Understanding the Reasoning Behind Basic Algebra

[installshu]

TL;DR

I’ve been thinking about how beginners learn algebra and wanted to get some opinions.

Hi everyone,

I’ve been thinking about how beginners learn algebra and wanted to get some opinions.

When solving a basic equation like:

2x + 6 = 14

it’s easy to learn the steps and arrive at x = 4. But is it more important for a beginner to understand why we can subtract 6 from both sides and then divide by 2?

Or is it better to first become comfortable with the standard methods and develop the deeper understanding over time?

I’d be interested to hear how others approached algebra when they were learning it.

 

[PeroK]

Why not? It gives you an equation you can solve, $2x =$ some number.

In this case, you could also divide by 2 first.

Apart from that, what else do you propose?

 

[Herman Trivilino]

But is it more important for a beginner to understand why we can subtract 6 from both sides and then divide by 2?

Depends on the learner's goal. Some learners just want to know how to make answers, others want to make sense. Regardless, there's nothing difficult about learning that if two expressions are equal and you subtract $6$ from both sides they're still equal.

There's a deeper understanding that you didn't mention. The real reason that $x=4$ is the solution to $2x+6=14$ is that $4$ is the number you multiply by $2$ and then add $6$ to get $14$.

 

[erobz]

Regardless, there's nothing difficult about learning that if two expressions are equal and you subtract $6$ from both sides they're still equal.

It's not difficult, but I do recall "getting it"(I learned algebra mostly as an adult though).

I teach my kids by balance scale type problems. There is an intuitive understanding about physical balance at a pretty young age. You might balance shapes with them, and you can use other balance scales "like two triangles balance a square" to give them an intuitive understanding about substitution. I think getting this abstract idea cemented in before they really even know numbers all that well is beneficial.

 

[phyzguy]

The fact that "if two quantities are equal and you do the same thing to both of them then they are still equal" seems pretty elementary. I'm not sure I would search for a "deeper understanding".

 

[Herman Trivilino]

It's not difficult, but I do recall "getting it"(I learned algebra mostly as an adult though).

But the concept is taught in elementary school, long before algebra. If you have an equal number of apples in each of two baskets and you remove the same number of apples from each basket, then there remains an equal number in each basket.

This is an intuitive concept already understood by your students.

My advice is that you focus on the meaning of the solution, not just on the steps taken to arrive at that solution. See the final paragraph of Post #3.

 

[kuruman]

One should not forget that algebra does not pop out of a vacuum. It is preceded by arithmetic which develops the necessary reasoning behind the algebraic equations. I remember word problems like

Problem
Bill and Jim together have 12 apples. Jim has twice as many apples as Bill. How many apples does each boy have?

For this kind of problem, part of my grade was writing down, longhand, my reasoning that will take me to the solution. For example "I need to split number twelve in two unequal parts, one of which is twice the other. I know that eight and four give twelve. Therefore Jim has eight apples and Bill has four."

Then in beginner's algebra I learned to "let" symbols stand for number. So I learned to cast the reasoning above in terms of symbols:
Let $B$ = the number of apples that Bill has. Then Jim's number of apples is $2B$. Since the sum is 12,
$B + 2B = 12$
$3B = 12$
$B = 4$

After I learned about systems of equations, I learned an alternative way to cast this problem as a system of 2 equations and 2 unknowns even more compactly
$\begin{cases} J+B=12 \\ J=2B \end{cases}$

My point is that arithmetic uses words as part of the development of the solution exactly in order to hone the students' reasoning and build the logical infrastructure. The idea that an equality is preserved when both sides are multiplied by the same number is (or should be) developed here. If Bill and Jim go to the orchard and each picks the same number of apples as they already have, Jim will still have twice as many apples as Bill.

Algebra is the shorthand notation that cuts down on the number of words by using symbols, but the reasoning is already there.

To @installshu : If you are wondering how to teach algebra to beginners, my recommendation is to create a few simple word problems and ask the students to provide solutions using arithmetic and their full reasoning shown with words. Then show them how the words and steps in their solutions can be cast in algebraic form.