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From a pedagogical point of view, when is the "right" moment to introduce letters and variables to school children?

Let's say we taught arithmetic, the four operations, did computation exercises, or even "disguised" equations, when in place of a variable we use a box, and ask what numbers should we put in that box.

How should we introduce the practice of using letters, to make the transition from arithmetic to algebra?

The context: I'm a high school teacher, I do my job pretty well and have good results with my high school students, but recently I've been tutoring a fifth grader, and for the future it is possible that I would have to teach also in middle school (grades 5-8). I find it is kind of hard to work with this student I tutor; he is having a hard time doing/understanding even simple stuff (simple for me, of course), and I find that variables/letters "intimidate" him.

How should I introduce and motivate using letters?

I feel that even high schoolers are sometimes "intimidated" by letters. I once had a student that said, "teacher, I really loved mathematics in the past, but only when we worked with numbers. Since we were introduced to letters in math, I don't like it anymore."

What should we do to avoid this issue? i.e., How should we introduce letters/variables in order for children to understand them better and not hate algebra?

Benjamin Dickman
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amarius8312
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  • Relevant: http://matheducators.stackexchange.com/questions/4174/literature-on-development-of-algebraic-thinking/ – Dag Oskar Madsen Dec 13 '15 at 12:03
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    Can you give an example of an equation with letters that intimidates him? – Amy B Dec 13 '15 at 12:30
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    Experienced teachers and textbook writers introduce the use of numbers at an appropriate pace. Done gradually, it can be less intimidating. So (until you are an experienced teacher) I recommend following them. – Gerald Edgar Dec 13 '15 at 14:27
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    Arithmetic is concrete, but algebra is abstract and requires a higher level of intellectual development. That's why California's attempt to mandate 8th-grade algebra was a failure -- many kids just aren't developmentally ready at that age. Was this the approximate age of the kid who said "I don't like it anymore?" My guess is that it's probably a great idea to do simple examples like 2+3=4+___, followed by 2+3=4+x. But keep the complexity minimal until they're older. –  Dec 13 '15 at 15:20
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    See "How do I teach algebra?" – Jasper Dec 13 '15 at 18:34
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    As one data point, Common Core standards specify this for the 6th grade: http://www.corestandards.org/Math/Content/6/EE/B/6/ – Daniel R. Collins Dec 13 '15 at 19:11
  • I'd wager/argue that talking about "variables" versus "unknowns" would add enormously to the difficulty, by creating confusion. Indeed, the notion of "variable" is much trickier than "unknown"... So, e.g., the title of this question should probably replace "variable" with "unknown"? If not, then I am confused... – paul garrett Dec 26 '15 at 15:53
  • I suspect you are making the classic mistake of assuming something easy for you is easy for a child. In the US, x variable is first introduced in 8th (or perhaps 7th) grade as part of pre algebra. Instead of asking here, you really ought to know what the normal timing is in your region. But 5th sounds early. And sounds like you are bringing in variables when NOT NEEDED to tutor the kid in long division, decimals, etc. (normal 5th grade topics). – guest May 28 '18 at 10:16
  • I seem to have managed fine with variables eventually but not at first. When the task is to just deduce statements from assumptions about what properties an unknown real number x satisfies, I think there's no problem. Even if the statement has less information than the original which I believe I once did. However, when they're asked to answer the question of what all possible values of x are, that's an entirely different question which might confuse people because it seems so much the same. In this case, when ever you correctly deduce from one equation from another equation, you can sometimes – Timothy Mar 11 '20 at 02:08
  • gain solutions but you can never lose solutions which is why you have to see which of the solutions you end up with are solutions to the original equation. Also, when the question is what the set of all solutions is, you're really not at all assuming x is an unknown quantity and it's an entirely different problem and it's very possible that the original equation has no solutions. However, if you are given the first type of problem that actually assumes x is an unknown real number and the equation has no solutions, you can totally correctly deduce that x is a specific real number that is not – Timothy Mar 11 '20 at 02:12
  • actually a solution and the teacher shouldn't take away marks for it but could instead discount the poorly formulated question that they asked. Although I understood what the book meant, Calculus is even more confusing. It had to introduce 2 variables, x and t, in order to give the definition of a definite integral in general. I think that it totally confuses a lot of people using variables in an entirely different more complicated way than just simply giving problems that assume x is an unknown real number. I think 3 variables would be used just to state the associative law of real number – Timothy Mar 11 '20 at 02:19
  • addition. It may potentially be very confusing how in order to state that all real numbers have the property that all real numbers have another property related to that specific real number, we need a second variable. Maybe we could instead state the associative law a being the statement that all real numbers have a certain property where each real number has that property if all real numbers have another property and each real number has that property if all real numbers have another property. I think that for jobs in general including job of educating, we should have something like a – Timothy Mar 11 '20 at 02:23
  • competitive market but not quite. It's to the benefit of everyone for everyone to work with everyone else as equal partners and each person will benefit more the more they benefit others. People will be able to pick which people to work with. Employers will be able to pick which people to give a job to but also employees will have a legal right to walk out of a job any time. Also, employers will have a right to only hire people who have already paid for their own training to not get less serious people. The public decides which companies to give how high a reputation. The ones where all – Timothy Mar 11 '20 at 02:27
  • the employers and employees cannot have a total income in any individual month that exceeds a certain amount other than having portions of a certain type of cost such as rent or dental covered without being terminated from employment and all money that anyone collected to charge people for their own training gets donated into the federal reserve and they don't compete to be the best and only decide not to work with companies that already have a flaw in the way they work will be the most successful companies. As they have a maximum income policy, the public will trust them more and be more – Timothy Mar 11 '20 at 02:34
  • willing to engage with them and then each person's best interest will be to work together to solve problems and avoid the risk of catastrophic consequences from not doing jobs that need doing and organizing the planning really well. Everyone who is available for the labour market will have a basic income. That way, people won't later create problems and go to great lengths to be secretive and compete to be the best at something or go to great lengths to make sure they are needed in a job. Also, now that people can better afford to quit jobs, managers will have an incentive not to have abusive – Timothy Mar 11 '20 at 02:38
  • power almost as abusive as dictatorship like I feel like may be going on in some workplaces. Now as a result of having many separate companies each of which has its own way, a much better education system may eventually come. If the public doesn't want to suffer extreme consequences later of greed and corruption, they're going to have to push for nonprofit organizations that insist on working only with other nonprofit organizations and will not engage with for profit companies. – Timothy Mar 11 '20 at 02:42

8 Answers8

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You don't need 1-letter variable names to do algebra. Basically, as soon as you start giving story problems to children, you need to start teaching algebra techniques. You can teach them as "easier ways" to solve the problems, that help kids "keep track of things" and "avoid mistakes".

Most computer programming languages (including spreadsheets with named cells) allow using long variable names. Languages like Logo have been successfully taught to primary school children.

The key is to clearly distinguish the long variable names from other text that might represent operations. Parentheses can suffice.

After not very long, children will get tired of writing out complete variable names. At this point, you can show them how to:

  • Draw a picture if it helps to understand the variables.
  • Explicitly abbreviate their variable names.
  • At the end of the problem, write a short summary of the answer, and circle it in a cloud. For example:

At the beginning of the problem:
(# apples) = (number of apples)
…
At the end of the problem:
There are 7 apples.

And later on:

At the beginning of the problem:
(a) = (number of apples)
…
At the end of the problem:
There are 7 apples.

And later on:

At the beginning of the problem:
a = number of apples
…
At the end of the problem:
There are 7 apples.

Jasper
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I'm not an expert, but I understand the curriculum in Russia has gone back and forth on this point, with a lot of debate.

Generally, grades 1 to 4 are spent on the basic arithmetic operations, while grades 5 and 6 are mainly focused on: familiarizing students with the algebraic properties of numbers in a concrete way, systematizing and extending knowledge of fractions and decimal numbers, and studying signed numbers. Algebra starts in earnest in grade 7.

In the earlier grades, letters are used freely to state the general properties of numbers, to state formulas, and to write down simple equations. But they are not used systematically to manipulate expressions and equations and isolate an unknown as is done in algebra. In the 1970s, there was an effort to introduce algebra in grades 5 and 6 (called grades 4 and 5 at that time, since school then started at age 7). But it was found that by turning the process of solving arithmetic problems into a kind of algorithm, it took away a lot of the opportunity for kids to experiment and develop intuition for numbers. So there was a kind of "back to basics" reaction in the 1980s on this point.

Let me give an example to show what I'm talking about. Your sister had twice as many books as you do until a friend gave her 6 books today, and now she has 24 books. How many books do you have? Before she was given the six books, she had 24 - 6 = 18 books. So you have 18/2 = 9 books.

If you compare this solution with the mathematically equivalent steps in solving $2x + 6 = 24$, the difference is that the steps have a concrete meaning in the first solution, but possibly not in the second. So the first solution helps develop intuition for numbers in a way that the second may not. In fact, the second kind of solution is more likely to make sense to you if you are already proficient in the first kind and can therefore make the connection between the two. There needs to be a phase in which kids are encouraged to think concretely about problems rather than turning them into equations and going on auto-pilot.

However, there is a role for equations where they clarify the meaning of an operation. For example, it's useful to say that subtracting $7$ from $12$ means finding the number $x$ for which $7 + x = 12$. Or that dividing $\frac{4}{7}$ by $\frac{3}{5}$ means finding the number $x$ for which $\frac{3}{5} \times x = \frac{4}{7}$. Using formulas like $d = vt$ or $A = \frac{1}{2}bh$ also accustoms students to the use of variables without turning everything into algebra.

David
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The letters intimidate him because he doesn't recognise them or their meaning. Only introduce them when he demonstrates that he is ready.

Ask him what his initials are. When he tells you, ask him if he would recognise he was being addressed if you used only his initials. When he (hopefully) says yes, then ask him if distance might have an 'initial' he could suggest so that when you used that letter, he would know you were referring to distance. Hopefully he will suggest 'd'. And so on.

You could even introduce the lesson by having every student only address other students using their initials... good fun, but it breaks down the resistance to the dreaded 'letters' :)

Then only introduce variables which have immediately obvious 'initials', such as p for price or pizza, h for height, l for length, a for adults, h for hours, etc, depending on the context.

Only after he has begun to associate letters with practical measurements and quantities should you mention that 'algebra is just the same... it just uses abbreviations to simplify things, just like his initials.

Any talk of x and y and z will re-introduce the fear of symbols. It's better to wait until he gets frustrated with 'all these words' and the idea of an abbreviation is his idea...

I often tell students that they have been using algebra 'for ages', only they didn't realise it. Then I tell them a word story like this: Jimmy goes to the bank and gets out enough money for three pizzas. On his way to the pizza shop, he finds 8 dollars on the footpath (sidewalk). When he gets to the shop he has 41 dollars. How much was each pizza going to cost him?

Almost all students who can add and subtract will get it correct very quickly.

Then say "I'm going to write the algebra abbreviation for that story, and write 3p + 8 = 41 on the board.

Say "if he hadn't found the 8 dollars, would he have more or less?' They all will say 'less'.

Then write 3p = 41 - 8

NB! don't introduce any rules about sides and signs and what you do to one side, etc, same for words like solve and expression and equation. These should only come after the process is understood using things they already know... or you'll turn him off all over again.

and ask how much money did he get from the bank? They will say '33'

3p = 33

The say 'So, how much was each pizza going to cost? They will say '11 dollars'. Trust them to refer to what they already do at the shops, not what you have to tell them about dividing, or whatever. Trust that their experience already contains algebraic processes, only they didn't call it that... but they were perfectly good at getting it right. Hint! money and food questions are very familiar to almost every student everywhere.

Then say 'That's all algebra is: an abbreviation for words, using a consistent set of rules.' They will get it... and then wait for them to ask about or discover or intuit or teach each other the rules. Teaching is as much about asking and listening and enabling as it is about telling.

Then make up similar stories that you read out and they write the abbreviations for. Include things like 'on the way he lost 12 dollars', etc.

I have done this for 25+ years and it it has 'worked' almost every time with almost every student, so long as they can add, subtract, multiply and divide competently. If not, then address that first.

Only introduce abstract concepts and 'letters' when students are more than comfortable... and you have exhausted most of the other letters and they are ready for some meat in the sandwich.

Hope this helps

KiwiSteve
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Sorry for adding another answer, I'm new here. This paper:

Carraher, D. W., Schliemann, A. D., Brizuela, B. M., & Earnest, D. (2006). Arithmetic and algebra in early mathematics education. Journal for Research in Mathematics education, 87-115.

is a good start. Lots of references to existing research. I can add a link but not sure if it's behind a paywall.

A quick glance tells me that the gist is that you can introduce variables as early as 9-10 years of age but that it's important to build algebraic thinking by looking for patterns in number properties first.

EDIT: Adding some detail as per Ben's comment.

This is something of a rabbit hole. It seems that very young children can solve equations with boxes, question marks, or even letters to represent unknowns, however, they may not be truly engaging with variables in and algebraic way. The question becomes whether they are actually operating on unknown values or using "pure-arithmetic" strategies like counting and working backwards. The authors present several sources claiming to show that young children (at least as young as third grade) can work with symbols for unknowns in an algebraic way, and that algebra and arithmetic are not entirely separate modes of thinking.

For anyone wanting answers to the posted question, I'd suggest reading the brief literature review in the above article. The article itself is really focused on using functional relationships to teach arithmetic, rather than variables in particular. The authors seem to feel that there is strong evidence that students can engage with variables in a meaningful way from about age 8 or 9, but that we need to be careful about what we define as "meaningful," since researchers who argue that younger children aren't "developmentally ready" may have somewhat rigid views about meaningful use of variables and the nature of the relationship between arithmetic and algebra.

Nat Stahl
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  • Ah, I've looked through this paper (link without paywall) and it seems the best as concerns the +50 bounty. You have only "a quick glance" in this answer now, so probably summarizing the content of the Carraher et al paper more fully would improve matters. [In which case, I will delete this comment...] – Benjamin Dickman Apr 26 '16 at 03:15
  • Interesting paper, thanks for the link. The thing I tend to be a bit skeptical for findings like this is: While a classroom full of students could communally answer these questions (i.e., perhaps requiring about 1-in-20 to suggest the answer to each part), do most of the students individually have the skill at the end (i.e., does it "scale down" to individuals)? There doesn't seem to be any testing of individuals at the end here to know one way or another. – Daniel R. Collins Apr 26 '16 at 22:17
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I'm a teacher at a small Christian school and so I fill a lot of gaps. I've taught 4th, 5th, 6th, 7th, Alg. 1, and Geometry, as well as tutoring everything in between up to Pre-Calc with great success.

My 4th graders just recently got introduced to algebra. I've found that at that age, a lesson dedicated to the subject matter is more than sufficient for mastery of the simple idea that a letter represents some unknown or variable amount.

Our curriculum is very good at building from the "boxes" mentioned in other's posts, and lends opportunity for a teacher to get students excited earlier in the year if the class seems ready, by telling them, "What you're really doing is Algebra! The only difference, is that in Algebra, a letter is used instead of a box!" This takes the intimidation factor out of algebra, makes the students feel smart, and excited about furthering their math knowledge. They beg for the lesson on algebra before you get there, and then as a teacher, you have an easy time keeping their attention and making sure correct steps are used in finding the correct answer that will build a foundation in 4th grade that they'll use all the way through Algebra.

I've also found while teaching higher grades, that students without this foundation have a much harder time of developing it after they've developed instead, an aversion to algebra in general, but a similar approach can definitely be done in a tutoring scenario.

Elem-Teach-w-Bach-n-Math-Ed
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    Let me add this. An answer was given here: "start with algebra the student can do mentally." Although these are the type of problems to start with, students should be required to show work on these. Let them know that they are easy problems intentionally, so that they can know that they did it correctly! In other words, do the problem mentally after showing the correct algebraic steps so that they see that the algebraic steps are valid. Then when they get to $5.7x+7.9=22.7$ where they can't do it mentally, the steps are ingrained, and they know both that the steps work and why. – Elem-Teach-w-Bach-n-Math-Ed Apr 29 '16 at 20:37
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Many elementary students have a problems with the equals sign. They may not be able to solve for the missing number 3+2=__+1. They see = and think "the answer is". Therefore they will fill in the blank with 5, which is of course incorrect. If you add to the difficulty of this problem with a variable 3+2=x+1, many students and teachers will think the problem is that there is an x and not realize the student just doesn't understand how equality works.

BTW this is why students often in a multi-step problem may write: 3+2=5-2=3+17 =20.

I suggest that you introduce variables first where they are the answer to the problem, so that you have 3(77) = x and 3(42-4.1)=x etc. If the student is overwhelmed by a letter, replace with a box and then show that they are the same.

Once the student is comfortable with x as the missing answer, go on to problems where one side of the equality is a single number and the other side is an expression containing a variable. Start with simple problems that the students can do mentally and then proceed to more complex examples. For example 3+x=5 and x- 7 =2. Proceed to more complex problems such as 3x+2 =8.

Finally make sure the students understands how equality works, by giving problems such as 3+11 = __*7. If the student understands that, then you can proceed to replacing these examples with variables.

Above all be patient.

Amy B
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    The issue around the equal sign is discussed in an earlier MESE thread; see, in particular, the answer of D. Hast here. But what does the mathematics education literature say about introducing variables and the age/stage at which it is appropriate? – Benjamin Dickman Apr 21 '16 at 06:49
  • @BenjaminDickman I was responding to the OP's comment that he is tutoring a 5th grader who is having a hard time understanding variables. He also asked how should we introduce the practice of using letters to make the transition from Arithmetic to Algebra. I was suggesting that first he make sure that the student understands the equal sign because without that basis the student will not understand variables. – Amy B Apr 22 '16 at 04:18
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I would start early with "boxes" -- just empty squares and playing "what goes in the box?" for simple equations. Works well for kids as young as second grade and definitely by 4th. It makes it much easier to introduce letters as variables later, with the understanding that they act like containers. And, as someone else said, you can bridge the gap by modeling story problems with known quantities (at least that's what my son gets in 4th grade).

I should note, I've used boxes for adults in GED class, equally effective. They really liked "what did I start with?" where I'd write [BOX]+7-2+4-9+6=12 or something and have them try to work out a general method.

Nat Stahl
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They use variables as soon as they start counting. We should introduce number machines ASAP to encourage thinking like a mathematician. The INPUT----> DO ----> OUTPUT relationship diagram is relevant at all levels of Maths and introducing algebra plus x, y graphs is more meaningful. This would also help with the true understanding of the equal sign. I am embarrassed as a maths teacher that variables are introduced in Science to elementary pupils but rarely mentioned in the Mathematics curriculum.

Tony Holmes
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