Announcement: GeoGebra seminar-workshop at UP NISMED

UP NISMED will run a two-day seminar-workshop in the use of GeoGebra in the teaching and learning of mathematics. Click image for details.

click this image for details


Doing problem solving

I like these graphs which show how a mathematician and a typical student solve a problem. The first graphs were from the post “Some research discoveries”. The last one is mine, on teachers time-line graph in doing problem solving.


Students: Read more of this post

What does it mean to understand mathematics?

I’m sharing in this post some of the “theories” underpinning the lessons, learning tasks, and math teaching ideas that I blog here. I thought, rather than talking about them in each post I make I will just refer the reader to this and to some other similar “theories” I will write about some other time. This is part of the literature review of my dissertation titled “A Framework of Growth Points in Students Developing Understanding of Function”. If you are a researcher and wants to see the references, you may download the dissertation here.

What is understanding?

Good ideas are networks

To understand is to make connection. The stronger the connection, the more powerful the knowledge. The more connected a concept is to other concepts, the easier it is to retrieve from memory and the wider the applications. In mathematics, this idea implies that one understands when he or she can make connections between ideas, facts or procedures (Hiebert & Wearne, 1991). In making connections, one not only links new mathematical knowledge to prior knowledge but also creates and integrates knowledge structures (Carpenter & Lehrer, 1999). Thus, the process of understanding is like building a network. Networks are built as new information is linked to existing networks or as new relationships are constructed (Hiebert & Carpenter, 1992). If one imagines a weblike structure, the mental representations constructed in the process of understanding can be thought of as nodes. These nodes are themselves “networks”. These smaller networks resemble what is called a schema in cognitive psychology which is a network of well-connected ideas, skills and strategies an individual uses in working with a particular task (Marshall, 1990).

The importance of the acquisition of cognitive structures (schemas) has been shown in studies of people who have developed expertise in areas such as mathematics, physics, chess, etc.  Read more of this post

Sorting number expressions

One of the ways to help students to make connections among concepts is to give them problem solving tasks that have many correct solutions or answers. Another way is to make sure that the solutions to the problems involve many previously learned concepts. This is what makes a piece of knowledge powerful. Most important of all, the tasks must give the groundwork for future and more complex concepts and problems the students will be learning. These kinds of task need not be difficult. And may I add before I give an example that equally important to the kind of learning tasks are the ways the teacher  facilitates or processes various students’ solutions during the discussion.

I would like to share the problem solving task I made to get the students have a feel of the existence negative numbers. We tried these tasks to a public school class of 50 Grade 6 pupils of average ability and it was perfect in the sense that I achieved my goals and the pupils enjoyed the lessons. This lesson was given after  the lesson on representing situations with numbers using the sorting task which I describe in my post on introducing positive and negative numbers.

Sorting is a simple skill when you already know the basis for sorting which is not case in the task presented here. Read more of this post


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