38

I'm confused. Are you really going to try to make this sort of distinction when teaching geometric figures to students "around 9-13 years old"? Students that age (and engineers my age -- much, much older) think that a triangle is a triangle. It's a polygon formed by three non-colinear points. A triangle has many ways you can think about it. ...


37

In the U.S. Common Core standards, functions are supposed to be introduced in the 8th grade, i.e., around age 13-14. So arguably age 15 is a year or two behind where they ought to be. The standard for the 8th grade says: Understand that a function is a rule that assigns to each input exactly one output. So honestly that really doesn't seem like a hugely ...


25

In my experience, students are often predisposed to "learn" by memorizing facts; that's how much of their early education worked, so that's what they're used to. When you give them a definition and then a bunch of consequences of the definition, they don't think "okay, I have a definition and then 87 examples of using the definition", ...


24

One encounters exactly the same issue teaching multivariable calculus when one treats integrals over three-dimensional regions and integrals over the surfaces that are their boundaries. In particular the word sphere is particularly confusing in this context. (Mathematicians use sphere to mean the the two-dimensional surface; colloquial speech and some ...


21

I think the distinction you are raising is not natural to students at this age. I teach undergraduates and graduate students, not elementary schoolers, but I find that it is not natural for undergraduates who have not had a theoretical math course. In my experience, students do not naturally think of geometric figures as sets of points. If $P = (-1,-1)$, $Q =...


16

First of all, you should test them on remembering the definitions. Second, there are probably a significant number of your students who do not understand the definitions. Suppose you gave them an example of a grammar that was not right-linear and the definition of a right-linear grammar, and asked them why, according to the definition, the example grammar ...


16

Functions are far broader and more applicable than you give them credit for. Consider the following: Country or state Capital Elevation (in meters) Bolivia Sucre 2783 Ecuador Quito 2763 Colombia Bogata 2619 Eritrea Asmara 2363 Ethiopia Addis Ababa 2362 Mexico Ciudad de Mexico 2216 New Mexico Santa Fe 2152 Wyoming Cheyenne 1856 Colorado Denver 1613 ...


9

Some abstraction is perfect for children. Mitsumasa Anno was a Japanese writer of children's books and he had many innovative approaches to introduction functions. This is from Anno's Math Games II. You can even see the introduction of function inverse.


8

I confess, when you start talking about 1-D triangles, my own first thought is "how can you have non-colinear points in 1-D?". So, I imagine most students that age will have a far more difficult time with that. Keep in mind age appropriateness. For 9-12 year old children, you are generally looking at a level of psychological development ...


8

Educator here who has worked with many students in the aforementioned age range (9-13) on triangles and squares. In my experience, it has never come up that a confusion between the boundary and the interior of a plane region was relevant to problem solving at that grade level. For these types of elementary shapes, the boundary and the interior completely ...


8

You asked: Students trying to learn all kinds of facts [...] by heart rather than recognizing that these are just consequences of the general definitions. [...] Are there any better techniques to get this point across? Or maybe some insight why this seems to be so difficult? The answer is simply that the students are unable to do basic logical reasoning. ...


8

Unfortunately, we don't have a set of universally agreed upon definitions in mathematics. It might seem like we do (or should), especially in Geometry with its long history and so much agreement, but the truth is that we use different definitions frequently. One proof of this is the differing definitions in your textbooks! That is just the nature of a ...


7

If 10-11 year olds can learn programming (which some have done even before Scratch was a thing), then it's hardly a leap at all to suggest that 15 year olds can learn functions since they are a common element of programming languages. Every student is going to be different and some are going to be better at math than others, but I think this notion that ...


7

Many of the geometric figures are so elementary that they are deeply rooted in daily language, and there seems to be no great solution. I agree with you here, and I think this is the key point. To me they are clearly well-defined: "Triangle", "square", and polygons in general, are bounded regions on the Euclidean plane, i.e., 2D figures. ...


5

This is something I know very well from the other side, as an amateur student of maths for the last 40 years or so. In my experience, reading a definition and actually understanding it are two very different things. Once you understand the definition at a deeper level, you very often forget how hard it was to get to that point. It simply takes a certain ...


5

I agree with the point that a lot of the answers here are making - some distinctions, while correct and important, are not accessible to the age group you're talking about - I also want to point out an important benefit of not making the distinction for them. While it is crucial in higher mathematics to be able to be extremely precise, it's also important to ...


5

Programmers consider the naming of things to be one of the three leading problems in our field. For the cases you describe, we do already have a well-established and widely-used set of terms that even non-computer users should recognize and be familiar with. A circle can be called a solid or filled circle, contrasted with wireframe or outlined circle. ...


5

Another example of a crystalized definition is Dedekind's approach to defining finite. First we define a set $S$ be infinite if it is equivalent to a proper subset, i.e., if there is an injection $f:S\to S$ that is not a surjection. Only then do we define what is means for a set to be finite. A set is finite if it is not infinite. "Hilbert's Hotel" ...


5

Your question makes me think of the definition of linear (in)dependence used in Lay's Linear Algebra. It does not feel natural, and yet it's the simplest definition to work with. I start with a more natural definition, that if one vector can be made as a combination of others in the set (ie it depends on those others), then the set is linearly dependent. ...


4

I can't spot in any of the other answers what I think is the main point. Neither teaching students to swap variables or teaching them not to switch variables is really the solution, as both of these simply train students to carry out a mechanical process without understanding what is going on. Either way, for the student the 'inverse' of a function remains '...


4

A triangle is born from three non-collinear points and the axiom that two points determine a line. In the context of neutral geometry, a triangle has no structure other than three lines and three points. In particular, there is no notion of the interior of a triangle without more axioms. In the real projective plane, one cannot define the "interior"...


4

insight why this seems to be so difficult? I have studied CS with a side-dish of math (some decades ago) and I was surprised by (but very much enjoyed) the actual theoretical aspects of CS. I generally didn't know what to expect from a CS curriculum except that in my case it was abundantly clear that I wanted to study whatever there is to know about ...


4

I've said in the past that the math discipline has a problem of jumping into the higher-levels of conceptual difficulty with exercises too fast. Let's say we take Bloom's Taxonomy as a model. Non-STEM teachers constantly complain that they spend all their time at Level 1, memorizing facts, and can't move past that. Math teachers, on the other hand, give a ...


3

I try to stress that definitions are just useful shortcuts, ways to put several disparate phenomena under the same roof (like context free grammars and the others in the classification), and show the similarities/usefulness of said definitions. One of the basic "proof techniques" taught in the discrete math course (first one that for our students ...


3

I would encourage you not to teach your students that an equals sign represents a question or implies that an action should be taken. Many students already struggle to interpret the equals sign as relational (indicating equality or balance) instead of operational (indicating an action to be taken). The Importance of Equal Sign Understanding in Middle ...


3

Most of pre 1900 mathematics can be done without the modern function concept. Hints that this was actually the case can be found in this hsm question Who first considered the $f$ in $f(x)$ as an object in itself, and who decided to call it a function? or if you skim through Leonard Eulers books on differentiation and integration (you'll have to look very ...


2

We shouldn't need to teach functions to 15 year olds, because ideally they should have already learned programming since primary school, including mathematical and general functions and inverse functions. Programming, including demos, games and robotics, is the best motivator to learn math in my opinion.


2

The wikipedia definition is the right one. A square is a rectangle. A rectangle is a trapezoid. Yeah, at times you can have specific/different definitions. But, the trapezoid one is pretty clear cut. You'd be hard pressed to find a (non contrived) theorem that applies to trapezoid that suddenly stops working because the shape is also a rectangle. It seems to ...


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