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I teach mathematics at a community college. The lecture resonated with me and I found it captivating. Here is a synopsis based off of memory: Masur discovere

I teach mathematics at a community college. The lecture resonated with me and I found it captivating. Here is a synopsis based off of memory: Masur discovered that there is something called the Force Concepts Quiz. He learned that students of physics across the nation did very poorly on the test. The test asks basic questions. For instance, a large truck and a small car collide head on. Which vehicle has more force imparted on it? This type of question requires no computations. It's easy if you really understand the concepts and hard otherwise. He teaches at Harvard and thought his students would do well. They didn't. They did terribly. What he found is that regardless of the quality of the teacher (note he is a very good lecturer), type of class, or size of class students universally did bad. It didn't matter the quality of the institution either. Conclusion, we are teaching physics wrong. He found that students who could do well on the Force Concepts Quiz could do tedious calculations and mastered that part of the course. Students who could master the calculus computations may or may not be able to pass the Force Concepts Quiz. Hence he decided to concentrate more on concepts and less on calculations. He does not lecture in the classroom. In my own experience I've found similar things hold in mathematics. For instance, I ask College Algebra II students to give me an example of an equation that has no solution. Most can't do it even though 1=2 is a simple example of such an equation.

qznc 9y ago
The question is if "1=2" is really an equation? Wikipedia says "an equation is a statement of an equality containing one or more variables", but your example lacks a variable. How about "-1=x²"? If x can be a complex number a solution is "x=i", but what if we only use real numbers? Wikipedia says "There are two kinds of equations: identity equations and conditional equations. An identity equation is true for all values of the variable. A conditional equation is true for only particular values of the variables." Does that imply that without a solution "1=x²" is not actually an equation? The two parts are not equal after all.
- yorwba 9y ago
  My personal definition of equation is just "an equation is a statement of equality", no variables required. But if you want to talk about solutions, you usually need variables. (The example "1 = 2" could be saved by saying it has no solutions, while "1 = 1" is solved by the empty assignment of variables.) The distinction between identity equations and conditional equations is basically one of quantification. (Quantification should really be introduced as soon as you have variables, but is unfortunately often left implicit.) An identity equation is essentially a statement of the form "for all valid assignments of the variables, ... = ...", where a valid assignment is constrained by conditions like "x is a real number". (Those are also often implicit when they really shouldn't be.) This means that "1=x²" is not an identity equation when x is taken to be real, but it is one when x is constrained to be a root of unity. (This is just another way of saying 1=x², but all identities are like that.) A conditional equation amounts to "there is a valid assignment of the variables such that ... = ...". Then "1=x²" is a conditional equation that can be made true on the real numbers. The solutions are variable assignments that make it true, in this case 1 and -1. If you constrain x to be an imaginary number, then it has no solution; the equation and the constraint on the variable contradict each other. If you are taking college level algebra, you should be familiar enough with equations to set up a contradiction like that.
- yequalsx 9y ago
  If you prefer, 1 + 0 x = 2. We agree that is an equation. It has no solution. Typically in math we don't write 0x in expressions. Suppose we start with 2x+5 = x - 4 + x. If we solve this by showing all work we'll end up at the end with 5 = -4 At this point students are told this means that there are no solutions. But what they miss is what an equation really means. Anyone who really understands the concept of an equation can easily give an example of an equation with no solution. An equation is a statement that two expressions are equal. What one tries to find is the solution set. That is, the collection of tuples (if more than one variable is present) that makes the expressions equal when evaluated at the tuple. An equation is really a question. How can these two expressions be evaluated at the same input to produce the same output? Lastly, We agree that f(x) = 1 and g(x) = 0 are valid functions. Here is the equation 1 = 0 stated in a more sophisticated way. f(x) = g(x) where f and g are defined above.
- gugagore 9y ago
  I agree that "1=2" would maybe cause more confusion. How about "x + 1 = x + 2".
TheOtherHobbes 9y ago
I suspect in context that's not a fair question, because students are probably be expecting a technical definition of no solution - maybe imaginary roots, maybe something else - instead of a much simpler inconsistent statement. Those two answers live in different conceptual categories, and most math lessons and problems live exclusively in one category and not the other. This is related to Force Concepts because physics-as-math is taught as a series of algebra problems with a rather distant relationship to lived experience rather than the other way around. Teaching physics as experience is harder than it looks. When I learned physics the course really tried to emphasise experience, but it did it in a stylised and abstract way with canned "experiments" using special equipment. This actually made it feel more distant and uninvolving, not less. I suspect to do a better job you have to get out the classroom. E.g. introduce force, momentum, and acceleration to kids on bikes and skateboards, not with wooden trolleys on ramps.
- yequalsx 9y ago
  It's a fair question in that a student's inability to give an example of an equation that has no solution indicates that we aren't really teaching students what an equation is/means. If one can't instantly give an example of an equation with no solution then they don't really understand what an equation is. Which means something is wrong with our teaching.

qznc 9y ago

The question is if "1=2" is really an equation? Wikipedia says "an equation is a statement of an equality containing one or more variables", but your example lacks a variable. How about "-1=x²"? If x can be a complex number a solution is "x=i", but what if we only use real numbers? Wikipedia says "There are two kinds of equations: identity equations and conditional equations. An identity equation is true for all values of the variable. A conditional equation is true for only particular values of the variables." Does that imply that without a solution "1=x²" is not actually an equation? The two parts are not equal after all.

yorwba 9y ago
My personal definition of equation is just "an equation is a statement of equality", no variables required. But if you want to talk about solutions, you usually need variables. (The example "1 = 2" could be saved by saying it has no solutions, while "1 = 1" is solved by the empty assignment of variables.) The distinction between identity equations and conditional equations is basically one of quantification. (Quantification should really be introduced as soon as you have variables, but is unfortunately often left implicit.) An identity equation is essentially a statement of the form "for all valid assignments of the variables, ... = ...", where a valid assignment is constrained by conditions like "x is a real number". (Those are also often implicit when they really shouldn't be.) This means that "1=x²" is not an identity equation when x is taken to be real, but it is one when x is constrained to be a root of unity. (This is just another way of saying 1=x², but all identities are like that.) A conditional equation amounts to "there is a valid assignment of the variables such that ... = ...". Then "1=x²" is a conditional equation that can be made true on the real numbers. The solutions are variable assignments that make it true, in this case 1 and -1. If you constrain x to be an imaginary number, then it has no solution; the equation and the constraint on the variable contradict each other. If you are taking college level algebra, you should be familiar enough with equations to set up a contradiction like that.
yequalsx 9y ago
If you prefer, 1 + 0 x = 2. We agree that is an equation. It has no solution. Typically in math we don't write 0x in expressions. Suppose we start with 2x+5 = x - 4 + x. If we solve this by showing all work we'll end up at the end with 5 = -4 At this point students are told this means that there are no solutions. But what they miss is what an equation really means. Anyone who really understands the concept of an equation can easily give an example of an equation with no solution. An equation is a statement that two expressions are equal. What one tries to find is the solution set. That is, the collection of tuples (if more than one variable is present) that makes the expressions equal when evaluated at the tuple. An equation is really a question. How can these two expressions be evaluated at the same input to produce the same output? Lastly, We agree that f(x) = 1 and g(x) = 0 are valid functions. Here is the equation 1 = 0 stated in a more sophisticated way. f(x) = g(x) where f and g are defined above.
gugagore 9y ago
I agree that "1=2" would maybe cause more confusion. How about "x + 1 = x + 2".

TheOtherHobbes 9y ago

I suspect in context that's not a fair question, because students are probably be expecting a technical definition of no solution - maybe imaginary roots, maybe something else - instead of a much simpler inconsistent statement. Those two answers live in different conceptual categories, and most math lessons and problems live exclusively in one category and not the other. This is related to Force Concepts because physics-as-math is taught as a series of algebra problems with a rather distant relationship to lived experience rather than the other way around. Teaching physics as experience is harder than it looks. When I learned physics the course really tried to emphasise experience, but it did it in a stylised and abstract way with canned "experiments" using special equipment. This actually made it feel more distant and uninvolving, not less. I suspect to do a better job you have to get out the classroom. E.g. introduce force, momentum, and acceleration to kids on bikes and skateboards, not with wooden trolleys on ramps.

yequalsx 9y ago
It's a fair question in that a student's inability to give an example of an equation that has no solution indicates that we aren't really teaching students what an equation is/means. If one can't instantly give an example of an equation with no solution then they don't really understand what an equation is. Which means something is wrong with our teaching.