Assignment, due Monday, 3/7/22: This week’s reading is on a crucially important (though often ignored) subject, the use of symbols in mathematics. Read the 5-page excerpt from Wu’s text and respond to the questions below by leaving a comment on this post.
- What do you think Wu means by “the basic etiquette of symbols”? Give one example.
- Choose ONE of the boldface words or phrases in the section. State the boldface word/phrase and the page number on which it appears. In your own words, explain the meaning of the word or phrase. Each student must choose a different word/phrase.
- Extra Credit. Respond to one of your fellow students’ comments. Do you agree? Disagree? Did their comments make you think or provoke additional questions? Reminder: Be respectful, be kind.
I think the basic etiquette of Wu Shuo’s symbol means that when we use an unknown symbol instead of a number, we should show what the symbol represents. For example, we have an example in class, Mrs.Bee is 38 years old and her son is 8 years old. The question is “Determine how many years from now Mrs. Bee will be three times as old as her son will be then.”
We write the formula: (38+n)=3(8+n),n=7.
The symbol n stands for n years from now Mrs. bee will be three times as old as her son will be then.
And after we solve the equation, n=7.
So, we know 7 years from now Mrs.Bee will be three times as old as her son will be then.
Variable and constant, Page301 Line7.
For example, I went to the supermarket to buy apples, which cost 4 dollars a pound.
I will spend 4 dollars to buy a pound of apples and 8 dollars to buy two pounds.
If I buy x pounds of apples, it will cost y dollars.
We can get the equation, y=4x.
x represents how many pounds I bought and y represents how much money I spent.
x and y are variables, and 4 is constant.
The distinction between variables and constants is an important one, and I like your example and your explanation. Wu also emphasizes that the word “variables” can put the wrong picture into students heads – the variable itself isn’t “in motion” or “constantly changing”, but instead is a letter that stands for one of many possible values, but only one value at a time. Great!
Wu defines “the basic etiquette of symbols” is identifying symbols with a context. For example, Wu describes the equation \begin{math} y= \sqrt{3x-7} \end{math} in four ways. one is For all real numbers x, we can find a real number y so that \begin{math} y= \sqrt{3x-7} \end{math}.
The phrase ‘Such a string of symbols has no meaning’ appears in the second paragraph on page 299. Presenting an equation without specifying symbols is hard to interpret. Identifying those symbols that Wu called the ‘basic etiquette in the use of symbols’ must be performed so that students are able to learn.
I agree with you that the symbols without explanation it not useful. All the equation is for use to solve the problem in the real world not to confuse people.
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It’s kind of amazing to me that my brain is so good at “assigning meaning” to things like \begin{math} y= \sqrt{3x-7} \end{math}, even when the variables are not further quantified — I think I must be using a combination of recognizing common usages and many years of making assumptions about what the variables mean. But without that experience, it is not nearly so clear. Great!
ps. I like your use of LaTeX for math symbols! To make it work (just as you have written looks good), you just need to add this shortcode at the beginning of your comment on a line by itself:
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I think you should be able to edit your comment and add the shortcode if you like…
I think the point that Wu is trying to emphasize is how we assign arbitrary symbols in lieu of numbers the student needs to first understand what that symbol represents. The erroneous part is that using symbols (or variables) in lieu of numbers to explain a mathematical idea (let’s say exponents) by telling students the n in x^n is an exponent where n can be any number is less likely to cause confusion since they now know that a number can be raised to any power.
On page 300, ‘has no meaning’ identifies how it is impossible to understand something as abstract as mathematics if we aren’t taught or have the pre-disposed knowledge of what that phrase is saying.
I used to teach younger kids how to read, and this isn’t any different from what Wu is saying (as reading like math is all about identification of well.. symbols). First, we learn the alphabet, then we learn the phonics (ie.. ea makes the ee sound) and use pictures to first represent words (a picture of a cat by the word cat) and then pictures that describe these sentences (“That cat sat on the hat” will have a cat sitting on a hat). These pictures of simple sentences help us understand how to read the sentence and what it means. In mathematics, it isn’t any different. If we don’t know how to ‘read’ (C1) as found on page 300, it is very difficult to understand what that ‘sentence’ is saying. We need to know what each symbol means and what it represents before we can go any further in understanding the topics we see in secondary mathematics
Not only do students need to know what the symbols are and what they represent they need to be able to know what to do with them. Take the word problem with Ms. Bee and her son the class knew what variables are and what they needed to represent. Yet the class still struggled with coming up with finding a solution our future students will face the same struggle and frustrations.
I find that what Wu is saying about symbols is largely what I have found in my experience of work with students. Oftentimes variables get introduced students develop procedural fluency but when you ask them what it means not as many student can answer. Students need to be able to understand why different symbols are being used and in what context. For example on page 300 the phrase “solving the linear equation 3x+7=5” this is a standard problem used in middle and high school. Knowing that the solving for x is actually finding all the values that when you plug them back in give you 5 back. The teacher is supposed to be able to help enable students to recognize and break down elements of the mathematical language.
I totally agree with you that students get confused with ‘x’. That’s why symbols need explanations for them to understand. Without it, more likely students will fail and hate math. As pre-service teachers, we should articulate symbols when we teach students.
By the “basic etiquette of symbols”, I think Wu means that it is vital for students to know how to label their variables. In other words, in order to approach your problem, you have to know what factors impact the use of a variable. Like for example in the reading Wu states, “the statement that mn = nm for all real numbers m and n, while in others, the symbols stand for the element in a set consisting of exactly one element”. This shows how the variables have a fixed value depending on the conditions. Without knowing this knowledge, it would be difficult to know which is the proper way to approach it.
On page 301, Wu uses the word “unknown” to describe a number. In this case it is a label placed in convenience in order to not waste time defining it. Since a number is unknown there is no point in acquiring relationships to the variable.
“In this case, it is a label placed inconvenience in order to not waste time defining it. Since a number is unknown there is no point in acquiring relationships to the variable.” YES
I agree we basically use symbols to save time and space
knowledge symbols in mathematics are crucial in order to represent something, however, Using symbols without knowing exactly what each variable really means is pointless. according to Mr. Wu, this is one of the “basic etiquettes in the use of symbols”. I totally agree with him because I also believe it’s more important to understand a mathematics expression than just solve the problem.
On page 300 He said “solving the linear equation 3x +7=5” where most the students can easily solve the problem without knowing why they are solving the problem and what those unknown values exactly represent. It’s crucial to know that X is independent and y is the dependent variable.
On page 299 He used the word “quantified,” which means each unknown variable must have an identity and representation. students who are able to describe all those unknown variables in his mathematical expression, most likely learning math in mature ways.