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"...anything that occupies your working memory reduces your ability to think."The only way to clear space in working memory is to store knowledge in long-term memory.
- Daniel Kahneman | Thinking Fast and Slow
This approach not only complicates the simplest of math problems; it also leads to delays. Under the Common Core Standards, students will not learn traditional methods of adding and subtracting double and triple digit numbers until fourth grade. (Currently, most schools teach these skills two years earlier.) The standard method for two and three digit multiplication is delayed until fifth grade; the standard method for long division until sixth. In the meantime, the students learn alternative strategies that are far less efficient, but that presumably help them "understand" the conceptual underpinnings.Once again, knowledge stored in memory is entirely different from knowledge stored on Google.
Memory consolidation refers to the idea that neural processes transpiring after the initial registration of information contribute to the permanent storage of memory.I don't know how much time the brain requires to consolidate memories, but I recall John Medina suggesting that the figure may be as long as 10 years. (That would jibe nicely with the 10-year rule for development of expertise, wouldn't it?)
Memory consolidation, retrograde amnesia and the hippocampal complex
Lynn Nadel* and Morris Moscovitcht Cognitive Neuroscience
What happens when you take long division out of the curriculum? Unfortunately, from personal and recent experience at Stanford, I can tell you exactly what happens. What I'm referring to here is the experience of my students in a differential equations class in the fall of 1998. The students in that course were the last students at Stanford taught using the Harvard calculus. And I had a very difficult time teaching them the usual content of the differential equations course because they could not handle basic polynomial manipulations. Consequently, it was impossible for us to get to the depth needed in both the subjects of Laplace transforms and eigenvalue methods required and expected by the engineering school.There is no just-in-time learning, and you can't catch-up.
But what made things worse was that the students knew full well what had happened to them and why, and in a sense they were desperate. They were off schedule in 4th and 3rd years, taking differential equations because they were having severe difficulties in their engineering courses. It was a disaster. Moreover, it was very difficult for them to fill in the gaps in their knowledge. It seems to take a considerable amount of time for the requisite skills to develop. [emphasis added]
Transcript of R. James Milgram
1999 Conference on Standards-Based K-12 Education
Suppose the educational system is drastically altered to reflect the structure of society and what we now understand about how people learn.Richard Clark, Paul Kirschner, & John Swellers, psychologists and education researchers:
Our understanding of the role of long-term memory in human cognition cognition has altered dramatically over the last few decades. It is no longer seen as a passive repository of discrete, isolated fragments of information that permit us to repeat what we have learned. Nor is it seen as having only peripheral influence on complex cognitive processes such as critical thinking and problem solving. Rather, long-term memory is now viewed as the central, dominant structure of human cognition. Everything we see, hear, and think about is dependent on and influenced by our long-termmemory.Here's a thought.
Putting Students on the Path to Learning
Richard E. Clark, Paul A. Kirschner, and John Sweller
American Educator | Spring 2012
...cognitive tasks can be completed only with sufficient ability to hold information as it is processed. The ability to repeat information [you have just heard or read] depends on task [difficulty]... but can be distinguished from a more constant, underlying mechanism: a central memory store limited to 3 to 5 meaningful items in young adults.Working memory can hold three to five items at once. That's it. That's the limit.
The Magical Mystery Four: How Is Working Memory Capacity Limited, and Why? by Nelson Cowan
Lefties should be particularly vigilant about the chairs/desks. I've heard stories about them having to take long tests on right-handed flip-up deskettes; a true nightmare. I know the registration forms for my grad comps asked lefties to identify themselves, because almost all of the seats in the auditorium had right-hand deskettes. They brought in as many extra lefty ones as they needed. Is there a similar question on SAT registrations?I have got to find time to write a quick post on working memory.
SAT math tries to trick students. You could say that the tricks relate directly to whether or not they really understand math. However, when you add in the time constraints, it really relates to preparation. Is preparation the same as mastery? Yes. Mastery of the test. Is this equivalent to mastery of math or whether you will do well in college math? Not necessarily. There are better ways of determining that than with the limited material included on the SAT. Why not just require students to take the Achievement Test? Look at the AP Calculus grade.and:
What is it about SAT-Math that is so important? They are trying to test something other than just math knowledge. They think that these tricky questions reflect on how well you think on your feet, but what it really does is test preparation and whether you have seen these questions before. The questions don't reflect on whether you have a wide body of knowledge and skills in math.
They create problems where you have to "see" the shortcut. You get problems with hidden 3-4-5 triangles. Add a time constraint and then what do you call those problems? It's not just about math knowledge and skills. The problem has to do with trying to determine the difference between aptitude and preparation. The tricks may have some basis in meaningful math, but that's not what they are trying to test.
It reminds me of questions companies like to ask at job interviews, like "Why is a manhole cover round", and "How many golf courses are there in the US.?" Preparation can make you look like you have a great aptitude. Preparation is directly related to math knowledge, and that is important, but identifying aptitude is an arms race for something like the SAT. That's causing the tricky problems, not any desire to test a breadth and depth of math knowledge.
In Dick Feynman's books, he talks about how he spent a lot of time in high school learning about all sorts of trick, lateral thinking problems. He would challenge people to ask him questions. There is nothing like preparation to make you look like a genius, although he really didn't need help with that. It really annoyed some of his colleagues.
My son will get to calculus in his junior year and he always gets A's. He still has to prepare for SAT-Math. He can't let others, with specific SAT-Math preparation, seem like they have a better aptitude than him.
They try to trick students in most questions....What bothers me the most are the shortcut problems where using standard math techniques cause you to take too much time. This is supposed to identify aptitude, but it really tests preparation for the test.
There are also the problems where using a brute force or direct counting technique works better than any applied math technique. In some cases, there is no math to apply. One question on a sample PSAT test asked for the number of positive integers less than 1000 which don't have a '7' as one of the digits. (notice - "don't have" and positive integer) This simply checks how well you work under time pressure. Nobody expects you to apply any fancy math to this problem. One of the answers was the "have" solution. This tests preparation and practice, not aptitude or math ability. There may be a correlation between the test and aptitude or math ability, but not to the resolution colleges use it to select students. At the top levels, it correlates to preparation. That's not necessarily a bad thing, but there are better ways of figuring that out.
[T]he findings from this study indicate that the three main components of the Baddeley and Hitch (1974) model of working memory are in place by 6 years of age. The capacity of each component increases linearly from age 4 to early adolescence.Children have lower working memory than adults, and lower working memory has ramifications for language learning and some cases of problem solving.
The Structure of Working Memory From 4 to 15 Years of Age
Susan E. Gathercole
University of Durham Susan J. Pickering, Benjamin Ambridge,
and Hannah Wearing
University of Bristol
Developmental Psychology 2004, Vol. 40, No. 2, 177–190
Directions:
Make each statement true by moving just one matchstick.
, Sian Bielock discusses a study in which more than 90% of adults with normal working memory correctly answered the first problem. Roughly the same number of people with damaged working memories also got it right.
writes that "working-memory differences across people account for between 50 percent to 70 percent of individual differences in abstract reasoning ability or fluid intelligence." 
by Sian Beilock, a terrific book. Mostly it's about why and how people choke under pressure, but I've been surprised at the number of ancillary topics that turn out to be related to choking -- including foreign language learning by grown-ups.Statement of Research Interests (pdf file)
Alan W. Kersten
This research has been testing one hypothesis for why adults have so much difficulty successfully acquiring a second language, namely the “Less is More” hypothesis of Elissa Newport (1990). According to this hypothesis, the reduced working memory capacity of children relative to adults actually results in better language learningby forcing children to focus on small chunks of language. Adults, on the other hand, can remember larger chunks of language, allowing them to memorize useful expressions in a foreign language (e.g., “Where is the bathroom?”), but making it difficult for them to extract the lower-level meaning elements from which those expressions are constructed. Adults are thus limited to the set of phrases that they have acquired, and are unable to recombine the lower level elements from which those phrases are constructed to express novel meanings. If this hypothesis is correct, one may predict that adults will learn a language better if they are forced to focus on small chunks of language rather than being allowed to learn entire phrases. We have tested this prediction using a miniature artificial language learning paradigm (see Kersten & Earles, 2001). One group of adults was presented immediately with complete “sentences” from this language, whereas a second group was presented initially only with individual words from the language. This second group was subsequently presented with incrementally longer chunks of language until ultimately they were hearing the same sentences that the other group heard all along. The group that was initially forced to focus on small chunks of language showed better ultimate learning of the word meanings and morphology of that language, consistent with the “Less is More” hypothesis. We are currently investigating whether starting small benefits the acquisition of a natural language with more complex grammar, namely French (Chin & Kersten, in press).
When we refer to "practice," it is important to be clear that it differs from play (which is done purely for one's own pleasure), performance (which is done for the pleasure of others), and work (which is done for compensation). Practice is done for the sake of improvement. Practice, therefore, requires concentration and requires feedback about whether or not progress is being made. Plainly put, practice is not easy. It requires a student's time and effort, and it is, therefore, worth considering when it is appropriate.
"when A.P. testing began in 1956, memorization was not yet a dirty word"In theory, the new A.P. courses are going to replace "memorization" with "critical thinking."
Rethinking Advanced Placement
By CHRISTOPHER DREW
Published: January 7, 2011
The sine qua non of skilled cognitive performance is the ability to access large amounts of domain specific information [i.e. knowledge]. For example, it is estimated that chess masters have access to as many as 100,000 familiar configurations of chess pieces (Chase & Simon, 1973). As another example, in order to make sense of what he or she is reading, a reader must have access to information gained from previously read text. This is particularly true when reading complex technical material filled with jargon.
summary of Ericsson, K. A., & Kintsch, W. (1995). Long-term working memory. Psychological Review, 102, 211-245.
David Zach Hambrick, 1998, gt8781a@prism.gatech.edu
When you encounter a new technology, try to think in abstract terms about what the technology permits that was not possible in the past. It’s also worth considering what, if anything, the technology prevents or makes inconvenient. For example, compared with a chalkboard, an overhead projector allows a teacher to (1) prepare materials in advance, (2) present a lot of information simultaneously, and (3) present photocopied diagrams or figures. These are clear advantages. However, there are also disadvantages. For instance, James Stigler and James Hiebert noted that American teachers mostly use overhead projectors when teaching mathematics, but Japanese teachers use chalkboards.33 Why? Because Japanese teachers prefer to maintain a running history of the lesson. They don’t erase a problem or an explanation after putting it on the board. It remains, and the teacher will likely refer to it later in the lesson, to refresh students’ memories or contrast it with a new concept. That’s inconvenient at best with an overhead projector.
33 James W. Stigler and James Hiebert, The Teaching Gap (New York: Free Press, 1999).
The question raised is: Can a marking that conveys a general idea be called writing, or must all writing represent specific units of speech?
To this question, the great linguist Leonard Bloomfield apparently gives his answer when he states, "Writing is merely a device for recording speech."
You recall that the hippocampus is wired to receive information from the cortex as well as return information to it. Declarative memories appear to be terminally stored in the same cortical systems involved in the initial processing of the stimulus. In other words, the final resting place is also the region that served as the initial starting place. The only separation is time, not location. These data have a great deal to say not only about storage but also about recall. Retrieval for a fully mature memory trace 10 years later may simply be an attempt to reconstruct the initial moments of learning, when the memory was only a few milliseconds old! So, the current model looks something like this:
1) Long-term memories occur from accumulations of synaptic changes in the cortex as a result of multiple reinstatements [spaced repetitions] of the memory.
2) These reinstatements are directed by the hippocampus [in the temporal lobe], perhaps for years.
3) Eventually the memory becomes independent of the medial temporal lobe, and this newer, more stable memory trace is permanently stored in the cortex.
4) Retrieval mechanisms may reconstruct the original pattern of neurons initially recruited during the first moments of learning.
[snip]
The day of a typical high-school student is segmented into five or six 50-minute periods, consisting of unrepeated (and unrelenting) streams of information. Using as a framework the timing requirements suggested by working memory, how would you change this five-period fire hose? What you’d come up with might be the strangest classroom experience in the world. Here’s my fantasy:
In the school of the future, lessons are divided into 25-minute modules, cyclically repeated throughout the day. Subject A is taught for 25 minutes, constituting the first exposure. Ninety minutes later, the 25-minute content of Subject A is repeated, and then a third time. All classes are segmented and interleaved in such a fashion. Because these repetition schedules slow down the amount of information capable of being addressed per unit of time, the school year is extended into the summer.
[snip]
In the future school, every third or fourth day would be reserved for reviewing the facts delivered in the previous 72 to 96 hours. During these “review holidays,” previous information would be presented in compressed fashion. Students would have a chance to inspect the notes they took during the initial exposures, comparing tem with what the teacher was saying in the review. This would result in a greater elaboration of the information, and it would help the teachers deliver accurate information. A formalized exercise in error-checking soon would become a regular and positive part of both the teacher and student learning experiences.
It is quite possible that such models would eradicate the need for homework. At its best, homework served only to force the student to repeat content. If that repetition were supplied during the course of the day, there might be little need for re-exposure. This isn’t because homework isn’t important as a concept. In the future school, it may simply be unnecessary.
Could models like these actually work? Deliberately spaced repetitions have not been tested rigorously in the real world, so there are lots of questions. Do you really need three separate repetitions per subject per day to accrue a positive outcome? Do all subjects need such repetition? Might such interleaved vigor hurt learning, with constant repetitions beginning to interfere with one another as the day wore on? Do you really need review holidays, and if so, do you need them every three to four days? We don’t know.
Years and years
Today, students are expected to know certain things by certain grades. Curiously absent from this model is how durable that learning remains after the student completes the grade. Given that system consolidation can take years, might the idea of grade-level expectations need amending? Perhaps learning in the long view should be thought of the same way one thinks of immune booster shots, with critical pieces of information being repeated on a yearly or semi-yearly basis.
In my fantasy class, this is exactly what happens. Repetitions begin with a consistent and rigorous review of multiplication tables, fractions, and decimals. First learned in the third grade, six-month and yearly review sessions on these basic facts occur through sixth grade. As mathematical competencies increase in sophistication, the review content is changed to reflect greater understanding. But the cycles are still in place. In my fantasy, these consistent repetition disciplines, stretched out over long periods of time, create enormous benefits for every academic subject, especially foreign languages.
Brain Rules, by John Medina
p. 141-142; 143-145
From Kevin McGrew's blog, a guest post by Walter Howe, Cognitive Load Theory for School Psychologists:
This theory suggests that learning happens best under conditions that are aligned with human cognitive architecture. The structure of human cognitive architecture, while not known precisely, is discernible through the results of experimental research. Recognizing George Miller's research showing that short term memory is limited in the number of elements it can contain simultaneously, Sweller builds a theory that treats schemas, or combinations of elements, as the cognitive structures that make up an individual's knowledge base. (Sweller, 1988)
The contents of long term memory are "sophisticated structures that permit us to perceive, think, and solve problems," rather than a group of rote learned facts. These structures, known as schemas, are what permit us to treat multiple elements as a single element. They are the cognitive structures that make up the knowledge base (Sweller, 1988). Schemas are acquired over a lifetime of learning, and may have other schemas contained within themselves.
The difference between an expert and a novice is that a novice hasn't acquired the schemas of an expert. Learning requires a change in the schematic structures of long term memory and is demonstrated by performance that progresses from clumsy, error-prone, slow and difficult to smooth and effortless. The change in performance occurs because as the learner becomes increasingly familiar with the material, the cognitive characteristics associated with the material are altered so that it can be handled more efficiently by working memory.
From an instructional perspective, information contained in instructional material must first be processed by working memory. For schema acquisition to occur, instruction should be designed to reduce working memory load. Cognitive load theory is concerned with techniques for reducing working memory load in order to facilitate the changes in long term memory associated with schema acquisition.
UNTIL recently, a person's IQ - a measure of all kinds of mental problem-solving abilities, including spatial skills, memory and verbal reasoning - was thought to be a fixed commodity largely determined by genetics. But recent hints suggest that a very basic brain function called working memory might underlie our general intelligence, opening up the intriguing possibility that if you improve your working memory, you could boost your IQ too.
Working memory is the brain's short-term information storage system. It's a workbench for solving mental problems. For example if you calculate 73 - 6 + 7, your working memory will store the intermediate steps necessary to work out the answer. And the amount of information that the working memory can hold is strongly related to general intelligence.
A team led by Torkel Klingberg at the Karolinska Institute in Stockholm, Sweden, has found signs that the neural systems that underlie working memory may grow in response to training. Using functional magnetic resonance imaging (fMRI) brain scans, they measured the brain activity of adults before and after a working-memory training programme, which involved tasks such as memorising the positions of a series of dots on a grid. After five weeks of training, their brain activity had increased in the regions associated with this type of memory (Nature Neuroscience, vol 7, p 75).
Perhaps more significantly, when the group studied children who had completed these types of mental workouts, they saw improvement in a range of cognitive abilities not related to the training, and a leap in IQ test scores of 8 per cent (Journal of the American Academy of Child and Adolescent Psychiatry, vol 44, p 177). It's early days yet, but Klingberg thinks working-memory training could be a key to unlocking brain power. "Genetics determines a lot and so does the early gestation period," he says. "On top of that, there is a few per cent - we don't know how much - that can be improved by training."
