Tag: problem solving

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 Algebra

Doing problem solving

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

This is how mathematicians solve problems:

This is how a typical student solves problems:

Here’s my  time-line graph of a typical teacher solving a non-standard problem in the class. I asked these teachers how they solve challenging math problems  and how long it usually take them to find the solutions. It actually resembles that of the mathematicians. For some reason when they do the problems in their classes  they present them like it’s magic, effortless.

Posted in Assessment, High school mathematics

Conference on Assessing Learning

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The conference is open to high school mathematics and science teachers, department heads and coordinators, supervisors, tertiary and graduate students and lecturers, researchers, and curriculum developers in science and mathematics.

http://www.upd.edu.ph/~ismed/icsme2010/index.html

Plenary  Topics and Speakers

1. The Relationship between Classroom Tasks, Students’ Engagement, and Assessing Learning by Dr. Peter Sullivan

2. Assessment for Learning: Practice, Pupils and Preservice Teachers by Dr. Beverly Cooper

3. The Heart of Mathematics Teaching and Learning: Assessment and Problem Solving by Dr. Allan White

4. Assessing the Unassessable: Students’ and Teachers’ Understanding of Nature of Science by Dr. Fouad Abd Khalic

5. Lesson Study in Japan: How it Develops Critical Thinking Skills by Prof Takuya Baba

6. Classroom Assessment Affective and Cognitive Domains by Dr. Masami Isoda

7. Assessment cum Curriculum Innovations by Dr. Ma. Victoria Carpio-Bernido

8. Strategies for teaching Mathematics to classes with Diverse Interests and Achievement – Having Problems with Problem Solving? by Dr. Peter Sullivan

9. Assessing Learners’ Understandings of Nature of Science – The New Zealand Science Hub by Dr. Beverly Cooper.

Aside from parallel paper presentations and workshops, there will also be parallel case presentations by science and mathematics teachers involve in Collaborative Lesson Research and Development (CLRD) Project of UP NISMED. CLRD is the Philippine version of Lesson Study.

Clickhere for conference and registration details.

Posted in Algebra, Geogebra, Geometry, High school mathematics

Teaching with GeoGebra: Squares and Square Roots

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This post outlines a teaching sequence for introducing the concept of square roots in a GeoGebra environment. Of course you can do the same activity using grid papers, ruler and calculator. However, if the students have access to computers then I highly recommend that you use GeoGebra to do this. In my post GeoGebra and Mathematics, I argued that the more the students understand the mathematics behind GeoGebra, the more confident they could become in using this tool. The earlier the exposure to this environment, the better. The way to do this is to integrate the learning of the tool in learning mathematics.

The figure below is the result of the final activity in my proposed teaching sequence for teaching square roots of numbers and some surds or irrational numbers. The GeoGebra tool that the students is expected to learn is the tool for constructing general polygons and regular polygons (the one in the middle of the toolbar).

Squares and Square Roots

The teaching sequence is composed of four activities.

Activity 1 involves exploration of the two polygon tools: polygons and regular polygons. To draw a polygon using the polygon tool is the same as drawing polygons using a ruler. You draw two pints then you use the ruler/straight edge to draw a side. But with Geogebra you click the points to determine the corners of the polygon and Geogebra will draw the lines for you. In the algebra window you will see the length of the segment and the area of the polygon. Click here to explore.

GeoGebra shows further its intelligence and economy of steps in Activity 2 which involves drawing regular polygons. Using the regular polygon tool and then clicking two points in the drawing pad, GeoGebra will ask for the number of sides of the polygon. All the students need to do is to type the number of sides of their choice and presto they will have a regular polygon. Click here to explore.

Activity 3 is the main activity which involves solving the problem Draw a square which is double the area of another square. Click here to take you to the task.

Activity 4 consolidates ideas in Activity 3. Ask the students to click File then New to get a new window from the previous activity’s applet then ask them to draw the figure above – Squares and Square Roots.  You can also use the figure to compare geometrically the values of \sqrt{2} and 2 or  show that \sqrt{8} = 2\sqrt{2}. This activity can be extended to teach addition of radicals.

Like the rest of the activities I post here, the learning of mathematics, in this case the square roots of numbers, is in the context of solving a problem. The activities link number, algebra, geometry and technology. Click here for the sequel of this post.

This is the second in the series of posts about integrating the teaching of GeoGebra and  Mathematics in lower secondary school. The first post was about teaching the point tool and investigating coordinates of points in a Cartesian plane.

GeoGebra book:

Model-Centered Learning: Pathways to Mathematical Understanding Using GeoGebra