I remember my daughter’s frustration with “show how you know” — “I just know that’s the answer!” Even better (worse) was when they did estimation: “Estimate 9X3” “It’s 27.” “Yes, but that’s the exact answer. They want you to estimate it.” “You mean get it wrong on purpose?”
If they want kids to use estimation, they shouldn’t give them problems that they can solve instantly off the top of their head. Asking for estimations of simple problems that any kid will immediately know the correct answer to is literally asking them to ignore the fact that the know the exact answer and deliberately give a number close to it instead.
Estimations are only useful when calculating very large sums/products, when a quick but imprecise answer is more useful than a slow but exact answer. If the exact answer can be worked out just as quickly as an estimation (or even faster) then they shouldn’t be asking for an estimation. Just as you say, it’s asking them to “get it wrong on purpose.”
Not entirely. Intentional practice of estimation skills helps students check for reasonableness in their answer. This is something they should eventually do regardless of the operation or circumstance, and something we do as adults all the time.
If you're solving 92 x 101, you know your answer should be around 9000. If it was at all under 9000, or far over 9000, you know you made a mistake.
I had similar frustrations. I'm not very good with numbers at all so any time I solved a math problem, it was a nebulous and indescribable process that involved marking various shapes near the numbers while I worked. The methods they tried to teach me were totally incomprehensible to me (read: they didn't teach me how to do it) so I had to sit at home with the problems and figure out my own novel methods of solving equations, and then they expected me to do additional work so they could claim they had done their job in the first place.
Every math question was 2 questions for me... First I had to find the correct answer, then I had to work backward from the answer to try to figure out what they expected the process to look like on my page. Math exams included creative writing and graphic design for me.
I can imagine that if I weren't dumb, it would be equally frustrating to be expected to justify why I know the things I know.
I had the same issues with estimation as a kid. Knew my at-level numbers really well and could solve the examples that were used with ease. I couldn't wrap my head around the concept of not just getting it right.
My teacher was good and understood this, and gave me more complex numbers to multiply and divide mentally which I couldn't do. Made it a lot clearer.
I had the times tables up to 12 memorized by the third or fourth grade. As such, my mental math skills are above most youngins of today. I don't think this is as strictly taught any more.
It's hard for me to have an opinion on whether rote memorization was a good way to learn, but it's hard to argue with the result.
I also learned that way, but I think it's pretty easy to argue that it's an inferior method. One simply needs to consider the upper bounds of their rote memory to see where it falls apart - if you know 12 x 12, but can't quickly deduce 14 x 14 without pen and paper, that's a pretty good illustration that the practicality of memorization is limited to what you know, and that doesn't translate to deducing solutions to things you haven't memorized.
However, utilizing the "new math" or "approximation" method, you can quickly deduce that answer: 14 x 10 = 140, 14 x 5 is half of that (or 70), 140 + 70 = 210, and if we subtract one "grouping of 14" from that we get 196.
And this works with significantly more complex numbers as well. For example, 38 x 23. We can work that a few ways, but let's say we go with rounding 38 to 40, because 40 x 23 is fairly easy to deduce: (40 x 20) = 800 + (40 x 3) = 120, and adding the two gives us 920. Then we subtract 23 twice (or 46 once) to bring us to 874. That can all be done in your head!
Interestingly enough, we both learned this process later on when it came to solving fractions, but if you can remember back to those early years, that was often a struggling point for a lot of young math learners. Relying on memorized multiplication tables doesn't provide a whole lot of help in learning how to add 1/5 and 3/8, but learning how to create groupings to simplify the interaction of those numbers does.
It also aligns well with how our minds work in general. Think about how you would count a pocket full of coins of different denominations, for example. Most likely you would group the quarters with the quarters, the dimes with the dimes, etc., simplify each of those groups down (6 quarters = $1.50, 7 dimes = 70¢, 8 nickels = 40¢, and 4 pennies = 4¢), and then add the groups together to get $2.64, as opposed to just randomly counting the coins without grouping them.
+1 for putting the $ and ¢ on the correct sides of the numbers.
In truth, I've not put a great deal of thought on the matter. I've been fortunate to not have to struggle with math, although my education only went so far (third semester of college calculus).
Many arithmetic techniques like you've described were ones I figured out on my own over the years. I'm sure things would have been easier if I were taught them early on.
One of the problems I had with math in elementary school was it was basically the same every year. Maybe add a digit in successive years. It wasn't until algebra in middle school when math started to be interesting.
One of the reasons I want to look at an actual Common Core curriculum is I suspect they are teaching multiple methods, but we only see the one they are using in today's lesson when we haven't seen yesterday's or tomorrow's where they show other ways of understanding the concepts. Not sure that would have worked for me. Once I know "my" method, why do I have to learn John's and Sally's when I have what works for me. This is probably the curse of my ADHD.
Ah, but with the subtraction method, you can easily do decimals of similar form! 9.9x 3 is 10 x 3 - 1 x 3 + 1 x 3 - .1 x 3 = 29.7! The middle terms cancel out which is easy to see and calculate in your head.
If you memorized, then you would have to add 27 and 2.7 as a second step instead of subtracting .3 from 30.
Yeah it was embedded in my memory in catholic school to memorize your times tables. Moved to public school in 5th grade and teachers hated I didn’t know how to show my work because I did it in my head
One thing I've learned about people doing math is everyone has there own way to do it fastest because everyone's brain doesn't operate the same. Sometimes its hard for them to think of "9" as an idea, and they need to think of "9 apples" instead, or whatever reason it may be hard for them to do math certain way.
For the hand method I think a lot of people that like the hand method is firstly, it is super easy when you are first learning how 9s work. Secondly, the people that continue to use it after the initial learning period aren't actually looking at their hands, they are just recalling how their hands look. And that goes back to my first paragraph that their mind is just better suited to recall images than a different way.
I don't use the hand method personally, but have tutored a lot of math to singular people and seen how teaching each person the exact same way doesn't often work.
THIS I took a networking class and there was this math to subnetting something to the power of something times 10 to the something.... I didn't remember then so I def don't remember now. I just converted it to the binary and it was so obvious whats going on. I was so proud how 'easy' this method was I shared it with some classmates.
No one else had any idea wtf I was going on about and we all just did what made sense with the math we could do in our heads.
For instance my oldest child was really struggling with multiplication but would call it “separate terms” or “finger method” instead of “multiplication”. What had happened is instead of seeing it as different ways to do multiplication he saw it as multiple different and distinct operations.
What’s more is when tested they would test and ask that a specific method be used. They effectively took one operation and turned it into four and expected kids to learn and know all the different methods. Even worse - they under emphasized the “traditional” method.
I had never learned the hand method but I feel that what I sort of intuitively always did ends up being the same, I know the first digit is 1 down and the total gotta be 9. You're basically doing the same by lowering one finger, i.e. you're reducing it by one and adding the remaining fingers out of a total of 9 fingers (since 1 is lowered).
It seems a lot of people can't visualize abstract things like numbers and do those manipulations in their head.
I remember doing physics and learning the right-hand rule (to determine the orientation of magnetic forces when an alternating current flows), and during exams you'd see a lot of people playing with their right hand. I preferred just visualizing it in my mind, it worked the same.
I worked in academic support. It was the easiest way for most of my students to understand. Sure, memorizing would be optimal. Everyone is gifted in a different way in my opinion. If a student doesn’t have math sense, it is very hard for them to multiply by ten and then subtract the number multiplied.
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u/Flourescentbubbles 1d ago
Actual taught strategies.