Category: Teaching mathematics

Posted in Algebra, Geometry, Misconceptions, Teaching mathematics

Mistakes and Misconceptions in Mathematics

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Misconceptions are very different from the mistakes students make. Mistakes are not consciously made. Misconceptions are. Mistakes are usually one-off, while misconceptions, the gods forbid, could be for keeps. Mistakes are made by a few, misconceptions are made by many and, repeatedly.

Students can figure out their mistakes by themselves because mistakes are usually due to carelessness. They cannot do the same for misconceptions. Misconceptions are committed because students think they are correct.

How can misconceptions be addressed? By undressing them, carefully exposing them until the students see it. It cannot be corrected by simply marking them x because misconceptions are usually made with full knowledge.

The following are common misconceptions in arithmetic, algebra and geometry:

1. Did we not learn that multiplication is repeated addition? So, -3 x -4 = -3 + -3 + – 3 +-3 = -12?

2. Didn’t we learn that to multiply fractions we simply multiply numerators and we do the same with the denominators? Didn’t the teacher say multiplication is simply repeated addition so {\frac{3}{5}}+{\frac{2}{3}}={\frac{5}{8}}?

3. Did not the teacher say x stands for a number? So in 3x – 5, if x is 5, the value of the expression is 35 – 5 = 30?

4. Did not the teacher/book say to always keep the numbers and decimal points aligned? So if Lucy is 0.9 meters and her friend Martha is 0.2 taller, Martha must be 0.11 meters in height?

5. Did we not learn that the more people there are to share a cake the smaller their portion? So {\frac{10}{16}}<{\frac{4}{5}}<{\frac{3}{4}}<{\frac{1}{2}}?

6. Did we not learn that by the distributive law 2(a+b) = 2a + 2b? So, (a+b)^2=a^2+b^2?

7. Did not the teacher show us that (x-3)(x+1) = 0 implies that (x-3) = 0 and (x+1) = 0 so x = 3 and x = -1? Hence in (x-3)(x+4) = 8, (x-3) = 8 and x +4 = 8 so x = 11 and x = 4?

8. Did we not learn that the greater the opening of an angle, the bigger it is? So, angle A is less than angle B in the figure below.

9. Did we not learn that you if you cut from something, you make it smaller? Hence in the diagram below, the perimeter of the polygon in Figure 2 is less than the perimeter of original polygon?

10. Isn’t it that the base is the one lying on the ground?

There’s nothing a teacher should worry about mistakes. There’s everything to worry about misconceptions. Good teaching practice exposes misconceptions, not hide them.

You might want to check out this book:

Posted in Teaching mathematics

To understand mathematics is to make connection

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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:

  1. Experts notice features and meaningful patterns of information that are not noticed by novices.
  2. Experts have acquired a great deal of content knowledge that is organised in ways that reflect a deep understanding of their subject matter.
  3. 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.
  4. 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).

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Credits

The image is from the post Good Ideas are Networks in  Slow Muse by Deborah Barlow.

A Framework of Growth Points in Students’ Developing Understanding of Function – PhD Thesis by the Author