The Value of N

I've been thinking a lot lately about the way math is visually presented to young students. Looking at equations and the meaning of equal signs, as written about in this recent post, I also began to wonder about the use of answer blanks, such as can be found in questions like this one:

5-3=___

There are reasons to write it like this, most obviously to give the student a place to put the "answer" or, better put, to complete the equation. However, following the equal sign with a blank might also be the cause of some confusion when students reach introductory algebra. They may have learned to associate the = sign, and/or the blank with "this is where the answer goes" rather than understand that this is an equation in which the value of each side must balance.

Consider the following way of representing the same question:

5-3= n

which could be followed with:

n = ___

or simply the words, "What does n equal?" or perhaps better still, "What is the value of n?".
(I chose the letter "n" because it can stand for the word "number", but any letter would work as well)

While there is still that problem of the equal sign followed by the blank, the way the first part is represented manages to help convey some information that is missing in the first example, such as "what are we looking for?" and "how can we represent the unknown number that will balance the equation?".

Maybe a picture would help get the idea across better (please forgive my crude drawings!):


Of course, if you have a balance and unit weights handy, you could always use those to help solve the problem.

I wonder if presenting simple arithmetic with a variable rather than a blank from the outset would help students better understand the concept of equation and equality better, and also predispose them to accept variable notation when it becomes more crucial in algebra.

If you choose to use this idea with your students, I'd be very grateful if you would let me know how it goes.

What is 4?

I'm currently reading The Glass Wall: Why Mathematics Can Seem Difficult by Frank Smith.

Early on, he describes how number is not the same as quantity. He uses 4 as an initial example, but then moves on to use a large number, somewhere over 7 hundred million, to demonstrate that the number is valid whether you have an associated quantity of something that it represents or not. Number is number. However we develop number sense, he argues, it occurs separate from natural language which tends to be ambiguous where math, buy its nature is not ambiguous (at least not to those who understand it!).

So, what in fact is 4? How do we truly understand the concept of 4 (or any other number)? Smith argues that a number can only be put into context when it is compared with other numbers. He sees mathematics as a separate existence than the rest of the world.

I'm not quite sure if this rings true for me or not, but it is an interesting thought to explore.

This also made me wonder what certain young mathematically inclined students might think about it, which led to the following idea for math enrichment.

Lesson Plan Idea: What is 4?

Students are challenged to brainstorm how they would explain the concept of 4 to people who had no numeracy (aliens, young children, etc.). They then move together into groups and share their ideas. The group chooses several to share with the class. One person (either the teacher or another student) plays the role of the learner while the students attempt to explain the concept. The learner should do their best to avoid using any previous mathematical knowledge and base their "understanding" purely on the information given by the students.

Class discussion should include:

•  counting--number/object connection; meaning of each number name
•  quantity--did they use concrete items (manipulatives) to demonstrate their ideas, and how successful might this be in getting the idea of number across without ambiguity (were the shape of items, colour, function or other characteristics confused with sense of number)
•  use of geometry and/or other drawings or models
•  other ways of relating the concept
•  would their system of explanation work for very large numbers, fractions, decimals, negative integers, zero, etc., and if not, how could they adapt or change it so that it will
•  was this difficult, and if so, why do the students think it was

The activity is open-ended and intended to provide deeper insights into the complexity of seemingly simple mathematical concepts.

Other related lessons might include:

• working with different number systems
•  working with different bases 
• writing computer programs to solve very basic mathematical problems (using a machine-based language or something without built-in mathematical algorithms that the students can access)
• a study of the historical use of "zero" and what it really means in mathematics (hint: it has a more complex meaning than simply "nothing")

For more math activities, see the Lemonade Math Page