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originally posted in: Texas was right (fixing US schools)
2/27/2013 6:57:01 PM
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[quote] There is a big difference between memorization and knowing something. [/quote] Bingo. My entire K-12 education was in public schools. While my particular district did a very good job in instilling the basic skills I needed, I still saw huge flaws in the system. The issue truly is that kids are expected to memorize rather than understand. This is an enormous problem in math education especially. How many tenth graders can recite the quadratic formula? Let's say 70% (probably generous). Now, of those 70%, what percentage can explain the derivation of the quadratic formula? I would venture to say less than 10%. Yet, deriving the quadratic formula is just completing the square on the standard quadratic ax^2 + bx + c. How many second year calculus students can recite the fundamental theorem of calculus? I would hope 100%. Now, how many of those students would be able to explain the conceptual framework of the fundamental theorem? I would venture less than 50%. "The area under a rate of change function tells how a function with that rate of change changes." While this may seem loaded, it really isn't. We don't need second year calculus students to be able to do formal proofs (as these proofs rarely confer intuition), but they should absolutely understand what the hell they are doing. They need to understand WHY setting the first derivative equal to 0 will indicate maxima and minima, and WHY taking the integral of a rate of change function divided by the interval will give the average value of the function. This intuition does NOT REQUIRE memorizing formal proofs. It only requires some basic critical thinking skills. With a conceptual framework, students can tackle much more difficult problems by applying knowledge. Memorization is completely useless for solving problems that are tougher than your basic textbook problems. And while I've only talked about math, this applies to all the sciences as well. What does F = ma MEAN? WHY do all objects fall to the ground at the same rate (related to the first question)? WHY do certain chemical bonds form and not others?
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  • One of the articles linked in there talked about connection. Like you said, the formulas are taught, but are never connected to anything real. They aren't connecting the activity to any greater purpose. For example, throughout high school, I had a handful of "major science projects" we all had to complete. One was observation. One spring, we had to keep a notebook of observations and we had to have so many a day. Not once was it explained why. No goal was given for them. Our grade was based on how many entries there were. There was a million and one ways that could have been used to teach us actual science. Have us observe the growth of specific plants. Make us observe the same things each year so we can start to look for trends. Give us some indication of how this activity is useful. Another was a toothpick bridge. We built them, and then broke them, and your grade was based on how much weight it held. No greater lesson was added about distribution of weight, structural design, or anything. Same with the mousetrap car. It was a state standard, you did it as quickly as possible and moved on. Jumping from project to unrelated project just to get items checked off a list just isn't a quality plan. They are introducing good things, and have picked good activities, but from what I have seen, really miss the opportunity to teach the lessons those activities are meant to cover.

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  • I've seen very similar issues in my education, but primarily in the actual textbooks we use. For instance, when I was in calculus I, I remember reading in the textbook that, in order to find inflection points of a function, apply to second derivative test (set the second derivative equal to 0 and ensure it changes sign at that point). The book went on to a formal proof, but made no attempt to confer INTUITION. The intuition isn't even difficult. When the second derivative changes sign, it indicates that the first derivative changes from an increasing to decreasing slope. This indicates that the original function, while still either increasing or decreasing in slope, is not increasing or decreasing as sharply, marking an inflection point. Similarly, the textbook said that, to find the average value of the function on an interval, take the definite integral of the function on that interval, and divide by the interval. It, again, went on a formal proof, but never conferred intuition. Taking the definite integral of a function indicates how a function with that function's rate of change CHANGES on that interval. Dividing this CHANGE by the domain over which it changes gives the average value. A very concrete example could have been shown. Let's say v(t) is a velocity function, and we want the average velocity from 0 to 10 seconds. Taking the definite integral from 0 to 10 of v(t) will give the total change in position (i.e., change in the function that has velocity as its rate of change). Dividing change in position by change in time gives average velocity (first year physics). This type of intuition is infinitely more valuable than some formal proof, at least in the early stages of learning calculus.

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  • Edited by M37h3w3: 2/27/2013 7:16:32 PM
    [quote]What does F = ma MEAN? WHY do all objects fall to the ground at the same rate (related to the first question)? WHY do certain chemical bonds form and not others?[/quote] Let's see if I can answer these and prove myself smarter than the average bear. F = ma is a mathematical way of stating that the [b]F[/b]orce an object has is [b]equal[/b] to it's [b]m[/b]ass [b]multiplied[/b] by how fast it's [b]a[/b]ccelerating. Why do all objects fall to the ground at the same rate? Because gravity what is pulling them to the ground and gravity is constant. It does not increase the more massive an object is. Why do certain chemical bonds form and not others? It's been a while since chemistry but IIRC it's because certain atoms are "missing" electrons and certain atoms have too many, and that by bonding with other atoms and "sharing" those electrons, they return to a state of equilibrium. Do I pass?

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  • F = ma indicates that, for a given force applied on an object, the object's acceleration will be inversely proportional to the mass of the object. What you said is 100% right, but the issue is that it is still just reciting a formula. All objects fall to the ground at the same rate because, while the force of gravity is definitely greater on heavier objects (i.e., they weigh more), their mass is also proportionally greater, and so the acceleration is the same (F/m = a). The force of gravity does increase the more massive object is. The simplest explanation for why certain chemical bonds form spontaneously while others do not is because the chemical enthalpies of the bonds formed are lower than the chemical enthalpies of the bonds that weren't formed. The only real exception to this would be if the process was not spontaneous OR if some rare entropy changes occurred in the process.

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  • Edited by M37h3w3: 2/27/2013 7:48:41 PM
    Of course the acceleration is going to be inversely proportional. If the Force is staying the same with objects of different masses the equation still has to balance. And that last one just flew over my head. Is that high school chemistry?

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  • Edited by Plasma Prestige: 2/27/2013 8:07:34 PM
    It is high school chemistry. To be fair, I was pedantic in my explanation. Enthalpy is the measure of the total internal energy of a system plus the amount of work done to allow for that system to exist. In a sense, enthalpy is the entire energy associated with a system. To change the enthalpy of a system, you need to either add or remove heat or have work done on or by the system. As a simple example, if I have a cylinder with a movable piston filled with gas, I can change the internal energy of the system (and therefore the enthalpy) by doing work (moving the piston), or adding/removing heat. Since it is exceptionally difficult to say for certain what the exact enthalpy of a system is, chemists and physicists instead measure the change in these state functions. In other words, we don't care what the actual value of the enthalpy is, we only care how much it changes. When you combust propane, the chemical reaction is as follows: C3H8 + 5O2 ----> 3CO2 + 4H2O Chemists assign delta H (change in enthalpy) values to reactions. While I don't remember the exact change in enthalpy, I do know it's negative because this reaction releases a lot of heat (is exothermic). To calculate the enthalpy change of this reaction, you can take the sum of the enthalpies of formation of the products and subtract from that the sum of the enthalpies of the reactants. When you do this, the answer will be negative, indicating that the sum of the enthalpies of the reactants was greater than the enthalpies of the products, meaning the products exist in a LOWER (and therefore more stable) energy state than the reactants. It all boils down to what is more stable.

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