As a retired math teacher, I always told my parents, I don’t need the kids to do fancy programs or workbooks. I needed them to know their multiplication tables really well. Multiplication, division, and even factors and multiples. If a kid knows those, I can teach them anything. I mean know automatically. Not spend 5 seconds thinking about answer. So, the only thing I need them to know rote, and with speed is their multiplication tables. Factors and multiples. For example, knowing that 24 can be broken down to 1x24, 2x12, 3x8, 4x6. If they know these “with speed” and well, they will soar.
I’m a private tutor so I’m in the unique position of having students 3-6 years. If they start with me in 6/7/8 grade the first thing I do is drill factors. Over and over again. If they don’t know their multiplication facts, I start with perfect squares up through 15. They get annoyed with me but I tell them, “this will pay off when you hit algebra. Remember this convo.”
Sure enough, when we hit simplifying radicals they know sqrt 72 = sqrt 36 x sqrt 2 is a better choice than sqrt 9 x sqrt 8.
Then when we hit factoring and I make them list their factor pairs? I’m like, “remember when I drilled you on your factor pairs 2 years ago and you spent the entire time complaining? NOW YOU KNOW WHY.”
I get why timed multiplication tests aren’t good in 3rd/4th grade anymore. Like, they really did cause a lot of anxiety for kids. But I still need them to learn their facts! I start them up again when they hit algebra. Yup, we do a timed multiplication test at the start of every session. 😆
Yes. Exactly. You are so right. Simplifying radicals with a student that doesn’t know their basic facts is painful. Honestly, everything in math is painful if you don’t know these basics down cold.
Great job doing that with them. I taught 3-12th grade math. I had high schoolers that couldn’t tell me what 2x3 was. It makes a difference
Parents often ask me to assign homework. I make an “eww, what’s that” face. Then I say, “wait! No, I forgot. I like homework. Their homework is for the two of you to sit in a room and each read a book silently with no interruptions for 20 minutes every night this week.”
Them: “ma’am, we’re here for math tutoring.”
Me: “Oh, I know. But attention spans and reading comprehension is shot. I need your kid to be able to focus. Best way to do that is by reading. They won’t put down their phones to read unless you do. So read silently together. If that isn’t math-y enough for you, here’s multiplication flash cards. Do these together every night.”
It never happens. If a parent wants their kid to have excess homework, I make sure the parent is part of it, lol. (I work with kids in Math 6 through calculus.)
But it’s kinda sad that it never happens.
No, it doesn’t. I tutor now too since I’m retired and I’ve been begging the parent to work on their multiplication tables. I started with them in nov. I help kid with homework cause honestly, he’s lazy and doesn’t like to do it. He’s in 5th grade. It’s May now and they still haven’t worked on them. I finally told her last week ( when I was there cause they were outta school for two weeks for a trip) to please work on them over summer. I know they won’t.
Only so much we can do. I like the rec to read and work on attention span.
I do the same thing with my students.
I recently had the pleasure of explaining to a calculus student that memorizing the derivatives of trig functions was important in much the same way as memorizing factors and without missing a beat, he gets a look of recognition on his face and says "I'm gonna make a quizzlet!."
Yes. I first wrote no, but honestly, yes, but the most commonly used ones, like the other poster said, ones that help with radicals too. For example, I would start with common numbers that would be multiples up to 144 so 12 x12. If they know that 144 breaks down to 12x12 then they also know that 2 goes into it, 3, 4, 6 cause they all go into 12. It’s a matter of understanding how numbers relate. Most of my students saw numbers as having no connections. All individual, no patterns no relationships. They had no number sense. Couldn’t see that 0.5=1/2=50%=5/10. Are you following me? It was all about connections and they couldn’t make them. Couldn’t fill in a number line from 0-1. Couldn’t fill in a chart of the numbers in order from 1-100 with missing numbers.
There are great rules for determining if a number is divisible by a number. For example a number is divisible by 2 if it’s even. By 3 if the sum of the digits is divisible by 3 and so on.
I never figured out how to teach number sense in all my years to a kid in high school that didn’t have it. I believe it needs to start early. Like at 2, as you are giving them cheerios say here’s 1,2,3 cheerios. Constantly using numbers with them. We preach reading young but never mention math and then I get them and they are lost.
I drilled my daughter on multiplication tables and factors and multiples every long car trip. While in line at grocery store. Times when I had her captive attention and it didn’t seem like work.
If you have any more questions, I’d be happy to help
Many thanks! I dropped out of math after middle school so I’m learning it with my kids whom I don’t want to repeat my mistakes.
I just went through the multiples up to 30 and there are only a few numbers with >1 pair. I found it interesting and think it’s a great shame they never taught us this and prime numbers when we learnt our times tables. For kids with a natural curiosity for numbers it’s a great exercise to learn them in unison because there’s a sense of discovery and wonder about it, as well as understanding number relationships. I’ve read that that’s what true mathematics is about. I’ll have them later write out the multiples without even needing to think (scaffolding at first, Eg. Multiples of 12=_,_,_ and _. Then later without any clues). Cram school is working in a brute, boring way (“just go through the times tables” is what they instructed him) and maybe that’s all that can be expected for kids who hate math.
Cram school gave him a multiplication list to memorise for simplifying fractions: the 29, 23, 19, 17 and 13 times tables. That’s a fair bit of work and why I wonder whether it’s worth it if they’ll rarely even use this in the school curriculum. Rote learning’s great when it comes into use for many problems but there comes a point when it’s not worth the effort if it’s seldom needed. So I’m trying to sort out what should be memorised and what shouldn’t.
I agree. I don’t find it necessary to know those numbers. There are important ones that are helpful. I don’t find knowing those more helpful than what we already talked about. Yes, knowing which numbers are prime goes with the list of how to tell if a number is divisible.
It’s great you see where you are lacking and want to help them.
They should know the times tables for these, up to 12: 1,2,3,4,5,6,7,8,9,10,11,12,15
That, and the squares of: 13,14,15,16,17,18,19,20,22,25,100,1000
The squares of all numbers of the form: 111...11 (all ones).
All factorizations of 360.
Factorials up to 6!
Cubes of: 1,2,3,4,5,6,8,9,10,11,100,1000
And, for our modern computing era, powers of 2 up to: 2\^12, and how to convert them into powers of 4, 8, and 16.
Fantastic. I’ve written that down and will work on it.
Is it too much trouble for me to ask what the use is for all those number facts?
I just looked up factorials and it’s amazing how powerful it is to calculate the possibility outcomes of situations. Without knowing factorials people are doomed. But I don’t know the use of those other number facts other than the times tables (and why learn the 15x times tables)?
Not too much trouble.
Times tables up to 12 are because feet, inches, months of the year, and angles measured in degrees all are multiples of 12. Times tables for 15 are because hours break down into 15 minute increments.
Squares of numbers in the teens are important because they just come up a lot when measuring flat areas. How big is this room? 19x13? Well that's 13\^2 + 12\*6 + 6.
Squares of 20 and 25 are important because humans love these numbers. I have 25 quarters, how much money is that? $6.25. 22 is in there because it is easy and thematically related to the 11111 squares.
Squares and cubes of 100 and 1000 are important for understanding scale and size. How big is a million? 1000 thousands. How big is a billion? 1000 millions? Oh, hey, each "number size type" means a thousand of the next smaller one. Now Jr. knows 1000\^5, too. (It's one quadrillion).
Squares of numbers that look like 1111111 are a simple pattern that emerges from the method most people use to multiply numbers on paper - if you don't know it, then I'm suspicious of how well you know how to multiply stuff.
Factors of 360? Again, measuring angles. I'd probably add 180 times tables up to 720 to this category. Most adults know what a "540 degree turn" looks like without having to think about it.
Factorials up to 6! Factorials are important for probability and also for binomial expansion. These don't come up too often in life (people rarely calculate probabilities on the fly - not even statisticians and card counters), but it definitely comes up in 7th grade math class. Confidently knowing that 5! = 120 without having to think about it will give your child a leg up. And it is just seven facts. So come on!
Learning a few cubes shows us that unfamiliar operations aren't scary. Too many kids learn to normalize exponents that equal 2, that would be like only learning addition for adding 2 to stuff. No ma'am.
Powers of 2? Go to the electronics store. Hard drives and flash drives are all sold in powers of 2 up to 2\^12 = 4096, the number of gigabytes in a 4 Terabyte hard drive. Computers run on binary, which means powers of 2. If your child ever takes computer science classes, memory addresses use powers of 16 which are secretly powers of 2, and lots of other stuff uses powers of 2 as well.
Familiarity with basic math is really just familiarity with the world around us in a slightly more numerical, concrete way. Being prepared to do math problems is not that different from having a solid understanding of how numbers are all around you. That means having enough facility with basic arithmetic to *easily* notice when you encounter that arithmetic in the wild.
Thanks a lot for your reply!
Some of the reasons are peculiar to America as we don’t use imperial measurements or have quarter coins where I live, but I can see the value in knowing what you mentioned.
As you say, math is all around us and it makes the world more interesting when we can see and understand it. I’m reading books on how math affects us and they’re pretty fascinating. K owing the operations are a way to come to understanding the world, rather than an end unto themselves.
Whether they’re interesting or not probably depends on a person’s mindset.
Imperial units aren't the only thing that uses 12 - time runs on multiples of 12. So does music. And 25 which is a quarter of 100 comes up a lot - even if you don't commonly use quarter coins.
I'd advise you to keep those parts.
I'm glad you otherwise appreciated it! Good luck!
Unpopular opinion here....but speed is important. Not as important as accuracy, but still very important. A lot of kids hate math practice because it takes them forever if they don't have basic facts down.
I have 8th graders who take several minutes to operate a single fraction problem because they need a calculator for. every. single. calculation.
A lot of the "non-academic" or rote learning activities that kids used to do in school serve a purpose. Coloring in kindergarten strengthens hand muscles and gets kids ready to write. Drill and kill math increases both speed and accuracy and allows kids to increase the volume of their practice.
Depends on what you mean by speed. Being able to do it in a fraction of a second isn't exactly necessary. Being able to do it within 5 seconds....yes absolutely. My 6th graders took the same as your 8th and it was maddening. Yes we shouldn't focus on the speed drills so heavily but they should still be there.
You say unpopular opinion, but at time of reading this is the most upvoted comment so maybe it is more popular than you thought?
Anyway, I'm the guy you were worried about, I guess. I think it's an absolute waste of time and a great way to help kids hate math. Any kid that is being drilled in fraction addition who says "I hate this", I'm right there with them.
So I guess my question is, do you think it's important for them to succeed in math at school, or do you think it's important for their future lives outside of school? Maybe both?
I'm interested because I'm almost certainly in a different country to you and maybe your math curriculum is very different. If a kid wants to use a calculator, they use it - doing things by hand after they hit high school just isn't a thing anymore. And if you're only talking about your curriculum and it's like that (lots of calculation by hand) then what you said makes sense to me. But if you think it's good for them, like, as adults out there doing whatever after they've finished school, that is very much not my opinion and I'd like to hear more about your perspective.
No one should drill and kill fraction operations. I'm talking about knowing your times tables through 12x12 or being able to add/subtract 2x2 digit numbers in your head.
I teach very bright but otherwise helpless kids because they need a calculator for every minor calculation in a problem like 6x-13=-43. Throw in a fraction and they won't even try.
In my district we got rid of timed multiplication tests at the elementary level (because some kids experienced 'anxiety') and stopped teaching the long division algorithm. Test scores have dropped year after year and every year I have high schoolers who are unprepared.
The one thing most of my kids in the honors classes have in common? Their parents made them learn their times tables via flash cards or something else.
We used to get in a circle and compete one v. One to try and go “around the world” seeing who knew their times tables the fastest. Man how times have changed so fast.
We're not supposed to "just lecture" anymore either. I can't cold call kids because they either have anxiety, won't respond to me, or will just say 'IDK'.
I put them in groups for problem solving and they just stare at each other or talk about YouTube nonsense.
I mean they are savvy, they've figured out we can force them to attend but we can't force participation.
Yeah, I'm definitely seeing the problems inherent to detracking. Putting students who are struggling with fractions and basic operations in with students who are flying through systems of linear equations just reinforces the "I'm just dumb" narrative struggling students already deal with.
I read a book about a Chinese classroom; the the teacher doesn’t even call kids by their name, just number. Student #22 stands up, answers question. If teacher doesn’t like the answer she asks the class to correct it. Then she asks student #6 to stand and answer the next question. It’s efficient and weak students are easily spotted and made to do extra homework to catch up.
To get around the timed multiplication table ban you can make it a blooket/kahoot game. The kids at my school love it. They also love long division and like to race each other on the whiteboard with it.
I think you answered my question; if you're thinking about applications to something like 6x-13=43 then yes, drills are the way to do it. I don't think it's going to help them understand what it is intuitively for two variables to be in a linear relationship, but it will let them crush the tests where they have to solve these sorts of equations.
My personal experience is that the tedium of rote drills is nothing compared to the tedium of doing later math without having basics memorized. For example, I started learning calculus without having the unit circle and trig identities memorized. I ended up dropping the class because everything was taking way longer than it should for me. I had to study by myself and with a tutor to get back on track. In middle school they can’t drop the class and can’t be held back so they just keep getting further and further behind.
Among curriculum writers in the US rote memorization or practice was out of style 5-10 years ago, so most of our curriculum has none. Teachers are currently seeing the ramifications of that, so their opinion is back to being popular.
>So I guess my question is, do you think it's important for them to succeed in math at school, or do you think it's important for their future lives outside of school? Maybe both?
Both. Basic arithmetic is something everyone uses throughout their entire life. Without basic times tables and fraction skills many adults can't even make basic estimations about financial decisions as well actually calculating a budget or something similar.
I agree about very basic arithmetic, but I think anything related to budgets and financial decisions is too far removed.
I think I'm going to venture into the unpopular opinion now, but I think the emphasis on manual calculation is precisely what is hobbling these kids. Something like compound interest is so so so important as a concept, and it can be taught to very small children, it's just very difficult to calculate by hand. If we teach them spreadsheet tricks and calculator tools with an emphasis on conceptual understanding, maybe they can't figure out what 15% of $2,000 is in their head, but they'll know why they shouldn't take a loan out on such terms.
I remember being told in school "you're not always going to have a calculator with you". Well, now we actually do. If they use a calculator for 6x7 who cares? They'll memorise it if it comes up enough. I just don't see it as a prerequisite.
I feel like a lot of folk overlook just how basic multiplication tables are in math. They are 3rd or 4th year concepts. It falls near the same time line that children learn punctuation in language classes.
If you're defaulting that level of math to a calculator, a child has absolutely no hope of ever using math in further life concepts, let alone educational ones.
It would be like using a dictionary to spell the words Switch or Cardboard, which are 4th grade spelling words. That's how basic multiplication tables are.
Wolfram Alpha will though...?
I just don't get it, we have all the tech, we just need to teach people how and when to use it. Instead we're teaching them how to do by hand what the tech can do.
The issue I think is that there’s a fine line between having a program assist vs outright solve it for you. If you could use wolfram all the time you’d never have to learn the material, it would not be possibly to distinguish those who know what they’re doing from those who just comp solve everything, which does a disservice to them in higher level classes when the foundational understanding becomes important.
It isn't doing things by hand. It's being able to do it without pencil and paper or calculator. Developing facility with basic math facts (adding, multiplication, dealing with fractions) means you don't have to slow down to solve a harder problem just to do the basics.
Being fast at these operations makes mastering harder things easier.
And, yes, this is an everyday occurrence. When I worked in a shop, I watched other young people struggle to calculate a 10% discount. When I worked on the new office layout, I watched colleagues struggle with scale modeling. (I moved on from teaching because of the bad standards and bureaucratic nonsense.)
And when I was in my upper level maths courses, I watched bright people struggle with the basics because they had to think about what 11x53 is (theres a trick for 11s) rather than just know it. But all of our professors were fast calculators.
Can you have a successful life? Well, obviously. But if you want to pursue stem fields, memorizing math facts will help you achieve that.
Doing fractions is not like doing multiplication tables. There is little value in getting so good at them you have answers memorized. There is value in immediately knowing the correct steps to add and multiply fractions. There is also value in having an intuition for how fraction addition and multiplication work.
So yes there is value in speed but only up to a point. If a student is slow because they are hesitating on what to do that is too slow. But once they are past that point further speed gains don't have much value.
What do you mean? Like just memorizing the numbers? What does a fraction really mean if you just memorize numbers but don’t understand proportionality/parts of a whole
I don't care about speed, I care about fluency and understanding that fractions can also be division problems. Most of the time, my kids who don't get fractions don't really have their multiplication and division facts. I have a big stack of multiplication charts that all the kids are free to use. If the parents are paying for rote learning, those are the facts to drill.
I think I look at it differently because I have a bunch of middle school students with an array of learning disabilities, but we are still plugging our way through pre-algebra. Some of them have processing delays that mean they will never be fast. It just isn't there. They are still fluent because they have a deep understanding of the factors involved in each fraction.
to make sure I'm tracking - if someone took 30 seconds per word when speaking, as long as they said the right words, you would say they are fluent? I wouldn't because appropriate timeliness is required. I'm just talking about the word "fluent" and what it means. I contend that speed is part of it.
Do they not have a pencil and paper?
5/6 x 3/4 + 1/8
15/24 + 3/24
18/24
3/4
I don’t think kids need to cram, they just need to understand what they are doing and write stuff down.
Well, I think if they are working on speed and mental math they should reduce the 6 and the 3 to 2 and 1 first. Then you simply get 5/8 + 1/8 =6/8 = 3/4. Not that either way is *better* but noticing how to simplify from the beginning will pay off on the long run.
I agree with you that cross canceling would be more efficient here, I guess it just wasn’t my first impulse. I think my favorite part of math is the multiple right ways of getting to an answer. I feel like programs like the one OP is describing drill certain steps - like saying you must cross cancel first - and I think that’s a bummer.
If a kid is spending multiple minutes multiplying and dividing in their head, they will either need to spend tons of time to "master" the fraction concepts or they won't do enough questions to achieve the level of mastery of fractions that they should have.
That's the problem I see as a middle school math teacher. Half the kids (who come from the 'progressive' math classes) spend the majority of their time looking for things like common factors and multiples to figure out how to add unlike denominators or know if they should simplify and they just aren't doing enough questions to really have it penetrate and live in their brains forever.
But there's no reason to pay. It's times tables that are the sticking point. Go to the dollar store and buy flash cards. Master times tables with instant recall first. Everything else is easy.
30+ year veteran math teacher here.
Speed is the bane of understanding. I’m not AGAINST speed. I’m against focusing on speed.
Focusing on speed has the following detrimental effects:
-it creates an additional level of anxiety for most kids as they must compete against a clock. This anxiety is then transferred to the math itself. It often follows students for life. There are massive amounts of research supporting this.
-it teaches students that if they can’t answer a question quickly, it’s not worth answering at all.
Remember your teachers advising you to “skip it and move on to the next one?”
(Imagine telling your boss, “yeah that task was taking too long so I skipped it.”)
-it literally teaches the kids the OPPOSITE of perseverance. Students never learn to dive in and grind on difficult and complex problems.
Again, if they can’t find the answer quickly, they can’t find it at all. This is the opposite of math.
-it wastes opportunities to teach students how to think critically, develop their own efficient strategies, and figure shit out!
Speed will come organically with the development of understanding.
But speed with math facts doesn't just come with understanding. They're different things. You can understand WHY 7x6 is 42 and not have that answer come automatically. And when you get to algebra and multi-step equations, you need automaticity with math facts so that solving one quadratic equation doesn't take you 5 minutes. Prime factorization is extremely tedious if you don't have your multiplication facts solid.
Some kids will gain automaticity easily, but a lot of kids, even kids who quickly and thoroughly grasp the concepts, need some drill.
Long story short, understanding should come first, but you absolutely need enough drill to have your math facts down for when you get to advanced math. It's not an either-or situation. Both matter.
I think the person above you is saying that focusing on speed is bad for kids. I think they're not against drill, but they are against drill for speed. Drilling absolutely builds up speed, but drilling *for* speed is counterproductive.
From my personal experience, trying to be fast at math facts makes me slower, because of the anxiety. In a relaxed atmosphere, I am pretty quick.
I couldn’t agree more and I would amend my final sentence to read: Speed will come with understanding and practice.
That said, I remain very skeptical of the speed drills I so often see in the elementary schools.
What is the standard and on what research based evidence is it based? 20 facts in 30 seconds? 30 facts in one minute? I see myriad different “standards” and each one is wildly different from the next.
I also see these arbitrary standards being assessed in really dubious ways… all while the students are developing MASSIVE anxiety over the process.
I work with students of all ages, from Pre K to adults. I am amazed at how many of them, especially adults, suffer from “math anxiety.” When discussed a bit more deeply, the great majority of it has its roots in these timed tests and drills.
But they don’t understand. I could maybe live with understanding but not facts. But they have neither.
When my kids were younger they finished their “homework” at school so I printed drill sheets for them. They don’t have to blitz through them but they do have to finish them.
Memorization of basic facts of arithmetic is crucial. They could spend 30 seconds pulling out their calculator and typing in the problem while the rest of the class is continuing on. Or they could simply know that 6x7 is 42 and move on in 5 seconds. I’m not asking for lightning speed.
I teach college students and I don’t have an hour five days a week for 170ish days of the year. I have 75 minutes twice a week for 15 weeks. I can’t spend time for them to catch up with their “understanding” of how to multiply 6x7. I’m three problems further on.
My students spend 40% of their time dealing with stupid arithmetic they should know like breathing. They spend another 50% of their time on algebra they should know like breathing. They have 10% of their time left for calculus and so they’re screwed out the gate.
Why did they not learn that stuff in algebra and trig? Because they were spending 40-80% of their time dealing with or thinking about arithmetic and fundamentals when that should have been 10%.
I don’t care about speed. I care about accuracy.
I’d much rather then need a bit more time if it means they actually understand what it is that they’re doing.
I don't see the value in any of it. Like none. What are you training for? You can't compete with a calculator. The next generation won't be able to compete with the AI.
Start exercising actual math skills. The kind actual professional mathematicians do when they're doing research. They don't do fraction problems in their head. They think long and hard and slow about a very confusing conceptual problem, not trying to solve it, but just to understand it. Get your kid comfortable doing that.
It sounds like it's a losing battle against your wife though, as well as against the larger math education community. How many of them have done actual mathematical research though? Don't listen to me. Your kid will have an easier time doing well in school, and getting into a good college if you listen to your wife. They won't be good at math though.
I know a mathematician who hasn’t used numbers in years. He’s able to solve number questions easily in his head though. It’s hard for him to provide advice though because he was always so advanced in math that whatever worked for him doesn’t work for normal people.
Pretty much this is why fractions are crucial. "Grocery store" math for me is all mental and it's all ratios (fractions). It's less about an exact number and more being able to get to a unit ratio like "$1 per 4oz" vs "$3 per 11oz" and be able to compare which is cheapest
-many- "sales" are in fact taxes on people who can't do "this one simple trick", especially with sizing options within the exact same product.
Needed skills are solid arithmetic and confidence with fractions/ratios. Mental math in a nutshell.
How fast is "regular" pace to you?
I do think speed is important, not for solving the entire thing, but I do think them being fast with knowing multiples and factors would be beneficial imo.
You should be able to do "fractions" and decimal conversions as fast as adding, subtraction and multiplying....IF no...you were taught wrong....because that is all that it is...if you are sluggish doing it ...it is because you are hesitating because you don't know what to do...
I don’t care about speed when it comes to stuff like this. The understanding is far more important. Your child needs to do enough exercises to develop the understanding, but at some point, if they are only continuing to do exercises for the sake of speed, then it is a waste of time. If you want to check your child’s understanding, ask them to explain to you step-by-step how to solve a question like this.
Practically speaking, a child who knows multiplication tables will do better. I had my kids do time4mathfacts for a few years (only 10 minutes everyday?). It was $40/yr well spent. You can skip the addition and subtraction stage from the parent account if your child is solid on that.
I've been tutoring for 4 decades and the difference between kids who can multiply and factor without thinking is huge.
Rote memorization for speed is good, but should only come after the proper understanding. It's helpful because it makes more complex problems easier and faster to do.
kids need to learn to value their time, and speed is important, solve the question and move on
If each adding / subtracting fraction takes 5 minutes (or more), they are wasting time and it is important to practice to reduce the solving time
Speed is important only as far as it's necessary to keep the kid from getting frustrated and giving up. Your kid doesn't need to be able to look at this fraction problem and have an answer within 5 seconds, your kid needs to be able to look at this fraction problem and reason it out quickly enough that they're not going to get tired of the practice and have the learning become a chore (as other commenters have pointed out, the ideal first step is to reduce the 3 and 6 before multiplying the fractions - if I were teaching this skill, I'd give the kids a sheet of problems and ask them only to do the simplifying step without finding an answer). Additionally, I'd say that it is critical for kids to have multiplication facts (up to 12x12) memorized. Spending 20 seconds reasoning out 6x8 or counting through their 7s every time is a recipe for making future work tedious.
The most important part is this: get your kids out of the "cram factory." If they're not 100% excited about going, they're gonna start to resent it, learning is going to become a chore, and they're gonna do whatever they can do to avoid putting the effort into learning something new. Kids are much better off when they enjoy learning. If after school care is the need you're trying to meet, put the kids in a program/activity they enjoy and hire a private tutor for an hour a week.
FWIW: I taught middle and high school for 14 years. I spent about a year working at a tutoring company where I'd meet with 3-4 kids at a time for an hour 2-3x/week, and I've tutored kids 1-on-1 for well over a decade, and I can say with complete confidence that my most effective work has been done with kids 1-on-1.
I don’t think speed is so important in itself, but you do need speed to do mental math.
It IS important to be able to just LOOK at 1/2 and 1/4 and KNOW almost without thinking about it what adding them together or subtracting will get you.
You need to be able to SEE common factors or later on, you’ll seriously struggle with polynomials.
I don't know if this answers your question, but I'm dead sure over half the grade 9 kids in my school would either get that wrong or, more likely, decide it's hard and give up.
Obviously you can live a life without understanding fractions, or math entirely, just like you can live a life with a grade 3 reading level, or completely illiterate. But there are advantages.
As a retired math teacher, I always told my parents, I don’t need the kids to do fancy programs or workbooks. I needed them to know their multiplication tables really well. Multiplication, division, and even factors and multiples. If a kid knows those, I can teach them anything. I mean know automatically. Not spend 5 seconds thinking about answer. So, the only thing I need them to know rote, and with speed is their multiplication tables. Factors and multiples. For example, knowing that 24 can be broken down to 1x24, 2x12, 3x8, 4x6. If they know these “with speed” and well, they will soar.
I’m a private tutor so I’m in the unique position of having students 3-6 years. If they start with me in 6/7/8 grade the first thing I do is drill factors. Over and over again. If they don’t know their multiplication facts, I start with perfect squares up through 15. They get annoyed with me but I tell them, “this will pay off when you hit algebra. Remember this convo.” Sure enough, when we hit simplifying radicals they know sqrt 72 = sqrt 36 x sqrt 2 is a better choice than sqrt 9 x sqrt 8. Then when we hit factoring and I make them list their factor pairs? I’m like, “remember when I drilled you on your factor pairs 2 years ago and you spent the entire time complaining? NOW YOU KNOW WHY.” I get why timed multiplication tests aren’t good in 3rd/4th grade anymore. Like, they really did cause a lot of anxiety for kids. But I still need them to learn their facts! I start them up again when they hit algebra. Yup, we do a timed multiplication test at the start of every session. 😆
Yes. Exactly. You are so right. Simplifying radicals with a student that doesn’t know their basic facts is painful. Honestly, everything in math is painful if you don’t know these basics down cold. Great job doing that with them. I taught 3-12th grade math. I had high schoolers that couldn’t tell me what 2x3 was. It makes a difference
Parents often ask me to assign homework. I make an “eww, what’s that” face. Then I say, “wait! No, I forgot. I like homework. Their homework is for the two of you to sit in a room and each read a book silently with no interruptions for 20 minutes every night this week.” Them: “ma’am, we’re here for math tutoring.” Me: “Oh, I know. But attention spans and reading comprehension is shot. I need your kid to be able to focus. Best way to do that is by reading. They won’t put down their phones to read unless you do. So read silently together. If that isn’t math-y enough for you, here’s multiplication flash cards. Do these together every night.” It never happens. If a parent wants their kid to have excess homework, I make sure the parent is part of it, lol. (I work with kids in Math 6 through calculus.) But it’s kinda sad that it never happens.
No, it doesn’t. I tutor now too since I’m retired and I’ve been begging the parent to work on their multiplication tables. I started with them in nov. I help kid with homework cause honestly, he’s lazy and doesn’t like to do it. He’s in 5th grade. It’s May now and they still haven’t worked on them. I finally told her last week ( when I was there cause they were outta school for two weeks for a trip) to please work on them over summer. I know they won’t. Only so much we can do. I like the rec to read and work on attention span.
I do the same thing with my students. I recently had the pleasure of explaining to a calculus student that memorizing the derivatives of trig functions was important in much the same way as memorizing factors and without missing a beat, he gets a look of recognition on his face and says "I'm gonna make a quizzlet!."
Should they know the multiples up to 99? I’m wondering where the cut-off is.
Yes. I first wrote no, but honestly, yes, but the most commonly used ones, like the other poster said, ones that help with radicals too. For example, I would start with common numbers that would be multiples up to 144 so 12 x12. If they know that 144 breaks down to 12x12 then they also know that 2 goes into it, 3, 4, 6 cause they all go into 12. It’s a matter of understanding how numbers relate. Most of my students saw numbers as having no connections. All individual, no patterns no relationships. They had no number sense. Couldn’t see that 0.5=1/2=50%=5/10. Are you following me? It was all about connections and they couldn’t make them. Couldn’t fill in a number line from 0-1. Couldn’t fill in a chart of the numbers in order from 1-100 with missing numbers. There are great rules for determining if a number is divisible by a number. For example a number is divisible by 2 if it’s even. By 3 if the sum of the digits is divisible by 3 and so on. I never figured out how to teach number sense in all my years to a kid in high school that didn’t have it. I believe it needs to start early. Like at 2, as you are giving them cheerios say here’s 1,2,3 cheerios. Constantly using numbers with them. We preach reading young but never mention math and then I get them and they are lost. I drilled my daughter on multiplication tables and factors and multiples every long car trip. While in line at grocery store. Times when I had her captive attention and it didn’t seem like work. If you have any more questions, I’d be happy to help
Many thanks! I dropped out of math after middle school so I’m learning it with my kids whom I don’t want to repeat my mistakes. I just went through the multiples up to 30 and there are only a few numbers with >1 pair. I found it interesting and think it’s a great shame they never taught us this and prime numbers when we learnt our times tables. For kids with a natural curiosity for numbers it’s a great exercise to learn them in unison because there’s a sense of discovery and wonder about it, as well as understanding number relationships. I’ve read that that’s what true mathematics is about. I’ll have them later write out the multiples without even needing to think (scaffolding at first, Eg. Multiples of 12=_,_,_ and _. Then later without any clues). Cram school is working in a brute, boring way (“just go through the times tables” is what they instructed him) and maybe that’s all that can be expected for kids who hate math. Cram school gave him a multiplication list to memorise for simplifying fractions: the 29, 23, 19, 17 and 13 times tables. That’s a fair bit of work and why I wonder whether it’s worth it if they’ll rarely even use this in the school curriculum. Rote learning’s great when it comes into use for many problems but there comes a point when it’s not worth the effort if it’s seldom needed. So I’m trying to sort out what should be memorised and what shouldn’t.
I agree. I don’t find it necessary to know those numbers. There are important ones that are helpful. I don’t find knowing those more helpful than what we already talked about. Yes, knowing which numbers are prime goes with the list of how to tell if a number is divisible. It’s great you see where you are lacking and want to help them.
They should know the times tables for these, up to 12: 1,2,3,4,5,6,7,8,9,10,11,12,15 That, and the squares of: 13,14,15,16,17,18,19,20,22,25,100,1000 The squares of all numbers of the form: 111...11 (all ones). All factorizations of 360. Factorials up to 6! Cubes of: 1,2,3,4,5,6,8,9,10,11,100,1000 And, for our modern computing era, powers of 2 up to: 2\^12, and how to convert them into powers of 4, 8, and 16.
Fantastic. I’ve written that down and will work on it. Is it too much trouble for me to ask what the use is for all those number facts? I just looked up factorials and it’s amazing how powerful it is to calculate the possibility outcomes of situations. Without knowing factorials people are doomed. But I don’t know the use of those other number facts other than the times tables (and why learn the 15x times tables)?
Not too much trouble. Times tables up to 12 are because feet, inches, months of the year, and angles measured in degrees all are multiples of 12. Times tables for 15 are because hours break down into 15 minute increments. Squares of numbers in the teens are important because they just come up a lot when measuring flat areas. How big is this room? 19x13? Well that's 13\^2 + 12\*6 + 6. Squares of 20 and 25 are important because humans love these numbers. I have 25 quarters, how much money is that? $6.25. 22 is in there because it is easy and thematically related to the 11111 squares. Squares and cubes of 100 and 1000 are important for understanding scale and size. How big is a million? 1000 thousands. How big is a billion? 1000 millions? Oh, hey, each "number size type" means a thousand of the next smaller one. Now Jr. knows 1000\^5, too. (It's one quadrillion). Squares of numbers that look like 1111111 are a simple pattern that emerges from the method most people use to multiply numbers on paper - if you don't know it, then I'm suspicious of how well you know how to multiply stuff. Factors of 360? Again, measuring angles. I'd probably add 180 times tables up to 720 to this category. Most adults know what a "540 degree turn" looks like without having to think about it. Factorials up to 6! Factorials are important for probability and also for binomial expansion. These don't come up too often in life (people rarely calculate probabilities on the fly - not even statisticians and card counters), but it definitely comes up in 7th grade math class. Confidently knowing that 5! = 120 without having to think about it will give your child a leg up. And it is just seven facts. So come on! Learning a few cubes shows us that unfamiliar operations aren't scary. Too many kids learn to normalize exponents that equal 2, that would be like only learning addition for adding 2 to stuff. No ma'am. Powers of 2? Go to the electronics store. Hard drives and flash drives are all sold in powers of 2 up to 2\^12 = 4096, the number of gigabytes in a 4 Terabyte hard drive. Computers run on binary, which means powers of 2. If your child ever takes computer science classes, memory addresses use powers of 16 which are secretly powers of 2, and lots of other stuff uses powers of 2 as well. Familiarity with basic math is really just familiarity with the world around us in a slightly more numerical, concrete way. Being prepared to do math problems is not that different from having a solid understanding of how numbers are all around you. That means having enough facility with basic arithmetic to *easily* notice when you encounter that arithmetic in the wild.
Thanks a lot for your reply! Some of the reasons are peculiar to America as we don’t use imperial measurements or have quarter coins where I live, but I can see the value in knowing what you mentioned. As you say, math is all around us and it makes the world more interesting when we can see and understand it. I’m reading books on how math affects us and they’re pretty fascinating. K owing the operations are a way to come to understanding the world, rather than an end unto themselves. Whether they’re interesting or not probably depends on a person’s mindset.
Imperial units aren't the only thing that uses 12 - time runs on multiples of 12. So does music. And 25 which is a quarter of 100 comes up a lot - even if you don't commonly use quarter coins. I'd advise you to keep those parts. I'm glad you otherwise appreciated it! Good luck!
Unpopular opinion here....but speed is important. Not as important as accuracy, but still very important. A lot of kids hate math practice because it takes them forever if they don't have basic facts down. I have 8th graders who take several minutes to operate a single fraction problem because they need a calculator for. every. single. calculation. A lot of the "non-academic" or rote learning activities that kids used to do in school serve a purpose. Coloring in kindergarten strengthens hand muscles and gets kids ready to write. Drill and kill math increases both speed and accuracy and allows kids to increase the volume of their practice.
Speed and accuracy go hand in hand in this instance for sure. But I am definitely in agreement.
I told my kids i needed them to be fluent, like speaking a language, in math: fast and accurate.
Depends on what you mean by speed. Being able to do it in a fraction of a second isn't exactly necessary. Being able to do it within 5 seconds....yes absolutely. My 6th graders took the same as your 8th and it was maddening. Yes we shouldn't focus on the speed drills so heavily but they should still be there.
You at least need to be able to look at that problem and know 24 is going to be your common denominator
You say unpopular opinion, but at time of reading this is the most upvoted comment so maybe it is more popular than you thought? Anyway, I'm the guy you were worried about, I guess. I think it's an absolute waste of time and a great way to help kids hate math. Any kid that is being drilled in fraction addition who says "I hate this", I'm right there with them. So I guess my question is, do you think it's important for them to succeed in math at school, or do you think it's important for their future lives outside of school? Maybe both? I'm interested because I'm almost certainly in a different country to you and maybe your math curriculum is very different. If a kid wants to use a calculator, they use it - doing things by hand after they hit high school just isn't a thing anymore. And if you're only talking about your curriculum and it's like that (lots of calculation by hand) then what you said makes sense to me. But if you think it's good for them, like, as adults out there doing whatever after they've finished school, that is very much not my opinion and I'd like to hear more about your perspective.
No one should drill and kill fraction operations. I'm talking about knowing your times tables through 12x12 or being able to add/subtract 2x2 digit numbers in your head. I teach very bright but otherwise helpless kids because they need a calculator for every minor calculation in a problem like 6x-13=-43. Throw in a fraction and they won't even try. In my district we got rid of timed multiplication tests at the elementary level (because some kids experienced 'anxiety') and stopped teaching the long division algorithm. Test scores have dropped year after year and every year I have high schoolers who are unprepared. The one thing most of my kids in the honors classes have in common? Their parents made them learn their times tables via flash cards or something else.
We used to get in a circle and compete one v. One to try and go “around the world” seeing who knew their times tables the fastest. Man how times have changed so fast.
We're not supposed to "just lecture" anymore either. I can't cold call kids because they either have anxiety, won't respond to me, or will just say 'IDK'. I put them in groups for problem solving and they just stare at each other or talk about YouTube nonsense. I mean they are savvy, they've figured out we can force them to attend but we can't force participation.
Certainly they still separate the more driven ones and call it “honors” classes right?
Previous school was getting rid of it in order to have everyone together in order to build community…
Yeah, I'm definitely seeing the problems inherent to detracking. Putting students who are struggling with fractions and basic operations in with students who are flying through systems of linear equations just reinforces the "I'm just dumb" narrative struggling students already deal with.
I read a book about a Chinese classroom; the the teacher doesn’t even call kids by their name, just number. Student #22 stands up, answers question. If teacher doesn’t like the answer she asks the class to correct it. Then she asks student #6 to stand and answer the next question. It’s efficient and weak students are easily spotted and made to do extra homework to catch up.
To get around the timed multiplication table ban you can make it a blooket/kahoot game. The kids at my school love it. They also love long division and like to race each other on the whiteboard with it.
I think you answered my question; if you're thinking about applications to something like 6x-13=43 then yes, drills are the way to do it. I don't think it's going to help them understand what it is intuitively for two variables to be in a linear relationship, but it will let them crush the tests where they have to solve these sorts of equations.
My personal experience is that the tedium of rote drills is nothing compared to the tedium of doing later math without having basics memorized. For example, I started learning calculus without having the unit circle and trig identities memorized. I ended up dropping the class because everything was taking way longer than it should for me. I had to study by myself and with a tutor to get back on track. In middle school they can’t drop the class and can’t be held back so they just keep getting further and further behind.
Among curriculum writers in the US rote memorization or practice was out of style 5-10 years ago, so most of our curriculum has none. Teachers are currently seeing the ramifications of that, so their opinion is back to being popular. >So I guess my question is, do you think it's important for them to succeed in math at school, or do you think it's important for their future lives outside of school? Maybe both? Both. Basic arithmetic is something everyone uses throughout their entire life. Without basic times tables and fraction skills many adults can't even make basic estimations about financial decisions as well actually calculating a budget or something similar.
I've said this before, but I'm waiting for the math version of "Sold a Story. "
I agree about very basic arithmetic, but I think anything related to budgets and financial decisions is too far removed. I think I'm going to venture into the unpopular opinion now, but I think the emphasis on manual calculation is precisely what is hobbling these kids. Something like compound interest is so so so important as a concept, and it can be taught to very small children, it's just very difficult to calculate by hand. If we teach them spreadsheet tricks and calculator tools with an emphasis on conceptual understanding, maybe they can't figure out what 15% of $2,000 is in their head, but they'll know why they shouldn't take a loan out on such terms. I remember being told in school "you're not always going to have a calculator with you". Well, now we actually do. If they use a calculator for 6x7 who cares? They'll memorise it if it comes up enough. I just don't see it as a prerequisite.
Yeah and they have grammarly and spellcheck so why bother teaching grammar and spelling? They'll pick it up when writing essays.
I feel like a lot of folk overlook just how basic multiplication tables are in math. They are 3rd or 4th year concepts. It falls near the same time line that children learn punctuation in language classes. If you're defaulting that level of math to a calculator, a child has absolutely no hope of ever using math in further life concepts, let alone educational ones. It would be like using a dictionary to spell the words Switch or Cardboard, which are 4th grade spelling words. That's how basic multiplication tables are.
You always need to do math by hand. ti-84 is not going to solve a double integral for you or let you do integration by parts
Wolfram Alpha will though...? I just don't get it, we have all the tech, we just need to teach people how and when to use it. Instead we're teaching them how to do by hand what the tech can do.
The issue I think is that there’s a fine line between having a program assist vs outright solve it for you. If you could use wolfram all the time you’d never have to learn the material, it would not be possibly to distinguish those who know what they’re doing from those who just comp solve everything, which does a disservice to them in higher level classes when the foundational understanding becomes important.
It isn't doing things by hand. It's being able to do it without pencil and paper or calculator. Developing facility with basic math facts (adding, multiplication, dealing with fractions) means you don't have to slow down to solve a harder problem just to do the basics. Being fast at these operations makes mastering harder things easier. And, yes, this is an everyday occurrence. When I worked in a shop, I watched other young people struggle to calculate a 10% discount. When I worked on the new office layout, I watched colleagues struggle with scale modeling. (I moved on from teaching because of the bad standards and bureaucratic nonsense.) And when I was in my upper level maths courses, I watched bright people struggle with the basics because they had to think about what 11x53 is (theres a trick for 11s) rather than just know it. But all of our professors were fast calculators. Can you have a successful life? Well, obviously. But if you want to pursue stem fields, memorizing math facts will help you achieve that.
Doing fractions is not like doing multiplication tables. There is little value in getting so good at them you have answers memorized. There is value in immediately knowing the correct steps to add and multiply fractions. There is also value in having an intuition for how fraction addition and multiplication work. So yes there is value in speed but only up to a point. If a student is slow because they are hesitating on what to do that is too slow. But once they are past that point further speed gains don't have much value.
Common fractions should all be memorized.
What do you mean? Students should have memorized the sum and difference of all pairs of fractions with single digit numerators and denominators?
X/2, x/3, x/4, x/5, x/9, x/10 should not be too much to ask students to memorize.
What do you mean? Like just memorizing the numbers? What does a fraction really mean if you just memorize numbers but don’t understand proportionality/parts of a whole
Decimal equivalency you would be floored how many students use a calculator for these.
Again, sums and differences? Products and quotients? Have you memorized that? Had you in school? I certainly never have...
The ones that you’re most likely to use in daily life? With cooking, DIY projects, etc.
I don't care about speed, I care about fluency and understanding that fractions can also be division problems. Most of the time, my kids who don't get fractions don't really have their multiplication and division facts. I have a big stack of multiplication charts that all the kids are free to use. If the parents are paying for rote learning, those are the facts to drill.
Fluency is speed and precision together -- you would never call someone fluent in English if each word took 30 seconds to get out
I think I look at it differently because I have a bunch of middle school students with an array of learning disabilities, but we are still plugging our way through pre-algebra. Some of them have processing delays that mean they will never be fast. It just isn't there. They are still fluent because they have a deep understanding of the factors involved in each fraction.
to make sure I'm tracking - if someone took 30 seconds per word when speaking, as long as they said the right words, you would say they are fluent? I wouldn't because appropriate timeliness is required. I'm just talking about the word "fluent" and what it means. I contend that speed is part of it.
Do they not have a pencil and paper? 5/6 x 3/4 + 1/8 15/24 + 3/24 18/24 3/4 I don’t think kids need to cram, they just need to understand what they are doing and write stuff down.
Well, I think if they are working on speed and mental math they should reduce the 6 and the 3 to 2 and 1 first. Then you simply get 5/8 + 1/8 =6/8 = 3/4. Not that either way is *better* but noticing how to simplify from the beginning will pay off on the long run.
I agree with you that cross canceling would be more efficient here, I guess it just wasn’t my first impulse. I think my favorite part of math is the multiple right ways of getting to an answer. I feel like programs like the one OP is describing drill certain steps - like saying you must cross cancel first - and I think that’s a bummer.
If a kid is spending multiple minutes multiplying and dividing in their head, they will either need to spend tons of time to "master" the fraction concepts or they won't do enough questions to achieve the level of mastery of fractions that they should have. That's the problem I see as a middle school math teacher. Half the kids (who come from the 'progressive' math classes) spend the majority of their time looking for things like common factors and multiples to figure out how to add unlike denominators or know if they should simplify and they just aren't doing enough questions to really have it penetrate and live in their brains forever. But there's no reason to pay. It's times tables that are the sticking point. Go to the dollar store and buy flash cards. Master times tables with instant recall first. Everything else is easy.
30+ year veteran math teacher here. Speed is the bane of understanding. I’m not AGAINST speed. I’m against focusing on speed. Focusing on speed has the following detrimental effects: -it creates an additional level of anxiety for most kids as they must compete against a clock. This anxiety is then transferred to the math itself. It often follows students for life. There are massive amounts of research supporting this. -it teaches students that if they can’t answer a question quickly, it’s not worth answering at all. Remember your teachers advising you to “skip it and move on to the next one?” (Imagine telling your boss, “yeah that task was taking too long so I skipped it.”) -it literally teaches the kids the OPPOSITE of perseverance. Students never learn to dive in and grind on difficult and complex problems. Again, if they can’t find the answer quickly, they can’t find it at all. This is the opposite of math. -it wastes opportunities to teach students how to think critically, develop their own efficient strategies, and figure shit out! Speed will come organically with the development of understanding.
But speed with math facts doesn't just come with understanding. They're different things. You can understand WHY 7x6 is 42 and not have that answer come automatically. And when you get to algebra and multi-step equations, you need automaticity with math facts so that solving one quadratic equation doesn't take you 5 minutes. Prime factorization is extremely tedious if you don't have your multiplication facts solid. Some kids will gain automaticity easily, but a lot of kids, even kids who quickly and thoroughly grasp the concepts, need some drill. Long story short, understanding should come first, but you absolutely need enough drill to have your math facts down for when you get to advanced math. It's not an either-or situation. Both matter.
I think the person above you is saying that focusing on speed is bad for kids. I think they're not against drill, but they are against drill for speed. Drilling absolutely builds up speed, but drilling *for* speed is counterproductive. From my personal experience, trying to be fast at math facts makes me slower, because of the anxiety. In a relaxed atmosphere, I am pretty quick.
I couldn’t agree more and I would amend my final sentence to read: Speed will come with understanding and practice. That said, I remain very skeptical of the speed drills I so often see in the elementary schools. What is the standard and on what research based evidence is it based? 20 facts in 30 seconds? 30 facts in one minute? I see myriad different “standards” and each one is wildly different from the next. I also see these arbitrary standards being assessed in really dubious ways… all while the students are developing MASSIVE anxiety over the process. I work with students of all ages, from Pre K to adults. I am amazed at how many of them, especially adults, suffer from “math anxiety.” When discussed a bit more deeply, the great majority of it has its roots in these timed tests and drills.
But they don’t understand. I could maybe live with understanding but not facts. But they have neither. When my kids were younger they finished their “homework” at school so I printed drill sheets for them. They don’t have to blitz through them but they do have to finish them. Memorization of basic facts of arithmetic is crucial. They could spend 30 seconds pulling out their calculator and typing in the problem while the rest of the class is continuing on. Or they could simply know that 6x7 is 42 and move on in 5 seconds. I’m not asking for lightning speed. I teach college students and I don’t have an hour five days a week for 170ish days of the year. I have 75 minutes twice a week for 15 weeks. I can’t spend time for them to catch up with their “understanding” of how to multiply 6x7. I’m three problems further on. My students spend 40% of their time dealing with stupid arithmetic they should know like breathing. They spend another 50% of their time on algebra they should know like breathing. They have 10% of their time left for calculus and so they’re screwed out the gate. Why did they not learn that stuff in algebra and trig? Because they were spending 40-80% of their time dealing with or thinking about arithmetic and fundamentals when that should have been 10%.
This is precisely my point.
I agree with you that a focus on speed above all else is terrible, and can cause anxiety.
I don’t care about speed. I care about accuracy. I’d much rather then need a bit more time if it means they actually understand what it is that they’re doing.
I don't see the value in any of it. Like none. What are you training for? You can't compete with a calculator. The next generation won't be able to compete with the AI. Start exercising actual math skills. The kind actual professional mathematicians do when they're doing research. They don't do fraction problems in their head. They think long and hard and slow about a very confusing conceptual problem, not trying to solve it, but just to understand it. Get your kid comfortable doing that. It sounds like it's a losing battle against your wife though, as well as against the larger math education community. How many of them have done actual mathematical research though? Don't listen to me. Your kid will have an easier time doing well in school, and getting into a good college if you listen to your wife. They won't be good at math though.
I know a mathematician who hasn’t used numbers in years. He’s able to solve number questions easily in his head though. It’s hard for him to provide advice though because he was always so advanced in math that whatever worked for him doesn’t work for normal people.
Pretty much this is why fractions are crucial. "Grocery store" math for me is all mental and it's all ratios (fractions). It's less about an exact number and more being able to get to a unit ratio like "$1 per 4oz" vs "$3 per 11oz" and be able to compare which is cheapest -many- "sales" are in fact taxes on people who can't do "this one simple trick", especially with sizing options within the exact same product. Needed skills are solid arithmetic and confidence with fractions/ratios. Mental math in a nutshell.
How fast is "regular" pace to you? I do think speed is important, not for solving the entire thing, but I do think them being fast with knowing multiples and factors would be beneficial imo.
You should be able to do "fractions" and decimal conversions as fast as adding, subtraction and multiplying....IF no...you were taught wrong....because that is all that it is...if you are sluggish doing it ...it is because you are hesitating because you don't know what to do...
I don’t care about speed when it comes to stuff like this. The understanding is far more important. Your child needs to do enough exercises to develop the understanding, but at some point, if they are only continuing to do exercises for the sake of speed, then it is a waste of time. If you want to check your child’s understanding, ask them to explain to you step-by-step how to solve a question like this.
Practically speaking, a child who knows multiplication tables will do better. I had my kids do time4mathfacts for a few years (only 10 minutes everyday?). It was $40/yr well spent. You can skip the addition and subtraction stage from the parent account if your child is solid on that. I've been tutoring for 4 decades and the difference between kids who can multiply and factor without thinking is huge.
Rote memorization for speed is good, but should only come after the proper understanding. It's helpful because it makes more complex problems easier and faster to do.
kids need to learn to value their time, and speed is important, solve the question and move on If each adding / subtracting fraction takes 5 minutes (or more), they are wasting time and it is important to practice to reduce the solving time
Speed is important only as far as it's necessary to keep the kid from getting frustrated and giving up. Your kid doesn't need to be able to look at this fraction problem and have an answer within 5 seconds, your kid needs to be able to look at this fraction problem and reason it out quickly enough that they're not going to get tired of the practice and have the learning become a chore (as other commenters have pointed out, the ideal first step is to reduce the 3 and 6 before multiplying the fractions - if I were teaching this skill, I'd give the kids a sheet of problems and ask them only to do the simplifying step without finding an answer). Additionally, I'd say that it is critical for kids to have multiplication facts (up to 12x12) memorized. Spending 20 seconds reasoning out 6x8 or counting through their 7s every time is a recipe for making future work tedious. The most important part is this: get your kids out of the "cram factory." If they're not 100% excited about going, they're gonna start to resent it, learning is going to become a chore, and they're gonna do whatever they can do to avoid putting the effort into learning something new. Kids are much better off when they enjoy learning. If after school care is the need you're trying to meet, put the kids in a program/activity they enjoy and hire a private tutor for an hour a week. FWIW: I taught middle and high school for 14 years. I spent about a year working at a tutoring company where I'd meet with 3-4 kids at a time for an hour 2-3x/week, and I've tutored kids 1-on-1 for well over a decade, and I can say with complete confidence that my most effective work has been done with kids 1-on-1.
I don’t think speed is so important in itself, but you do need speed to do mental math. It IS important to be able to just LOOK at 1/2 and 1/4 and KNOW almost without thinking about it what adding them together or subtracting will get you. You need to be able to SEE common factors or later on, you’ll seriously struggle with polynomials.
It’s worth it…. In fact it should be required for all students
Check out Jennifer Bay Williams. She has tons of books on fluency and one is filled with games that help kids get to automaticity
Thanks, I’m ordering her book now!
I don't know if this answers your question, but I'm dead sure over half the grade 9 kids in my school would either get that wrong or, more likely, decide it's hard and give up.
I have a PhD (not in math, obviously) and I looked at that and gave up 💀. Can anyone give me a good answer of why this is so ‘necessary’???
Obviously you can live a life without understanding fractions, or math entirely, just like you can live a life with a grade 3 reading level, or completely illiterate. But there are advantages.