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Readings for Oct 28: Relationships in Learning
Submitted by kbrennan on October 21, 2009 - 6:09pm.
Continuing the theme of the social nature of learning, Debbie, April, Rita, and Takashi provide reflections and questions on this week's reading about situated cognition. Please post your response by Monday at 5pm, and add at least one follow-up comment (on someone else's response) by end of day Tuesday.
It's fine to keep the posts short (just a couple of paragraphs). What's most important is communicating your ideas clearly.
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Hi Everyone,
Here are some key take-away points from the article to help guide your reading:
Knowledge, learning, and cognition are situated
Learning is a process of enculturation
Authentic activities vs. traditional classroom activities
Cognitive apprenticeship
Tasks:
Please visit this website (http://moderator.appspot.com/#15/e=eb5b0&t=eb5b1) and post and/or vote on comments, topics, or questions that you want to discuss in class on Wednesday.
There are no comments/topics/questions listed here? Are we supposed to propose our own? I'm interested in how math is taught, I'll post it -- if we are supposed to choose from a preselected set feel free to delete mine.
Are we supposed to propose our own?
Yes, please do. This would allow us to gauge everyone's interest(s) and give us a starting point in our class discussion.
Just some clarifications about moderator. Please post on moderator any comments, topics, or questions that came to mind while doing the reading. These could range from things that you disagreed/agreed with, areas you'd like clarifications about, topics you would like to hear other opinions on, quotes that struck a chord, etc. It is totally open. While you're there, vote on any topics that you would also like to talk about. You can vote up (the check mark) or down (the X) as many topics as you would like.
I’m completely in sync with Florence and Danny 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.
My experience is quite similar to yours, Lass. My grasp of mathematics crumbled once I hit trig, but I still managed to pull good grades in my high school trigonometry and calculus courses since I knew how to navigate the academic culture. (I think I felt a bit like a criminal at the time, to tell you the truth... because I was well aware of my faulty understanding!)
Much as you described, college calculus turned me on my head. Since I handled high school calculus without issue, I figured it would be more of the same... but I was so wrong. I actually failed college-level single-variable calculus the first time through. It wasn't until I retook the course (with a different instructor) that I started to understand what calculus was all about and, furthermore, what trig was all about. My first calculus instructor was a brilliant math scholar type, while the second instructor focused on situating calc/trig concepts in a real-world context to which I could relate. I found this instructor's teaching style so effective that I was able to recover much of my missing trigonometry knowledge while also falling in love with calculus. I, too, wish that my high school math teachers had approached teaching in the way Schoenfeld, Lampert, and my second calc instructor have.
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.
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 trigonmetry to do some of the calculations. Through minimizaiton/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.
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.
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:)...
Florence Gallez
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.
April Lee
Dear team of moderators: I need to remove my vote on the question of assessment, which I didn't intend to vote on. I already had voted on a question. I don't see how to do this.
Thanks!
Florence Gallez
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.
Florence Gallez
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 Adv Chem 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 adv chem concepts to something I could relate with at the moment.
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:
1. Introduce the concepts of an Atom: Electrons, Protons, Neutrons
2. Talk about different elements and how they relate to their constituent parts.
3. Talk about attraction/repulsion, using images/models, and talk about different types of bonding.
4. Use computer simulations to show 2 atoms bouncing around and when the randomly hit each other in the correct configuration have them bond.
5. Using this model you can then add catalysts, add heat, talkabout endothermic and exothermic reactions etc... all with very simple atoms and molecules.
6. 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.
7. 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).
8. After that organic chemistry can be introduce, but with a focus on actual practical biology. Talk about dna replication, the Krebb cycle, DNA replication, etc...
I think chemistry needs to be taught with significant phsyics 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.
/rant :)
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.
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.
Florence Gallez
"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?"
Not all knowledge is conserved, or conservable, in situations like career changes. Until fairly recently I had to be prepared to evaluate if people I met with were likely to harm themselves or someone else in the near future. Thankfully, I haven't needed to make use of that skill since I came to the Lab. (Well, perhaps that will change if Wednesday's class discussions get really animated...;) )
However, to make a strong counter-argument against situated cognition as a style of learning, one would have to prove that non-situated learning is a better way to learn skills like the ones you describe as valuable and transferable: abstract thinking, rapid problem solving, etc. On the contrary, I think that these process based abstract skills arise out of contextualized, situated experiences. While the content of a particular experience may not be transferable, the process usually is. When you learned how to use the esoteric in-house software you (probably?) got a bit better at learning software and using it solve problems in general. That knowledge of process is transferable, even if the associated knowledge of content is not.
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.
I kind of agree with you here Victoria. 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. (Sorry it's after 5PM, so I'm late responding. Rough week!)
I appreciate the comments made by Florence, and Lass about their struggles with math: as someone who had a difficult time grasping the concept behind borrowing in subtraction in the 4th grade, I can sincerely say, I feel your pain!
In spite of my efforts to understand the material presented, I always felt as if I'd missed that one class where all the mysteries of mathematics had been revealed. To echo Florence's sentiments, I viewed math as something that was alien, unrelated and irrelevant to my life; I really had no interest in knowing where 2 trains (one leaving from the East Coast and one leaving from the West Coast, traveling at variable speeds), would meet!
It wasn't until my Structures class in Architecture school that I acquired an appreciation and working knowledge of math; those mystifying word problems about various modes of transportation were replaced by columns and beams supporting varying loads. I still struggled but there was a purpose to it all which kept me motivated and engaged.
To this day, I credit my Structures class with providing me an authentic experience that finally led me to an understanding of math.
The curriculum of my undergraduate computer science program was relatively theoretical. While I did a lot of programming in my classes, it was focused on representing theoretical algorithms and data structures. As much as I appreciate having this theoretical foundation, I feel that the curriculum left little room for exploration of practical concepts and applications. After graduating from college, my peers and I found that we were hitting walls when trying to find employment; employers looked for implementation-oriented programming experience and most could have cared less about our theoretical grasp of computer science.
So, in this case, I retained the knowledge learned in class, but to this day am struggling to apply it to real-world situations. I don’t think of myself as a ‘computer scientist’ (despite having a degree in the field) and I am frequently flustered by how my practical expertise pales in comparison to that of many self-taught programmers.
I do think that it’s important for computer science students to have a strong theoretical foundation and I was happy with the structure of my first few computer science classes (which did emphasize programming for the sake of understanding concepts). However, I think that subsequent upper-division classes should have further cultivated theoretical understanding in the context of building useful applications, using frameworks, and interfacing with developer communities.
I was recently asking one of our group’s UROPs about her experiences in the computer science program here at MIT and I was fascinated to hear how much the structure differed from my program. From what she told me, it sounds like MIT has done a reasonably good job of maintaining a relevant curriculum that emphasizes application just as much as theory.
your experience reminds me of my confusing points. My major now is Architecture design. However, there is a tremendous gap between academic training and practice, not about the skills, but the way of thinking. At school, all the studio works are highly focusing on the concept and how to make argument for your design. Sometimes i think students are just designing for the concepts, not for the buildings. Most architectural professors do not have a lot of practical experience either-- there are very few of them who have more than three buildings built in the real world.
So, sometimes i just think as a architect student, the way we are trained is like playing a LEGO game -- it is just a game, all the things we proposed are based on our too conceptual basis. And the way we solve it is even more conceptual.
Kathleen
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"?
Like Lass, Kent and others, I would say that math is the domain in which I have retained the least knowledge. Like Lass, I was considered an excellent student in math in elementary and high school. I was tracked into the most advanced math classes and earned good grades in all of them, right through AP Calculus. And like Lass, I "shopped" a freshman calculus class during my first week at college, only to discover that I was not comfortable around real mathematicians. (Unlike Lass, I didn't take the course!) You've probably heard the adage, "use it or lose it", and I think this applies to my experience of losing my hard-won math knowledge as the years have passed. In terms of making the math-learning experience more "authentic", I think I disagree (as usual) with much of what's been posted. I will post more about that shortly.
Here is a link to the Feyman thoughts on textbooks: http://www.textbookleague.org/103feyn.htm
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.
Marie
My undergrad background is industrial design (product design). we had a class called material study (i can not 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, these 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 experience at the company really helped me to understand the materials and how to apply them.
huang zhe
Even though I agree with most of what has been discussed … I want to provoke a little bit in order to open the discussion (particularly on the topic of “math” teachings):
According to the text conceptual knowledge can be understood as a “set of tools.” Yet, are these tools used in the same way they were supposed to be used? In other words: after “learning” math in a "non-authentic" environment should we judge our knowledge on math just on how good we are able to solve a math problem?
My argument is kind of confusing --I know--let’s try with an example. If I remember well, the first sport I practiced was Judo. In Judo you learn about equilibrium, how to move your body without falling, how to coordinate your body and how to use the weight of your opponent in your advantage. I practiced Judo for at least 5 years… from 8 to 13 years old. I remember repeating tedious postures and movements forever… and after I while I realized I was bored and that I didn’t want to do Judo anymore. That’s how I stopped practicing Judo, and I can say after a year or two I “forgot” everything I knew about that sport. I started to practice other sports, particularly soccer which I really liked (real football) and after a lot of years I discovered a hidden truth: I realized I was good in soccer mainly thanks to Judo… the way in which I moved, I keep my balance, I protected the ball, I run, jumped and confronted other players without falling was, actually, learned not by playing soccer: it was learned during my earlier days practicing tedious Judo postures.
Now… coming back to the initial questions: what are the tools we really learned in school?? Let’s put it now in a positive way: Did the tedious “classroom” experience of math give us some tools to confront other scenarios apart from solving math problems? Did the tedious and formal repetition of Judo postures let us play soccer?
AGNE
Daniel -
I like your analogy here, which is essentially a "making lemonade out of lemons" scenario and speaks to the transferability of skill sets across very different situations. It also illustrates the persistence of learned experiences (even those from "non-authentic" environments) that come back to us in different ways.
If knowledge is indeed a set of tools, I'd like to think it's similar to a Swiss Army knife that's capable of providing a multitude of functions, some which were originally envisioned but many which were never imagined.
I think you are talking about the transferability of knowledge, one of the high goals of teaching and learning. Can you take knowledge that you learned in one context and apply it in another context? If so this is evidence that you have truly learned something in a deep and applicable way. The fact that we can transfer knowledge learned in non-situated ways, outside of community, might be more attributable to our cognitive capacity to abstract and transfer knowledge, than to the ways in which that knowledge was taught. In other words our teachers and school systems were just lucky that we had the capacity to learn and abstract at that point in our education. So yes, I think the repetition and exposure helps, but doesn't necessarily engage you, help you learn deeply, or help you reach your potential.
Daniel,
You verbalize something that I've been cogitating this whole time. Sometimes the repetitive, boring exercises are good for us. Two examples:
1. One of my professors at Harvard also works for an educational software company, and he showed us the research behind one of his products that helps kids learn basic arithmetic facts. Basically, the research shows that kids who don't just know/memorize the basics (2+2, 2x2, etc) have to spend too much cognitive energy trying to calculate things which prevents them from engaging in higher-level thinking. They also found that putting this math practice into games wasn't an effective way to teach these skills--unfortunately, not everything can be fun.
2. When I was learning piano as a child, I started every lesson with an exercise from either Hanon or Czerny, which were basically scales. At the time, I thought they were really boring and I had no idea how Hanon exercises related to the fun music I got to do after I got through the warm-up. In the end, these exercises gave me the proper skills to control my hands, develop good form and do complicated things in rhythm. This kind of focused practice couldn't really come from playing through other pieces.
Could these exercises be incorporated into a situated activity? I'm not sure, but I also wonder if it's so bad to work on these perfect these skills in an isolated setting.
When I fist 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 were 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.
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!
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.
Kronick, (why can't we see people's full names on this blog!), you raise some very interesting questions. You're right, choosing one particular culture in which to situate certain knowledge is a problematic way of looking at knowledge. Why should one have to commit to learning differential equations, for example, as an engineer would? Why should one have to become enculturated in that particular domain, learn that particular jargon and adopt those particular beliefs?
Apprenticeship is a wonderful way to learn, but there's a reason why society has evolved to the point that we no longer apprentice children to master carpenters, blacksmiths or rocket scientists, but put them in school instead. The reason has to do with commitment. We no longer think it's fair or just to commit a child to one profession, one trade, one domain or one way of knowing about the world. When children are apprenticed - say, in carpet-making - we are rightfully outraged and call it what it is - child abuse. We send children to school because school is about giving all children a knowledge of basic skills (reading, writing and 'rithmetic), and an introduction to all the various domains of knowledge that are available to them. Apprenticeship works better when learners are mature enough to make informed decisions about committing themselves to one particular domain.
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 loosing my skills slowly but surely. I wonder why this happens. Was it because the learning process was not contextual enough to retain the activities ?
As an undergraduate student, I studied electrical engineering; an area of interest for me since childhood. My previous experience in the area in some cases provided a wealth of knowledge, which gave context to the topic being conveyed. In other courses, my previous experience was of no use, which certainly affected my performance. It is interesting that the classes which I had difficulty with were in many cases incremental in complexity relative to other classes I had taken and should have been relatively easy to comprehend. It was not until later, when working in industry that I had a full appreciation for the topics which were then situated in a culture. Among engineers who have had a similar experience, we have discussed the need for an introductory engineering class which is not a bottom up approach, but rather a top down view of engineering. Such a class would show advanced topics of engineering at work and part of a larger system. This is in many ways what my childhood experiences in engineering provided.
I agree that school often don't show the top down approach or the "big picture." Fortunately, gradate school forces this top-down approach. The mentality is that you have an application/problem, and the classes that you take are like tools to help you solve your problem; so you are constantly thinking about how you can apply this technique or algorithm to improve your research.
Kathleen
Back to math... It seems to me that there is a contradiction between the situated, negotiated, useful learning/knowing of math ("math in the real world") that some of us have been posting about and "authentic" math. That's because so much of the exquisite beauty of math lies in the abstractness of it.
One of my most vivid memories from childhood is of the moment when I "figured out" what multiplication was, and how to do it. I was 6 years old, and my teacher, parents and older sister had all told me that I would learn multiplication when I was old enough to understand it (shades of Piaget) - in second grade. I didn't want to wait a whole year, so I just thought about it and thought about it until I had reasoned my way to an understanding. I knew in my heart that I was right, but to be sure I ran to my mother shouting things like "Two times two is four, right?" and Three times four is twelve, right?". She was quite startled - no one had taught me multiplication, but I understood it. My understanding wasn't situated in any concrete mathematical exploration involving manipulatives like Froebel gifts, Lego blocks or cut up drinking straws. I hadn't been trying to resolve any "real world" math problem. I didn't use math, per se, to master this fundamental math concept. I used a word. The word was "times". I just kept thinking about what a number "times" another number could possibly mean, until it became very clear in my mind. I suppose I was having, as Duckworth would say, a "wonderful idea".
Another recollection from childhood is of watching a documentary series called "The Ascent of Man", hosted by the mathematician Jacob Bronowski. Bronowski believed that the arts and the sciences were two sides of the same coin. In one episode he explained the mathematical basis of music, which had a profound effect on me and my way of thinking about almost everything. I began to be curious about the mathematical and physical underpinnings of everything with which we come into contact. Math and science have always seemed to me to exist on a practical, "real world", problem-solving, engineering level - to be sure - but also to exist as a kind of invisible and silent web or net holding our whole enterprise (life, the universe and everything...) together. This has to do with abstraction.
Brown, Collins and Duguid maintain that taking math out of context makes it harder to learn, less meaningful and less transferable. They give word problems as an example of the way math is taught in schools, divorced from the authentic context of true mathematical inquiry. I would argue that word problems are an attempt to connect math to authentic inquiry. Lampert's way - asking her students to make up the story to go with a math problem - is better pedagogically, because the 4th-graders are more engaged when it's a problem of their own making. But it's still a word problem. What makes it "authentic" is that the children care about the outcome. They care how many butterflies are in the jars, and they're interested in investigating all the ways one could "decompose" the multiplication, because they have imagined or created the jars and the butterflies. Students are always more engaged, they have more intrinsic motivation, and their learning is more authentic when they have some creative control, and some control generally, over what, how and how fast they learn. Good teachers know and enable this.
John Seely Brown and his colleagues wrote about situated cognition quite a long time ago. The theory has been absorbed into schools in the past two decades. For example, in the early 1990s, there was a then-experimental series of films made for classrooms, featuring the adventures of a young man named Jason, who had no end of adventures involving calculating the speed of his motorboat, the speed of the river, the price of gas for his boat's gas tank and the number of miles to the next dock with a gas pump. Now there are countless multimedia resources available to help kids apply math concepts to "real" problems. Word problems were just the low-tech solution to that quest. And there are many good teachers helping their students to discover interesting problems to solve and, in some cases, taking them out of the classroom to help them work out real-world solutions. If you click on the link below, and then select number 5, "Bungee Jump", you'll see a video that shows one such teacher. I've seen another video of a teacher exploring the same concepts with her students in an even more engaging way. She had her students use water balloons instead of weights, and the class dropped their bungee-jumping water balloons off the roof of their school - alas, I couldn't find a link to that video, but this one can give you an idea of how teachers can and do pull this real-world applicability of math concepts into their classrooms.
http://www.learner.org/resources/series34.html?pop=yes&pid=925#
In terms of what I would change to ensure better learning and retention of math - and every other - subject matter? 1. Situated learning, for sure, when it's possible - (I can't think of too many serious mathematicians who want troops of elementary school apprentices in their labs and offices). 2. Allowing students to help set the agenda for learning and to frame some of the questions that need exploration (a la Duckworth). 3. Encourage students to make things that require them to learn and understand - whether it's building a robot, baking a cake, shooting and editing a video, or writing and publishing their own magazine. 4. Encourage them to follow their passions in learning. 5. Facilitate and enable collaboration and sharing. 6. Tailor the scaffolding and the instruction to the individual, as much as possible. 7. Make every learner feel valued and bright.
Sorry for the long post...
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.
A few of my responses to some of the comments posted so far:
Some posts really resonated with me: Lass' 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.
Emily 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 Ducks' 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.
I am not sure I am on the same page as Kathleen with regards to making things such as baking cakes having as much educational value as making more complex systems such as robots orother crafts enhanced by technology - or even studying more traditional subject matters. My recollection of cake-baking activities in primary school is a messy, noisy affair, with kids screaming and flour everywhere. Having no special interest in cooking, to this day I still don't know how to bake a cake. Frankly, I am not sure what I learned in these cooking activities at school.
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 do 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. Do math lend themselves better, more easily to such research? I'm just curious...
Florence Gallez
Great little video about how the math curriculum should be structured (3 Minutes):
http://www.ted.com/talks/view/id/587