1. Compare and contrast problems with:
I started by finding definitions of the four terms according to Merriam-Webster:
Problem- a question raised for inquiry, consideration, or solution b : a proposition in mathematics or physics stating something to be done
Exercises- practice - drill – training
Puzzles- to offer or represent to (as a person) a problem difficult to solve or a situation difficult to resolve :challenge mentally; also:to exert (as oneself) over such a problem or situation <they puzzledtheir wits to find a solution
Projects- a task or problem engaged in usually by a group of students to supplement and apply classroom studies
Based on these Merriam-Webster definitions, I would say that a problem is the purpose or goal. An exercise is what you do to perfect or practice reaching that goal or, in math, finding the solution to the problem.
A Problem and a puzzle could be the same thing, or more specifically, a problem may be in the form of a puzzle. It is something which begs to be solved.
Just as a problem and puzzle can be one and the same, I see a project and an exercise as being similar; however, to me, a project seems to be greater in scale. Perhaps an exercise is a small task and a project may involve more steps in order to complete it.
2. Which aspects you absolutely must keep, no matter what, so the task is still problem-solving?
According to one website I found, there are Seven steps to problem-solving:
There are seven main steps to follow when trying to solve a problem. These steps are as follows:
1. Define and Identify the Problem
2. Analyze the Problem
3. Identifying Possible Solutions
4. Selecting the Best Solutions
5. Evaluating Solutions
6. Develop an Action Plan
7. Implement the Solution
It would seem to me that you must keep the first step for you must know that you have a problem in order to fix it. I also feel that the second step is essential so that you can consider the real or perceived depth of the issue. I feel it’s important to analyze the problem, though I think this can be easily combined with step 1. If step 1 is done effectively, step 2 could easily be combined with it; thus, eliminating a step. Likewise, I think steps 3, 4 & 5 could be combined. As you come up with a possible solution, it could examine its feasibility and rank them in order of likely success. If time is of the essence, then combining steps is possibly, but the act of brainstorming solutions and likely outcomes must be kept. For, to go blindly without consideration, would certainly lead some dead-ends or wrong turns which would cost time. One the best solution is identified, Step 6 should be obvious and so, I think you might be able to jump to Step 7.
3. Do you agree kids of any age can engage in real mathematical problem-solving? Why?
This is an interesting question from a special education perspective. My initial reaction was that based on my own daughter, I would say no, I don’t think all children can engage in real mathematical problem-solving. One could assume that her intellectual disability prevents her from having much understanding of math; however, now that I've taken this course, I have a different perspective of what math is. She is able to complete puzzles. She can sort shapes. She can count to 30, and she recognizes and can name numbers. She counts the bus steps as she descends, and she sings songs that have numbers in them. Based on that, I would have to say that I would change my mind and say yes, children of any age can engage in real mathematical problem-solving. Certainly a toddler can count their toys. Certainly an infant can figure out how many toes will fit into their mouth—even with limited spatial ability! I think “real” math problem-solving is dependent on the age and the mathematical “need”.