The phrase ‘problem solving’ has different meanings in mathematics education. Even its role in mathematics teaching and learning is not clear cut. Some view problem solving as an end in itself. Others see it as starting point for learning. Nevertheless, here are some of the types of problem solving tasks we would see in textbooks and teaching. They are arranged according to cognitive demand. Remember that learners will only consider something a problem if they experience some sort of barrier in a situation they are curious about. There’s a difference between an exercise and a problem. Continue reading “Types of Problem Solving Tasks”
Tag: problem solving
Making Sense of Power Function
The power function, 
With graphing utility, it is no longer as much fun to graph function. What has become more challenging is interpreting them. Here’s are a set of tasks you can ask your learners as review for function. You can give it as homework as well.
Consider the sets of power function in the diagrams below. Answer the following based on the diagram
- What are the coordinates of the points of intersection?
- Why do all the graphs intersect at those points?
- When is
?
- When is
?
- Why is it that as the degree or exponent of x that defines the function increases, the graph becomes flatter for the interval -1<x<1 and steeper for x > 1 or x >-1 ?
- Sketch the following in the graphs below:
,
- Why is it that power function with even exponents are in Quadrants I and II while power function with odd exponents are in Quadrants I and III? Why are they not in Quadrant IV?
What other questions can you pose based on the graphs above? Kindly use the comment section to suggest more questions. Thanks.
My other posts about function
Top 5 Best Math Education Sites and Blogs
Math teachers serious at improving their craft should find a wealth of resources in the following math education sites:
1. The Klein Project blog is a collection of vignettes written for secondary school mathematics teacher. The blog is unique in the sense that unlike other blogs for teachers, “the vignette is not about pedagogy, but inspires good teaching. It is not about curriculum, but it challenges teachers to reconsider what they teach. It is not a resource for classroom use, but source of inspiration upon which teachers can draw. The goal is to refresh and enrich teachers’ mathematical knowledge.” Each vignette starts with something with which the teacher is familiar and then move towards a greater understanding of the subject through a piece of interesting mathematics. It will ultimately illustrate a key principle of mathematics.
Here is a list of interesting vignettes from the blog:
- Public-key cryptography
- Benford’s law: learning to fraud or to detect frauds?
- What is the way of packing oranges? — Kepler’s conjecture on the packing of spheres
- Matrices and Digital Images
- Fair voting: the quest for gold
- How Google works: Markov chains and eigenvalues
2. The NCETM Portal contains excellent resources and support tools for math teachers continuing professional development.
My personal favorite in the portal is their collection of research study modules. I also highly recommend the Personal Learning section which includes the Professional Learning Framework, Self-evaluation Tools, as well as a Personal Learning Space for anyone registered with the NCETM, which is free. You can use these self-evaluation tools to check your and your understanding of the mathematics you are teaching and to explore ideas on how to develop your practice. Click How confident are you to teach mathematics for sample questions.
3. NRich – is a collection of resources for teachers, students, and parents. It is hosted by the University of Cambridge. I love this site because it promote learning mathematics through problem solving. The following description about their resources in the Teaching Guide page should be enough make you signup to them. It’s free!
At NRICH we believe that:
- Our activities can provoke mathematical thinking.
- Students can learn by exploring, noticing and discussing.
- This can lead to conjecturing, explaining, generalising, convincing and proof.
- In a classroom, the students’ role is to focus on the mathematics while the teacher focusses on the learners.
- The teacher should aim to do for students only what they cannot yet do for themselves.
4. Math Education Podcast is a collection of interviews with mathematics education researchers about their recent studies. This is hosted by Samuel Otten of the University of Missouri. For math education students and researchers, this site is for you.
5. The Math Forum @ Drexel – offers a wealth of problems and puzzles, online mentoring, research, team problem solving, and professional development. The site need no introduction. Their most popular service is Ask Dr. Math.
Levels of Problem Solving Skills
Here is one way of describing students levels of problem solving skills in mathematics. I call them levels of problem solving skills rather than process of reflective abstraction as described in the original paper. As math teachers it is important that we are aware of our students learning trajectory in problem solving so we can properly help them move into the next level.
Level 1 – Recognition
Students at this level have the ability to recognize characteristics of a previously solved problem in a new situation and believe that one can do again what one did before. Solvers operating at this level would not be able to anticipate sources of difficulty and would be surprised by complications that might occur as they attempted their solution. A student operating at this level would not be able to mentally run-through a solution method in order to confirm or reject its usefulness.
Level 2 – Re-presentation
Students at this level are able to run through a problem mentally and are able to anticipate potential sources of difficulty and promise. Solvers who operate at this level are more flexible in their thinking and are not only able to recognize similarities between problems, they are also able to notice the differences that might cause them difficulty if they tried to repeat a previously used method of solution. Such solvers could imagine using the methods and could even imagine some of the problems they might encounter but could not take the results as a given. At this level, the subject would be unable to think about potential methods of solution and the anticipated results of such activity.
Level 3 – Structural abstraction
Students at this level evaluates solution prospects based on mental run-throughs of potential methods as well as methods that have been used before. They are able to discern the characteristics that are necessary to solve the problem and are able to evaluate the merits of a solution method based on these characteristics. This level evidences considerable flexibility of thought.
Level 4 – Structural awareness
A solver operating at this level is able to anticipate the results of potential activity without having to complete a mental run-through of the solution activity. The problem structure created by the solver has become an object of reflection. The student is able to consider such structures as objects and is able to make judgments about them without resorting to physically or mentally representing methods of solution.
The levels of problem solving skills described above indicate that as solvers attain the higher levels they become increasingly flexible in their thinking. This framework is from the dissertation of Cifarelli but I read it from the paper The roles of reification and reflective abstraction in the development of abstract thought: Transitions from arithmetic to algebra by Tracy Goodson-Espy. Educational Studies in Mathematics 36: 219–245, 1998. © 1998 Kluwer Academic Publishers. Printed in the Netherlands.
You may also be interested on Levels of understanding of function in equation form based on my own research on understanding function.
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