First, I need to vent about the awful first sentence:
A common view is that students learn maths best when teachers give clear explanations of mathematical concepts, usually in isolation from other concepts, and students are then given opportunities to practise what they have been shown.
I agree that clear explanations and practice are important to learning maths. I consider myself a more traditional teacher who uses worked-examples and leading questions to develop mathematical concepts. However, my beef comes from the phrase "usually in isolation from other concepts." Why would any teacher agree that maths is best learned through isolated ideas?
For example, in my maths class for pre-service teachers, we recently discussed the number of divisors a number has. We considered two small example first: 12 and 16.
12 = 2^2 x 3 and 16 = 2^4
12 has 6 divisors (1,2,3,4,6,12) and 16 has 5 divisors (1,2,4,8,16)
Then I asked "Is there any connection to the number of divisors and the prime factorization?" A difficult question, so I allowed them to ponder a bit. We eventually connected the number of divisors to the powers of the primes in the factorization through addition of one. For example, 12 has (2+1)(1+1)=6 divisors and 16 has (4+1)=5 divisors. We discussed why this always works by considering the combinations of the powers, then made a generalization so we could find the number of divisors for larger numbers. And voilà, the concept of prime factorization has been connected to the concept of divisors. I don't believe that good mathematics teaching happens inside boxes.
So why does the author set up this idea that individuals believe that concepts are taught in isolation? Well, it is a straw man that leads into the research that he is doing on open-ended questions - the way I think you are teaching maths is incorrect, here is the way I am teaching maths. In case you missed my earlier post on poorly-designed open-ended questions, you can revisit it here. Essentially, the way he is teaching mathematics is by using open-ended questions such as
"The minute hand of a clock is on 2 and the hands make an acute angle. What might be the time?"
or
"What are some times for which the hands on a clock make a right angle?"
The main issue I have with open-ended questions is that they are often poorly thought out. For example, if we consider the first question, what time signifies a degree value of zero between the two hands (we need some kind of reference point)? Is it at 12:00? If 12:00 is zero degrees, then technically, any time after 12:15 is obtuse because we need that initial time as a reference point [here, I am assuming that we are measuring angles in a clockwise fashions, rather than the usual anti-clockwise fashion]. However, I have assumed that the hour hand doesn't move (see next paragraph)! So if we assume the hour hand moves discretely, and not continuously, then 12:15 fits the bill for 90 degrees, making any angle bigger then this obtuse. This means any time after 12:30 must be a reflex angle! So there are no acute angles if the hour hand is on 2. This question isn't mathematically sound until more information is added. Questions similar to this may develop misconceptions in your students: when given a reflex angle, they may measure the incorrect portion or fail to add the multiples of 360 degrees.
In our second question, again using 12:00 as a starting point of zero degrees, we might consider that 12:15 must be the 90 degree angle. But again, this technically isn't the case since the hour hand will have moved one-fourth of the way to 1 by the time we get to 12:15, so the hands at 12:15 are actually a little less than 90 degrees. (If interested, you can show that the time the hands will make 90 degrees is actually 12:16 and 4/11ths of a minute). Because of the argument above, this will be the only time we have a 90 degree angle! All others will be 90 + 180k degrees (k is a positive integer). Again, this open-ended question may result in the misconception that the hour hand doesn't move at all.
This article adds to my belief that open-ended questions are poor ways to teach concepts to students who are not well-versed in the content material, since poorly-designed questions may result in mathematical (or real-life) misconceptions. If these misconceptions occur during the vital younger ages and are stored as memories, it can take a significant amount of energy and time to remove these misconceptions from the long-term memory and rework in the correct concepts. The easiest solution: don't use open-ended questions, or use them sparingly.
You're right about that phrase. Where the heck did that come from? The whole point of conventional instruction is to present material in an orderly, sensible fashion. Almost by definition that means that material you learn today is connected to that which you learned yesterday and will be relevant to what you learn tomorrow. Mathematics is intrinsically hierarchical and those paying attention to that structure will teach in a way that makes sense of it.
ReplyDeleteIn contrast, those who hold to the doctrines associated with pedagogical constructivism, social constructivism, discovery and inquiry learning, generally believe that students should be exposed to problems without prior preparation and mastery of relevant tools and background -- in fact the whole point is for students to discover these things themselves!
The phrase appears to be an example of projection; it is quite the opposite of the truth, in the dichotomy being expressed.
Yes, I was rather taken aback by that particular phrase. The author is using the phrase to set up the dichotomy (like you mention) that clear explanations and practice are "bad" and his open-ended problem solving is "good". However his open-ended problems are not clear, and there is likely little to motivate why we are connecting angles to the clock. Students likely do not know anything but digital clocks, and angles are better connected to ideas of shapes/geometry. It seems as if the author is actually doing what he deems is "bad" practice - isolating concepts in a box.
DeleteI observe that most open-ended questions being tossed about as exemplars by the math gurus today are simply classical, well-defined problems with something missing or deliberately left out. I dislike them because they do not specify what is wanted. Hey! I've got a great idea! Let's expect children already struggling with a subject to simultaneously deal with unclear expectations and poorly worded task assignments!
ReplyDeleteThis wording "what might [some value] be?" Really REALLY bothers me. What is being asked? Will a single example suffice? Or is the student expected to produce an expression that includes every possible answer? In mathematical terms, the LATTER is almost always the desired end. But in these questions it seems almost universally the case that the intention is for each student to come up with a single answer and then maybe compare and say "WOW! We got DIFFERENT ANSWERS! How unexpected, and enriching! I never would have expected there might be MORE than ONE answer to a math question ... even though the last million questions this teacher asked us was just like this! Let's share how we got our answers and come up with deep insights!"
For the question "two numbers add up to 100 ... what might those numbers be?" If it is asked in an early-years class then the mathematically correct answer is, the first number is anything, say A. The other number is 100-A. The "fuzzily-correct" answer is "87 and 13?" The question mark is that the student is asking whether this "fits the rule" the teacher has in mind. But the mathematically correct version of that form of answer would be "0 and 100" or "1 and 99" or "50 and 50". Not because the student is being a wise guy, but because the appropriate mathematical habit to develop is when producing an EXAMPLE or counterexample of something (which is what is being asked here) always focus on obtaining one as simple as possible. So the answer 87+13, while perfectly correct, is not good mathematical practice. I expect all the strong students to give 0+100 (or something like it) and weaker students to pick a number and do a calculation or, worse, guess and check. After a couple of dozen questions in which all the relatively strong students produce mindnumbingly obvious solutions to these open questions a reasonable teacher would conclude that it has next to no educational value, and move on to questions that lead students to do more educationally interesting and challenging tasks.