More than I ever learned in high school....

Gilbert, M.J., Coomes, J., (2010). What Mathematics Do High School Teachers Need to Know? Mathematics Teacher, 103, 418-423.

People assume that if you can do, you can teach. Gilbert and Coomes claim this is a fallacy, and that teaching mathematics effectively requires much more mathematical knowledge than is necessary to merely solve the problem. They list and explain several types of knowledge that a teacher needs in addition to Common Content Knowledge – the ability to solve a problem.
A Teacher must be able to do more than simply mark “right” or “wrong”, as Students will approach a mathematical problem in varied ways and analyzing the methods used by students and allows teachers to assess student learning. This requires that teachers understand and are able to interpret each of these solutions and determine how the student was thinking so that they can best help the student to improve.
For example, it is important for a teacher to recognize common errors and to realize why a student might make these errors. This allows the teacher to provide the student with guidance on which part of their concept model is incorrect, so that they can adapt it accordingly. To increase understanding, perhaps a teacher can teach with these possible errors in mind, outlining why a student should avoid certain techniques and so on. Teachers must also recognize how each approach relates to the other tactics that students use, to help the students develop connections, and allowing the teacher to determine when the students are ready for new material. Additionally, they must understand how these solutions connect to instructional goals, so that they can help their students to achieve them.

I think this is actually one of the main reasons why so many students struggle. I haven’t worked in a high school, but in working with elementary school teachers in their mathematics, I find it is often assumed that a teacher can teach mathematics because they can do the mathematics. But many teachers are simply on a hunt for correct answers – if correct answers are being produced by the majority of students, the teacher feels they have done a good job and their work is done. However, in my observations, it is never this simple. Some students manage to extract a correct answer from dubious methods and are never able to correct their misconceptions because their lack of understanding is rewarded with a pass. Others think deeply about the material but make some trivial calculation error. Their thought processes are correct but they abandon them after being told they are doing it wrong. Others’ responses show that the students have a serious misconception that needs to be addressed, and as such, telling a student to go back and check their work will not be very constructive.
Students always initially try to understand the material. They take the information provided, build a mental model, and apply it as best they can to the problem. Then they are told, no, that answer is not right, but they are confused as to why. Gradually they come to the conclusion that math just doesn’t make sense. Few children can survive years of this treatment and still be willing to actually think mathematically. If a teacher only understands how they would do it themselves, then they will not be able to teach much to the students who don’t think exactly the same as them…. i.e all students.

Readers: Have you ever found this to be an issue? I'm curious....

Gordon, S. P. (2006) Placement Tests: The Shaky Bridge Connecting School and College Mathematics. Mathematics Teacher, 100, 174-178.

The main point of this article concerns the transition between high school and college mathematics. Most incoming freshman are placed in a class determined by their results on placement tests. Gordon argues that this approach is extremely unsuccessful and contributes to the unpopularity of .He believes that since the changes implemented by NCTM’s Curriculum and Evaluation Standards for School Mathematics, Colleges no longer understand what students have been taught in their schools. With the increased emphasis on conceptual understanding, the greater use of technology, and the reduced emphasis on algebraic manipulation and procedural proficiency, freshmen are equipped with a very different set of skill, skills that many would argue to be preferable in a college setting. However, these Placement tests – used at almost every college for more than 20 years- test the degree to which students have mastered basic algebraic manipulation, and little else. Students are often placed far below where they are conceptually, and the colleges complain about how students are placing lower and thus how unprepared freshman are. On the other hand, some students from more traditional high schools do very well on the tests and are surprised to find that the course requires much higher levels of thinking and reasoning then they are able. Gordon suggests combining placement tests with other indicators of ability, such as ACT or SAT scores, portfolios of work, and semesters of math taken.

The more I look at this article, the more I feel that the author is making a big deal out of a small issue. There are no clear statistics to show that so many students are being misplaced. Gordon uses only anecdotal evidence and the vague data collected by angry high school teachers who would have liked to see their students placed higher. I would like to see, at the very least, quotes from the students themselves showing their negative feelings about their placement, preferably some statistics about the portion of students who feel misplaced. The students should be the judges based on their own knowledge of their own skills.
Secondly, from my own experience, I do not feel that students will be able to succeed in college mathematics without the algebraic manipulation skill that Gordon so casually dismisses. If students cannot, for example, quickly factor polynomials then they will have a tough road ahead in college mathematics. If students do not have these skills, perhaps it is better for them to take remedial or other math classes so that they are more prepared.
Gordon also claims that students perform poorly on the test because their understanding is more conceptual than procedural. From our reading in class, particularly the comparison between the 2 schools Amber Hill and Phoenix Park, I am convinced that if students have a solid conceptual understanding of mathematical concepts, then they should not have problems with these questions, in fact, they should score just as high or higher on the tests.

We learn something new every day...

In our learning, we are all like scientists. We create (or construct!) a model in our mind based on whatever information or experience is available to us. Then we act as if it is true until we discover a contradiction, and at this point we revise our model until it again fits with experience. This process is ongoing, and as such there is never a point when we are not learning; whether or not our sensory experiences oppose or support our theories. Another implication of constructivism is, because knowledge is built from experience, that there may be many different models of the same concept in a classroom . Knowledge cannot simply be transferred from teacher to student, students must build it from information provided them. If we are to be believers in this theory of Constructivism, then we know that there is no “truth” out there, just a set of theories that have not yet been contradicted.

My first thought about applying constructivism in our schools is this: “If we are never sure if our knowledge is correct or not, who are we to be the teachers?”. Then I decided that this discussion would probably not be very helpful, so I have decided to talk instead about feedback. If students are to be able to learn, they must be able to test their theories and concept models against “truth” (or whether it agrees with our supposedly more complete knowledge) and it is our job as educators to provide them with the opportunity to do so. This will be most effective if done promptly, while the construct is still fresh in the student’s mind.

The Dangers of Math Magic

In the first paragraph of his paper, Erlwanger states that his purpose is to show the IPI system to be is unsuccessful because of the way in which it presents Mathematics to the learner. The main point is that the answer to the questions “What is the subject of Mathematics?” and perhaps more importantly “What is not Mathematics?” (according to Erlwanger himself) is in disjoint with the subject being taught to students such as Benny. The paper is, in a sense, simply Erlwanger’s own version of our own first blogs, albeit a little longer. For example, Erlwagner finds that Benny believes in a kind a Math “Magic” – if he applies the correct rules, rules which some guy devoted his life to creating, he will be able to produce the correct answer. The task is simply to find the rule that works for that particular problem, and as such, Benny has no trust in the consistency of Math. He has no idea whether his answer is right until the mysterious answer key hanging over his head deems to respond with a “Yes” or “No”. In contrast, Erlwagner sees Math as a rational subject, one in which separate concepts, such as fractions and decimals, can be related and answers can be verified according to common mathematical sense.

I think that Erwanger’s definition of Mathematics is still extremely applicable to learners today. In my own experiences in the schools, I find that teachers hand out Math worksheets and explicitly state “Don’t show any of your messy working on the paper”. How can a child conclude anything from this other than that a correct answer is all that is required from them? This encourages the kind of rule application favoured by Benny: Use any method that seems to work, regardless of whether or not it makes any kind of Mathematical sense! Teachers need to try and gain an understanding of how a child produces answers, and why he thinks the way he does.

Instant vs Delayed Gratification

According to Skemp, when we speak of understanding a Mathematical concept, there are two different types of understanding that we could possibly be referring to. The first, termed Instrumental understanding, is knowing what to do in order to get a correct answer. The second, Relational understanding, expands on this: it is knowing what to do, and the reasoning behind why this produces a correct answer. Both provide methods for solving Mathematical problems and both produce correct answers. The benefit of learning Math instrumentally is all to do with ease and speed. Given the right set of rules, a child can quickly, and correctly, produce a set of correct answers. It is easier for the child to understand what she or he has to do. However, there are drawbacks, many of which can be avoided by using relational teaching and learning. For every new set of problems, an instrumental learner must learn a new set of rules. On the other hand, a relational learner can adapt his knowledge of previous concepts and derive a new method. He is also more likely to remember something that he understands. Consider the 2 words "Happy" and "T8grP". Which are you more likely to remember tomorrow? The one that makes sense to you, that you can imagine using correctly in situations, that you feel confident adapting for a variety of uses. The learner will enjoy mathematics all the more for being able to solve problems without being dictated to at every step.

I have too much fun with numbers....

What is Mathematics? For me, Mathematics is the way we express our trust in the universe. Every time we use numbers to express the happenings in the world around us, we are declaring “Yes universe, I believe that you make sense.”

I’m going to talk a bit about what I call “Mathematical sense”. I see so many students in schools that simply don’t have it. They add together 2 numbers and get a number smaller than either. They count …97, 98, 99,100, 200, 300… For them, and the numbers have no meaning to them. They represent nothing, except a lot of wasted time, trying to reach some unknown goal along this dark unknown path that their teacher is trying to push them along. Mathematical sense is the art of knowing, of feeling, that when we are solving mathematical problems, we are not just shifting around numbers because the teacher told us to, but that something deeper is coming into play, something powerful, something that is fundamentally true and right.

Personally, I like working through lots of examples in Mathematics. The more I play with numbers, the more develop my mathematical model of the world – my knowledge of how numbers are supposed to work. I think too often we push students straight to the more difficult questions, the ones that will challenge them and “stretch” their knowledge or understanding, without allowing them proper time to get really comfortable with the material.

I think that also, too many of our public school teachers hated math in school and consequently hate to teach it. Our children are taught from a very young age that Math is boring, difficult, nerdy and completely detached from real life. Far too often I have heard statements such as “I’m sorry class, you’re not behaving, so we’ll just have to do Math sheets instead of our colouring”. An effective bribe, but one that places Math firmly in the undesirable category. We have developed a culture where Math is not respected as it should be.

I cannot think of one person who would stand out in public, give a small burst of apologetic laughter and say “Sorry, I’m just not any good at reading”. So then do we hear the same thing about Math every day? Why are people so proud of their mathematical inability? Why is it socially acceptable, desirable even, to declare a complete revulsion of numbers! Rise up Flatlanders! Rise out of obscurity and the shout to the world “Hurrah! I Love Math and my world is beautiful!”. My dear Flatlanders, lets go out and teach them.

Much love,

Emily the circle