Has math instruction actually changed in the last 40 years?

Has math instruction actually changed in the last 40 years?

Summary: I started my last two master's classes last night (!!).


I started my last two master's classes last night (!!). Since the classes consist largely of teachers, we usually end up having some pretty interesting discussions about teaching approaches to translating the master's-level content to secondary education.

The first professor of the evening (analytic geometry, although the class looks like it's going to get into some fairly wide-ranging content) posed a good question: Are we teaching geometry any differently than we did 40 years ago when he took it in the 10th grade? Or had it introduced to him the 5th grade? Same question for the other core math subjects.

We know that we're being assessed more rigorously and supposedly being held more accountable. I can't speak for math teachers outside of the States (although I'd especially like to hear from my international readers tomorrow), but I know that we're supposed to be using all of the data from the countless standardized tests to which we now teach to drive instruction.

However, my professor pointed out that as he has worked with both primary and secondary math teachers over the years (his area of specialization is math instruction, aside from a love of linear algebra), the fundamental way in which we teach mathematics hasn't changed much.

We don't teach formal proof for most kids, we drill kids less on basic arithmetic, and we spend a lot of time accommodating special needs, but have we figured out a better way to help kids understand math? I'm inclined to agree with my professor.

We've integrated technology to varying degrees and with varying degrees of success. However, we still basically cover the same content at the same times, though perhaps with less rigor. My grandfather dropped out of school in the 8th grade to take over the family farm, but was still vastly more capable of balancing his books than the majority of students I see in the 9th and 10th grades. To be fair, my classes are heavily loaded with inclusion students, but I wouldn't want most of them balancing my checkbook, let alone keeping records for a business.

Forgive the rant that seems to be building here, but I'm frustrated. We put men on the moon 40 years ago, every student has a graphing calculator with more processing power than the computer I used in college, the students are tested up the wazoo and teachers are evaluated until they don't have time to teach. Yet the biggest complaint from professors who teach undergraduate math is that students lack fundamental understanding of conic sections, functions, and basic calculus. Students struggle to relate geometry with trigonometry and periodic functions with the real world. I had to cut a third of the curriculum in my physics class to backfill the necessary math.

Where are we going wrong, folks? How can we use all of the technology and data at our hands to teach kids better? Share any innovative approaches you have below and let us know what's working and what isn't.

Topic: CXO

Christopher Dawson

About Christopher Dawson

Chris Dawson is a freelance writer, consultant, and policy advocate with 20 years of experience in education, technology, and the intersection of the two.

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  • It depends....

    In my kids' school district, math teaching is geared toward taking the district assessments and focuses specifically on learning what will be on the next assessment. Assessments are given quarterly. While this school district didn't exist 40 years ago, I don't remember taking any math test outside of school until I took the SAT in the mid '70s. I remember more being taught how to figure things out and being exposed to a broader range of topics and techniques than my kids are being exposed to today.
    • Least common denomination

      [i]In my kids' school district, math teaching is geared toward taking the district assessments and focuses specifically on learning what will be on the next assessment.[/i]

      And, at least in my kids' district, once they have kids up to the tests, they ignore them so that they can spend time on bringing up the ones who [u]aren't[/u] up to spec on the tests.

      As the saying goes, "no child left behind" can't happen unless no child gets ahead.
      Yagotta B. Kidding
      • no child left behind

        In the school district I grew up in, which by the way has changed for the worse, [i]no child left behind[/i] means that they check the buses after everybody has got off to make sure there isn't some kid who fell asleep.

        Getting back to math, our school district also spends extra time on kids who aren't up to spec since, as is probably the case with you, allocations are based on what percentage passes the tests. But there are some exceptional teachers, like my daughter's 2nd grade teacher, that will spend extra time with kids who pass the tests easily and make an effort to see that those kids don't get completely bored with math.
  • To answer your final questions...

    I am really not sure where using technology better comes in. But I do have a degree in Math, I believe in teaching fundamentals and teaching why things are the way they are. As a simple example, I have seen kids who know the multiplication tables, yet don't fully understand what 3 X 8 means. They will know it is 24, but not that it is 8 + 8 + 8 or 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3.
  • Computing power

    Dare I suggest that if the computing power isn't part of the solution, it's part of the problem?

    If your objective is to program Mark One Wetware with fundamental mathematics, offloading the maths to external tools is Not Going To Help. Your grandfather could do his books because he didn't have bookkeeping software. That may or may not have been better; I'm not at all sure that the ditch-diggers of the 17th Century were better off than I am for being able to shovel tons of wet dirt every day.

    Well, you are inheriting kids who [u]know[/u] that math is for machines to do. Mental arithmetic is, after all, a waste of time that teachers only make kids do because they hate kids -- once anyone is out of class they'll never add two numbers again in their lives (just ask them!)

    Mathematics hasn't really changed much at the secondary level in my lifetime, and I'm not aware of any major revisions to the wetware. Unless you're planning to do something about one of them, the only remaining room for improvement is in the communication channel between the material and the student (and that includes the teacher.)

    Student, teacher, log. Which one are you going to change?

    PS: I'm actually a big fan of individualizing instruction as much as possible. Some kids can't get enough formal math, others are only in it to satisfy some imposed requirement. Letting the latter get in the way of the former borders on abuse, but is all too common.
    Yagotta B. Kidding
    • Computing power

    • I second that.

      [i][b]Mathematics hasn't really changed much at the secondary level in my lifetime, and I'm not aware of any major revisions to the wetware.[/b][/i]

      I would dare say that mathematics hasn't changed much at the secondary level in [i]anybody's[/i] lifetime.

      As important as it is to know what works, it's even more important to know what does NOT work. Allowing calculators in the classroom (at the lower levels) does not work. Dumbing down the curriculum -- as in the case where children are now being taught to calculate fractions by shading in blocks (I'm not kidding) -- works only as a passing [i]illustration of the concept[/i] of multiplicating fractions, not worthy of the inordinate amount of touchy-feely time that's actually devoted to it. Depreciating rote memorization of basic math facts not only does not work, it's deplorable. You should never have to grope for the answer to "what's six times seven?"

      University requirements aside, the present approach to math instruction has given us store clerks who have a mental meltdown if you hand them a five-dollar bill and two pennies to cover a $4.97 purchase. All you want is a nickel back, but instead you wind up remedial math at the 7-11.

      I'm a parent of twin sixth-graders. I know how I was taught, and I've seen how my kids are being taught. I spend far too much time at home supplementing what's not in the curriculum with what should be. ABSOLUTELY math instruction has changed in the last 40 years, and for the worse.
  • RE: Has math instruction actually changed in the last 40 years?

    First thing to do is LOSE THE CALCULATORS! They were forbidden when I was in high school (early 80's) and now they are required as early as 7th grade. Teach the kids that they don't need a calculator to do math. Also the most common complaint I hear about math instruction is "where will I need this, how does it apply to the real world?" I had the same questions as a teenager, and I like math. When there is a clear application, they understand so much more quickly.
    • I disagree...

      We've *trained* our kids to think that way - always questioning "how am I gonna use this?" They (and we) need to fall in love with the purely theoretical first, and then we'll see imagination providing new ideas and solutions to existing problems! I distinctly believe that is part of the turn-off to math and science these days - they are taught as all too practical, and seem often to be leading one to become just another cog in the machine of practicality, instead of being allowed to wonder and marvel at something.

      Example: How many kids don't aspire to be an astronaut? And do you think they perceive the practicality of that in 4th grade? It's because they see pictures of the planets and galaxies and are captivated by the beauty of something that awes them...we are forgetting the need in the human soul for seeing beauty and marvelling at things! It is true that we have many practical applications resulting from our space programs...but that is not fundamentally why kids are captured by the "being an astronaut" idea. And I think it's clear that those who actually become astronauts as adults do so because of the "bigness" of what they can do...the frontieriness of it all.

      If we teach kids to think and love the theoretical, they can find that same sense of wonderment in mathematics. If we don't...they become bored with mere calculations.
  • RE: Has math instruction actually changed in the last 40 years?

    Practical math has changed. My kids (13 and 16) have never been taught the basic skills of making change, balancing checkbooks, or the basic math of business.
  • All the computing power is great...

    ...but you still have to teach the concepts. The calculators may even be a distraction in the early stages. I still have the sneaking suspicion that my high school chem teacher did the right thing all those years ago by banning calculators and making the class use slide rules (30 years later, I still have the slide rule).

    For the stuff that's typically taught in high school math, a calculator is really only good for two things:

    1. Arithmetic, which the students are already supposed to know.

    2. Graphing, and then it's probably better to make students learn how to do it manually (so they know what they're doing) first, so they understand what it is that the calculator is doing.

    Pencil and paper is still the student's best friend in high school math.
    John L. Ries
  • Logic and philosophy

    Kids aren't thinking theoretically or abstractly enough precisely because the focus has become too much on "technologizing" the learning process. Ironic though it sounds, we need to de-emphasize computers (and graphing calculators during the teaching of foundational material.

    Furthermore, you mention not teaching formal proofs as much. This is a fundamental problem in *all* of education - the lack of instruction in classical philosophy and formal logic (syllogisms), etc. It makes all the difference between ending up with thinkers and idea people and those who can simply regurgitate facts, punch in numbers in the calculator and follow already-defined processes.
  • An ugly five letter word


    It seems teachers just expect kids to remember things "by exposure." And for some of the more gifted children that may work. But for the vast majority of student, a far greater proportion of wrote learning is necessary than currently happens.

    Concepts are only part of education, not all of education. Without the raw facts wrote provides, the concepts are useless. It's like trying to teach composition to a person who can't spell or punctuate.
    • Another five-letter word: irony


      Ah, English. One of the homonyms that you just have to learn by [u]rote[/u] is the difference between brute-force memorization by repetition and the past tense of the verb "to write."

      Normally I don't do spelling flames (esp. on someone I suspect isn't a native English speaker) but in this case it really is a marvelous example of the limitations of the technological assists.
      Yagotta B. Kidding
  • We have changed, but I'm not sure it helped.

    We are teaching Algebra to 7th graders now, but when I was in Junior High school, NO ONE took Algebra 'til 9th grade. Then in 9th grade we moved to a town where the upper half of the classes took Algebra in the 8th grade, so I was a year of math behind the people I competed with for scholarships 4 years later. That discussion continues today, as many say absolutely no 7th grader should ever do Algebra due to developmentall issues, blah, blah, blah. We put the best 5-10% of the 7th graders in Algebra, and they blow the curve for the 8th graders (not that I use curves)
    but I doubt we should try that with the majority of 7th graders.

    But I am not convinced this whole "move them ahead faster" way of doing it helps; as I teach Algebra, I find that those kids who struggle with math facts (times tables, fractions, etc) just get even more frustrated when they try algebra, regardless of their grade level, GPA, etc.

    We usually teach ideas and concepts, teach how to work things out by hand, then show them how to use the calculator to do it faster; simply because we have to teach so many things to prepare for the end of course testing driven by NCLB, or state requirements. We never got to work at anything until it becomes easy and natural. We ask 15 year old basketball players to shoot 100 baskets a day, but how many math problems do they do?
    They spend 2 hours a day in sports practice, how much time do they spend working on math?

    You want to change the math scores to match the other countries' then do away with 5th and 6th grade sports and make them do homework; just like those countries do. In fact I find it funny that German kids go to school for about 75% as many hours as our kids do, but because they don't do all the feel-good stuff we require, and because they are expected to do 3-4 hours of homework every day,they get farther, faster.
    But then how would we produce he athletes we need to keep the NBA or NFL going? Guess our priorities are a little askew...
  • What does it mean to be mathematically proficient?

    Interesting discussion. There are so many aspects to this issue too. I work with a non-profit that helps teachers transform their instruction in math and other topics. We have a podcast on what it means to be mathematically proficient: http://www.idra.org/Podcasts/Resources/Five_Dimensions_of_Mathematical_Proficiency/
  • Pleasing Shapes

    Totally True, my son was learning to create rectangles out of 24 squares (2x12, 4x6, etc). The final question was 'which is the most pleasing'! At least he "feels good" about math.
  • Reduce over the assessment and reporting now required

    Then when you give a math test get rid of multiple choice
    and require students to show work on a paper test. Then
    have teachers grade not just on correct answer, but on
    correct method. Teach students to check answers. If you
    solved for x, but x back in the equation and see if it

    Lastly, we need to teach kids how to learn and what
    methods of learning work best for them. Kids have not
    idea how to learn if we don't teach them. Oh, but that is
    not a requirement of NO CHILD LEFT BEHIND.
  • Sneaky way...

    ...to get instructional feedback from parents. Mind you, we're not your students' parents, but I suspect that a high percentage of us either have or have had children in high school (not to mention that we've all been there ourselves).
    John L. Ries
  • RE: Has math instruction actually changed in the last 40 years?

    Its been a few years since I've been in a classroom, but I suspect that this hasn't changed. Students simple do not spend enough time doing math. There is no substitute for memorizing multiplication tables, working problems and memorizing derivative/integral relationships. There is no shortcut. The student must spend the time and too few are willing to do that today. Generally the more the individual student needs to spend the time the less likely they are to do it. This is the primary reason tutoring companies are successful. The parents are paying money for the student to spend the time, under supervision, so they do and their grades improve.