Dec 19 2013

Math topics that need to be put out of their misery. Part 1: Writing algebraic expressions

Last month I wrote a post called ‘The Death of math’ which got a lot of attention as I described how I’d improve math teaching in this country by significantly reducing the number of topics taught and by making math optional after 8th grade.  A line that got quoted a lot when people tweeted about this post was “If I were made ‘Math Czar’ I would gleefully chop at least forty percent of the topics that are currently taught from K to 12.”

The fact is that the original intent of the common core, was to address this very concern.  The problem, though, is that they didn’t actually cut anything, as far as I can tell.  Why this happened is tough to say.  One complaint that teachers have about the common core is that they were developed by a very small group of people without input from teachers.  Perhaps that small group didn’t have the heart to cut any topics.

Or maybe the committee, even if it was small, was unable to come to agreement on which topics should be cut.  You see, the issue isn’t so much the size of the group or whether or not they got input from teachers, the issue is that the ability of that group wasn’t up to the task.  Perhaps it would have been better if the task were left up to one very clever individual who understood the needs of teachers and of students and could be trusted to do this right.  So, yes, I’m suggesting that rather than whatever group they formed, the common core standards would have been much better if they had just hired one person, namely me, to do the whole thing.

The first thing I would do if I were redesigning the math curriculum for this country would be to make a list of all the topics we teach and decide which can be cut.  Over the years, topics just seem to get added and added to the curriculum, which is why math books are traditionally giant monstrosities.  Rarely do you hear about anything being eliminated from the curriculum.  I can think of only a few examples, like calculating square roots by hand, the log tables, and The law of tangents.  All three of these topics, I think, were considered less important now that calculators have gotten so cheap.  As I think about what needs to get cut, I’d, ironically, put two of those three topics back into the curriculum, but don’t worry, I’d cut way more than I’d add.

For a topic to be worthy of inclusion in a course of Mathematics, it must have at least one of the following three characteristics:  1)  It must be thought provoking, 2) It must be beautiful, or 3)  It must be useful.  Percent problems, for example, are somewhat thought provoking, not very beautiful, but extremely useful so those would remain.  Calculating Pi would be an example of something that is thought provoking and beautiful, but not particularly useful.  Not many things would have all three properties, maybe teaching some types of encryption could come close.  If a topic satisfies NONE of these three characteristics, it has no place in the math curriculum and must be ‘retired’ I think.  Unfortunately many, if not most, of the topics we currently teach would fall into this category.  I’m going to write a series of approximately ten posts in which I go into detail about some of these ‘stale’ topics, and why I think they can be safely eliminated and nobody will really miss them much.

You might wonder how it is that as a math teacher I find a way to teach some of these topics when I resent them so much.  Well, for one thing, most of the classes I teach are for my ‘math research’ class where all the topics were chosen by me.  But still, I teach ‘normal’ math too, and I suppose that I’m a bit like a musician in a wedding band when it comes to teaching topics that I don’t like very much.  Surely musicians in wedding bands get pretty tired playing Earth, Wind, and Fire’s ‘September,’ yet when that song is requested they still play it as well as they can, giving it their own personal touch.  So when I have to teach something, I still try to give it my best effort and find something thought-provoking in it to make it worthwhile.  For some topics, this is not easy.

For the first topic on my list that I would love to cut from the math curriculum altogether is something that doesn’t even have an official name.  Basically it is the skill of converting an Algebraic  sentence from words into symbols, and looks like this from an actual modern textbook:

 

 

In case you’re wondering what the purpose of this topic is, it is for eventually solving algebra word problems that require manipulating variables on both sides of the equal sign.  For instance:  “In fifteen years, Mary will be twice as old as she was five years ago.  How old is Mary now?”  So this would get converted into the Algebra problem x+15=2(x-5), which becomes x+15=2x-10, which becomes x=25.  So Mary is 25 years old now.  Fifteen years from now she will be 40 years old which is twice as old as the 20 years old she was five years ago.

So you might think that I am opposed to contrived word problems like the question about Mary.  Actually I don’t mind that question as some kind of riddle that students can apply things they have already learned for.  But the process of spending a few days mastering the step where they convert sentences into symbols, so they can learn to do the age riddles in a mechanical mindless way is something that we have been torturing students with for too long.

Now the hope of the common core is that it would eliminate the requirement for mindless activity like this, but it is very clearly part of the sixth grade standards under 6.EE.A.2a:

CCSS.Math.Content.6.EE.A.2a Write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation “Subtract y from 5” as 5 – y.

Clearly, it is going to be a long time before students and teachers, alike, are rescued from this mind numbing topic.  It doesn’t make kids smarter.  It doesn’t make them ‘college or career ready.’  It just makes them want to get math over with as quickly as possible so they don’t have to think about it ever again.

Next time:  Absolute value

14 Responses

  1. We had a good discussion about math curriculum deadwood at Andrew Gelman’s site.

    http://andrewgelman.com/2011/11/05/deadwood-in-the-math-curriculum/

  2. NewarkTFA

    Just a question from a curious English teacher: Is it your idea that it would make more sense for students to just work on this skill in the context of solving actual problems rather than “scaffolding” by pre-teachiing this concept before they actually need to apply it?

    • Not answering for Gary, but as another math teacher, I’ll just say that in my experience, “pre-teaching a concept” if the students have no context to apply it in is very frequently an exercise in futility. And the fact that we continue to do that in much of the math we teach is a huge, huge problem in my opinion.

    • ARA

      I’m a history teacher and this is my question, too. I would have thought that teaching this skill would be necessary for students to be able to translate word problems (and real life math problems) into equations. I was quite good at math (according to standardized tests), but bad at word problems. I actually still do this translation in my head when I have to do percentages.

  3. Shannon

    But then what will they put on standardized tests???

  4. tlmerrie

    I loathe absolute value.

  5. jj cm

    Am curious which of those three topics you would reinstate, as I teach square roots of 4-6 digit numbers to my fifth and sixth-graders and think it’s so valuable for geometric, algebraic, and number concepts…not to mention it is self-checking and most of the kids enjoy getting the hang of it!

  6. Toby

    I’ll take the bait: the natural exponential function meets all three criteria, though probably not often when taught at the secondary level.

    I expect “thought provoking” is going to be the most problematic to apply. It’s hard to keep it from being conflated with difficult or complicated.

  7. Ted

    Mary? How about Plaxico, Picabo, or Aishwarya?

    The quotient of a number and 8 =?

    and? and? and? WT….

  8. Ted

    Mary? How about Plaxico, Picabo, or Aishwarya?

    The quotient of a number and 8 =?

    and? and? and?

    Incorrect syntax for subtraction or division.

  9. Ted

    The quotient of a number and 8 =?

    and? and? and?

    Incorrect syntax for subtraction or division.

  10. Audrey

    You could also address the absence of important topics. Probability is missing from the 8th grade math standards.

  11. J Choi

    The most intriguing new standard is buried in this one, from the grade 1 standards:

    CCSS.Math.Content.1.OA.A.1 Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations “with a symbol for the unknown number to represent the problem.”

    (Quotes are mine.)
    Does that really say to teach symbolic representation to first graders?
    This phrase appears in each of the standards from grade 1 – 6.

    Does anyone know if there is a textbook that does this?
    Are any districts actually teaching this?
    I have to see this in practice…

  12. J Choi

    Actually, the ability to translate an understanding of words into an understanding using symbols is absolutely critical to algebraic understanding and development. I agree that it is typically done in a very boring or contrived setting. I am actually glad to see that it is introduced as a grade 6 requirement. However, my personal work shows that this can easily be done at grades 3 or 4. The trick is to adjust the phrase or problem to the students’s level of ability. However, prior to this, it is absolutely essential that the student grasps two other concepts. The first is that a symbol (variable, letter, shape, blank space or question mark) can actually be used to stand for a quantity. This concept is essentially NEVER taught to any student below grade 5-6 in almost any curriculum that I have reviewed. The understanding of a variable introduced in grades 5-6 is very weak, and is used sporadically, and inconsistently. This action sets the student up for failure in grades 7-8 as they attempt to learn algebraic thinking. In almost all cases, the concept of a variable is introduced by the teacher, and not by the text. Why? Because it is a necessary part of math and algebra. And everyone (teachers, textbook authors, standard writers, and even early education algebra experts…) ASSUMES that the student has grasped this idea by grade 5 or 6. You don’t believe me, right? Grab you favorite texts, grades 1-6, and search for the first use a variable. Then watch if it was just simply used without a solid introduction, or were there exercises to introduce the student to what it means and how it is used. It turns out that that it is trivially easy to teach the concept of a variable to first graders. I have done it many times. If the usage is maintained CONSISTENTLY during grades 1,2, and 3, these students can, in fact, write the dreaded algebraic expressions described above… in grade 3. (some by grade 2). This is, in fact, why I previously posted regarding textbooks that conform to the new CCSS standard. It actually works. The issue I am currently studying, is that most textbooks read the above standard CCSS.Math.Content.1.OA.A.1 as “add or subtract within 20”. Not one that I have seen addresses the “use of symbols”. Note that adding to within 20 is an arithmetic concept, which is in fact, already taught, whereas the representation with symbols is an algebraic concept, and is not taught by any school, anywhere prior to grade 5-6. This sharp transition to “algebra” in grade 7 – symbolic and conceptual thinking at a time of intense peer pressure causes system-wide failure at the seventh grade level – leading to remedial 8th grade work. Students who have already made the transition in the upper elementary years (3-6) are well prepared to transition to the conceptual work of algebra I and II.

    Dr. Jonathan Choi
    Worcester, MA

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By a somewhat frustrated 1991 alum

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