I remember back in 2007 feeling that my TI-89 was almost unfair. Pretty sure I used it for the ACT, and you can solve something like 60% of problems with no effort if you just learn how to graph. And now you can use Desmos on the ACT.
Nowadays, most students in higher level math are equipped with calculators that can just solve things. You don’t need to learn how to convert fractions to decimals, or work with a percentage, or even do algebra (80% of working with calc 1 students can often be “yeah here’s how to put it into solver”).
I’m very torn on this. On one hand, I think that doing it by hand is the only way to develop on understanding of what it all means. There’s patterns to what a base system mean that you start to “get” once you’ve done enough borrows and carries. Small and consistent practice in the small skills adds resonance to the major skills you are building to.
On the other, there are things like dysgraphia that just there’s no reason to not work around. Some people can’t hold onto times tables. There are amazing ways to do multiplication that are slow but work for people (draw a rectangle - 3x4 best for demo purposes - have person count squares, bam, you have now outdone their 2nd grade math teacher.) Why bar someone who can’t memorize things but can understand why things work from further study of math?
I do sorta wish that the SAT kept its no calculator section though. It would be interesting to make a bunch of adults take the modern tests and compare their scores to they got 20+ years ago…
Most of what you’re describing is algorithms, and algorithms are useful, but they are overly emphasized in education, in my opinion. Some people memorize what the product of 2 twelves is, other people need to use an algorithm to calculate it using special properties of graphical and symbolic notations of a domain specific language.
Is memorizing your times tables “bad”?
In some ways no. Knowing many number facts speeds up your work.
In some ways yes. Relying on number facts exclusively and never learning algorithms reduces your ability to deal with problems that rely on facts you haven’t memorized.
Algorithms, effectively, produce facts. Some algorithms are incredibly complicated and require major shifts in symbolic representation of concepts you already have symbolic representations for. This can be incredibly difficult for some thinkers, who nonetheless would be good at math if they were not required to make this shift until they were cognitively ready for it.
Calculators basically implement algorithms and algorithm catalogs. They help you produce facts.
In some lessons, learning to derive the fact from specific algorithms is the point of that class. Using a calculator in this context is preventing you from the learning objective.
In other lessons, the facts are more important than the algorithm, usually because the facts are used as inputs to other algorithms.
Once you have this perspective on learning, it’s hard to avoid the inevitable conclusion : our math education is terrible. It forces cognitive behaviors that many people are unprepared for, it provides no support for the diversity in cognitive development, and in the end the tests are graded primarily based on whether you provide a fact correctly. The incentives are off. The scaffolding is off. The sequencing is off. The focus is off. Math education is a disaster.
But not because of calculators
In my practical pedagogy, I try to emphasize heuristics more than algorithms or “facts.” My goals vary by student, but there are good higher order thinking patterns that can be taught with any math concept.
I agree with you that math education is messed up. The way to fix sequencing is to align it to the science sequence, ideally teach it simultaneously. Partial credit for common error categories, based on work shown. I’ve become the evil math teacher that docks points for not including units in geometry (I promise I say it like forty times while they are testing!) because I want them to connect what the math is saying to eat it would look like.
I hated learning the calculator stuff in high school, because it was never explained what the calculator was doing… How the hell does x=1 make a line, when it’s a literal one dimensional data point!
It wasn’t until I was in my late twenties before watching a Numberphile video that explained exactly what a damn parabola was!
Calculators are just a symptom of a more systemic problem, like memorizing times tables. You need to teach how those numbers/lines came to be, not just ‘memorize this list that has nothing to do with 98% of the math you’ll be expected to do as an adult…’
How the hell does x=1 make a line, when it’s a literal one dimensional data point!
I think that indicates that you weren’t properly taught whenever you were learning lines. Did you continue to be confused about vertical lines when you were learning about asymptotes and rational functions?
memorize this list that has nothing to do with 98% of the math you’ll be expected to do as an adult…
Should the only things we learn at school be directly involved in what we do as an adult? Do you “use” what you know about Tom Buchanan or Huckleberry Finn in your day to day life, or were there bigger picture skills that you were reading books to develop?
A line (to me) is defined as two or more points that connect. On point it’s one point, and no line can be made until another point is defined to connect them. I was never taught the parabolic function that turned that single point into a line, the calculator did that for me.
And to your second “point”, how is learning to do the math, rather than just memorizing a list of answers, “directly involved in what we do as an adult”?
A line (to me) is defined as two or more points that connect.
And to your second “point”, how is learning to do the math, rather than just memorizing a list of answers, “directly involved in what we do as an adult”?
The heuristics are what is valuable - the ability to look at a problem, break it into smaller chunks, and the select the correct algorithm to process that smaller chunk.
Eg, cleaning my apartment: I divide the apartment into sections (bathroom, living room, kitchen, etc), then apply the correct cleaning algorithm to each room.
It’s the higher order thinking skills that really matter, and those are trained and exercised in math classes.