark!Delete
This is such a pithy post Simon and you've left us much to chew on. This quote leaves me with more questions a than answers, “Children are forced to use too empirical a mode of thought.”ReplyDelete
I think there are many variables at play here in regards to the types of relationships fostered between the student, manipulative, and teacher. I'm left trying to identify the role of each piece. Manipulatives are used to make sense of the math in non-routine situations. If the math is routine, is the use of manipulative still an expectation? What is the role of the teacher when thinking is routine? Both the teacher and manipulative would seem to be a mute point.
I’m thinking that both teacher and manipulative are required when an exploration is required. Comes back to what Joe said about moving students too quickly away from the concrete. We ourselves need to be comfortable with exploring the messiness of math. If we’re not, then we look to move students through the learning and away from manipulatives, and never allow them to wrestle with their own understanding.
I've been thinking about Joe and Mark's comments and, now, your comment Graham. I think you're right, there's just so many variables! Trying to describe a way to teach is a bit like giving directions to a place - and the number of times people have given me directions and I've got completely lost! Words like "turn left at the first turning" seem simple enough at the time, but then... is that little road a turning?ReplyDelete
And that's partly my frustration with Goutard's book. To get a really good grip on a way of teaching, I feel like I want videos of lessons, or transcripts of conversations, photos of written work and the like, as well as the intentions behind the lesson and some description and justification of approaches. It's what's so good about our blogs - we start to get a real feel for lessons, even though the classrooms are thousands of miles away.
Simple binaries like "use more manipulatives" are good as first approximations, but like "turn left" they don't cover the complexity.
I've just been reading about how the physicist Feynman competed with an abacus-seller. It's a nice story (although the elements of speed and competition are a distraction here) and it captures something. Ultimately we want to know our way around the landscape, to know some landmarks that help us do that.
But our manipulatives are not abacuses. We're doing something different with them. At their best, they're a ladder leading up to a more abstract understanding. And I think Goutard has it right, that they are at their best a ladder that's in the hands of the student, obviously with the teacher there too proposing, probing and pointing from time to time. Her experience with very young children gave her the appreciation that play and experimentation were going to be at the heart of the process.
But again, I feel my attempt to encapsulate something is too abstract, could mean too many things.
And that's why I love seeing actual lessons and activities, and reflections on them on you guys' blogs!
Simon, I am in way over my head with this crowd. Many great comments.ReplyDelete
1) I don't think Goutard meant the C-rods when she was worried about keeping children in the concrete. I think her reference in your quote is related to the traditional math curriculum where we subject children to endless repetitions of counting and circling objects on worksheets or even counting manipulatives.
She is, however, concerned that the rods not be used as mere calculation tools. That misses the genius of the rods altogether.
Gattegno sets out a path in what he calls, "The Subordination of Teaching to Learning," but it is hard to wrap your mind around at first. This is a great book from the perspective of various teachers: https://issuu.com/eswi/docs/gattegnoeffectwebbookcolor
also this book has changed how I think about my own education and the students in my care http://www.amazon.com/How-We-Learn-Should-Taught/dp/0956875505.
The subordination of teaching to learning seems much easier to implement one on one. As a homeschooling parent/tutor, that leaves me with an advantage. Gattegno said that the only thing that can be educated in awareness. My blog post series has helped me more than it's probably helped anyone else. In thinking through the posts, I see how he is creating awareness in each carefully chosen game/activity and how that is going to come back around later on. TIn the recap of chapter 2 Gattegno's Textbook 1, I will be addressing awareness more an what I've learned and the mistakes I've made. I am also looking forward to our training at the Bronx School of Better Learning http://www.bronxbetterlearning.org/, which is a charter school based on Gattegno's work. Maybe I will have something more intelligent to say then.
Thanks Sonya. I don't believe that "over my head" bit though!Delete
Yes, I'd like to read some of Gattegno's thoughts more generally on learning, and I've heard people refer to the subordination of teaching to learning. Something to look forward to!