Teaching through mathematical investigation allows for students to learn about mathematics, especially the nature of mathematical activity and thinking. It also make them realize that learning mathematics involves intuition, systematic exploration, conjecturing and reasoning, etc and not about memorizing and following existing procedures. Although students may do the same mathematical investigation, it is not expected that all of them will consider the same problem from a particular starting point.
What is essential is that the students will experience the following mathematical processes which are the emphasis of mathematical investigation:. In this kind of activity and teaching, students are given more opportunity to direct their own learning experiences. Note that a problem solving task can be turned into an investigation task by extending the problem by varying for example one of the conditions.
To know more about problem solving and how they differ with math investigation read my post on Exercises, Problem Solving and Math Investigation. Some parents and even teachers complain that students are not learning mathematics in this kind of activity. This goes without saying that teachers should try the investigation first before giving it to the students.
I think mathematical investigation is constructivist teaching at its finest. For a sample lesson, read Polygons and algebraic expressions. View All Posts. We also have to note that those young ones have their own way of doing things; which in some ways may simplify processes.
Naci John. Only those topics in math that involves generalizing and conjecturing using the mathematics students at that grade level know may be taught by math investigation. The students may see patterns and be able to state a conjecture but they will not always have the capability to explain or justify it mathematically.
Doing math investigation is more for developing disposition to think mathematically. I find this article very informative. I think reconciling LS and MI will be a great boost in the teaching strats of teachers and mathematical abilities of the students. In elementary school I had to take investigations. In all lessons standard addition is mentionedONCE and not truly taught. This lleaves students with lower grades on timed tests because they are rushed due to the innefficient method they first learned.
Many students who used the program in elementary school needed help with math in middle school. So answer this, why do we continually use investigations? To me, Math investigation is an ally of constructivism. Students are engaged in tasks that allow them to put in more on what they know as they tinker on a math problem. They do exploration and research to be able to shed more light to the problem.
Creative and critical thinking come to the fore. The investigation may take time. But, at the end of the process, students defend what they got, defend it, and share it. The first is that in future employment, students may have to produce information, consider different outcomes and put together a coherent report describing the field and the possibilities.
Suppose we tried that where would it lead? But investigation is the lifeblood of mathematics. It is what research mathematicians spend their lives doing. Children can get some idea of the feel of mathematics and what mathematicians do, by engaging in investigations.
This should be done gradually. Start first with relatively simple one-off problems and then work on harder problems and their extensions, until the students are capable of tackling investigations that may take a week to complete. And the final advantage of such investigations is that it gives students the opportunity to practice a range of skills from all areas of the curriculum.
By controlling the investigation you can to some extent control the skills that the students will use. However, you should expect them to come up with the unexpected. What is Mathematics? Below we produce a diagram that shows the way that mathematics develops and we hope in the process that you will see how this applies to problem solving and more particularly to investigations. The bones of mathematics are in the diagram above. All mathematics starts with a problem that is either a well known one or invented by someone on the spur of the moment or after a great deal of thought.
To make any progress on a problem usually requires a lot of playing around with the ideas and making up lots of examples. We call this experimenting. If we refer back to the postage stamp investigation, the problem was posed for us. However, it is worth noting that it was originally a problem that was posed by mathematicians. The person who solved it was Sylvester at the end of the 19 th Century. He undoubtedly spend some time experimenting in the way we did above.
He would have needed to have guessed the pattern 2n — 2 and then m — 1 n — 1 for the onwards numbers, somehow. To do this he would have needed data that could only have been obtained by experimentation. But, finally, as a consequence of his experiments he would have arrived at the guess m — 1 n — 1. Actually mathematicians prefer to call their guesses, conjectures.
Generally it is not realised that mathematicians do a lot of conjecturing, guessing. In fact it is one of their major tools. However, on a difficult problem they make a thousand guesses before they make the right conjecture. So how do they know if a conjecture is false?
They find a counter - example. This is an example that runs counter to their conjecture. Suppose Sylvester had conjectured 3 n — 2 — 1 for the onwards number. If you find a counter-example, then you need to think again, maybe experiment again, to find a new conjecture. But Sylvester, at least, was able to prove his conjecture. Now proofs are actually what separate mathematics from other disciplines. No other discipline proves things without any doubt. These proved things are generally called theorems.
As far as the stamps go the theorem would say that, given m, n with no factors in common, any amount of postage from m — 1 n — 1 onwards can be made. Having produced a theorem what mathematicians do next is to try and generalise or extend their result.
For instance, going from 3c and 5c to 3c and nc is a generalisation. This is because 5 is a special case of n. If we know what happens with 3 and a, we can certainly find out what happens with 3 and 5. But we can generalise 3 and n further still. After all the result for 3 and n is a special case of the result from m and n. So m and n generalises 3 and n. Summarising, then, a generalisation of a problem is a bigger problem that contains the original problem as a special case.
That leads us to extensions. These are also problems that grow out of other problems but here the link is often less strong. For instance, with 3c and 5c stamps we know that we can add any non-negative multiple of 3 to any non-negative multiple of 5.
This is a similar problem but the original problem is not a special case of this problem. The squares idea is therefore an extension of the first problem.
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