Teaching Mathematics: What, When and Why

An in-depth examination of mathematics education, topic by topic

How to Introduce Algebra?

It was February 1957. I still can remember the feeling of panic.

Let me back up. My chemical experiments with hydrochloric acid and Vegemite had persuaded my mother that I had a future as a pharmacist. Pharmacists need to read Latin she told me, so when I returned from my first day of school, enrolled in the French form, she wrote a note to have this corrected. Thus I found myself the next day in an algebra class not knowing I had missed the first such. “Go on with the problems in your book” said the teacher. Problem 1 was 2x + 3x. I stared at this in panic; maths had become gibberish.

It worked out for the best. Looking at the work of the boy next to me, I found that the answer was 5x. The clouds cleared and I had the beginnings of a new career, looking for ways introducing mathematics ideas in a natural way, building on relations to the real world, or on questions raised by previous mathematics. I guess if the boy next to me was equally lost then perhaps I might have become an unemployed Latin scholar.

Our chief problem in teaching mathematics, I feel, is the top-down approach. We present some abstract idea, spend years developing it, and then perhaps, some years later, show it may be useful. And to fill the emptiness we add Problem Solving exercises where some artificial problems make the students even more doubtful. It’s called New Math(s).

I was lucky; in my youth the mathematics syllabuses were controlled by academics with a deep knowledge of both pure and applied. Teaching followed the historical development, an organic, bottom-up approach, all except that damned algebra.

Until recently I had no opportunity to teach the beginnings of algebra. I now have the luxury of face-to-face tutoring at this level and here I report my attempts to tailor the teaching to the interests of the student.

Method A

Here the algebra formalism is introduced to find an “unknown”. Such problems were studied in the Egyptian Middle Kingdom, though without the use of letters to stand for numbers. Some more contemporary problems might explain the approach.

XXX drives to Bendigo each month to play cello in the orchestra. Bendigo is x km from her home.

  1. How long is the round trip to Bendigo and back?
  2. Kyneton is half way to Bendigo. How far is Kyneton?
  3. How long is the round trip to Kyneton and back?
  4. In one year, XXX makes 12 trips to Bendigo and back, plus 6 separate trips to Kyneton and back. What is the total distance of these trips?
  5. Those trips give a total distance of 4590 km. What is x?

The drawback with this approach is that to show the advantage of using algebra, you need problems where the answer is a fraction and cannot be easily guessed. This requires a formal method for solving equations. It seems that although a worthwhile part of the syllabus, this approach is not suited to the first meeting with algebra?

Method B

Here we use algebra as a mnemonic to remember how to calculate something. For example the area of a rectangle is length by width, or LW for the lazy. After several examples familiar to the student I dived into some deeper waters.

There is a plethora of formulas used in the sciences that encapsulate interesting stories. The one I found best is the formula for “Reynolds number” for liquid flow in a pipe. At low speed the water particles in a pipe follow fixed steady paths, laminar flow. When the speed is increased there may be a sudden jump to turbulent flow where the particles follow convoluted chaotic paths. The power needed to push the fluid jumps also. Modern science does not understand this transition although it can be predicted if you know the diameter of the pipe D, the stickiness (viscosity) of the fluid \mu, its density \rho, and speed u. The Reynold number is then D \rho u/ \mu. Transition from laminar to turbulent flow occurs when the Reynolds number is about 2000.

I wrapped the lesson in a story about a Civil Engineer who is selecting a pump to send water at a certain rate. So the student had a glimpse of the actual work of an engineer. But the more exciting aspect was a window into the greatest unknown in physics.

Method C

I currently have a student who is deeply interested in computer code, from the viewpoint of making games of course, but has neglected the standard primary syllabus. So I am trying to craft a path into algebra using his interest. Code contains variables which at some stage may be assigned a value, and decisions are then made using criteria based on those values. It is a form of algebra. Here is an example that I intend to try at the Year 7 level.

If the number assigned to the year (eg 2024) is divisible by 4 then that year is a leap year, except that if the number is divisible by 100 then the year is not a leap year, except that if the number is divisible by 400 then the year is a leap year. Write a function that enters the year number and prints out a decision as to whether the year is a leap year.

Not only does this code require a symbol representing the year, it does require some careful consideration of nested conditions, very algebraic.

My Questions

Most teaching does not have the luxury of a bespoke approach for a single mind. Which of these would fit your situation? Or is there another approach that I have missed? What is your experience as a teacher of beginning algebra? What happened when you were a student? Or should we just keep algebra for the few and leave the majority in blissful ignorance?

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tom

25 responses to “How to Introduce Algebra?”

  1. tom Avatar
    tom

    If your comment includes mathematics symbols or formulae then to invoke latex preface the material with the word “latex” and wrap it all in $ signs. For example, wrapping “latex E=mc^2” between two $ signs gives E=mc^2.

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  2. Red Five Avatar
    Red Five

    Nice one Tom. I feel Method A has the most chance of working longer-term for the majority of students because it feels quite natural (unlike a lot of the convoluted rubbish quasi-applications I often see early in an algebra course…)

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    1. tom Avatar
      tom

      Thanks Red Five
      I hope to keep the conversation positive, but I am interested in these “quasi-applications”. Examples?

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      1. Red Five Avatar
        Red Five

        Textbooks are full of them. My favourite is Marty’s cheese shop example (it features in his webinar presentation Mathematics in Hell from 2020 but it may also be in some of his other posts as well).

        What I mean is that textbook exercises on Algebra, especially in early levels seem to go out of their way to pretend there are applications when there need not be until more of the skill has been learned.

        Questions such as “Peter has $x$ pencils, Paul has twice as many pencils as Peter and Penny has five less pencils than Paul. How many pencils does Penny have?” The answer intended is $2x-5$ but without specifying that $x$ is an integer, 3 or greater, then there are clear issues with the question!

        I think it better to leave these quasi-applications out of the learning and then when algebra can actually be useful, show students how to use it well.

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  3. Yes More Mr. Nice Guy: Tom Peachey’s New Blog – BAD MATHEMATICS Avatar
    Yes More Mr. Nice Guy: Tom Peachey’s New Blog – BAD MATHEMATICS

    […] first topic post is How to Introduce Algebra? Tom also has a suggestion post, where readers can indicate topics that they would like to see […]

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  4. Vicky Avatar
    Vicky

    This is a comment and a suggestion all in one! I like to introduce algebra by using manipulatives such as algebra tiles and lots of visual representations. You can use these to model a problem and to help you solve it. So my suggestion for a post is looking at manipulatives and their use/non-use in secondary maths.

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    1. tom Avatar
      tom

      Thanks Vicky
      As you can see I’m a newchum teaching algebra, have never heard of this approach. It sounds worthy of a separate thread, but I am impatient to learn more. Can you describe a typical lesson?

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      1. vicky Avatar
        vicky

        See my blog here https://mathsmanipulatives.com/algebra-tiles-part-1/ or my learning sequence here https://calculate.org.au/2019/03/27/algebratiles/

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      2. tom Avatar
        tom

        Good site vicky. Using physical objects has appeal, the later videos in your blog worked nicely. At which year level have you tried these?

        I assume that the manipulatives are just used to develop the manipulation of expressions/equations. How do you first introduce this algebra business?

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  5. Red Five Avatar
    Red Five

    So… my LaTeX didn’t seem to format correctly there for some reason. Any ideas, anyone?

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    1. tom Avatar
      tom

      Blame my Scotish ancestry, I went with the cheapest WordPress deal and the LaTex editor is a little clumsy. Instead of $x$ pencils, try x pencils. No space between the $ and the ‘latex’.

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      1. tom Avatar
        tom

        Damn, it has compiled the latex. If in doubt read the first post in this thread.

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  6. Potii Avatar
    Potii

    Mike Clapper from AMT had a good series of lessons on using “box stories” to create algebraic equations and “backtracking” to solve them. I’ve has seen such a strategy before but his sequence of lessons make a lot of sense.

    For example:

    Say we wanted to create the equation 2*3 +1 = 7.
    Using the “box stories” we would go 2 –> multiply 3 –> add 1 –> = 7

    Pronumerals can then be introduces once student get the hang the the method to create algebraic equations.
    After that we can “backtrack” and reverse the operations to deconstruct the equation to get the value of the pronumeral.

    What I like about it:
    – abstract but approachable
    – easily relates to orders of operations
    – complex equations can be systematically broken down

    What was not so good:
    – Students can struggle to link this method to the conventional way of solving an equation
    – doesn’t address concept of equality well

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  7. Anonymous Avatar
    Anonymous

    I disagree. I don’t think an applied problem is the best way to start.

    And I say this as a much more applied person than a pure mathematician. Have done mechanical engineering, chemistry, business, and applied psychology (in the work world and in school). I probably know way more than the average math teacher (often lacking a general engineering background) about what subjects will end up using math.

    The reason, I say don’t start with a physical example is WORD PROBLEMS ARE HARD. Don’t start with an LRC circuit for second order ODEs with constant coeffiricents. Don’t make me wonder what a dashpot is. Start with the algebraic. Learn the techniques. And then (after mastering just mving x’s around….and I mean demonstrating in several problems) move to word problems. The exact same thing is true in algebra. Master the x pushing first.

    This is one of the reasons I hate almost all PDE textbooks. They start off with a heat transfer (or even worse, statics) problem…and derive the math procedures en passant. Way too much cognitive load for a new learner.

    Consider freshman college chemistry. An applied algebra class if ever ther was one (stoichiometry, equilibria, gas laws, etc.) The point is they’ve already HAD and mastered algebra in dedicated math classes, with x pushing, before they have to do applied algebra problem with the chemical content larded on.

    You might say word problems are no big deal. But you’re wrong. Lot’s of evidence on this point.

    Start off easier.

    Oh…and make sure they master solving for one equation with one unknown at first order. Doing it with each step written down and laboriously showing how you subtract the same thing on each side of the equation and no thing at a time. Don’t combine x and number at the same time. Don’t “throw something across the equals while changing the sign”. That can come later. For the start, keep it simple, explicit and step by step. Algebra is a challenge. Give the kids a chance by being step by step and by grooving the brain to new methods like Umbridge getting letters on Harry’s hands.

    Start with something extremely simple. 2 + [box] = 5. Then 2 + x = 5. And eventually 2x-7=34-13x. And step by step in terms of getting harder (not just the three shown). And lots of practice along the way.

    P.s. Reynold’s number?! Cripes. Ai yi yi! And I say this as someone who’s worked at Langley on a wind tunnel.

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    1. tom Avatar
      tom

      Thanks Anonymous, I was hoping for a contrary post, perhaps not as persuasive as this.

      I agree that WORD PROBLEMS ARE HARD for the current generation. These days they are add-ons to a more abstract presentation of the main idea. There was a time when word problems were ubiquitous, often the lead-in to a new topic. I witnessed their slow death caused by (i) the introduction of new math, and (ii) an argument that such problems are unfair to the large contingent of students in many Australian classes that do not have English as a first language.

      Let’s consider the example that I gave in the initial post. I have used it or similar examples with just 4 students, all in Year 5 or 6, only one of which was an exceptionally fast learner. I found that all students easily followed the first such example and then were able to perform steps like 1 to 4 on similar problems. In under an hour all were able to formulate an equation with guidance like that given in the example. Notice I was concentrating on the setting up, not the solution. Solving the equation was more difficult, especially if the solution was not an integer. A follow up session was required to teach the standard method for solution of linear equations. And yes, some years of practice are advised like with any language, but here we are discussing the introduction to algebra, how to make it meaningful from the beginning.

      Ai yi yi! The Reynolds number business was only tried the once, with the exceptionally fast learner (ELF) when she was in Year 6, or was it 5. This strikes me as one of the most successful lessons in my career. ELF was quite surprised to learn of the transition to chaotic flow and excited to learn that it is unexplained. We discussed what factors might affect the transition: the pipe diameter, the fluid speed, fluid stickiness and heaviness. Then I revealed Reynolds formula – no attempt to derive it from theory, just presented as an experimentally verified result. If you think this is beyond primary school children then I suggest you try it. But work up from reality and common sense; it will not work with a top-down approach.

      There are people who learn more easily with the abstract approach, but in my experience a very small percentage. Such people either win a Fields medal or become maths teachers. They might write a text book on partial differential equations without an introductory link to the historical origins, but have trouble finding a publisher?

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  8. Dr Mike Avatar
    Dr Mike

    I can’t agree more. Mathematics is the first and foremost language. A language that describes nature/life around us. Yes, mathematics can be abstract, but so can linguistic constructions. However, first people need to learn how to speak to describe life around them..

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  9. marty Avatar
    marty

    Hey, where did everybody go?

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  10. Terry Mills Avatar
    Terry Mills

    A few years ago, I gave a lesson to Year 7 students on algebra by considering how to calculate the date of Easter Sunday for any given year. They had previously had only a few lessons on algebra. Since then I have used the same idea with Year 8 and Year 9 students. The lesson was based on Fitzpatrick, T., Martin, M., Mills, T. (2009). Easter algorithms. Vinculum, vol. 46, no. 3, pp. 16-17. It seems to work well.

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  11. Terry Mills Avatar
    Terry Mills

    I recall how I was introduced to algebra. My father explained it to me when I was in Grade 6 by writing in the sand with a stick on the bank of Deadman’s Creek.

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    1. tom Avatar
      tom

      Do you remember how it was presented – the content I mean – not the stick. Can we add the method to the list of approaches as the Deadman’s Creek method? Or was it a variant of the current list?

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      1. Terry Mills Avatar
        Terry Mills

        I don’t remember the details.

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  12. Terry Mills Avatar
    Terry Mills

    I have been introducing students to algebra using material from Todhunter, I. (1870). Algebra for beginners. Macmillan. Despite the title, this little book would be useful for students from Year 7 upwards.

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    1. tom Avatar
      tom

      Google has “offered” the book by making a digital copy

      https://archive.org/stream/algebraforbegin00todhgoog/algebraforbegin00todhgoog_djvu.txt

      but the equations are mangled. I see Amazon have a reproduction for $261.

      Ah! here is a proper copybooks.google.com.au/books?id=KGRkAAAAMAAJ&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false

      with a downloadable pdf under the cog icon.

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  13. Terry Mills Avatar
    Terry Mills

    I just made a post but it did not appear.

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    1. tom Avatar
      tom

      Sorry Terry
      I should pay more attention. WordPress deemed your post a spam for no apparent reason. It should now be visible.

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