Tag: K-12 math

Posted in Math blogs

Math Problems for K-12 with solution

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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.

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”…