Inspired on the point of mathematical pitfalls...
Unless you have a photographic memory, I find that the most common mistakes and errors are missed assumptions in algorithm processing, or thinking to apply a step wise solution with a number of sequential steps but losing track of things like carrying signs. Write out the problem as completely as needed from step to step, and generally with a little added work rewards less overall work in re checking works when errors result. In fact, sadly enough while having some informal experience in dealing with order of math operations, I couldn't recall this explicitly in the formal sense introduced until I were underway in college level math. Most of this practical sense and experience in math operations handling in lower levels of education were likely due to the process of instilling order of operations through memorization, or repetitive problem solving, instilling a sense about the order of operations...or maybe I weren't paying attention personally so closely back in those days as a kid which could be another likely answer here.
Generally the larger an expression or solution becomes, the more sequential steps required in solving a problem, means the greater likelihood of errors being made in solution work. Thus it should be priority to really write steps out, and paying attention closely to signs and proper operations (including operation order) when solving problems. I've seen some use accounting type practices in writing steps, meaning that you don't write things out to erase them, but instead by way of mathematical accounting (this must be something of business/accounting hand me down practice to mathematics), leaving negated/removed elements with something of sign indicators which say in effect what has occurred. Thus if a mistake is made, for instance, in a given cancellation, the step is clearly shown in logic...I've likely resorted to this sort of error of omitting the logic demonstrating this step, and I've seen others do it I don't know how many times, and it may not be very good practice if you want to have a thorough handling in shown of logic. The more clearly you can see where a mistake has occurred means that all steps in logic are ledger ed, likely leads to the best outcome in determination of where mistakes have occurred. Generally I see this as standard practice in any event usually in teaching methods so I wouldn't worry about so much the educators, as much the co educators potentially in home. Then as I am not sure kids learn these days as once were common practice, for instance, cursive handwriting, you know where this form of calligraphy instruction originates right?! :) Well actually at one time back in the days of yor, before businesses had automated print machines they relied upon legible handwriting for business, accounting, and so forth reasons, hence the necessity of instruction in school, that I imagine these days is something of an art losing ground, enough of that for an aside.
Personally, I think it could be a lacking error in mathematics education, and this relates to applied mathematics, practical application and math projects. Often times, while our educational systems set about teaching math in rote ways, through memory instillation, through limited task handling, often times this could involve for any problem, less than 20 or so steps, anything relating to longer work could be delegated as project work, and often times is not given...one because of the time investment that could be considered for a teacher assigning, I imagine, and guiding students in their work for a given problem, and two because it seems more likely that text books should follow the standard methods for applied teaching. On the other hand, maybe it is that students could find greater self motivation if having, under instructor/course guidance, projects demonstrating any learned method. If not readily to underscore real world application to learned principles. Really I think this is true at various levels of math...laughing...I don't know how many times, I hear people say 'yeah, I learned that in school but I never use it...' often times never used in my mind is equivalent to...never taught to relate world experience to mathematical models and creating structured framework. Sure in construction/carpentry, for instance, you might get by as I think could be common, through some manner empirical methodology...often times, instead of resorting to equations, or mathematical models/geometry/trigonometry, master carpenters may actually use tools which judge real world geometric space (for instance a tool that literally measures the angles between corners that one can fix, and project/duplicate onto another surface for templating purposes), and then having recorded these values, apply these results in determining things like lengths of cuts and relationships otherwise, it makes sense this way, but then as to the mathematical sides of things, when given a clean slate to imagine work, mathematics can provide a framework for creating incredibly complex structures that could be hard to construct without some degree of literacy, or at least a fluency in speaking and thinking about a world with such language (according to one's profession). Of course, sometimes one should be left with mock up models when the math for some odd reason fails, and likely failure I imagine could have occurred for the central big errors mentioned above leading to enough anomaly. At least one could offer even if while the model works in the theory, but never is applied quite exactly as in real world application as one should hope, at least you were in proximity to your results?!
Take time to learn something new. If you are an adult and co educator, I actually enjoyed the experience, with some patience, in puzzling over a bit of differed mathematical logic that I hadn't encountered before. For instance, a base 10 geometric table, that I hadn't seen which provided a relation to additive and multiplicative operators, but using a visual geometric spatial algorithms in applying operator algorithms. Of course, this particular method, weren't stressed above any other computational algorithm, or at least a given central algorithm were stressed first and foremost...and in this other case, a lot of this means having the willingness and patience and learn what someone is learning that you hadn't learned as an adult co educator being educated. As in teaching these steps by the way, I avoided the mathematics legalese manner of speaking with base 10 and the use of operator in teaching here...Likely it seems as we age, maybe our tolerance to new ideas diminishes, and especially resistance to change in learning is through the common culture practice that changes to culture legacies could be seen as threatening, and the interpretation is that a given practice is more likely as seen as pointless...of course, if we thought to pay for some monthly subscription to a web site like luminosity, and thought differently of the mental gymnastics that we might be demanded of us, if we thought this were somehow increasing our potential IQ, we might see these sorts of learning processes differently, as potentially beneficial?! I think unfortunately, also, it is that while some professions make use of continuing education as common practice, there must be a general cultural sense that once we've left school in a way, we've left another life behind in so far as experience and practice are concerned. The other reason why it might not be as difficult for the students of new educational systems as we suspect but really for the parents, or this is merely to say that the adults neither help in reinforcing through self fulfilling ways the likelihood social outcomes in a given educational context. As it turns out, as in the above puzzling visual algorithm mentioned above, while initially there were resistance met, instead a group of adults including myself would turn their attention in attempting to understand the puzzle, and once the system were understood, there were really an enjoyable success in teaching something that everyone seemed to learn something from. Think of it this way, adults in past times may be as willing, if given the right context to learn with their kids new things, take for instance that board game that nobody knew how to play?! I couldn't write this likewise, without saying personally there weren't times in having been frustrated and so forth, it happens, and then likely it seems taking a breather and gaining a little bit of space from problem work can help where a little bit of re group pays off. I think I am projecting my biases here, and my apologies if others had did exactly the things they thought were right and hadn't accomplished on this end.
I think unfortunately, and I must confess to a bit of possession on this subject matter, that our cultural ownership, a place of meritocracy is such that, I am to wonder with some level of suspicion and bias that weren't outright at time authoritarian in our degree of ownership on subject matter, or at least the belief that a greater authority resides by way of the lens that we possess on the matter of education...hence, the engineer that writes about his outrage and misgivings in understanding a given mathematical framework...although I'd admit I tend to be more shameless on the subject matter of not knowing (or actually having these sorts of misgivings, the rise of territorial ism!) , or sitting there listening to the math instructions from peers at such an age, and having gone through so much countless instruction otherwise. Again, my apologies for expressing these thought out projections and biases, or mere suspicions.... :)
And to think for the voyager 1 or whatever satellite spacecraft out there having exited the solar system, being reduced to a series of pictographs which indicate not only using things stable galactic celestial bodies (I believe pulsars or something or other for honing in or something like triangulating our solar's systems galactic position), but also the stability of radioactive decay(?...I think for the deduction of the records angular velocity...don't quote me on that) I think it is this of some common isotope or something or other. There's a lot of reasoning to be done for interpreting that puzzle for some civilization out there...sorry digression...at least surpassing common core architecture?!
I think unfortunately, and I must confess to a bit of possession on this subject matter, that our cultural ownership, a place of meritocracy is such that, I am to wonder with some level of suspicion and bias that weren't outright at time authoritarian in our degree of ownership on subject matter, or at least the belief that a greater authority resides by way of the lens that we possess on the matter of education...hence, the engineer that writes about his outrage and misgivings in understanding a given mathematical framework...although I'd admit I tend to be more shameless on the subject matter of not knowing (or actually having these sorts of misgivings, the rise of territorial ism!) , or sitting there listening to the math instructions from peers at such an age, and having gone through so much countless instruction otherwise. Again, my apologies for expressing these thought out projections and biases, or mere suspicions.... :)
And to think for the voyager 1 or whatever satellite spacecraft out there having exited the solar system, being reduced to a series of pictographs which indicate not only using things stable galactic celestial bodies (I believe pulsars or something or other for honing in or something like triangulating our solar's systems galactic position), but also the stability of radioactive decay(?...I think for the deduction of the records angular velocity...don't quote me on that) I think it is this of some common isotope or something or other. There's a lot of reasoning to be done for interpreting that puzzle for some civilization out there...sorry digression...at least surpassing common core architecture?!
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