HW solutions got posted online

I had a cheating case last semester where it was abundantly clear that the students had a copy of a previous year’s homework solutions.  They even had the same typos.  They denied it and I had no proof, other than everything being identical, but I did search around and found that although that specific solution hadn’t been posted online, everything for my second semester class had.  (And a couple of assignments from the first semester class had as well.)

It looked like nothing important had been posted to chegg, but a lot of stuff was on course hero.

Irritatingly, I even recognized the names of the former students who had posted the solutions.  One of them had even worked for me as an RA!

If you’re teaching a class that has regular homework assignments, then you might want to take a moment to google your name and the class number or the full class number (Ex. ECON 411) just to see if your stuff has also been posted.  Then check out the major sites (course hero, chegg… I’m not sure what else) and search through them for your name or the course number.

It is relatively easy to put in a take-down notice– you don’t even need to sign up for the site.  You do have to give them your professional information and you have to say yes, this is mine, but that seems reasonable– if they really want my university number it’s on the university website for anyone to see.  Heck, it’s on the syllabi of mine they have posted.  (I don’t mind them having the syllabi, so I didn’t put take-down notices for that.)

For one of my requests they sent an email asking me to click a link or something, but for the others they just took the solutions down after a couple of days and informed me of the fact.

Do you teach any classes where students can copy solutions?  How do you deal with cheating?  How do you deal with solutions being readily available?

Books of fun math games

This post was in my drafts from 2013.  Amazon links are sponsored, though we don’t get a ton from them so definitely buy elsewhere if you can support a local bookstore.

In order of difficulty:

Math for Smarty Pants is a book from the 1980s that’s still in print because it’s still good.  It’s definitely for elementary school kids.

Sadly you can only get Aha! by Martin Gardner used.  It is a fun book of math puzzles for all ages, including adults.  There’s comics for kids and text for adults.

Gotcha! by Martin Gardner is even more fun because it introduces paradoxes, which are mind-blowing.

You can also hit the Car Talk archives for their weekly puzzlers.

We never really fell in love with Family Math, but some people like it a lot.

Related:  tangrams are fun geometry puzzles

Dragonbox apps for your favorite tablet device are also really great

DC2 has discovered Square One on youtube.  They don’t have every episode, but what they do have is fun.

Do you have any fun math suggestions?

Complaint about AP stats: one tail hypothesis testing

Apologies for people who don’t care…

But DC1 needed help on hypothesis testing, which is totally understandable since it’s non-intuitive for a lot of people (including me!), particularly the one-tailed tests.

So, most of us who use stats regularly are used to the idea of two-tail t-testing– that’s when we want to know if there is actually a relationship between two variables or if it’s just random chance making it look like two variables are related.  This is kind of the essence of regression analysis– we want to know if we can be 95% ok with the idea that the coefficient we are getting is different from zero.  Two-tailed t-tests are used for a lot of other things besides regression, like seeing if a variable is different from a mean or two means are different from each other, but the basic idea is we want to know if things are the same or different.  We don’t assign any direction to the difference– we don’t say which is bigger or smaller, just that the sample was drawn from a distribution with the same/different mean, or the two samples were drawn from one distribution or from two separate distributions.

Hypotheses then look like:

H1: μ1≠μ2

H0: μ1=μ2

With a one-tailed test you are applying a direction– you’re saying that it could be equal, but it could also be going the other direction.  Your alternative (H1) hypothesis is that μ1 is bigger than μ2.  (Or that μ1 is smaller than μ2, depending on what you’re trying to show.)

So the hypothesis should look like:

H1: μ1>μ2

H0: μ1≤μ2

There are a number of different ways to write out H0.  You could write > but then cross it out (I couldn’t find that character in word).  I tell my students they can write “We cannot say that μ1>μ2,” because that’s really the point.  We don’t know if μ1=μ2 or if μ1<μ2 or… even if μ1>μ2 and we just have too small of a sample size to say for sure.  (My students generally will be doing practical t-testing for non-academic employers rather than writing up fancy scientific papers that require formal hypotheses, so it’s more important for them to understand what they’re doing and what their outputs mean than it is for them to follow a specific jargon structure.)

Imagine my astonishment when the AP review sheet DC1 had showed this instead:

H1: μ1>μ2

H0: μ1=μ2

Surely the teacher made a mistake, I thought.  But no!  This is how all the AP stuff is.  This is genuinely what they’re teaching and testing for AP stats.

The problem is that when you formulate it like this, you’re not allowing for the possibility of μ1<μ2, and you’re making kids think that you’ve actually proven the null hypothesis, that you can really say that μ1=μ2 when really all you can do is say you’re not sure if μ1>μ2.

I took a picture and texted it to my friends without comment and got replies like, “What fresh hell is this?” and “!!, No!  less than or equal.” And they were astonished to find it was AP Stats and for real, not just a mistake in the notes.

No wonder some of my students who come in to my class after having taken AP Stats never really got hypothesis testing the first time around.

Selection into the sample on the variable of interest: A Rant

First, a disclaimer:  I dislike mike the mad biologist.  I blame men like him for Trump being president because of his relentless attacks on ‘this woman, I would support any woman for president, just not this one” and the fact that he’s one of those assholes who does absolutely zero work and again, attacks people who are actually doing something as always doing the wrong thing.  Those men make doing things that they claim to support more difficult and emotionally draining while not lifting a finger themselves.  The only reason I ever look at his blog is because he’s on Bardiac’s blogroll and occasionally he’ll have one of those post headlines that I have to click.  Previously I thought he was just an unselfaware misogynist blowhard.  Now, I’m realizing that he’s … also not got very good statistics training.  (Interestingly, I’ve seen a recent survey that finds that people who are explicitly racist also just tend to be wrong about other things unrelated to their racism.  Not saying that translates to implicit misogyny, but… )

Ok, so here’s the post in question.  In it, he claims that a survey of people who aren’t getting vaccinated proves that time pressure and inability to take a day off work are not reasons.  (Therefore any policies targeting making getting vaccinations easier or getting time off work are wasted effort.)

The problem?  If you click on the survey, it says it has a 6.1.% weighted rate for taking the survey.  (So only about 6 out of every 100 people they sent the survey out to actually responded.)  Also, it is an online survey.

If that 6.1% were a randomly selected sample of a general population, there wouldn’t be a problem.   The problem is when the selection into the sample is based on the outcome that you’re measuring.  In this case, if you’re measuring people who don’t have time or ability to get vaccinated, well, likely they don’t have time or ability to take a survey either.

The Census Bureau isn’t stupid– they know this is a problem and they have lengthy documentation about the non-response bias in the sample generally.  They make it clear what you can trust the results for and what you can’t, as well as the limits of their weighting schemes.  The survey isn’t completely useless, but it is only externally valid for the groups that were surveyed!

I had been planning to use this little example of “sample selection on the Y variable” in my stats class this fall, but now I can’t because his response was so ironically ignorant that I have to blog about it instead.

Here’s his response:

The low income people who are supposed to be burdened by the time constraints also don’t report access as an issue compared to other factors. Are there any data that could convince you, or will the answer always be the same?

So– I guess I was right about his complete lack of self-awareness.  Can you imagine being convinced to make a huge policy change based on one extremely selected survey?  The people who ran the survey would never ever want you to make a policy change based on this result!

The answer to the question of what would be convincing (taking it seriously rather than just an accusation of me being set in my ways):

1.  A nationally representative sample that found the same result.  The US government has some of these, where you are required to take the survey and they have much better response rates (not perfect, but much better).  This survey is not one of them.
2.  A sample representative of the population who we are trying to target with our policies (ex. a state going into a few factories with onsite vaccination clinics before expanding the program).
3.  Multiple biased surveys that are biased in different ways (that is, are not biased on the outcome variable of not having enough time).

No good policy maker would make policy based on such flimsy evidence as the survey mike the mad biologist presents.  In fact, we rarely make big changes based on the result of any one study, unless it is the only study available or is the only well-designed large experiment available.  And even then, good policy makers keep their eyes out for new evidence and try not to do huge national things when the evidence is scant.  Ideally we’ll have a largescale randomized controlled trial, but failing that we’ll take a series of mixed methods– qualitative information, event studies (these two are the easiest and cheapest to do but can be biased depending on how they’re done), natural experiments, and so on.  Ideally we’ll have information about heterogeneity– we think, for example, that the effects of the Affordable Care Act and the effects of universal health insurance were different for Oregon compared to Massachusetts compared to Wisconsin or Tennessee.  And that could be because they have different populations and different starting environments, or it could be that each of these states had a different methodology used to study it with different biases.

Unlike Mike the Mad Biologist, every single thing I do (in research and in teaching!) has the potential of helping or harming someone’s life.  I have to be extremely careful.  I don’t make policy recommendations until the bulk of evidence supports those recommendations.  Because, getting back to that first disclaimer– I’m actually out there doing stuff, not just complaining about the people who do things.

So yeah, I teach my students about how not to use samples that are selected on your variable of interest.  It’s a more challenging concept than people say, lying about their weight or height, but it is an extremely important one.  I have a lot of students who go out and design/make/evaluate policy when they graduate.  Hopefully the lessons I give them remain with them.

Ask the grumpies: Computer educational programs that DC2 (grade 4) loves (and others zie’s ok with)

Parent to two virtual learners asks:

“Have you or your readers been using any computer programs to supplement virtual learning at home? I need to work full-time and my kids (elementary schoolers) seem to get done with their work in 2-3 hours each day. They don’t want to spend all their time reading books and I don’t want them to spend all their time playing video games or fighting with each other. Do you have any educational games and similar things to recommend? Anything non-electronic that doesn’t create too much of a mess?”

DC2 loves Prodigy to pieces. It’s basically an RPG-ish game where your choices don’t matter much, but in order to fight monsters you have to get a math problem correct. The math problems are very similar to going through Khan Academy– they get harder and you master a specific skill and then they give you a new skill. I think it’s free and they make their money with in-game purchases that DC2 would really like to have but we have not been doing.

DC2 loves Epic! though sadly we have to pay for it now because this year’s teacher didn’t sign up for it like last year’s did. (Technically zie can sign in as guest with hir class from last year, but that seems wrong somehow.) Epic has a lot of fun books, including a TON of comicbooks to read. All the Big Nates, even some that are so old or new that DC1 doesn’t own them. (There are Bill Clinton jokes(!))

DC2 likes DuoLingo just fine. Zie is doing Spanish with it, mostly on weekends. There isn’t much playing with it if it’s not required like there is with Prodigy.

We’ve recommended DragonBox before– DC2 did a spin through again before school started this past summer.

Similarly to DuoLingo, DC2 is fine with Khan Academy. Zie is mostly going through the math part.

DC2 is supposed to go through a math thing called xtramath, but finished last year, so after zooming through addition and subtraction, hir teacher asked hir to play Kakooma instead. DC2 is not a fan of the timed aspect of it, but did enjoy it at first.

Instead of Epic!, DC2’s teacher is doing Reading A-Z, which has more educational stuff than does Epic and has step readers in both English and Spanish. One not so great thing is that it tested DC2 in English and is now only allowing hir to read within a few levels of hir tested level in Spanish. Hir Spanish level is not as high as hir English level! There’s no Phoebe and Unicorn here, but there’s lots of non-fiction.

With all the additional laptop stuff, DC2’s computer class has a lot of typing. They use Typing Agent through Clever. This is not as much fun as typing of the dead, but it’s more focused for if you don’t know how to touch type (as opposed to just needing practice to get your speed up).

In terms of exercise, we have two games for the Switch right now that DC2 has been using. Ring Fit Adventure is a good exercise game, much like Prodigy except you do exercises to complete an attack. Zie has also been doing Just Dance 2020 and now I have Baby Shark stuck in my head for all eternity even though I’m fairly sure I’ve only heard it twice.

DC2 is also getting into Minecraft Coding Academy, though that occasionally requires assists from DH. Similarly DC1 (in high school) is working on USACO coding projects (zie isn’t actually signed up, just going through the problems)

I’m probably forgetting something, but these are the things that have been popular enough (or required enough by class) to come to mind.

In terms of non-electronic stuff:

DC2’s art teacher requires them to draw daily in a composition book. We’ve gotten some calligraphy pens and so on to make that a little more interesting. They’re both also into origami, but that makes a mess.

DC2 does a page of Singapore Math each day.

We’ve downloaded a bazillion children’s classics from Gutenberg to DC2’s kindle (Oz is very popular with both kids, also E. Nesbitt).

We’ve increased the chores they have to do– doing laundry has been added to their list which previously only had laundry folding and removing things from the dishwasher. Now there’s 15 min of cleaning their rooms each day, though sometimes we add an additional 15 min to DC2 because zie is so talented at making messes. DC1 cooks a meal a week (usually on Sundays) and DC2 helps. They also pick out one meal each week for menu planning (DC1 makes hirs).

DC2 also has regular minecraft playdates with friends which isn’t exactly educational but is social at least. DC2 and hir frenemy are actually much more mature playing on minecraft together than playing on the playground. Now that it’s cooler outside they’re getting kicked out for exercise again. Oh, and they’ve started teaching themselves contact juggling (using a weighted rubber ball, not one of DH’s actual clear contact juggling balls because there’s a lot of dropping). I’ve been trying to get DC1 to pick up pencil twirling because zie fidgets by worrying things until they break off and zie needs to stop doing that, but so far no luck (zie is also uninterested in fidget spinners). DC2 has been working on juggling one club, but that somehow morphs into all the juggling clubs being all over the floor of the living room.

Grumpy Nation: What educational stuff do you recommend for when you have to work but your kids are out of schoolwork?

Ask the Grumpies: Schools in the SF Bay area that are good for mathematically advanced kids?

Mover:

I am moving to the SF Bay area for a new job from a city across the country.  My six year old is currently several years ahead in math at school and I would like to find a school supportive of continued enrichment/acceleration.  Any words of wisdom?

What we’ve had to do when doing sabbatical moves is to call up the school districts of all the places we’ve been thinking of moving to and just ask what is done in DC’s situation.  This has been very informative as some districts are much better than others.  In our case, we didn’t ask about single-subject acceleration which is what it sounds like you need, but instead whole-grade acceleration.  I know we posted on our process, but I cannot for the life of me find that post.  Essentially we looked at a map of a reasonable commute to my sabbatical place and called all the school districts and explained and just asked what their general policy was for our situation.  Some said they would obviously put DC1 in the next grade, some said they’d test DC1, and some said they’d put DC1 in the grade for hir age level.  You would likely want to be asking about single subject acceleration.

In terms of the Bay area specifically, I don’t know if this is still true, but Berkeley schools have always had a reputation for being anti-intellectual.  They’re very into “letting kids be kids” which means bored out of their skulls.  There used to be a forum that you could find online with lots of parents complaining about it.  Sunnyvale is another district that is not great in that respect.

In terms of out of school math enrichment:  Math Circles are great.  DC1 started going in middle school.  Our DC2 hasn’t gone to one yet, but DC2 is only doing third grade math right now.  These are usually on Saturdays.  They get to do fun stuff that often isn’t done within school.

Good luck!

Do any members of grumpy nation have experience or insight with single-subject acceleration at the elementary school level in the greater SF bay area?

Replacing a misogynistic mnemonic

It is easy to get backwards when doing t-tests, especially when you’ve first started.  You have to remember that big t and small p-values mean to reject.  Of course, if you’re a little dyslexic (undiagnosed) like me, when you get confused, you can go back and re-figure out that you want small amounts in the tails and the t-slice to be far away from the mean making it large etc., but that is really time consuming, especially if you’re in the middle of a lengthy problem set or a timed exam.

Many years ago, one of my students shared a dirty misogynistic mnemonic.  When p is low, she said, reject the ho.  This is clever and funny because p is both probability and slang for penis.  Ho is both how a null hypothesis is written and a derogatory term for female prostitute.  When one’s penis is at low mast, it makes sense that said penis-holder might not be purchasing the services of a prostitute.  And it rhymes.

But it’s both dirty (I got into trouble for saying “prick” meaning “jerk” early in my classroom career) and one should not be using derogatory terms for prostitutes or women anywhere, much less in an institution of higher learning.

More recently, I was explaining my conundrum in office hours and one of my sunshiny students came up with a much better mnemonic.  It isn’t quite as clever, but it’s just as memorable and it still rhymes.  “When p is lo, reject H-O!”  Like a cheerleading chant (aich – oh).

It makes me much happier.

What are some good non-racist, non-misogynist, non-ablist, non-patriarchy mnemonics that you know?

Ask the grumpies: Math practice and enrichment for different kinds of learners

Natasha asks:

I have a kid who is […] having a tough time with math (3rd/4th grade): he grasps new concepts just fine, does well on tests… and then 2 weeks later he can’t remember any of it! His school math program seems to fly from topic to topic, and even though his teacher assures me that even if he missed something this year, all the same or similar topics will be revisited next year, I worry that he hasn’t had the chance to master the basic concepts. It’s more of an issue with retention of the material than understanding the concepts. I know you love math – do you have any suggestions as to what books or methods may be helpful to practice 3rd-4th grade math? I believe it is so important for kids to get solid foundation at the elementary-school stage.Teachers simply shrug and say it’s the student’s responsibility to practice old material (well, I do agree with that) and point to Khan academy. The school is using the Envision Math program. I am terrible at explaining but love doing math puzzles and fun problems together with kids – and that doesn’t seem to be enough.

On the flip side of the coin – I have a second grader who is doing really well in math and needs more challenge. The teacher gives her additional (optional) higher-level worksheets, but my daughter doesn’t seem to be thrilled about those and prefers to read or draw. We are doing some fun logic and puzzle games at home, but maybe you have additional advice on fun math activities (books, games, workbooks) that provide additional challenge without being too much like homework?

Let’s start with the older child.  There are two potential things that could be going on.

The first is that your kid is a normal kid who is good at cramming for the test and then forgetting after.  This habit is so normal that much of the US math curriculum just assumes it will happen– that repeating topics thing they’ll be doing next year even has an education jargon term.  It’s called “spiraling”.  The best math curriculum for this specific problem is called Saxon Math, which is not the most exciting math program (it can be enervating for gifted students), but does an excellent job of repeating and integrating concepts throughout the year and not doing the standard focus and forget.  There’s a good research base behind Saxon Math working well for average to below-average math students (less well for high ability and gifted).  If you’re attached to a university library, you could probably check out a textbook for 4th grade to see if it is helpful.

The second potential problem is one that I saw highlighted when I did a quick google of the Envision Math program (which I hadn’t heard of before this query).  Apparently Envision Math is  shallow (or at least that’s what people complain about along with it being repetitive) so it is natural not to remember the concepts– there’s not really anything to remember.  If what people say online is true, it’s all surface with no roots.  If you want to grow roots and approach math from a completely different angle, you can’t go wrong with Singapore Math.  That’s exactly the opposite solution of what my initial thought was, but after having read a few of these links of people complaining, I’ve reconsidered.  Another benefit to Singapore Math is that it ISN’T the same as what’s being taught at school.  Being able to do the same math multiple ways is valuable both because it keeps you from getting bored, but also because it gives a much greater context and understanding to how this magical world of numbers and mathematical concepts actually works, how it’s put together.  You start seeing the full 3-d math forest and not just the shadows of the math trees.  Those Aha! moments have always been my favorite part of math tutoring and teaching.  Singapore Math also has a strong research base, although most of this research is done on the full population of students, not any specific group.

Given my morning’s research, I take back my initial recommendation about Saxon and suggest starting with Singapore instead.  They have placement tests he can take to see which books to start with.  You will need two workbooks for each year (ex. 3a/3b) and the textbook is useful.  We didn’t find the home instruction guide or teacher’s guide to be useful– it was essentially a lot more examples and activities for the teacher to demonstrate, but your son is already getting the concepts, so the textbook and workbook should be enough.  It probably does not matter which of the three series (US/Core/CA) you use as long as you’re consistent.  We use the US editions because the other two didn’t exist when DC1 started and we wanted to reuse the textbooks.

If he also needs to know his addition/multiplication facts, we don’t really know any solution for that other than practice.  Flashcards aren’t much fun, but they do cement facts and make later math easier.

Turning to the younger daughter.

Second grade is the perfect year for Math for Smarty Pants.  In another couple of years you can get used copies of Aha! and Gotcha! by Martin Gardner which are super fun.  She may enjoy tessellations coloring books (and creating her own using graph paper!) or folding 3-d geometric shapes.  I am having a really time finding anything on amazon, but somewhere out there, there should be workbooks that show you how to use a compass to create a triangle and then other 3-d geometric shapes from that.  A quick google finds lots of the basics with “compass and straight-edge construction” (and some youtube videos where people put together the already made forms) but with cardstock, tape, and something to score with you can make really elaborate 3d designs.  Origami is another fun math craft– DC1 has been watching youtube videos to make shapes, but there’s also a lot of great books out there.  Tangrams are perfect for this age group.  This classic set from Tangoes is my favorite (mine from childhood was black, my kids’ is blue), but DC1 also really enjoyed a magnetic set that comes with a book that is occasionally available from scholastic.  I found the rubix cube super frustrating, but now there are online videos showing you how to solve it so it’s more fun.  DC1 also really enjoyed maze puzzle balls (and saved up allowance money to buy a second)– but I also find these frustrating.  I think it depends on your agility not just the thinking things through thing.  (And, as we’ve mentioned before, DragonBox is fantastic.)

If your son is willing, there are a number of card games that secretly practice concepts that they might be able to play together.  I tend to like the ones that Scholastic sells off and on– they have a really good one called money madness that was a money addition/subtraction game that we liked a lot.  Our kids recently each won the raffle for the university’s math day and got math games.   The one currently spread all over our dining room table is a simple memory game called rat-a-tat cat, and the one neatly stacked in a tin is 7 ate 9 which is a fast little addition and subtraction game.  They’re probably too simple for your kids.  :/

Our math tag has a bunch more suggestions for enrichment at various stages, including items our readers have recommended in the comments sections.

Best of luck!

What enrichment would the Grumpy Nation recommend for these ages?

RBOC

• DC1’s algebra teacher quit to join administration a couple of weeks after the second semester started (after a week long absence).  I guess we’ll have to keep a closer eye on the rest of the semester since algebra is so fundamental and I’m pretty sure zie doesn’t now how to factor polynomials even if zie does know how to multiply them.
• Super bummed that Teen Vogue is no longer doing a print edition.  The last few issues were AMAZING, including one guest edited by HRC.  Irritatingly, they switched DC1’s subscription over to Allure magazine “the magazine for people who care about beauty” or something like that.  Full of “beauty tips.”  This month’s issue was on nudity and had a nearly naked airbrushed stereotypical model on the cover.  Completely not appropriate for an 11 year old or really anybody.  And very different from a magazine that features people like Malala Yousafzi on the cover.  I will be getting a check for $2 in the mail for my cancellation– Teen Vogue should have been charging more.  It’s a different market and was worth much more than the ridiculous $10 for 2 years or whatever it was I paid.
• Forgive me, for I have referred to a paper about fertility as “seminal” in published work.  Next up:  referring to a paper about religion as “canonical”.  And a paper about building cities as, “ground-breaking” (or should I save that one for agriculture?).
• it is weird to me that my kids have had macarons before having had macaroons.
• DC2 has moved onto chapter books at school.  Zie is in love with the Geronimo Stilton that DC1 read maybe once or twice.  They have such different taste in books.  Really the only commonality is that they both love Jim Benton, author of Franny K Stein and Dear Dumb Diary.  I so wish we had Scholastic so I could indulge in buying sets of series we don’t have (like Thea Stilton!)
• Preserved walnuts are really good.  If you ever get the opportunity to try/buy them, take it!
• My cholesterol is fine this year (whew!), so maybe all that additional lunchtime walking did some good!  Unfortunately, it doesn’t seem to have helped with my vitamin D levels (which may explain the fatigue I’ve been having), so my doctor wants me to go from 2000 iu to 5000 iu.  I’m going to compromise and do 4000 iu because that means I can have a 2000 iu when I brush my teeth and keep another bottle of 2000 iu in my office when I get my mid-day slump.
• It isn’t a bargain if you can’t afford it.
• We owed an additional $2846 in taxes this year, not counting the estimated taxes for this next year.  [Update:  We forgot a whole ton of donations– didn’t go through the school email folder or the check register, so it’s actually $100 less than that.  With the additional donations, we’re just a little over the standard deduction.  Also turns out there’s no point for us to declare a home office since we don’t get anywhere near the minimum for it to count for us– We make too much and our house is too big and too cheap.]
• DC2’s school was having a performance for parents/relatives and one of their dances had them shooting with finger guns.  This disturbed DC2 enormously given that they started practicing right after the FL school shooting.  Thankfully someone decided to change that number to something in less bad taste.

Ask the grumpies: Math for ages 0-5 for kids who love math

Leah asks:

How did DC2 learn so much math? And when did you start? We’ve started discussing addition using finger counting or counting treats, but I’m not sure my little gal is picking it up yet. I sometimes wonder if I should be doing something more formal (or trying more) or if it’s fine to just chill. I do fractions and percentages when we cut nails (1 nail done, that’s one out of ten, or 1/10th, or 10%, etc).

So, this is based on a comment from a post about how my DC2 is age 5 in kindergarten and is doing multiple digit addition and subtraction with carrying and borrowing as well as some simple multiplication (no times tables memorization yet).

First off– I can’t take much credit for the multiplication.  DC2’s Montessori taught all the “big” kids multiplication.  This is pretty standard in a lot of Montessoris and I think it is part of the curriculum, though I do not actually know how it is taught.

Here’s some suggestions from people in the comments:

Becca says:

If you don’t know about Bedtime Math yet, get the app or the books :-)

A big part of very early math is pattern recognition. Grouping items according to different criteria, making designs with blocks or beads are good things to do.
The vocabulary of positions (over/under) and sizes (bigger/littler) and so on can also be good to get down early.
Other stuff, from a pretty evidenced-based group: https://www.zerotothree.org/resources/299-help-your-child-develop-early-math-skills

I’m also a firm believer in counting during swing pushing at the park. It gave me something to do, and gave Roo exposure to numbers bigger than 100 (ok, so we may have both had an inordinate patience for swinging).

I have to admit that we own the first Bedtime Math book (on Laura Vanderkam’s recommendation, along with Family Math, if I recall correctly), but we haven’t really used it.  DC1 already owned Aha! and Gotcha! (and had kind of outgrown Math for Smarty Pants), so we briefly looked through it but really had outgrown it.  I haven’t dug it out of DC1’s bookcase to try with DC2.  Maybe I should.

We have two different sets of brightly colored manipulables that I will dig out to play math with the kids with.  One set is a set of pixel-blocks that DC1 loved to play with.  Zie has always been into small things (and not into putting things into hir mouth), so pixel blocks work well for that (not safe for many small children!).  DC2 prefers a set of bigger circular pieces that DH initially bought to use as game pieces for game design (I can’t easily find them on amazon, but there are a lot of reasonably priced options if you search for manipulatives).  We also have lots of fun toddler sorting games because apparently I never grew out of them.  (I may be messy and disorganized in most of my life, but I find sorting to be extremely soothing.  This is part of why my bookcases and spice cabinet are beautifully alphabetized.)  Back when we had access to swing sets (our town has removed them all for “safety”/lawsuit reasons), we definitely counted pushes.  Once my kids were able to talk, I would ask, “How many pushes do you want this time?” and then I’d count out that many pushes and ask again.

omgd says:

I started trying to introduce fractions by talking about sharing. As in, “There are 6 apples and your friend takes half. How many do you have left?” She doesn’t really get thirds or quarters yet, but I think it’s because of the vocabulary.

I have to admit, I haven’t really thought about teaching fractions other than what DC2 is getting in hir brainquest book.  They will become more prevalent in the Singapore book a book or three from where DC1 is right now [Update:  the day after I typed this, DC2 had to color in halves and quarters in hir Singapore Math 1b book, but only for a couple of pages].

Ok, now back to me:

When my kids are bouncing off the walls someplace that they shouldn’t be bouncing off the walls, we practice counting.  When counting is too easy, we practice skip counting.  When skip counting becomes too easy, we will practice multiplication.  Then division.  I use this technique with my brilliant but overly energetic nieces and nephews who are too excited at being with extended family to be controlled by their parents.  (Back when I flew Southwest, I would keep the small children I invariably ended up sitting next to given my need for a window seat occupied by figuring out what their math level was and teaching them the next thing.  There are kids who learned long division from me because I wanted them to stay still!)

I LOVE Singapore math SO much.  It’s really great because it sneakily builds up to future concepts.  Examples are chosen specifically to help the subconscious pattern-match to figure out new things that won’t be introduced for chapters.  It is lovely.  Plus they teach a lot of really great mental math techniques that those of us who are really comfortable with use automatically (things like realizing that 10-1 = 9, so sometimes it’s easier to mentally add 10 and subtract 1 than it is to add 9 directly).  I am extremely impressed at how much facility DC2 has with numbers right now. Here’s me talking more about the workbooks the kids do.

DC2 had learned the borrowing and carrying from Brainquest (and me)– we spent about a month slowly cranking through double and triple digit addition and subtraction.  There are a lot of problems on a page and I would have hir just do 3 a day once we got to carrying and borrowing.   But zie wasn’t really facile with it until we got Dragonbox Big Numbers which is an enormously fun and addicting game (I finished it, but I still sort of wish I could be picking apples now.  It is a really great game.)  DC2 sped through it (as did DC1 and I– I finished first, then DC2, then finally DC1 sometime after that English project finished [for those who are curious, it wasn’t interpretive dance next… they’re doing another powerpoint (or, she suggests, PREZI UGH) use MOTION!… and a bunch of other suggestions that are super bad powerpoint etiquette].)   By the end of Big Numbers, DC2 was a multiple digit addition and subtraction wizard.

DC2 is mostly through DragonBox Numbers right now and is really good at it, but it’s not really as much fun as Big Numbers was, and it’s got some bugs which are irritating.

And, as I said earlier, I do break out the manipulables a lot.  Sometimes we use them to illustrate a particularly tricky workbook problem, but sometimes we just have fun doing number patterns.  We’ll also do patterns with fingers.  I really like playing games with these and making 10s.  So you start associating 3 and 7, 4 and 6, and so on.  We can also do grids of squares and rectangles with the manipulables to get used to multiplication (which I did more with DC1 than with DC2 because DC2 came home from preschool one day completely understanding multiplication).  There are a lot of fun ways to mix and match numbers and different colors to get an understanding of the patterns (and the beauty) of mathematics.

We also give the kids an allowance at a pretty early age, at first so they can get familiar with money and learn the denominations of coins and dollars.  (After the sticking random things in mouths stage though!)

Later on, I will introduce Hard Math for Elementary Students, but DC2 isn’t ready for that yet.  DC1 is really enjoying Hard Math for Middle School right now, as well as Saturday Math Circle, and zie just started doing every other week competition-based Math Club once a week after school, though zie is skipping the competitions this year/semester.  (Mainly because the first qualifying one is at the same time as a birthday party!  But also partly because zie does math for fun, not to compete.)

Later on, DC2 will also get introduced to Martin Gardner and Aha!  and Gotcha!  But not yet.

Should you be doing more or is it fine to chill?  I’m sure it is fine to chill.  But I can’t not teach math anymore than I can not drink water or keep from breathing.  It’s my nature.  It’s what I do.  And I gotta say that counting/practicing tables is the best for getting kids to behave while waiting for food at a restaurant, though occasionally it does get you dirty looks from other people who think you’re somehow harming your precious child or doing this to show off and don’t realize how much the alternative would interfere with their dining experience.  (Pro-tip:  It is often more fun when you trade off saying the next number, especially when sometimes you get it right away and sometimes you pause dramatically to think for a bit.  This also helps them to notice that skip counting by 2 is literally skipping every other one, and that skip counting by 10 is the same as every other 5.  It’s pretty amazing when they make that Aha! on their own.)

Oh man, I love math so much.

Grumpy Nation:  What are your math teaching tips?