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Background information

I'm currently teaching common core geometry, which assumes that a student has algebraic knowledge coming in. Clearly, we shouldn't expect students to retain everything from their algebra class before they take geometry. I've found myself reteaching the entirety of the class what slope is and how to calculate it. Ok, I can understand if about 80% of the students forgot entirely how to create a triangle on a graph and count how long its sides are. That's fine. But there are other concerning observations about some students in my classes, some of whom apparently failed algebra 1 but are being passed along into my class anyway...

  • Forgetting which axis is x and which is y, and had difficulty plotting (x,y) coordinates at the beginning of the year.
  • Not being able to determine the length of an unknown line segment -- (2) and (?) next to each other, with a total length of (11). She guessed 5.5. Even after I told her the smaller segments add up to 11.
  • Not being able to solve the equation x * 1.2 = 24 with any amount of prompting until I literally rewrote the equation as 1.2 x = 24 (I had suggested that she do so herself, but she didn't and we had been talking about this equation together for several minutes).
  • Not being able to determine whether a shape looks like it has been rotated or reflected, even though they demonstrate knowledge of what the words "rotate" and "reflect" mean (spatial skills).
  • Could not graph an equation in the form y=mx+b at the beginning of the year, and still require prompting and guidance to do so.

I guarantee that some of the students mentioned above will have extreme difficulty in geometry class. Not only do we use algebra in most sections now that it is "common core," but some of them are missing the arithmetic reasoning that they need for geometry. Several failed algebra 1, but passed the "EOC" exam, so they got placed in my classes anyway. They will struggle to pass my class as much or more than they struggled to pass algebra.

Main question(s)

  • At what point is it a disservice to pass a student into a class they are not ready for?
  • How does someone determine the prerequisite knowledge for a math class while recognizing that the knowledge that would be best for students to have is not necessarily the knowledge they will enter with?
  • Is having students repeat low-level math classes like algebra before moving on to geometry beneficial, or should students be able to succeed at common core geometry without algebraic background?
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    $\begingroup$ What is EOC? One thing to keep in mind is that there are really two separate effects to consider when deciding whether to hold a student back. (1) What is the positive or negative effect on that student? (2) If the student is allowed to move on and take the next class, what is the positive or negative effect on the other students in that class? There are a lot of studies that seem to show that holding students back is a negative thing for those students. But letting them continue is likely to severely degrade the quality of education experienced by the more competent students. $\endgroup$ – Ben Crowell Nov 2 '15 at 16:58
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    $\begingroup$ @BenCrowell EOC means End Of Course test. $\endgroup$ – Benjamin Dickman Nov 2 '15 at 19:55
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    $\begingroup$ Here is some info about the outcomes when students repeat Algebra I: usnews.com/news/articles/2014/12/16/… . AFAICT there simply is no evidence that anything at all can be done for students who fail the first time around. In another era, we would have placed them on a track that didn't require algebra. $\endgroup$ – Ben Crowell Nov 2 '15 at 21:30
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    $\begingroup$ Well, then the question becomes: is it algebra that those students are missing, or something before algebra? If it's something BEFORE algebra, of course repeating algebra won't fix those skills. That line of discussion almost deserves its own question... $\endgroup$ – Opal E Nov 2 '15 at 21:56
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    $\begingroup$ @OpalE: It's also possible that some students who fail Algebra 1 are perfectly competent with previous stages of math such as arithmetic, but are incapable of abstract reasoning, such as manipulating equations in which the values are not known a priori. In Piagetian terms, these could be people who are at the concrete stage of development, but have not yet reached the formal stage. California, for example, tried to mandate algebra for every student in 8th grade, but later gave up. Realistically, many kids at that age are still playing with toy trucks. $\endgroup$ – Ben Crowell Nov 3 '15 at 3:48
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Social promotion is an interesting topic, and I know my personal views are unpopular in my school. So they probably will be here too, but I'll go ahead.

Due to developmental issues with children I think that social promotion is acceptable before high school. It should be coupled with remediation, but I think there are too many issues that can arise from having a contingent of older students in lower grades. I think that it is part of our responsibility in high school to end social promotion. It is our responsibility to hold students to a higher standard and should start holding them back is they do not meet expectations. Making them repeat courses or take alternative tracks. I have taught a lot of first year classes, and I have come to expect a wide range of abilities in the students sent to us and that's OK.

Now, what we've done within our school to deal with the ranges is a little different. We've found that meeting the minimum requirements of a class is not always a good predictor of success in a future class. So we've added that in order to take the next class in a sequence a student must have received a C or better. Students who end up below this threshold may repeat the class, and we're trying to create more avenues for students to receive remediation without repeating the whole class. Some students who do not meet the grade requirement are passed on by teacher recommendation.

We've determined the standards necessary for the classes that follow and we are trying to improve the ways we can use these as objectives instead of a raw grade requirement. I think these learning objectives will eventually replace the raw grade requirement, but for now using the grade is a lot easier for guidance and parents to understand. Also it could allow us to better deal with students who generally understand the material, but for one reason or another have critical gaps in their knowledge.

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There is a point at which material is so far beyond students' skills that they learn less. Imagine a freshman calculus student plopped into a graduate topology seminar: they will learn almost nothing.

Consider Csikszentmihalyi's flow model:

enter image description here

We'd like students to be in that upper right corner, where they are highly skilled (relative to the task) and highly challenged.

When students are so unprepared that promoting them to the next class will cause them to learn less than keeping them back, they should be held back.

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    $\begingroup$ +1, though I'm not sure whether having students in a state of flow is desirable. At the least, I think that having them constantly be in flow may close them off to some of the new material; similar remarks distinguishing orthodox creativity and radical creativity can be in found in work by the Cropleys. (Though I don't wish to conflate flow and creativity; for more, here is Csik's TED talk on flow.) $\endgroup$ – Benjamin Dickman Nov 2 '15 at 20:00
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    $\begingroup$ @BenjaminDickman yeah, I think flow is the wrong word and maybe even the wrong idea entirely, but it's the best I could come up with. I'd really like to reference some research into the diminishing returns of difficulty--up to a point, students learn more if we challenge them more, beyond that they learn less--but I don't have anything more than my own experience. $\endgroup$ – petehern Nov 3 '15 at 21:16
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Indeed it is futile to expect students to successfully catch up with the prerequisite mathematical understanding for a course if they are too far behind. This is because many students think that they would rather focus on getting their grades for the current course despite not having any proper foundation in prerequisite courses. (This is real and happens even in top universities.) So it is a disservice to allow students to take a class they are not ready for, because it is a waste of everybody's time (especially the poor teacher's).

Certain fundamental concepts cannot be neglected. Basic arithmetic and algebra need to be there, otherwise you get the kind of problems you described in your question. I would say that if students magically pass a geometry course without being able to solve quadratic equations, then I can safely conclude that they have nearly zero understanding of geometry as well.

As for determining what concepts are really necessary prerequisites, that is a very vague question that has to be dealt with on a case-by-case basis. I would think that even very basic Euclidean geometry requires propositional logic, arithmetic and algebra. Anything more, including dealing with degenerate cases properly (such as in Pascal's theorem and Desargue's theorem and their converses), would require a firm grasp of full first-order logic. In general, all areas of mathematics require rigorous logical reasoning, which is somehow not taught in most high-school curricula.

One solution is not to force everyone to proceed through education at a fixed pace, but rather take as many or as few classes as they want to each term, and require that students are allowed to take a class only if they understand the core concepts in each prerequisite class (this may be assessed by grades as in BBS's answer).

In any case, the classes must ensure that students are not simply taught about mathematical objects and techniques, but rather taught why one should want to know about the objects in the first place, and how to devise the techniques without knowing them, otherwise most students will form a justified opinion of mathematics classes that they are useless (because they don't know when to use the stuff) and meaningless (because no one ever explains to them the reasons for what is done).

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  • $\begingroup$ Funny story: I found out recently quadratics are not part of the algebra I class at my school AT ALL. (Maybe they plug them into a graphing calculator once.) So, either I teach them about quadratics (not in my purview as a geometry teacher) or... they end up passing a geometry class without them. $\endgroup$ – Opal E Mar 24 '16 at 19:53
  • $\begingroup$ @OpalE: Sad funny right? I personally always try to teach at least the bare minimum necessary to bridge any gap. Pure Euclidean geometry usually does not need quadratic equations, but when it does, it's easy enough to just show how to solve it by completing the square, without even calling it a "quadratic equation". =) $\endgroup$ – user21820 Mar 25 '16 at 2:04
  • $\begingroup$ They may not have ever multiplied binomials, depending on where they took algebra 1.... Also common core standards involve "optimization" problems (maximize area given perimeter, etc.) I'm not sure what to do. $\endgroup$ – Opal E Mar 25 '16 at 3:50
  • $\begingroup$ @OpalE: Area of rectangle I presume? If so, then even better because completing the square makes the maximum area and when it is attained obvious. But if you're talking about area of an arbitrary closed rectifiable curve then that's only possible with some heavy real analysis. $\endgroup$ – user21820 Mar 25 '16 at 4:23

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