I wouldn't doubt that math students fail because they don't understand the reasoning behind the mathematics. But why does that happen? I imagine there are many different reasons and that these reasons differ from student to student, but many of these reasons stem from students' rote learning of mathematics from previous courses, students' lack of experience with genuine reasoning in mathematics, students' difficulties with understanding algebraic language, students' difficulties with understanding what algebraic expressions mean, and students' boredom and/or fear of mathematics. How many elementary school teachers in this country actually have a deep understanding of the mathematics they teach? How many of them enjoy mathematics? How many of them have any idea of what mathematical reasoning is about? How many realize that mathematics is not all recipes and algorithms and memorization? Relatively few. Even if a math teacher has all this understanding, that does not mean that the teacher will avoid teaching mathematics as a bag of tricks, as a series of disconnected problems and topics, as pure or almost pure memorization, etc. All or nearly all the elementary math books I have are not true math books but are really cookbooks disguised as math books. All or nearly all the reasoning behind the mathematics is gutted from these books.
When students are presented with mathematics as this kind of garbage, they have serious difficulties in learning the reasoning behind mathematics. Without that reasoning, the mathematics becomes meaningless mumbo-jumbo to them. And most students suffer for years through math classes that teach this kind of garbage, and they develop a great fear of mathematics--along with strongly negative opinions about mathematics and their own abilities to learn it. Even worse, their opinions about mathematics are greatly distorted because their views of mathematics are greatly distorted.
These kids then fail to learn algebra because learning algebra requires a correct mindset to begin learning abstract reasoning. But all that previous garbage taught to them does not give them a correct mindset they need. They cannot go into a beginning algebra class thinking that mathematics is nothing but meaningless mumbo jumbo, uncreative, a bag of tricks, a series of disconnected topics, etc. and then expect them to learn algebra--especially if the teacher makes little effort at the beginning to help students clear their minds of all this junk. How many algebra teachers do that? How many algebra textbooks address these issues? From what I have seen so far, I would bet that relatively few do this.
How can we expect students to learn to reason about algebra if they do not understand algebraic language and how to make sense of algebraic expressions and equations? In fact, how do we expect them to do that if they cannot make sense of numbers? The students I've seen who struggle the most to learn algebra struggle with something more basic than arithmetic: the meanings of numbers and the meanings of the various arithmetic operations on these numbers. Most of my remedial math students have little understanding of what a fraction means and the conceptual meanings of fraction arithmetic (including trouble with performing this arithmetic). Some of them have this trouble even with whole numbers. So if they have trouble interpreting arithmetic expressions and operations, it should be clear that they will struggle mightily to understand the basic meanings behind algebraic language and algebraic expressions. If they can't make sense of this stuff, they won't learn algebraic reasoning. Yet most algebra teachers go on trying to teach this reasoning to such students because they cannot afford to take the time to do otherwise. No matter how high or low the standards are for students in such a course, the outcome is disastrous (maybe not in terms of grades but definitely in terms of whether students learn any real algebra or not). The results are even more disastrous if the algebra course presents mathematics as the same kind of garbage as their previous courses because that kind of garbage does not promote any mathematical understanding or reasoning of any kind. The algebra textbooks I've seen do present mathematics as this same kind of garbage, so I imagine that most math students learn garbage mathematics all the way through school and even in college. And if they continue to see mathematics as this kind of garbage, they have no chance of learning anything of value. Throw all the reasoning you want at them, but the problem will not go away if we cannot help these students see that mathematics is nothing like the garbage they had seen before.
On 3/13/2010 at 4:52 pm, Robert Hansen wrote:
> The short answer: They don't get it. > > The long answer... > > What is it that they don't get? They don't get the > chain of reasoning involved in mathematics. Imagine > being in a class in a subject where each new year you > are presented with refinements and extensions of last > year's reasoning, that you didn't get. Anxious? Yeah, > you are going to be more than a little anxious. And > this goes on for years. > > The instant that the mathematics has exceeded their > ability to follow this chain of reasoning is their > departure point. Unfortunately, that instant becomes > apparent when they start algebra however by simple > reasoning it must have occurred much earlier. I > haven't looked enough at pre-algebra and arithmetic > but whatever you see on day one in algebra was > already fated in those previous 8 years. Algebra > classes generally start off with a lot of new > notation and language and that is fine IF the > pre-algebra course brought the students into the > kiddy pool of algebraic reasoning worthy of all these > new refinements to mathematical reasoning. If > however, these kids are showing up with nothing more > than a class schedule then you are obviously going to > have a problem. How can you expect to refine > reasoning that doesn't even exist yet? > > Filling them with motivation by telling them about > all the jobs they can do if they learn algebra is > hardly addressing the problem. All you are doing is > telling them all the jobs they cannot do. Engaging > them in activities that have math in them may sound > nice but still does not directly address the problem > of not getting it. And if you are not careful with > the activities and design them poorly, all you will > do is make "not getting it" less stressful. These > activities then become nothing more than palliative > care for a dying patient. I have done enough tutoring > to look at these activities and tell when one author > is actually still trying to win the game and another > has given up and chosen the palliative route. > > And Jonathan will make this exact same choice many > times over, every semester, with the students in his > class, probably more so than others because of the > nature of his students. Blaming all the teachers > before him that were faced with the very same > situation as he is being faced with now is a foul in > my book. He should think deeply after he has tried to > reach a student and failed about what actually > failed. And he will find that no matter how much he > talked about algebra, talked about great > mathematicians or did fun activities, at the end of > the day when he presents an actual algebra problem > requiring nothing but algebraic reasoning, the > student still does not get it. > > I understand the reasons for these motivational and > engagement methods but let's be real, now that you > got their attention, you need to get them reasoning. > And in all of my studies of this problem, they seem > to be showing up for algebra class, but they are > showing up entirely in the wrong frame of mind > without prerequisite reasoning experience. And > reasoning is so personal that it is very difficult to > keep up with each and every student to see if they > are using their head enough. As a father I can > monitor my son much more closely and in a more > personal manner than any teacher but even I have to > keep on my toes. Since I am very mathematical it's > easy for me to spot reasoning flaws. > > Unless you only teach off of AMC exams (which is > impossible in the beginning), no matter how you teach > math, there will always be a rote path of some sort > available to the student. You have to make sure that > the student doesn't rely on this rote-ness and > instead connects to the mathematical reasoning. Why > are there rote paths? As Jonathan stated, there are > techniques and algebraic procedures and you cannot > escape that. But those techniques and procedures are > supposed to be the culmination of chains of > reasoning, not the steps to bake a cake. The learning > cycle is supposed to bring the student to the edge of > solving a particular type of problem and the > "technique" is actually someone else's solution and > the student is supposed to go "So that's how they did > it!" They are not supposed to memorize the technique > they are supposed to get the reasoning behind the > technique and own it for themselves. Lou asked me to > critique traditionally paced curriculums and I said > "They expect the student to run a perfect race." As > they move through the material they expect the > student to keep up with the reasoning and by doing > this you are not teaching technique, you are > developing their reasoning skills. Once you lose that > connection then the student has no choice but to fall > back to rote learning and while baking a cake may > only require one to memorize a sequence of steps, > problem solving requires reasoning strategy and you > will fail without it, unless of course the test is > "fixed" to pass them anyways. > > > So the number one reason why kids fail math is very > simple and any "mathematical" teacher knows this. The > kid fails get the chain of reasoning. And all these > new "proxy" methods of teaching fail to address this > problem or they avoid it entirely with intentional > deliberateness because they don't want to face > reality. > > Math is about mathematical reasoning. Everything else > that is discussed here are refinements to that > reasoning. Word walls and journals are so secondary > to the problem at hand that I will not even comment > on these things any more. Teaching kids algebra when > they are 6 years old is retarded but typical of many > approaches to teaching math that appear to be very > non-mathematical in their reasoning. In fact, it is > very evident that the people coming up with these > "theories" are not very mathematical at all. Kids do > not fail algebra when they are 14 because they didn't > learn algebra when they were 6. They fail algebra > when they are 14 because they didn't learn arithmetic > when they were 6. They fail because their ability to > reason mathematically isn't even in the ballpark when > they step into the class. > > I do not preach one curriculum for all students. I > would love that the world be so nice. But the fact is > that many students do follow the traditional pace and > their reasoning keeps up with the math. And it is a > fact that many do not. This trend seems to appear > very early but I do not put any weight into any trend > before third grade because everyone develops on a > different timeline. After third grade though, if you > want to lick this thing, then you need to focus on > reasoning ability and not be shy about it. The test > can be done in person and be free response in format > and it should consist entirely of problems with a > range of reasoning difficulty appropriate for that > age. No preconceptions of how reasoning works, just > problems that require reasoning. And once you get the > results, take it from there. If you don't do that > then you are not being sincere about this at all. I > would never treat a cancer patient with a drug that I > thought might work, and walk away feeling good as if > I did something, without ever knowing if it did work. > > > Every teacher with any mathematical ability and > reasonable experience knows the problem and we most > certainly have the ability to create tests of > mathematical reasoning ability (I am not talking MC > tests). We are just not in the right mindset to > accept the results and deal with them as best we can. > Instead we come up with an endless series of theories > with some slight obfuscated measure of success in > some slight obfuscated and ridiculous measure of math > and announce to the world that we have licked it. It > would be comical if it wasn't connected to such a > serious subject.