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?

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  • 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:

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?

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Excel vs. traditional math

Excel makes it a lot easier to brute force problems.

DC1 was doing some Hard Math for Middle School students problems with different bases.

An example of one of these:  (p.14, #8):  Find all the values of A for which the base 63 number A7894321 (base 63) would end with a zero if it were written in base 10.

The first step to approaching this problem is the same for both approaches– you have to go through the first step of mechanics of turning something base 63 into base 10.  That means realizing each place value means 63 instead of 10 and expanding out. So:

1 + 2(63) + 3 (63^2) + 4 (63^3) + 9 (63^4) + 8 (63^5) + 7 (63^6) + A (63^7)

In Excel, what you then do is you take =1 + 2(63) + 3 (63^2) + 4 (63^3) + 9 (63^4) + 8 (63^5) + 7 (63^6) and fill that down 10 rows.  Then you put in another column that is nothing but =  (63^7), then a third column that is 0, 1, 2, … 9.  In the fourth column you type = A1 + B1*C1 (or whatever your top 3 cells are), and fill down.  A final step is to take those responses which are unhelpfully in exponential “E” format and paste them as values and look at each one to see which end in 0.  (Only 6 in case you were wondering.)

To do this by hand, you will do something called “casting out 10s”.  You only care about the last number, so you don’t have to do all the math, just the last part.  So instead of the full equation above:  1 + (2*3) because you only care about the last digit, + 3 * 3* 3 = 7 + 27 but we only care about the last digit, so we cast out the 3 tens and are left with 4 to which we add 4*3*3*3, and so on.  In the end you will end up with: some number + A*3^7 mod 10 = 0, solve for A.

I use Excel a lot in day-to-day life.

Do you?

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What does statistical significance mean?

One of my students sent me this article because we spend some time in class covering Type 1 and Type 2 errors.

All the .05 threshold means is that you have a false positive 1/20 times.  A .005 threshold would say you’re getting a false positive 1/200 times.  So by moving to a .005 threshold, you’re less likely to get a false positive.  That’s good, right?  In common parlance, we’d be less likely to send an innocent person to jail.

Well, that depends.  At the .005 threshold, we’re more likely to get a false negative than you would at the .05 level.  That means we’d be more likely to get a guilty person go free.  (Indeed, the only guaranteed way to send no innocent people to jail would be to send nobody to jail.  I, for one, am happy that folks like Charles Manson are behind bars.)

It isn’t as easy as saying, oh we should just switch to .005.  When you adjust the p-value you’re making trade-offs between type 1 and type 2 error.  With a lower p-value threshold you’re going to be getting a lot more false negatives even with fewer false positives.  What we always need to be cognizant of when we’re doing policy is that significance isn’t everything– we also have to think about what the damage is if this information turns out to be incorrect.  For example, doctors recommend that pregnant women should heat up cold cuts if they’re worried about listeria, which is a very low probability event but if it happens it’s horrible.  It’s pretty easy to avoid room temperature cold-cuts for 9 months, so unless there’s some other difficulties attached to diet, women will probably follow this recommendation.  (And if one accidentally eats room temperature coldcuts while pregnant, one shouldn’t freak out because the probability of getting listeria is very low!)  But if we’re talking about something like doing chemotherapy or surgery, that’s a much more onerous action and we might want to be more sure we need it before going ahead with it.

Another thing to note is that the article talks about how physics and genetics have already made this switch, while most social sciences haven’t.  One big difference between the fields that have made the switch and the fields that have not is how easy it is to get large samples.  A larger sample size will make it so your sample behaves more and more like the population that you’re trying to study.  We can reduce both Type 1 and Type 2 errors simply by increasing the sample size.  So why don’t we do that?  Well, it turns out that increasing the sample size can be very very expensive when you’re dealing with people and behavior.  Sometimes doing the study with a large enough sample to get 80% power and an alpha of .005 might be more expensive than just throwing that same money at the intervention you’re trying to decide about, whether or not it actually works.  There probably is some resistance because people in these fields want to be able to publish their 5% results, but that’s not the main or only reason we haven’t yet made the switch.  Research is complicated and expensive and we have to make trade-offs.

The context for these really does matter, and you shouldn’t necessarily put off making policy choices just because your sample size is too small to get significance (or to make policy changes just because you have significance).  You always have to be aware of the costs and the benefits.


(Incidentally, in case he comes across this, Hi Dan!  I’m assuming that the reporter greatly simplified your arguments here because I know you must know this stuff.)


More on math and perfectionism

Combating perfectionism and its sequelae is an ongoing battle at houses with gifted youngsters.  It is hard to provide continual challenges for smart kids that allow for failure but also allow for recovery from said failure.  When life gets too easy, failures seem to become that much more devastating when they do occur.

I really like math.  And math is nice because it comes in different levels which can provide different kinds of challenges and generally there’s going to be a solution.

We really enjoyed the workbook, Hard math for elementary students, though when I say “enjoyed” it was kind of a love-hate relationship for DC1.  There were sometimes tears.  But in the end, zie always triumphed, and that was exciting for DC1 and created true pride (though an odd consequence was that when DC1 cranked through a page easily, zie decided that page was too easy!).  It truly was a hard math book.  We were thinking of going through it again, but DC1 hasn’t wanted to.  Since DC1 just got into brain teasers and is spending hours on them on hir own, I ordered Aha and Gotcha and am going to let hir explore by hirself.

One of the really good parts of math for perfectionist people is that sometimes in order to get things right, you have to get them wrong a lot first.  There’s a method of solving things called “brute force” in which you just methodically try all of the possible answers to see which one(s) work.  You *have* to get things wrong.

The game Mastermind is another example of needing to get things wrong in order to find information that gets to the right answer.  You guess and then get feedback that helps you guess again until you narrow down the answer.  The game just isn’t that much fun if you guess right on the first try.  This game too initially caused tears in DC1, but coming back to it later it has been fun.

Finally, a fun (free, online) game recommended by school is fire boy and water girl.  This is another one where you learn about the world and have to try again and again in order to get the solution.  This one has never caused tears to my knowledge, though zie has stopped playing in frustration and come back later, which is totally valid.

It would definitely be nicer if there were never tears, but the pride that happens after figuring out something that previously seemed impossible might be worth it.

Do you have any suggestions for challenges, math or otherwise?


Programming for kids

I think there’s a part one to kids programming from a couple of years ago, but I cannot find it anywhere in the archives, so maybe it was a discussion in the comments section of another blog.

Computer programming is fun and extremely important.  Even as a social scientist, having a modicum of programming knowledge makes life a ton easier.  Doing statistics requires data cleaning and statistical programming knowledge.  Just knowing what is possible — being able to think in a way that allows programming to make life easier– means big efficiency improvements.  And that’s just social scientists.  Engineers and scientists often have to deal with much more complex coding structures.

Added to that, having a good background in programming also means that your code is much easier to read, understand, and to pick up later and figure out what the heck it was you were doing.  I can usually tell when someone has programming background because they do useful things like comment their code or put carriage returns between sections or indent their code properly when doing loops.  I *wish* more of the people I work with had programming backgrounds!  (She says, after spending a day putting in comments, carriage returns, and tabs so as to be able to read a program before adapting it for this year’s dataset.)

Anyhow, our first foray into programming a couple of summers ago was to try out Scratch.  Scratch was a lot of fun, but it’s more of a toy that teaches some programming structures (ex. loops) than actually teaching programming technique.  I know there’s a lot of thought that playing is the right way to go with programming, and I’m not against playing at all, but there’s a *lot* to be said for getting good technique in while you play.

So now that DC1 is 8 and has spent a couple of summers playing with Scratch, we decided to try something more systematic in.  After some amazon searching, we settled on Python For Kids: A Playful Introduction to Programming. We didn’t want to make this a chore like DC1’s homework books (which sometimes cause angst) and we didn’t want a time limit like piano practicing, so we just said ze has to do some each day, but as much or as little as ze wants.

So far it seems to be working out.  It takes the best parts of Logo (remember the turtle who made boxes?) and combines them with Python.  DC1 is pretty excited about it, and occasionally asks DH for help.  If you don’t have a professional programmer in your house, the same publisher also makes a Python book for parents, Teach Your Kids to Code.

Do you use programming or programming techniques for your work or hobbies?  Any suggestions for introducing people to coding?