But perhaps not! Suppose that you had latex gloves (or vinyl gloves for those LaTeX averse like me) – that way there are no intrinsic differences due to thumbs, for now. If you put one on each hand, and dye the palms one color and the backs another color, you could tell the two apart. But suppose the dye bled through the gloves, so that the inside were colored as well. You could turn the right-handed glove into the left-handed one by flipping it inside out!

Since the orientation is not about shape, but what is inside or outside, we can stick our glove on top of a Cartesian space, with the tip of the third finger on the z-axis. At the tip, the normal vector to the surface of the glove points upwards (so it is positive) – we can define this as right. But if we flip it through itself past the xy-plane, the new outward direction points in the negative z-directions – so it is left-handed.

But what about thumbs. Topologically, it feels like thumbs make no difference. And if you put a glove on the right hand, put your fingertips together, and you could have someone flip the glove over to the left hand. The thumbs would stay in the right position, I think – it would be kind of weird if the thumb sheath suddenly jumped to the back of the left hand.

]]>Any well-taught mathematics class, regardless of content, is valuable to our students. It provides them the opportunity to learn to make distinctions, to use precise notation, to reason soundly, to justify steps, procedures, and arguments, to discover, to problem solve, and to prove. In short, while the mathematical content will fade with time, they have learned to think more clearly for life.

The single most important change we can make to our curriculum to benefit our students would be to add Statistics as a graduation requirement. A citizen of todayâs society is well served by knowing more about data collection, data analysis, data use and misuse, and statistical inference.

]]>On a different note, I hate the fact that certain colleges expect students to have taken calculus in high school. It becomes the criteria from which students and schools make decisions about curricular options. Ideally, students would make those decisions based on what is best for them personally, but that is not to be. I do think that having seen calculus once, often in a smaller setting, it is easier to succeed in it later in college.

]]>I think being more helpful by being less helpful is a great idea. A way to foster an increase in their willingness to engage with material over a prolonged period of time, even if no answer is in sight. I can, as teacher, establish a bigger playing field, limits further off.

The presi talked about motivation. I have worked in schools where motivating the students was the measure of the teacher. This motivation was measured under a classroom management rubric. This is not the sense that the presi had in mind.

The presi talked about how needing rational functions in some future class was poor motivation for learning about them in this one. This is motivation in the sense of providing context in which learning the material has some meaning to the students. The motivation is provided by the context set up by the teacher. Some teachers are more successful at this than others, regardless of the textbook or the curriculum or whether the teacher is traditional or adventurous, linear or nonlinear. But here especially I think being more helpful by being less helpful is a rich avenue to explore.

]]>Teaching a 9th grade course allows for some special âelectivesâ work since there are 4 experimental education trips per year. Each trip, approximately 75% of the students remain and we have the freedom to introduce various topics. Some examples from prior years include: constructions, calculator programming, intro to excel, logic puzzles, and Cramerâs rule. These periods away from the regular curriculum are meaningful in several ways. Students are introduced to topics that wonât be tested so learning is less stressful and some topics tie well into enriching the curriculum. Hence, I agree we should introduce elective, where possible.

Currently, algebra/geo classes are covering quadratic functions. One lesson was on completing the square when a = 1. I spent a second day on CTS when âaâ was not equal to 1. For the third day, I had students CTS for x^2+Px+R= 0 then for HX^2+x+1=0. Completing the square with variables posed more of a challenge but the students saw the process was the same. Both problems were a warm up to solve Ax^2+bx+c=0, which is the derivation of the quadratic formula. My best attempt to be âless helpfulâ worked for the majority of students but there were some who got bogged down in notation and detail. If I had only given them the third example to attempt on their own, the breakdown of those succeeding might have been closer to 50/50. I feel it is important to introduce the quadratic formula like this, but for some the exercise did not work. Hence, with a younger age group, there has to be balance with how much subject matter is approached in this manner because students can quickly become overwhelmed.

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