As I’ve worked in elementary education I became convinced that while the 5th grade math that I taught wasn’t hard to perform, explaining the thinking and principles behind them can be challenging. Getting students to think about bigger mathematical ideas instead of just performing an algorithm and coaxing students into developing thinking patterns when reading a text, require critical thinking.
I loved Seymour Papert’s, influential book, Mindstorms because it digs deep into the processes that children apply when using Logo to build. It explains the importance of discovery and theorizing when learning. I think it demonstrates the criticality of the young years for developing a child into someone who believes they can learn all things. It shows the difference between learning by imitation (not a bad way to learn) and learning by creation. The book shares a learning environment that is a beautiful marriage between mathematics, psychology, and cognition.
Papert talks about discovering gears as a child and how they captured his fascination. From this interest in gears he latched on to mathematics and he used this model he’d developed in thinking about gears to learn about math. He talks about creating learning spaces where students can have these types of personal, emotional, engaging connections to some object to learn around. He suggests that, though all students won’t necessarily be interested in gears, the computer offers a tool that can manipulate things to interest any child. He says,
“This book is the result of my own attempts over the past decade to turn computers into instruments flexible enough so that many children can each create for themselves something like what the gears were for me.”
“In many schools today, the phrase “computer-aided instruction” means making the computer teach the child. One might say the computer is being used to program the child. In my vision, the child programs the computer . . .” 5
Papert was onto the philosophy of growth vs. fixed mindset ahead of his time. I remember a teacher once telling me that when a student failed a reading level on a test and had to read another test, she’d tell them that they read it so well they get to read another one! Kids of nearly every grade level can tell when they’ve tanked something and the message that we send if we can’t talk about failure is that it’s such a big deal, so horrible a thing, we can’t talk about it. I think fear of not being loved is at the root of most insecurities and to be unable to talk about failure implies that perhaps someone is less lovable if they can’t perform a particular task, not to mention the diminishing of academic ambition. Paper tells us,
“School teaches that errors are bad; the last thing one wants to do is pore over them, dwell on them, or think about them.” 114
“Deficiency becomes identity: “I can’t learn French, I don’t have an ear for languages;” “I could never be a businessman, I don’t have a head for figures;” “I can’t get the hang of parallel skiing, I never was coordinated.” 42
“In particular, our image of knowledge as divided up into different kinds leads us to a view of people as divided up according to what their aptitudes are. This in turn leads to a balkanization of our culture.” 171
“Thus deficiency becomes identity and learning is transformed from the early child’s free exploration of the world to a chore beset by insecurities and self-imposed restriction.” 8
I remember discovering the importance of the words and explanations in my math book in jr. high geometry. I’d always been a great reader, but felt that I wasn’t a math type of person. I’d been looking at examples of math problems without reading the words, my greatest ally. As an adult studying for the GRE I knew my strengths and had the confidence of a thousand academic successes in a variety of areas to realize that I could access anything. My math study as an adult was a completely different experience. I found success and pleasure in my growth.
“First, relate what is new and to be learned to something you already know. Second, take what is new and make it your own . . .” 120
” . . . what shows up as intellectual weakness very often grows, as Jim’s did, out of intellectual strengths.” 45
“One does not expect anything to work at the first try.” 101
“Here too I was developing a way of thinking that would be resonant with Piaget’s. The understanding of learning must be genetic. It must refer to the genesis of knowledge. What an individual can learn, and how he learns it, depends on what models he has available. This raises, recursively, the question of how he learned those models.”
I struggle a bit with the exploratory method of learning, at least in a school setting. Presenting models to be imitated is the general rule. If we want something to improve we have to be able to measure improvement. If we are going to measure improvement we’d better have some goals for where we’re starting and where we hope to get. If we hope to get to a specific place, specific steps have to be taken to get there. You can’t skip fractions because you were more interested in multiplication and still think you’re going to be able to do algebra.
Universities will be the body to lead the change. When they begin looking for thinkers and problem-solvers and ask for evidence of those kinds of skills, schools won’t be held hostage to college admissions requirements in teaching x number of things by x date. Thinking will be more highly valued than measurable skills in particular areas.
I think that teachers of the future will largely be trained in psych and tech. Computers are very good at teaching students things that computers are good at, but when students train the computers they develop both the hard skill, soft skills and synthesizing skills they’ll need to be successful in a post internet world. The collaboration and focus on problem solving with groups and as individuals develop people who will find success in any scenario or career path regardless of their initial knowledge in the area.
Papert said a lot of great things. Below are some favorite quotes I hope to return to:
“The discrepancy between our experience of ourselves and our idealizations of knowledge has an effect: It intimidates us, it lessens the sense of our own competence, and it leads us into counterproductive strategies for learning and thinking.” 172
“Imagine a suggestion that we invent a special language to help children learn to speak.” pg. 35
“Very few with the imagination, creativity, and drive to make great new inventions enter the field.” 37
“This book is an exercise in an applied genetic epistemology expanded beyond Piaget’s cognitive emphasis to include a concern with the affective. It develops a new perspective for education research focused on creating the conditions under which intellectual models will take root.”
“The social construction of the individual is as a bundle of aptitudes.” 43
deaf children cannot have language because they can’t hear – cars aren’t horseless carriages
“When a teacher tells a student that the reason for those many hours of arithmetic is to be able to check the change at the supermarket, the teacher is simply not believed. Children see such “reasons” as one more example of adult double talk. The same effect is produced when children are told school math is “fun” when they are pretty sure that teachers who say so spend their leisure hours on anything except this allegedly fun-filled activity . . . and I think it introduces a deep element of dishonesty into the educational relationship.” 50
“But now that we can purchase electronic calculators cheaply we should reconsider the need to expend several hundred hours of every child’s life on learning such arithmetic functions.” 51
“Most parents have very little idea of why anyone should know this, they become indignant when their children do not. They assume that there must be a profound and objective reason known to those who better understand these things.” 52
“Some of the most personal knowledge is also the most profoundly mathematical.” 54
“If children could see Descartes’ invention of coordinate geometry as something not totally alien to their own experiences of daily life, this could not only make Descartes more meaningful but, at the same time, help the children come to see themselves as more meaningful.” 97
” . . . a central theme was how debugging is facilitated by the use of an appropriate description of a complex process.” 111
“People are capable of learning like rats in mazes. But the process is slow and primitive. We can learn more, and more quickly, by taking conscious control of the learning process, articulating and analyzing our behavior.” 113
“The child did not yet know how to say it, but what had been revealed to him was that he and the teacher had been engaged together in a research project . . . Discovery cannot be a setup; invention cannot be scheduled.” 115
“She emerged when she was ready, several weeks later . . . ” 119
“The experience, lacking immediacy, is slow to change the student’s intuitions.” 124
“The dynamic visual effects of a TV show, an animated cartoon, or a video game now encourage them to ask how they could make what they see.” 129
“Thus, the importance of studying the structure of knowledge is not just to better understand the knowledge itself, but to understand the person.” 164
“What is a man so made that he can understand number and what is number so made that a man can understand it?” 164
“Even when the process looks identical, is there any reason to think that the underlying processes are the same?” 164
“Some researchers try to make programs be intelligent by giving them such quantities of knowledge that the greater part of solving a problem becomes its retrieval from somewhere in the memory.” 166
“The theme of this book has been the idea of exploiting this special role by giving children access to computational cultures.” 170
“Not all programming languages embody this theory of pure procedures. When they do not, their role as metaphors for psychological issues is severely biased.” 171
“The dynamics of lift are fundamental to flight as such, whether the flyers are of flesh and blood or of metal. We have just seen a principle that may be fundamental both to human and artificial intelligence: the principle of epistemological modularity.” 171
“We have already seen that despite their experience of themselves as theory builders, children are not respected as such.”
Their own thinking is much more like the mathematician’s than either is like the logical ideal.”
“But we could hardly ever learn a new idea if every time we did we had to totally reorganize our cognitive structures in order to use it or if we even had to insure that no inconsistencies had been introduced.”
“The act of learning is itself a local event.” 172
“In our culture number is richly represented, systematic procedure is poorly represented.” 175