Problem Solving Using Multiple Representations: Fill & Download for Free

In the U.S., 25 years ago math at the elementary level focused on solving problems with the four operations: addition, subtraction, division, multiplication by memorizing the steps. For example, long division was given a formula: Divide, Multiply, Subtract, Bring Down and students dutifully followed that formula. They also did not really understand the concepts behind what they were doing, but if the steps were followed, they arrived at a correct answer. “Word” problems were added in on occasion to give a context to the problems. (David had 12 cookies. He gave his three friends the same number of cookies. How many did each friend get?)Now, we have “Common Core” math which has a significantly different approach. Here, students are to learn the concepts first and the “formula” is last. This approach requires much time developing students understanding: what does it mean to divide a number? Students are asked to show their understanding of concepts with multiple representations - draw a picture, use a number line, use an area model, create arrays. Students must write to explain their math thinking every step of the way, telling what and why with each step of problem solving. They might be doing multiplication in 3rd grade, but they are not taught the standard algorithm until 5th grade! In the meantime, they are solving the problem with those multiple representations and explaining their thinking with writing.This is the area model for multiplication:Students use this model for a couple of years before they learn this method:When we get students from Asian countries, in general, they are one to two years ahead of our U.S. students in math. They struggle with having to explain their thinking in writing which Common Core teaching demands. But, they are usually moved ahead to more advanced math. (Note also, U.S. students have the same struggle and do not like writing about their thinking. They can get a correct answer, but if their writing, or use of multiple representations is not used, they are marked “incorrect.”This has been my experience.

TL;DR: As you solve more and more problems, compare different problem-solution pairs and try to abstract common problem structures and abstract solution schemas for future use.--Cognitive science research into analogical learning and problem solving may suggest some useful tips for this question. We know that analogical comparison of cases is a powerful means of abstraction. In fact, it is often touted as a crucial mechanism by which humans develop concepts over their lifespan [*], from concepts as "rudimentary" as "animal", to concepts as sophisticated as "nonlinear causal systems". Basically, analogical comparison involves aligning two knowledge representations, then noticing where their structure matches/overlaps (what relations between objects, and relations between relations, e.g., causal structures) while ignoring structurally irrelevant details.One of the classic examples in the analogical problem solving literature is the radiation problem and its analog "generals" problem. My memory fails me on the details of each, so I'm going to quote Wikibooks[**]:Radiation problem:"As a doctor you have to treat a patient with a malignant, inoperable tumour, buried deep inside the body. There exists a special kind of ray, which is perfectly harmless at a low intensity, but at the sufficient high intensity is able to destroy the tumour - as well as the healthy tissue on his way to it. What can be done to avoid the latter?"Generals problem:"A General wanted to capture his enemy's fortress. He gathered a large army to launch a full-scale direct attack, but then learned, that all the roads leading directly towards the fortress were blocked by mines. These roadblocks were designed in such a way, that it was possible for small groups of the fortress-owner's men to pass them safely, but every large group of men would initially set them off. Now the General figured out the following plan: He divided his troops into several smaller groups and made each of them march down a different road, timed in such a way, that the entire army would reunite exactly when reaching the fortress and could hit with full strength."It turns out that the General's "convergence" solution can be applied to the radiation problem with success: use multiple rays to converge on the tumor from multiple directions so they concentrate on the tumor with enough intensity to destroy it without damaging the healthy tissue in the rays' paths. Comparing these two solutions allows one to abstract a general "convergence" schema that can be applied to other problems with a similar abstract structure.As the example illustrates, the process of analogical comparison can facilitate the development of increasingly abstract schemas that can be remembered and applied across a range of seemingly disparate problems to be solved, due to your ability to connect them via deep causal structure. It has been shown to effectively facilitate "far transfer" (that is, transferring knowledge from one problem to another across very different domains and across long time scales)[***]. There is some debate in the literature over how much of this transfer has to do with transferring case-specific knowledge/strategies, and how much of it has do with activating and applying abstract schemas extracted from individual cases. This is an area of ongoing research. In any case, we do know that analogical comparison can help tremendously.So much for the theory. Now onto a practical suggestion, where we cognitive scientists can often be out of our element. I will throw out an idea that might leverage the power of this process to improve one's abstract problem solving skills.1) Create a running "case library": that is, for each problem that is solved, store it somewhere explicit (e.g., Evernote). Write down what the basic problem was, what the solution(s) was/were, and why it/they worked.2) Periodically, go back and review your case library. During review, use analogical comparison to look for deep structures across your cases that you can abstract and use for other problems.2a) For this step, you might make it as explicit as possible by writing out and drawing/sketching the abstract schematic solution, and also linking it to the specific cases that exemplify it.I would hazard a guess that this process would support the development of a robust abstract problem-solution knowledge base in memory that can be flexibly applied to new problems. If you try this, or have done something similar to this, I would be curious to hear how it worked out. :)Good luck![*] Gentner, D., & Colhoun, J. (2010). Analogical processes in human thinking and learning. In B. Glatzeder, V. Goel, & A. von Muller (Eds.), Towards a theory of thinking (pp. 35–48). Heidelberg, Germany: Springer (http://groups.psych.northwestern.edu/gentner/papers/Gentner-Calhoun_in_press.pdf)[**] http://en.wikibooks.org/wiki/Cognitive_Psychology_and_Cognitive_Neuroscience/Problem_Solving_from_an_Evolutionary_Perspective[***]Gick, M. L., & Holyoak, K. J. (1983). Schema induction and analogical transfer. Cognitive Psychology, 15(1), 1-38. (http://deepblue.lib.umich.edu/bitstream/2027.42/25331/1/0000776.pdf)Kurtz, K. J., & Loewenstein, J. (2007). Converging on a new role for analogy in problem solving and retrieval: When two problems are better than one. Memory & Cognition, 35(2), 334-341. (http://business.illinois.edu/loewenstein/papers/Kurtz&Loewenstein Mem&Cog07.pdf)Loewenstein, J., Thompson, L., & Gentner, D. (1999). Analogical encoding facilitates knowledge transfer in negotiation. Psychonomic Bulletin & Review, 6(4), 586-597. (http://business.illinois.edu/loewenstein/papers/Loewensteinetal PBR99.pdf)

As a hunch, I’d say that the things that clearly sets apart the reasoning of those with very high IQs from that of most people would include, in no particular order:A much stronger role of connections, associations and analogies. Basically, it is a higher-order form of pattern-detection where one sees that the logical structure or functioning of one system is similar or even identical to that of another, seemingly unrelated, system. For example, a city’s vehicle traffic, information packets traveling through the Internet, a crowd moving through a sports stadium, and the dissemination of information through social media can all be seen as following similar dynamics.Always attempting to reduce any and all knowledge to its bare-bones basic concepts and principles from which everything else can be derived. As a simplistic example, one could understand Physics as the study of the basic interactions between space, time, and matter through the fundamental concepts of force, mass and energy. “Field” is not a fundamental notion, as it can be understood as energy and force operating in space.Rapidly fragmenting knowledge into its basic concepts and endlessly combining and recombining them between each other using multiple different mental tools.The use of a “mental whiteboard”, usually with visual-spatial mental representations, though it can also be done, to a lesser degree, with verbal content. It works as if one had plenty of “scratch paper” to quickly “see” sets of ideas and their interrelations.Trying to find a new and more convenient viewpoint or perspective from which to think of a problem, such as thinking of trajectories in urban traffic in terms of time instead of distance, thereby generating a completely different map of the city.Note that all of those things relate to one another and are generally used mixed together.Naturally, I am super simplifying things so as to avoid a lengthy and cumbersome text, but I think one can get the picture.This is by no means an exhaustive list, but rather some of the first things that come to mind.