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 showing their full reasoning with words. Then demostrate how the words and steps in their solutions can be cast in algebraic form.

(Edited for clarity)

 

[Gavran]

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.

People accept mathematical concepts much better if they can see their practical applications and relevance.

 

[PeroK]

People accept mathematical concepts much better if they can see their practical applications and relevance.

Some people. I don't care.

Apart from mathematics, chess is a good example of something where people become hooked without any need for applications or relevance. Music as well.

What are the practical applications of Tchaikovsky's 6th Symphony?

 

[kuruman]

What are the practical applications of Tchaikovsky's 6th Symphony?

Out of curiosity, I asked the exact same question on the web. I got an unexpectedly interesting summary with details gleaned from various sources across the web. There is more there than I imagined. Alas, I cannot share any information because I have been warned by the mentors not to quote "unacceptable references or topics" and I do not wish to be shunned as a repeat offender. Nevertheless, my unfortunate experience should not prevent you from exploring your question in the privacy of your own home.

 

[Mark44]

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?

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".

Another way to say what @phyzguy said, more-or-less, is that if you apply certain operations to both sides of an equation, you get a new equation that is equivalent to the one you started with.

By "certain operations" I mean operations that are one-to-one, such as adding or subtracting the same number to/from each side, multiplying both sides by the same number, and a few other operations. Operations that are excluded include squaring both sides and others.

By "equivalent" I mean two equations that have exactly the same solutions. As an example of why squaring both sides is excluded, consider the simple equation $x = 4$. Obviously, 4 is the only value of x that makes this equation a true statement. If we square both sides, we get $x^2 = 16$. In this latter equation both 4 and -4 are values of x that make the equation a true statement.

 

[kuruman]

Another way to say what @phyzguy said, more-or-less, is that if you apply certain operations to both sides of an equation, you get a new equation that is equivalent to the one you started with.

All that is true, but it does not address OP question which was whether the rationale behind the rules for algebraic manipulations needs to be explained right from the start. At that point all one needs to show is that whatever rule one applies to find a value for $x$ can be verified by going back and inserting that value in the original equation.

I view these rules are just algorithms that expedite the work when doing algebra. IMO, there is no need for a "deeper" understanding.

Let me illustrate why with two examples drawn from with my schooling in arithmetic.
(a) I remember my third grade teacher asking us to memorize the Pythagorean multiplication table so that "when you buy four pieces of candy at the store, you don't get cheated." I listened to my teacher and memorized things like $9\times 7=63.$ Never in my life did I question this result and added nine sevens longhand to verify the "deeper meaning" that it is sixty three and I don't plan on doing it now that I think about it.
(b) In sixth grade (pre-calculator days), I learned how to find square roots of (non perfect square) four-digit numbers to any decimal point accuracy longhand. I didn't know why it worked, but I was satisfied that it did because I always checked my work by squaring the number I got. That was then. Now, I don't even remember how to use the algorithm. Even if I did, I feel no desire to figure out how it works because I see absolutely no reason to do so.

In short, I think that it is more important for algebra beginners to ensure that the rules are applied correctly and that each step follows from what has already been said than to understand the reasoning behind these rules. It's a drill. Soldiers who need to learn how to walk in unison don't need to understand that it's static friction that propels them forward.

 

[DaveC426913]

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?

I'm a visual learner. I struggled with high school concepts of math such as algebra and calculus untiI I could envision them.

It took me a while to intuit a formula such as y=x²+2x+1. It's easier when you look at distance/speed and acceleration graphs.

The ² bends the line, the 2x tilts the line and the +1 raises the line.

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

That's what homework is for. The repetition instills a cause-effect relationship.

 

[symbolipoint]

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

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

When solving a basic equation like:

2x + 6 = 14

Basic properties of numbers, which are clearly taught during the first few weeks of "Elementary Algebra". They are both taught, AND practiced.

 

[symbolipoint]

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

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

Truthfully those two need to be learned together; mostly.

 

[symbolipoint]

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.

What you say does not work for everybody. None of that took shape in my intellect UNTIL Introductory Algebra in high school.

 

[symbolipoint]

Some people. I don't care.

Apart from mathematics, chess is a good example of something where people become hooked without any need for applications or relevance. Music as well.

What are the practical applications of Tchaikovsky's 6th Symphony?

Some of Mathematics honestly IS practical. Certain people, depending on their jobs or professions (or leisure activities) really do use mathematical reason and skills. INCLUDING Algebra (introductory level even, if not higher).

 

[symbolipoint]

Kuruman, #10; I know what you are talking about.

 

[symbolipoint]

By "certain operations" I mean operations that are one-to-one, such as adding or subtracting the same number to/from each side, multiplying both sides by the same number, and a few other operations. Operations that are excluded include squaring both sides and others.

What were these called? Axioms Of Real Numbers?

 

[symbolipoint]

I'm a visual learner. I struggled with high school concepts of math such as algebra and calculus untiI I could envision them.

It took me a while to intuit a formula such as y=x2+2x+1. It's easier when you look at distance/speed and acceleration graphs.

I do believe that I am also a visual learner. All mathematics instruction was less than fair-effective to me, UNTIL Introductory Algebra in high school. Tough to explain! Number Lines helped. Two-D graphing (for learning about lines in cartesian system) helped. The so-called Number Properties were mostly very easy to learn and just seemed simply logical.

 

[Mark44]

By "certain operations" I mean operations that are one-to-one, such as adding or subtracting the same number to/from each side, multiplying both sides by the same number, and a few other operations. Operations that are excluded include squaring both sides and others.

What were these called? Axioms Of Real Numbers?

No, these concepts are somewhat related to the real number axioms but go a fair amount beyond them. The usual operations in the axioms are addition and multiplication, as well as the concepts of additive and multiplicative inverses. What I was talking about is a bit more advanced than that and is more related to operations other than addition and multiplication, such as function inverses. It's also related to mathematical implications and proofs -- that would be my guess. One-to-one-ness would be presented in college algebra or precalculus, and would involve functions and whether they have inverses (that are themselves one-to-one functions).
One example of a function pair that goes beyond addition and multiplication is exponentiation and taking the log of a quantity, with both exponentiation and taking the log using the same base.

 

[symbolipoint]

somewhat related to the real number axioms but go a fair amount beyond them. The usual operations in the axioms are addition and multiplication, as well as the concepts of additive and multiplicative inverses. What I was talking about is a bit more advanced than that and is more related to operations other than addition and multiplication, such as function inverses.

I can only remember finding rules which I thought of as Properties of Real Numbers or Axioms of Real Number, in an old Intermediate Algebra textbook, with Lial & Miller, the authors. The properties (or Axioms) were limited to Addition, and Multiplication, and gave rules for Inverses and some properties like Commutative, Associative, and Distributive. Also additive and multiplicative inverses.

I would have to try to dig-up the book to be absolutely certain.

edit: authors reference adjustment

 

[Mark44]

I'm a visual learner. I struggled with high school concepts of math such as algebra and calculus untiI I could envision them.

"A picture is worth a thousand words."
Virtually all calculus textbooks include lots and lots of graphs. Algebra textbooks maybe not so much, as solving an equation such as $2x + 4 = 10$ can be solved in two steps.

It took me a while to intuit a formula such as y=x²+2x+1. It's easier when you look at distance/speed and acceleration graphs.

The ² bends the line, the 2x tilts the line and the +1 raises the line.

First off, this graph is a curve, not a line. Second, this is not a good way to understand what the shape of this graph looks like, aside from being mostly wrong -- the 2x term doesn't "bend" anything and the "+1" doesn't raise anything. See below for why I say this.

I taught a precalculus class in a community college (first two years) for almost 20 years. Our approach to problems such as this was to look at a related function in its simplest (i.e., untransformed) form and determine which transformations have been applied to the simple form that result in the graph we're investigating.

For this problem, $y = x^2 + 2x + 1$ can be written in a factored form as $y = (x + 1)^2$. This graph is identical to the graph of $y = x^2$ except for a shift to the left of every point by 1 unit. Although it seems counterintuitive to say that the "+ 1" in the factored form causes a shift to the left, not that in this parabola the vertex is at (-1, 0) rather than at the origin for the graph of $y = x^2$. Likewise the point (0, 1) is on this graph rather than the point (1, 1) on the graph of $y = x^2$.

Other transformations that we studied included reflections across either the vertical axis or horizontal axis (or both), as well as compressions and expansions, which change the basic shape of the graph.

Students were able to apply these concepts to a variety of "simple" functions such as polynomials, trig functions, and log and exponential functions.

 

[Herman Trivilino]

I listened to my teacher and memorized things like 9×7=63. Never in my life did I question this result and added nine sevens longhand to verify the "deeper meaning" that it is sixty three and I don't plan on doing it now that I think about it.

Hmmm... I wonder then how you ever found out that adding nine sevens is equivalent to multiplying 9 by 7.

Was it ever explained to you? I don't see how you could have ever discovered it on your own without doing the addition.

My point is that somehow you became aware of the deeper meaning, and it's become part of your worldview.

 

[DaveC426913]

First off, this graph is a curve, not a line.

Without the x² it would be a straight line. The exponent causes it to curve - in this case, into a parabola.

Second, this is not a good way to understand what the shape of this graph looks like, aside from being mostly wrong

You are quite confident about my learning experience, and about my understanding of what that graph looks like.

-- the 2x term doesn't "bend" anything

That's not what I wrote. 2x (two times) tilts the line (AKA changes its slope).

(I was oversimplifying by combining it all in one equation - mea culpa).

and the "+1" doesn't raise anything.

Yes it does.
y=2x will pass through the origin 0,0.
y=2x+1 will pass though (0,1).

I taught a precalculus class in a community college (first two years) for almost 20 years. Our approach

Apparently there is only one way for a student such as myself to learn, and I guess it's your way?

 

[Mark44]

Properties of Real Numbers or Axioms of Real Number, in an old Intermediate Algebra textbook

What you're describing are the field properties -- they wouldn't cover what I was talking about, as that was more about the properties of functions as well as steps that are reversible/not reversible in implications. These concepts probably wouldn't be covered in an intermediate algebra book.

 

[Mark44]

Without the x² it would be a straight line. The exponent causes it to curve - in this case, into a parabola.

But the equation does include the squared term, so the equation does not represent a straight line. It's not helpful to think of the graph in terms of a line.

You are quite confident about my learning experience, and about my understanding of what that graph looks like.

Your understanding leaves something to be desired. On the contrary, I'm not that confident about your learning experience, for the reasons I laid out earlier and reiterate below.

That's not what I wrote. 2x (two times) tilts the line (AKA changes its slope).

Apologies. I confused what you wrote: namely, the bending vs. tilting. The 2x term in the equation $y = x^2 + 2x + 1$ does not "tilt" anything. Again, thinking about this equation in terms of lines does not help your understanding. Granted, the difference between $y = x$ and $y = 2x$ is that the latter has a greater slope, but we are not dealing with lines in the quadratic equation under discussion.

(I was oversimplifying by combining it all in one equation - mea culpa).

Oversimplifying, indeed.

Yes it does.

(The above is in reference to you saying that the "+1" term raises the graph.)

y=2x will pass through the origin 0,0.
y=2x+1 will pass though (0,1).

True enough for this example, but that's not the equation in question. The constant term in $y = x^2 + 2x + 1$ does NOT raise the graph, The most reasonable interpretation of the difference between the graphs of $y = x^2$ and $y = x^2 + 2xs + 1 = (x + 1)^2$ is that every point satisfying the latter equation (shown in unfactored and factored form) is shifted one unit left relative to the graph of $y = x^2$.

Apparently there is only one way for a student such as myself to learn, and I guess it's your way?

I never said there's only one way, but there are multiple ways that lead to erroneous understanding. As an exercise, see if your method works in a slightly more complicated polynomial $y = x^3 + 3x^2 + 3x + 1$.