Reflection and Questions
Posted by DL, AL, RC and TI
Here are some key take-away points from the article to help guide your reading.
Knowledge, learning, and cognition are situated
- Knowledge is partly the product of the activity, context, and culture in which it is developed and used.
- Concepts and knowledge are developed through activity.
Learning is a process of enculturation
- To learn to use tools as practitioners use them, a student, like an apprentice, must enter that community and its culture.
- Given the chance to observe and practice in situ the behavior of members of a culture, people pick up relevant jargon, imitate behavior, and gradually start to act in accordance with its norms.
Authentic activities vs. traditional classroom activities
- Authentic activities are simply defined as the ordinary practices of the culture.
- Traditional classroom tasks tend to fail to provide the contextual features that allow authentic activity.
- Cognitive apprenticeship supports learning in a domain by enabling students to acquire, develop, and use cognitive tools in authentic domain activity.
- Apprenticeship helps to emphasize the centrality of activity in learning and knowledge and highlights the inherently context-dependent, situated, and enculturating nature of learning
- Cognitive emphasizes that apprenticeship techniques actually reach well beyond the physical skills usually associated with apprenticeship to the kinds of cognitive skills more normally associated with conventional schooling.
Tasks for this week:
- Taking into consideration our reading, think back to a past classroom experience where you either did not retain the knowledge learned in class, or did not know how to apply this knowledge to other areas. Describe briefly why that was, and how would you do things differently to make the experience more "authentic"?
- Visit the class Google Moderator page and post and/or vote on comments, topics, or questions that you want to discuss in class on Wednesday. [The results are given below.]
|QUESTIONS posted to Google Moderator page||VOTES|
|1. "If schools adopt a cognitive apprenticeship model, should the way students are assessed change as well? In what ways should it change?" [RC]||8||1|
|2. "Since in a cognitive apprenticeship model, each individuals' learning result will be very different according to the different "project". How could student acquire the knowledge systematically, if schools turn to a cognitive apprenticeship model." [ZH]||5||0|
|3. "How should math be taught once the mechanics have been learned." [DG]||5||2|
|4. "After reading several papers on the subject of situated cognition I'm still puzzling over how a learner moves from situated knowledge to abstractions, which are still incredibly important to disciplinary cultures. (Related to Question 3)" [SL]||4||0|
|5. "Is there proof that encountering a concept in a situated way is more effective than pre-teaching the concept and then using it in context?" [VC]||3||1|
|6. "Craft apprentices are great at perpetuating their traditions but they shy away from new ways of doing things. Are cognitive apprenticeships similar? When faced with new, complex global problems, will learning via old frameworks provide new solutions?" [SK]||2||0|
Student Reading Responses
Posted by SL
I'm completely in sync with FG and DG this week. Mathematics, or more specifically, Calculus was a subject that I never grasped in school. Geometry made perfect sense to me. Algebra made a lot of sense too, and I think word problems helped in this case, though unlike Danny I was never very good at decoding problems. Trig, however, was more problematic. I understood some of the basic underlying principles, but mostly memorized formulas and tricks. My teacher in high school thought I was a terrific math student, and I made great grades in all mathematics related courses, but it was because I was good at navigating the school culture rather than assimilating the math culture. I had no real understanding of the practical applications of Trig, and that lack of understanding continued into my Calculus class and worsened. Again, I made great grades in Calculus, but had not a clue about what I was doing or why.
When I got to college I thought that Calculus would be a simple course for me, as I had done so well in high school. So I signed up for the two entry-level Calculus courses. I was very wrong about my capabilities, my memorization and tricks did not serve me well in that environment. I was so bad at Calculus that I had a private tutor, who practically gave up on me. I did pass the courses, but not with flying colors. And I never unlearned the bad habits of school culture, so never really understood the underlying logic and concepts.
It wasn't until maybe 6 years ago when I realized that Calculus had more to do with flows and things happening over time that I began to understand the big picture. This was in conjunction with my entry into the world of making things. All of a sudden I was desperate for the tools and capabilities of calculus and trig, they totally made sense in an engineering environment. I am perfect fodder at this point for just-in-time learning in these mathematical disciplines. If I had it to do over again, I would have immersed myself in a project-based, fabrication or programming learning environment to contextualize my mathematical skills and understanding, and to inspire myself to learn more deeply, more geekly. I loved the two examples in our reading of the magic square, and the story problems. I just wish my teachers in high school had been as visionary in their approach to teaching.
Posted by JC
I have a similar experience. I managed to get through a math minor in college; I quit at this point when my pattern matching was no longer enough and I was attempting to understand the underlying concepts. I find that going back through some math for engineering coursework makes so much more sense now; either it is the way the material is presented or I'm deliberately trying not to just pattern match (this is tough to do as brains are made for pattern matching).
I had an amazing math logic professor in college though; I loved that class. He was passionate about when he taught and brought the material in context of the real world (like Danny had suggested). Whatever methods he used worked for me. I really wish all of my educational experiences were this clear.
Posted by DG
I've since mastered this, but the readings discussion of mathematics struck a chord with me. When learning basic arithmetic, I remember my fellow students dreading the word problems. Once the mechanics are learned (symbol manipulation), it then becomes important to know how to bind the symbols to real world values. This often caused difficulty in class, and we were normally assigned a few word problems (not enough, imho). These problems could be easily completed, because I knew all the information needed was in the problem, and there were very few problems with unneeded complexity it became a process of decoding the problem.
To fix this issue, I would advocate "world problems", problems that exist in the world and need to be solved. The students are given a challenge to build/design a bridge to get from point A to point B over a ditch. By manipulating the tools an materials that are available we can force them to use different types of operations. For instance if they only have 12" ruler they can use addition and multiplication to determine the distance. By providing various sized pieces of lumber, they will work with subtraction and addition. By adding constraints, we can force them to use trigonometry to do some of the calculations. Through minimization/maximization requirements we can force them do even more advanced analysis.
The key is to teach them the tools and then give them a problem in which the tools are required, but their use is not specified.
Posted by JC
I relate to your response and like the suggestion of math via real world problems. This is similar to one of the case studies outlined in the paper. I wonder how different the grade school math experience would be given a more tangible understanding of math.
Posted by FG
I think the idea of using 'world problems' is great, and as I wrote in my earlier post, I think this method would have made a significant difference in my assimilating math concepts in primary and secondary school.
However, I am not sure anyone could 'force' me to do anything:)...
Posted by AL
I really like this idea of applying math to "world problems." I see that you have posted a question on the moderator site-perhaps we should explore this idea of world problems further in our discussion.
Posted by FG
To answer the first question:
Well, it will have to be math for me. All of it: algebra, geometry, statistics, algorithms, trigonometry... the whole family. I struggled with these abstract concepts from day one throughout my six years of primary school and six years of high school. I think it was a painful experience for my math teachers too:) All of them - those at school and those who would give me private lessons on weekends, holidays and the summer months, just to keep me afloat and help me pass the exams. Our attic at home is now full of notebooks filled with equations and little drawings of animals, coins, banknotes and buckets, anything that can be counted and all sorts of 'real world' objects, like the jars and butterflies in Brown, Collins and Duguid's paper. But like them, I also thought that filling those buckets endlessly and calculating these distances which I would never cover had little connection and relevance to the real world, at least to my world as a child and teenager.
One of my biggest stumbling blocks in my learning math [or more accurately non-learning], is that throughout those years of study I never understood what was the point of doing math, what were the real-life applications and uses for them. This was never explained to me. I feel all I wanted was an explanation of why math mattered and how they mattered in the real world. 'When will I need this as an adult?' was often my question to my teachers.
So I guess the argument of our authors this week, which stresses the need for the use of real-life applications that directly speak to the child/teenager makes sense. Only, designing such 'real-life' examples and case studies is a challenge in itself, since to some extent they are already being used in the traditional system of education [with objects and 'real' problems, narratives, etc]. As is argued in our two papers, embedding these mathematical concepts in the child/student's everyday life seems to be taking this methodology one step further, and I think it is definitely a worthy experiment. I certainly would have loved to have had those methods at my disposal when in school.
Posted by JC
When I was in high school, I struggled in Advanced Chemistry class. I recall the concepts being so abstract that it was easier to pattern match/memorize than to truly understand. I happened to have a very patient teacher who spent time working with me, answering questions, and explaining the concepts in as many different ways as was possible. He allowed me to have epiphanies, no matter how little. And he encouraged me to keep trying; "sometimes we need to review concepts many times from different angles before we understand". Despite attempts and the paper trail showing I survived the course, my patterns only lasted long enough to get through the course. This experience was frustrating enough to prevent me from wanting to continue studying chemistry. The Advanced Chemistry concepts were being taught in isolation when I was in high school. I don't recall any tangible comparisons to put the concepts some context. This made them extremely difficult to memorize as well.
I recall wishing for better visual representations for the concepts being taught. It was not enough having a chalk board. Perhaps today's 3D animation methods would have been helpful. I imagine visuals beyond static 2D to help; however, I am not coming up with any ideas for how to equate Advanced Chemistry concepts to something I could relate with at the moment.
Posted by DG
I've been thinking about chemistry on and off for the past couple of months, and I think it would be a lot more intuitive if probability was core to the explanation. I envision it would looks something like:
- Introduce the concepts of an Atom: Electrons, Protons, Neutrons
- Talk about different elements and how they relate to their constituent parts.
- Talk about attraction/repulsion, using images/models, and talk about different types of bonding.
- Use computer simulations to show 2 atoms bouncing around and when the randomly hit each other in the correct configuration have them bond.
- Using this model you can then add catalysts, add heat, talk about endothermic and exothermic reactions etc... all with very simple atoms and molecules.
- The simulations can be scaled up and we can start using real science terms to talk about the reactions and describe what we know about the interactions.
- At this point make a departure to industrial manufacturing techniques to understand how chemicals are produced in the real world, design considerations for chemical plants (this reaction releases hit, so we should put it near one that requires it, etc).
- After that organic chemistry can be introduce, but with a focus on actual practical biology. Talk about DNA replication, the Krebs cycle, DNA replication, etc...
I think chemistry needs to be taught with significant physics and biology content so that the material can be appropriately situated.
When we think of addition and subtraction we are not learning merely how to manipulate the symbols we have an intuitive understanding of the symbols from years of counting things around us. In chemistry we're quickly introduce to the elements/molecules and then jump right into manipulating them without truly understanding what is going on. Before the advent of computers this was how it had to be done, but now that we can simulate/see atoms bouncing around on a computer screen there is no reason that is where we shouldn't start.
Posted by JC
I would sign up for your Chemistry class now:) The visual and practical aspects of your proposed method sound ideal for me. Unfortunately in the era I was taught, this was not possible (as you mentioned). So in no way am I blaming my favorite HS teacher; he did what he could with what he had. Chemistry makes sense when presented visually; I experience this from time to time when I need to look up a concept.
Also, your description of how physics and biology provide the "context" in which chemistry can be taught makes sense and resonates with the paper.
And I appreciate the rant.
Posted by FG
As a follow up to my earlier comment, I thought I would add a note on the general proposal for context- and culture-based learning, or as our authors call 'situated cognition.'
As said, I believe that concrete, real life and realistic applications and examples would have helped me grasp and master mathematical concepts much better than the way they were taught by my teachers in school.
Having said this, I wonder if the purely situation-based model is not slightly reductionist. I have too many memories of learning, at school and in a new job, skills and how to use tools that I never had further use for in subsequent years. This is especially true for in-house technical tools and computer programs used in companies that are so specific that you never encounter them later on, outside of a given company.
I would also think that the crucial skills in today's competitive marketplace are those that are transferable - that is, those that are general enough so that the person will be able to apply them with ease to many different situations and projects and make versatile use of them. Example of these are abstract thinking, the ability to summarize, write with impact, think fast and consider multiple scenarios, problem-solve efficiently and creatively, adaptability and vision, among others. The ability to learn fast, on the job, is also highly valued.
When these more general skills have been integrated into one's learning, we become self-sufficient. These skills become part of the fabric of the person I am, part of my personality, they become 'second nature' - in other words, I can apply them easily to any new situation or task.
And isn't it what the nurturing culture and human resources that the papers describe are supposed to do - support the newcomer in his learning period and help him learn fast the specific skills and requirements of a particular job? Is there a need to develop an entire educational system around them, since they are already performing their main purpose of training and forming the new recruits to help out in their adaptation to the new culture of the company or learning environment?
Each culture, school, learning group, each new job and company has its own very specific mentality and set of practices, requiring very specific tasks and skills - as the authors of 'Situated Cognition and the Culture of Learning' make clear. But what happens then, when one leaves a specific group to enter a new one? Or start a new activity? How will people know how to transfer easily from one to another if the skills he/she learned in the previous position were applicable only in its context?
If the context and community of a learning environment or new activity help in the learning curve - great! But if they don't, we still have to learn the given skills or perform the given tasks, right? I am tempted to use the supporting situation and resources as just that - help resources. But I am not sure about devoting a whole educational approach to this practice.
The culture-centered approach to learning and knowledge also raises the possibility of agenda-based teaching. The authors are right to point out that any specific learning or work environment comes with its own culture, beliefs and narratives developed over years in a closed community. This makes me think that this type of environment is likely to produce very subjective notions of what needs to be learned, how, etc. I would be hard-pressed to find such narratives and conversations devoid of subjective thinking and clear personal purposes.
These are just my initial thoughts and immediate reactions to the readings - I initially thought that teaching a skill or subject should be as objective as possible. But then, there might be some benefits to a more directed, subjective approach and guidance in a learning situation. Perhaps the learner learns better/faster if aided by some opinionated teacher/guide, rather than some neutral and less involved person.
As far as I am concerned, the debate is open on this one.
In any case, I agree with Duckworth: let's keep our educational programs 'unexpected' - I certainly embrace improvisation and the creative use of tools and situations from the real world in education.
Posted by VC
Being asked to re-imagine the teaching of something you don't understand is a funny thing! I only made it through AP Physics my senior year of high school because I was already friends with the smartest people in the class and the teacher allowed for partner tests. My engineer father, my friends, and the teacher all put in extra time and effort to try to explain things, but it never clicked. Like Lass, I learned to leverage school culture to succeed. If asked to list Newton's laws or define velocity, I could certainly do it, but I couldn't figure out the word problems. I still don't understand what happened; I was always fascinated by the class lectures but simply could never apply the concepts to labs or my homework. Perhaps I had trouble visualizing the different forces acting on an object and my problems snowballed into a comprehension disaster. If I recall correctly, my teacher tried his best to make the experience "authentic" with lots of NOVA videos and demonstrations, and I don't know if there was anything else he could have done to make it clearer.
The main point of the Brown, Collins, and Duguid reading-that concepts should be taught in the contexts in which they will be used-seems like common sense, but my experience with high school physics definitely shows that there are limits to this style of teaching. I don't think I failed to understand physics for a lack of effort on my part or my teacher's; perhaps (and here comes another Lass comparison) we eventually develop the maturity or cognitive space to rethink how we understand abstract concepts like calculus and physics. (I've been meaning to watch the introductory physics lectures on MITOpenCourseWare to see if my time of revelation has come.) This is not to say Brown, et al., were wrong about promoting situated learning, or even that they suggested that their method would solve all educational problems, but in my case, I really wonder whether there was anything else that could have been done.
Posted by SL
I kind of agree with you here, VC. We do have to reach a certain maturity, perspective and cognitive capacity to be able to absorb this kind of knowledge. In my case, I guess I just wasn't ready. There are important subjects and abstract understandings that don't seem to have easily applicable contexts. How do you teach these abstractions? Physics is a discipline that in some cases has great contexts, and in other cases not so. How about general or special relativity? While relativity is about space and time and gravity and motion, the deeper you dig, the more complex and abstract the concepts become. Same for quantum mechanics. On the surface these are elegant, simple ideas, but the deeper you dig, the complexities and abstractions can be overwhelming. These are concepts that go way beyond normal applications and contexts, yet are really important. And this kind of knowledge points to a moment in the situated cognition discussion that I always trip over. How does situated knowledge translate into conceptual knowledge. I'm not yet satisfied with the answers to that question. Seems a bit of hand waving happens at this point in the discussion. Learning does happen through experience, apprenticeship, and community, but there's something else at work as well that allows the cognitive construction of abstractions. How that happens still eludes me.
Posted by DG
Richard Feynman's thoughts on textbooks can be found here:
- Feynman, Richard. "Judging Books by Their Covers." In Surely You're Joking, Mr. Feynman! New York, NY: W. W. Norton, 1997. ISBN: 9780393316049.
Posted by MN
Taking into consideration our reading, think back to a past classroom experience where you either did not retain the knowledge learned in class, or did not know how to apply this knowledge to other areas. Describe briefly why that was, and how would you do things differently to make the experience more "authentic"?
During the summer of 2007 I took a class at the Harvard Summer School, called "Quantitative Methods for Economics". This was the first applied math class I ever took. The class was very theoretical and did not relate to real-life practices or applications in business and economic contexts. Besides the fact that the content itself was much more advanced than what I could have absorbed, I had particular difficulties in figuring out what the formulas and theories meant without understanding why they existed. I simply memorized the formulas, but they were naturally short-lived.
Because of this experience, as a read through the Situated cognition and the culture of learning by J. S. Brown, et al., I could particularly connect to his idea of inherently context-dependent, situated, and enculturating nature of learning.
Posted by ZH
My undergrad background is industrial design (product design). we had a class called material study (I cannot really remember the exact name). On the class, the professor introduced many different materials and the different qualities of them. He explained the materials by analysis its chemistry components and structure. However, to a designer, this level of introduction would not make a lot of sense to help them use the materials in a smart way for a product, since we could not really understand those materials just through a serious of lectures and homework. And understanding the chemistry components was also not necessary.
After graduation, I worked for a design company. In the studio, there is one big room which is used for showing different kinds of plastics. Every design team has an engineer who knows the materials very well. When we start to work on the different projects, we have chances to know some specific materials deeply. For example, one project is to design a sports watch for NIKE. NIKE provided couple of rubbers with different types. We just played around and tested those rubbers. Those experiences at the company really helped me to understand the materials and how to apply them.
Posted by JP
When I first took a programming language course in college, it was a painful experience and I had to drop the class. I learned C programming language with a thick, almost dictionary like, reference book. The process was mainly memory based learning. I had to memorize all variables, keywords, famous algorithms and their uses without proper explanations. The instructor taught that programming is a style (culture) and it forced users to follow its unique way.
Several years later, I met another challenge, learning object-oriented-design in architectural contexts and in Java programming language. The purpose of the class was teaching a new paradigm, looking at architecture in terms of architectural elements rather than their composite form. The concept was interesting enough for me to dig into programming language yet the class did not provide enough context.
I collected as many introductory programming books as possible and learned programming concepts rather than specific uses of programming. What I found was each programming language was not unique one, rather all languages connected with each other and had history. One language has multiple predecessor languages and descendents. Most of the vocabulary was shared in the family of similar languages and also styles of usages were identical. Once I was familiar with the culture of programming language, the learning process became incomparably easier.
The making of programming language is dependent on the activities of professional programmers; the language is developed to support programmers by making their works effortless and collaborative. Once learners understand the culture of programming, it will be much effective to learn programming.
I had a couple of chances to teach programming languages by using Rhinoscript and Scratch last winter. Students understood the programming culture not when I explained but when I showed how I used the language; the way I consulted reference material, used sample codes and made code structures. Students had a problem when I talked about Array but as soon as I showed how I use it, most of students showed good understanding. I guess that this process proves the cognitive apprenticeship of this week's reading.
Posted by JL
In the previous blog subject, I wrote about my frustrating experience with learning vocabulary words. This is a subject in which this paper addressed, and I wholeheartedly agree with their approach. Vocabulary words are best learned through context and experience. I would never memorize SAT words of the week; I would immediately forget all of them the next day.
Math was another area where I did not know how to apply my knowledge. Oddly enough, math was my favorite subject, and I always felt comfortable with doing math problems. However, once the math problems were not in the form "Solve y=3x+5," I did not know how to solve them. Critical thinking problems that explained the math problems through words and scenarios were always difficult for me to solve since they were not in this nice clean form that textbook problem sets were in. I also never understood the "why" in math. For example, why do we need so many different forms of representation like the coordinate system? Why polar coordinates? Why are we learning different graphical functions? Teachers never gave me the answer to the whys or the bigger picture.
I do not agree with the order in which math is taught. Currently, it is taught in levels of complexity and logical progression. To make the experience more authentic, I would teach math through application and scenario. You can teach money through a bank simulation, percentages through clothing sales, probability through playing cards, quadratic equations through rocket projectiles and many many more. Show the students how we use math and make it interesting and relevant to life!
Posted by SK
In the spring of my Freshman year at MIT, I took an introductory differential equations class. It wasn't required for graduation or my major; I took it primarily because I liked the teacher. The focus of the class was on identifying and solving different types of differential equations. Because (as others have similarly expressed) I was good at "school culture," I got an A in the class but promptly forgot how to solve even the simplest of equations.
The strange thing is, I did retain a lot about the nature of differential equations from the class - convergence, stability, initial conditions and chaos, for example. This has been helpful in reading about systems theory and cybernetics, two topics I have come across only relatively recently. In many ways, an understanding of differential equations in this sense is much more helpful with my chosen path (architecture) than an understanding of solving the equations themselves.
Brown, et al., write: "Many of the activities students undertake are simply not the activities of practitioners and would not make sense or be endorsed by the cultures to which they are attributed." The problem I find with their argument here is that many activities and units of knowledge cannot be attributed to only one "authentic" culture. Choosing which culture in which to situate the knowledge is difficult and potentially as misleading as situating knowledge in "school culture." From my example, would an authentic situation been learning differential equations as a mathematician uses them? As an engineer? As an architect? Each group's application is very different and focuses on different aspects of the concepts entirely. To complicate matters further, my chosen context of architecture does not even traditionally incorporate the concepts of differential equations into its practice - how would this knowledge have been situated there? Yet many truly innovative practitioners in the field have found creative inspiration by taking concepts from outside their field and developing architectural ideas around that. In some ways I feel like learning differential equations in an "inauthentic" situation allowed me to generalize and re-work some concepts while cast aside others that weren't relevant because I did not come to associate the field with a specific practice.
Posted by JP
Your comment on retention is quite interesting. The purpose of contextual learning might be maximizing the retention rate after learning. Still you are distinguishing something that students know and that solve problems. I had also similar experience; I took optimization class last semester and could solve most of the problem sets by myself, however I am losing my skills slowly but surely. I wonder why this happens. Was it because the learning process was not contextual enough to retain the activities?
Posted by DK
I have to admit that I can't precisely recall what I learned well in school. Art history and Languages are probably the best examples. I still understand Art and speak a few languages like German, French and English. I have not retained Latin, Russian or Italian.
Humanities and some basics of sciences have remained somewhere in my brain. I am mentioning what I forgot rather than what I learned because this was 20 to 30 years ago and I actually can't point to anything that I would have learned well in school. My entire school education from age 13 on was formal as I was in a very conservative school in Austria.
My university education was also formal, but I since I was convinced that my university was very bad I started to explore the real world very early on and was able to quickly build up knowledge as a product of the activities in offices, context of professional practice and culture of architecture. I had to learn the language of the profession, learn to use tools as architects do and was able to enter that community and its culture.
Authentic activities would be drawing, building models make images, think about the content conveyed in renderings and critiquing what was produced by me and others. I worked in 7 offices before I graduated and lived in 4 different countries while being a student - yes we didn't have tuition, education is free in Austria. So I am the perfect example of an Apprentice and was in fact able to actively learn and acquire knowledge in a highly context-dependent discipline.
Posted by FG
A few of my responses to some of the comments posted so far:
Some posts really resonated with me: SL's realizing that she needs now the math skills she was taught in school and had difficulties mastering in order to design, program and build as part of the activities we are doing here at the Media Lab. I am also telling myself right now, 'God, I wish I had been told back in school that one day the math I am trying to learn will be wonderfully useful one day, and will have plenty of great uses and applications.' This would have been such a great motivating factor, I suspect it would have helped me understand better those abstract concepts I hated in the first place.
EL feeling 'like a criminal' also resonated with me, as this is exactly how I feel right now with my Python/programming classes whenever I am not getting it and everyone else does, I feel like I am kind of 'tricking' the teacher and TAs by not understanding as fast as I should and not always acknowledging it.
I can definitely relate to DK's forgetting what he learned in his school years, especially - again, math and the subjects I didn't like or was struggling with, like physics, chemistry and all math-related courses. I feel that a day after school ended, when I was 18, all math concepts leaped out of my head with the resolve never to return.
As a final note, I have to ask: what's up with math?:) Judging by the amount of responses mentioning math to highlight both positive and negative experiences, I am tempted to ask what makes that subject so special that it is embedded in our memory and subconscious, as well as, it seems, in our immediate everyday life experiences to a higher extent than other subjects we studied in school? Examples, stories and case studies abound and surpass the number of examples for say foreign languages, English [as mother tongue], history or geography? Is it easier to remember our math learning experiences than other subjects?
Why also does math seem to have attracted so much more research and observations from academics and researchers in the field of education, cognitive behavior, and child development, to name a few. Papert and Minsky come to mind, but there are many others. Does math lend themselves better, more easily to such research? I'm just curious...
Posted by DG
Great little video about how the math curriculum should be structured (3 Minutes):
"Arthur Benjamin's formula for changing math education." TED Talks, 2009.