Posted in Curriculum Reform

The nature of math vs the nature of school math

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The mathematics experienced by students is so much different from the real nature
of math. What a tragedy!

What is the nature of mathematics?
  1. Mathematics is human. It is part of and fits into human culture. It is NOT an abstract, timeless, tensely, objective reality…
  2. Mathematical knowledge is fallible. As in science, mathematics can advance by making mistakes and then correcting them…
  3. There are different versions of proof or rigor. Standards of rigor can vary depending on time, place, and other things. Think of the computer-assisted proof of four color theorem in 1977…
  4. Empirical evidence, numerical experimentation and probabilistic proof all can help us decide what to believe in mathematics…
  5. Mathematical objects are a special variety of a social-cultural-historical object …They are shared ideas like Moby Dick in literature and the Immaculate Conception in religion.

The above description of the nature of mathematics is by Reuben Hersh,  from his article “Fresh Breezes in the Philosophy of Mathematics published in American Mathematical Monthly Aug-Sept, 1995 issue. He is also the author of the now classic What Is Mathematics, Really?.

What is the ‘nature’ of school mathematics?

The following is a 2002 critic of the US k-12 mathematics by Paul Lockhart in A Mathematician’s Lament.  It’s also true in my part of the globe.

The Standard of K-12 mathematics according to Lockhart:

LOWER SCHOOL MATH. The indoctrination begins. Students learn that mathematics is not something you do, but something that is done to you. Emphasis is placed on sitting still, filling out worksheets, and following directions. Children are expected to master a complex set of algorithms for manipulating Hindi symbols, unrelated to any real desire or curiosity on their part, and regarded only a few centuries ago as too difficult for the average adult. Multiplication tables are stressed, as are parents, teachers, and the kids themselves.

MIDDLE SCHOOL MATH. Students are taught to view mathematics as a set of procedures, akin to religious rites, which are eternal and set in stone. The holy tablets, or “Math Books,” are handed out, and the students learn to address the church elders as “they” (as in “What do they want here? Do they want me to divide?”) Contrived and artificial “word problems” will be introduced in order to make the mindless drudgery of arithmetic seem enjoyable by comparison.

ALGEBRA I. So as not to waste valuable time thinking about numbers and their patterns, this course instead focuses on symbols and rules for their manipulation…. The insistence that all numbers and expressions be put into various standard forms will provide additional confusion as to the meaning of identity and equality. Students must also memorize the quadratic formula for some reason.

GEOMETRY. Isolated from the rest of the curriculum, this course will raise the hopes of students who wish to engage in meaningful mathematical activity, and then dash them. Clumsy and distracting notation will be introduced, and no pains will be spared to make the simple seem complicated. This goal of this course is to eradicate any last remaining vestiges of natural mathematical intuition, in preparation for Algebra II.

ALGEBRA II. The subject of this course is the unmotivated and inappropriate use of coordinate geometry. Conic sections are introduced in a coordinate framework so as to avoid the aesthetic simplicity of cones and their sections. Students will learn to rewrite quadratic forms in a variety of standard formats for no reason whatsoever. Exponential and logarithmic functions are also introduced in Algebra II, despite not being algebraic objects, simply because they have to be stuck in somewhere, apparently.

TRIGONOMETRY. Two weeks of content are stretched to semester length by masturbatory definitional runarounds. Truly interesting and beautiful phenomena, such as the way the sides of a triangle depend on its angles, will be given the same emphasis as irrelevant abbreviations and obsolete notational conventions, in order to prevent students from forming any clear idea as to what the subject is about. Students will learn such mnemonic devices as “SohCahToa”…

Posted in Elementary School Math, Number Sense

What are fractions and what does it mean to understand them?

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Negative numbers, the irrationals, and imaginary numbers are not that easy to make sense of for many students. But this is something understandable. One only needs to check-out the histories of these numbers. The mathematicians themselves took a long time to accept and make sense of them. But fractions? How can something so natural, useful, and so much a part of our everyday life be so difficult? Didn’t we learn what’s half  before we even learn to count to 10? I’m sure this was true even with our brother cavemen. So how come the sight of a fraction enough to scare the wits out of many of our pupils and yes, adults, too?

Fractions are used to represent seemingly unrelated mathematical concepts and this is what makes these numbers not easy to make sense of and work with. In mathematics, fractions are used to represent a:

  1. Part-whole relationship – the fraction 2/3 represents a part of a whole, two parts of three equal parts;
  2. Quotient – 2/3 means 2 divided by 3;
  3. Ratio – as in two parts to three parts; and
  4. Measure – as in measure of position, e.g, 2/3 represents the position of a point on a number line.

Of these four, it is the part-whole relationship that dominates textbooks. For many this conception is what they all know about fractions. While it is also the easiest of the four to make sense of, students requires series of learning activities to fully understand part-whole relationship . Crucial to this notion is the ability to partition a continuous quantity or a set of discrete objects into equal sized parts. Below are sample tasks to teach/assess this understanding. They call for visualizing skills.

Of course understanding fractions involve more than just being able to use them in representing quantities in different contexts. There’s the notion of fraction equivalence, which is one of the most important mathematical ideas in the primary school mathematics and a major difficulty. This difficulty is ascribed to the multiplicative nature of this concept. There’s the notion of comparison of fraction which includes finding the order relation between two fractions. And if your students are having a hard time on comparing fractions you can check their understanding of equivalence of two fractions. It could be the culprit. And let’s not forget the operations on fractions. An understanding of the procedure for adding, subtracting, multiplying, and dividing of fractions depends on students’ depth of understanding of the different ways fractions are conceived, on the way fractions are used to represent quantities, on the idea of equivalent fractions, and on order relation between fractions, and  many others such as the meaning of the operations themselves.

A study has been conducted categorizing students levels of conception of fractions, at least up to addition operation. Just click on the link to read the summary.

 

Posted in Algebra, Math blogs

Math Teachers at Play Blog Carnival

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Math  Teachers At Play- Blog Carnival #49 of Let’s Play Math is now live in TeachBesideMe. Go check-out the fabulous submissions and of course the photos and images.

Mathematics for Teaching will be hosting the 50th edition of MTaP. You may use the  Math Teachers at Play Blog Carnival — Submission Form to submit your posts or email it to mathforteaching@gmail.com. MTAP 50 will go live on 2nd week of May.  Looking forward to your great articles on teaching and learning mathematics . Thank you.
Posted in Algebra, GeoGebra worksheets, Math Lessons

Teaching maximum area problem with GeoGebra

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Remember that classic maximum area problem? Here’s a version of it: Pam wishes to fence off a rectangular vegetable garden in her backyard. She has 18 meters of  fencing wire which she will use to fence three sides of the garden with the existing fence forming the fourth side. Calculate the maximum area she can enclose.

This problem is usually given as an application problem and is solved algebraically. For example if x is one of the two equal sides to be fenced then the area is the function f(x) = x (18-2x). The maximum area can be found by graphing or by inspection. If students have done a bit of calculus already then they can use the first derivate to solve the problem. But with free technology such as GeoGebra, there should be no excuse not to make the teaching of this topic less abstract especially for Year 9 or 10 students. It need not be at the end of the chapter on quadratic but as an introductory lesson for quadratics. Here’s a GeoGebra applet I made which can be used to teach this topic more visually and conceptually. Below is the image of the applet. I did not embed the applet here because it takes a while to load. Click maximum_area_problem worksheet to explore.

Here’s my suggested teaching approach using this applet. Students need to be given a bit of time exploring it before asking them the following questions:

  1. Pam wishes to fence off a rectangular vegetable garden in her backyard. She found fencing wires stored in their garage which she will use to fence three sides of the garden with the existing fence forming the fourth side. How long is Pam’s fencing wire? What are some of the sizes of gardens Pam can have with the fencing wires?
  2. If you were Pam, what garden size will you choose? Why?
  3. What do the coordinates of P represent? How about the path of P, what information can we get from it?
  4. As the length of BD changes so does the length of the other two sides. What equation will describe the relationship between the length of BD and EF? between BD and DE? between BD and area BDEF.
  5. What equation of function will run through the path of P? Type it in the input bar to check.
  6. What does the tip of the graph tell you about the area of the garden?

Feel free to use the comments sections for other questions and suggestions for teaching this topic. How to teach the derivative function without really trying is a good sequel to this lesson. More lessons in Math Lessons in Mathematics for Teaching.