Week 13 Problem-solving (April 9 - 15)


This week's content theme is problem-solving. 

TASK

  1. This part of the task is about finding similarities in different things, and differences in similar things. Compare and contrast problems with:
    1. Exercises
    2. Puzzles
    3. Projects
  2. Sometimes you can't implement all aspects of problem-solving because of time or other constraints. Which aspects you absolutely must keep, no matter what, so the task is still problem-solving?
  3. Jerome Bruner famously said, "We begin with the hypothesis that any subject can be taught effectively in some intellectually honest form to any child at any stage of development." Do you agree kids of any age can engage in real mathematical problem-solving? Why?

Discusión de la Tarea


  • Amanda Graf   18 de abril de 2012 a las 20:47

    1. Exercises, puzzles and projects all have one major theme in common: they are all a legitimate approach to solving math problems. However, while they may all get you the same answer to a math problem, they are three VERY different approaches, in my opinion.

    When I hear the term math "exercises" I think of your basic, daily math problems that you independently solve. When you physically exercise, you focus in on a particular muscle in your body that you want to strengthen and you work it out a little to do just that, strengthen it. Same thing with math exercises. Maybe one day you want to strengthen your multiplication skills so you do 25 multiplication problems. You may show your work, you may not. Your simply exercising and strengthening a skill you already have, so showing your work may not be neccessary.

    Math "puzzles" are more in depth then just exercises. Math puzzles may demand things from you that you're not typically used to exercising (thus differentiating it from math exercises) or perhaps it expects you to utilize the combination of several of your math skills to come up with the answer. It's usually a problem with a longer process then simply a math exercise. Like when you're playing with an actual puzzle, you have to put all of the pieces together to get the result: a complete picture. Same thing with math puzzles, you're expected to put several math skills (or puzzle pieces) together to come up with the solution (or the completed puzzle with a picture). 

    Math "projects" are even more in depth then puzzles. When I think of a math project, I don't think of just solving a math problem, I think of looking into the problem, figuring out WHY the answer is whatever it is. For example, instead of simply saying that 4x3=12 (which would be just a math exercise), a math project based on the concept of 4x3=12 would expect the student to explain the process of how you come to the conclusion of 4x3=12.

    2. Problem solving is the process of looking at a problem and coming up with a solution to it. What process one uses to come up with the solution depends on the student and what kind of learner they are and if they understand the problem. Therefore, I think, that what can be cut out of a problem solving process really depends on the student. For example, looking at the problem of 10x3=?, if a student knows off the top of their head that 10x3=30 then, great, problem solved. They can cut out the actual process of coming to this conclusion. However, if a student needs to cross multiply to come up with the answer, then they can cut out the process of counting by 10's on their fingers. Or, maybe, a student needs something more visual and needs to draw 10 sets of 3 tally marks and then count up all the tally marks to understand. Therefore, they can cut out cross multiplying. Basically, what I'm trying to say is that unneccessary procedure when problem solving should definitely be cut out if there are time restraints, however, what exactly that "unneccessary procedure" is really depends on the student.

    3. Yes, I believe that kids of any age can be engaged in mathematical problem solving because I believe that mathematical problem solving is exercised by humans from day one, even before they are aware of what "math" really is. For example, I work at a preschool and I am often in the infant classroom. For snack I may give them cheerios and David constantly throws most of his cheerios on the floor the minute I put them on the tray of his high chair by waving his hands back and forth through the pile of cereal. To prevent him from doing this, I've started only putting three cheerios at a time on his tray. He sees that he has less and he doesn't want to throw them on the floor because he thinks that's all he's going to get and if he throws any on the floor he won't have any to eat. Of course, once he eats the three cheerios I give him three more until he is done eating, but he doesn't realize this. What he does realize, though, is that I've started giving him less cheerios then normal so he therefore doesn't throw them on the floor. This simple realization of what is less and what is more is math right there, a basic form of it, but it's his math skills developing none the less.

  • Keisha   17 de abril de 2012 a las 13:03

    1) A definition I found for problem solving is: the thought process involved in solving a problem.

    Problem solving and exercises seem to be very similar to me. Math exercises usually require you to solve a problem of some sort. It's a way to practice what you've learned in class and perfect it.

    In order to complete puzzles you to need to use problem solving skills. Both puzzles and problem solving require you to think logically and outside the box. To me puzzles are basically problem solving but are modified to be more exciting and engaging for students.

    At first it was confusing for me to find similarities or differences in projects and problem solving because projects are like the same thing as exercises. But then I thought about it and projects give you the opportunity to be creative. You can make it your own by coming up with your own problem to solve.

    2) Aspects you must keep in order to make sure problem solving are still problems solving is to first identify the problem. It's not problem solving unless you're actually finding the problem. After finding the problem you have to come up with ways to eliminate the problem which will bring you to the solution.

    3) Yes I agree that kids at any age can be engaged in problem solving. Kids are very smart and we shouldn’t underestimate them. I have twin cousins that are 2 years old and the toys that they have required them to problem solve. They have blocks that you have to stack on top of each other but can only be done by matching the shapes. Just by seeing how their mother put the blocks together they quickly understood and figured how to do it themselves. They also problem solve when one of them notices that the other has more lollipops. We don't realize how early children actually start learning how to count or problem solve.

  • Carolyn   16 de abril de 2012 a las 17:46

     

    Exercises-To me exercises can be just about anything to practice a particular skill. I think generallly you have a problem is you have to exercise it. If students are just learning multiplication then there problem is to figure out how to multiply and then to exercise it through whatever problems the teacher assigns.

    Puzzles-Puzzles are a problem! I love them but they are very challenging and I see them as a problem that can have more than one answer. I was actually in a gifted classroom the other day where they had different stations with diferent puzzles. They were all challenging in their own way but some had multiple solutions while the others only had one. I think this relates to problems by showing that some have multiple answer and one solution is not always right.

    Projects-Projects I think are similar to puzzles in that there can be multiple answers or just one depending on the project. Projects use a lot of brainstorming, planning and time before arriving at a solution.

    2. Problem solving obviously must have a problem, it does not always have to be clear but there has to be one to be identified, the most common first step in problem solving. I really think that is all you need because maybe the solution to the problem is that there is no solution, whether it be because there is not enough information or simply cannot be solved.

    3. "Real" mathematical problem solving is what is tripping me up. What is that? I think all kids are capable of solving problems with some guidance. And I don't believe it matters what subject it is in because I think the approach is more important. Even though the content is different the approach is normally the same in problem solving, its the methods that get a little tricky. For example if a student can identify the problem and a way to figure it out but does not have the method such as, division skills, to figure it out, then obviously they won't be able to do it. 

  • Laura Haeberle   16 de abril de 2012 a las 10:43

    1. I think of problems as issues with many potential solutions, depending on one's creativity. Often, they're generated with a goal for the solution. For example, someone could have a problem of having a cluttered room. They could solve this in many ways, but the ultimate goal of the solution is to have a cleaner room or process of cleaning.

    Exercises stem of a proposed problem, but there is typically only one solution. Moreover, the solution needs to be reached quickly, as there can be multiple exercises or drills. Whereas, with problems, you normally have time to consider all possible options.

    Puzzles seem like a more tricky version of problems. With puzzles, there is still one answer that is expected, though you will need to think creatively to obtain it. You are forced to use the mental process of problem solving, however, there is a feeling that one person already knows the answer and you simply need to think as they do. 

    Projects rely on creativity similar to problems. Projects can actually be the solution to problems sometimes. In the cleaning example, doing a weekly cleaning project could fix the issue of clutter. With projects, there is no real "answer." It's more of a representation of what you know, what you've researched, what you enjoy, etc. It's more about the expression of a topic more than solving for an answer.

    2. To figure out the most key aspects of problem solving, it's helpful to look at what problems, puzzles, exercises, and projects all require. The first step in problem solving is to identify the problem, which is needed in all of these. For projects, you could also think of this as understanding the goal. To me, this means you have to either know the question you're being asked or simply understand the assignment. You can't provide an "answer" any other way. I think you also need to choose a solution for each, and stress that process. In your mind, there may be multiple options. Perhaps there's multiple choice, or you're considering different clutter solutions, or you have too many interests to select one project topic. In the end, you need to evaluate the options and select the best one, based on your goal or the question itself. Lastly, you need to talk about evaluating the process at the end. You can't become better at problem solving if you don't reflect! Even when doing exercises, maybe a multiple choice math quiz, I would always check over my answers at the end. Your thought process may have changed, and reflecting helps you see which answers worked well and which didn't. This helps you learn for next time.

    3. I definitely agree that children frequently engage in math problem solving. It begins at a very young age as well. For example, in an infant classroom, the class may have five children but only four toys. The children must figure out a way to divide the toys, figuring out which can be used by more than one child at once. Of course, this could end in children fighting over toys, but it all comes down to simple math of one-to-one correspondance. As children start learning math in school, they're forced to use their knowledge of math to scaffold to more advanced material. They are synthesizing information, problem solving with new material. I think problem solving is so critical in a classroom and needs to be taught and experienced. It relates to all areas of life and helps the children make real world decisions.

  • Carolyn Lesser   14 de abril de 2012 a las 11:36

     

    1. Math Projects

    • Long
    • Bring many ideas together
    • Fun
    • Something you are interested in
    • Research
    • Learning
    • Showcase results
    • Work together

    Math Exercises

    • Shorter problems
    • Specific problems for the day
    • Practice
    • learning

    Math Puzzles

    • fun
    • challenging
    • work together
    • competition

    2. In problem solving you have to keep certain aspects in order to actually be able to do the problem. First you have to be able to look at the problem and have some kind of idea of what the problem is. If you can’t figure that you it will be difficult to find a solution. The next thing you must keep is figure out a way to solve the problem. There are many ways to solve certain problems but you just need to figure out one that works for you. The next necessary step for problem solving is combining everything you know about the problem and making plan and actually doing the problem.

    3. I believe that anyone can problem-solve. Maybe not to the extent of some but people are problem-solving constantly throughout their life and even their day. Numbers and shapes surround us and with numbers and shapes there is problem solving. My three year old nephew is problem solving when he is learning to count and putting his blocks together so that fit in the right place. He has to think about and plan his blocks out. If he gets half through and they all fall over he has to go back through and see what he did wrong so that he can do it right the next time. I have seen him do this a few times, getting it correct the next time around. I think the youngest can do some kind of problem-solving which is pretty amazing. I never really thought about until now but we can do some complex things even a small child.  

  • Kathy Cianciola   14 de abril de 2012 a las 10:16

    1. Exercizes, Puzzles and Projects

    I think of exercises as practice and review of some method that's already been studied.  Puzzles involve being given a fixed problem to solve, and figuring out solutions to that problem.  Projects, like puzzles, involve figuring out solutions, but I think of projects as being more creative and open-ended than puzzles. A project is your own from beginning to end and you decide where it's going to go step by step.

    2. Problem Solving, and Which Aspects To Keep

    In problem solving you must identify or figure out what the problem is before you can solve it. Try to think (brain-storm) about possible solutions, choose a solution, and plan a method of attack, and finally work out the solution to the problem.  I really don't see how you could skip any of these steps if you want to effectivly solve the problem. I guess if I had to pick the most important steps they would be, identifying the problem, and working out the solution.

    3. Do you agree kids of any age can engage in real mathematical problem-solving? Why

    Yes.  I think that children begin solving math problems at a very early age.  It probably begins with fingers and toes, as Sandy mentioned.  Then at around preschool-age, as children become more focused on themselves, before they are even able to count, they will begin to identify, "Tommy has more match-box cars than I have."  The young child will easily recognize that a grouping of cars is more than just a single car, or a pair of cars (especially if he/she is the child with less).  Isn't it interesting that even at that age children are applying the mathematical concept to what is important to them personally. They are "making math their own."

  • SandyG   12 de abril de 2012 a las 17:13

    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

    (http://www.pitt.edu/~groups/probsolv.html)

    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”. 

  • Maria Droujkova   11 de abril de 2012 a las 09:40

    There was a question about this week's task on problem-solving. I will elaborate. 

    The goal of the first step is to compare problems with other closely related entities. This will require the analysis of different aspects, say, of problems and exercises - so you can compare. What are similarities between mathematical problems and exercises? What are differences? My hope is that in the process of this analysis, your understanding of different aspects of problem-solving will grow more refined. This usually happens in comparison tasks. A good place to start is to figure out how to define problems, exercises, puzzles and projects in mathematics, maybe by looking at some media on the subject.

    As a result of Step 1, you will probably have a definition of problem-solving that includes many aspects, such as "not knowing ahead of time what methods you will use" or "heuristics." While Step 1 is more about analysis, Step 2 is more about synthesis. Now that you have listed some aspects of problem-solving, which ones you consider most important, and why?

    There are a lot of materials available on mathematical problem-solving, from the old classic "How to solve it?" by Polya to the Common Core writings that came out in the last year. Wikipedia does not have much on mathematical problem-solving, but the article about puzzles is a good start. Let me know if you would like more references or materials on the subject.