Math Problems for K-12 is my new site that contains problems with solutions, explanations and common errors students commit in solving the problem. Sample students answers are shown with the corresponding marks. The posts are written for students but teachers, I’m sure, will also find them helpful. You can say this is the student version of Math for Teaching blog. The problems are categorized according to math area and year level. Here is a sample problem for middle school algebra. Click the image to go to the site.
And here’s another for high school mathematics. This post links equation solving and graphing functions, a key concept in algebra.
If you are a teacher and wishes to contribute a problem you have done with your class, feel free to share it here together with students’ solutions. It would be great if you can also show how you marked it together with comments. It is through assessment and marking that we communicate to students what we value. Email me at mathforteaching@gmail.com so I can invite you as author. Thank you.
I’ve written a number of posts the last couple of months which I published in other sites. They are problem solving tasks mostly in geometry using GeoGebra and a few on function, trigonometry and calculus. May I share 10 of them here. The first six are teaching resources which I posted in AgIMat, a site about science and math teaching resources. The last four problems are in Math Problems for K-12 to help students in their revision. Both sites are new ones. I hope you subscribe and promote them in your social networks. Thank you.
I’m sharing in this post some of the “theories” underpinning the lessons, learning tasks, and math teaching ideas that I blog here. 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.
Good ideas are networks
Understanding as making connection
To understand something is to connect it with other ideas. The stronger the connection, the more powerful the knowledge. The more connected an idea is to other ideas, the easier it is to retrieve from memory and the wider the applications. In mathematics, this 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.
Bransford, Brown & Cocking (1999) summarised the key principles of experts’ knowledge. Some of these are:
Experts notice features and meaningful patterns of information that are not noticed by novices.
Experts have acquired a great deal of content knowledge that is organised in ways that reflect a deep understanding of their subject matter.
Experts’ knowledge cannot be reduced to sets of isolated facts or propositions but, instead, reflect contexts of applicability: that is, the knowledge is “conditionalized” on a set of circumstances.
Experts are able to flexibly retrieve important aspects of their knowledge with little attentional effort (p. 19).
Von Glasersfeld (1987) described understanding as a “never-ending process of consistent organization” (p. 5). It is not an all or none phenomenon hence “it is more appropriate to think of understanding as emerging or developing rather than presuming that someone either does or does not understand a given topic, idea, or process” (Carpenter & Lehrer, 1999, p. 20).
Conceptual vs Procedural Knowledge
Related to the notion of understanding is knowledge of concepts and procedures. Conceptual knowledge in mathematics is “knowledge of those facts and properties of mathematics that are recognized as being related in some way” (Hiebert & Wearne, 1991, p. 200). It is “knowledge that is understood, … a knowledge that is rich in relationships. … A unit of conceptual knowledge is not stored as an isolated piece of information; it is conceptual knowledge only if it is a part of a network” (Hiebert & Carpenter, 1992, p. 78) [italics, mine]. This implies that the quality of conceptual knowledge is a function of the strength of the connection or relationships between the concepts involved.
Hiebert and Carpenter defined procedural knowledge as a sequence of actions and as such, the connection between concepts involved is minimal. An example of procedural knowledge is knowledge of standard computation algorithms, which consist of a step-by-step sequence of procedures of symbol manipulation. Hiebert and Carpenter argued that procedural knowledge could contribute to mathematical expertise only if it is related to conceptual knowledge: “From the expert’s point of view, procedures in mathematics always depend upon principles represented conceptually” (p. 78).
This is my third post on constructing test items based on TIMSS Assessment Framework. My first set of examples is about assessing understanding of zeros of polynomial function and the second post is about graphs of rational functions. Of course there are other frameworks that may be used for constructing test item like the Bloom’s Taxonomy. However, in my experience, Bloom’s is not very useful in mathematics, even its revised version. The best framework so far for mathematics is that of TIMMS’s which I summarized here.
Here are three examples of trigonometry test questions using the three different cognitive levels: knowing facts, procedures and concepts, applying, and reasoning.
Knowing
If cos2(3x-3) = 5 , what is the value of 1-sin2 (3x-3) equal to?
a. 0
b. 1
c. 3
d. 4
e. 5
Applying
Given the graphs of f and g, sketch the graph of f+g.
Reasoning
Which of the following functions will have the same set zeroes as function g, given that g(x) = sin kx and f(x) = k?
