Doing math multiple ways

On gifted forums, sometimes parents complain that the teacher says the kids have to do something X way, but DC gets the right answer doing it a different way.  So why should they have to do it X way when Y way is obviously working?

It’s kind of reminiscent of the argument that elementary schools no longer need to teach math because we have calculators now.

I disagree with that sentiment.  It’s important to do math multiple different ways.  There’s value in learning a different way to get the same answer.  You get a better understanding of how numbers (and later, symbols for numbers) are put together.  That leads to more accurate math, better estimates, faster calculations even without a calculator or pencil, and a greater knowledge of the possibilities of what can be done.

Even if we have computers that can do calculus, it’s still important to know how calculus works, because you know what is possible, you have ideas about what to try for things… and that’s even ignoring that math just makes you smarter.

DC1’s school just switched from Saxon math to Chicago math, but we’re doing Singapore math at home.  I’m glad ze’s learning the traditional computational methods at school (and we practice them in hir Brainquest workbook during summer and on the weekends), but I love love love that Singapore math looks at the same things in a different way.  For example, we just hit multiplication of 2 or 3 digits by a 1 digit number.  The traditional method ze’ll learn in school (and practice in brainquest) is to start with problems that don’t require any carrying.  Probably lots of x2 and x3 simple problems (23 x 2 = ?, 12 x 3 = ?), in order to cement the idea of multiplying the ones digit and then the 10s digit (and then the 100s digit another day).  Eventually they’ll introduce the concept of carrying (23 x 4 = ?).  (Then next year, the mechanics of double digit multiplication.)

The Singapore method, instead starts with some pictures.  It says, you remember when you learned multiplication how that was like having 3 rows of 4 balls?  And 3 * 4 = 12?  Well, what if, instead of each ball being worth one, that each ball is worth 10.  So you have 3 rows of 4 (10) balls.  (In pictures this is more obvious than in words.)  They’ve done the 10 ball representation previously with place value and with skip counting and x10s, so they’ve seen this idea before multiple times.  So 3 * 40 = 12 tens, and they know that 12 tens = 120.  Then they move on to 3 * 400 with the same pictorial representation.  Finally they finish up with 6 sample problems:  5*9, 5*90, 5*900, 9*5, 9*50, 9*500.  These last problems are set up in a way such that there’s pattern matching insights there for students who are good at getting insights from pattern matching, but it isn’t forced on kids who aren’t.  (At this point DC1 asked if 50*90 = 9*500 and 5*900.)  The next day moves on to 2 and 3 digit times 1 digit without carrying, but teaches it using these insights with the distributive property (13* 2 = 10*2 + 3*2), and this is not the first time they’ve seen the distributive property either– they’ve worked a lot with it with addition.  By the time Singapore math kids get to algebra a lot of tricky algebra concepts should seem pretty obvious.

I believe there’s value to being able to do math with both of these techniques.  They each provide different insights to how numbers are put together.  They each have different numerical problems for which they are the faster and easier method of solution.  In addition, the standard US method tends to be easiest when one has a pencil handy, whereas Singapore math is often best for mental math.  It isn’t that one technique is better than the other (though I confess that Singapore is more beautiful and I can see the sneaky ways it’s introducing higher level math while working with simple numeric problems, something beautiful in itself).

Being able to use multiple methods is even more valuable, however, than the sum of being able to use two individual methods.  Because of the insight given by seeing two different ways to solve the same problem, I would argue that the value of learning a second method isn’t even multiplicative, but instead exponential (or maybe factorial…)  Each new way provides a deeper insight into the magnificent world of numbers.

And, with that pattern matching turned on… if there are multiple ways to get to the right answer in math, maybe there’s multiple ways to get to a solution in other kinds of problems too.  If everyone had that particular insight, then maybe government policy wouldn’t be quite so messed up (a long shot, perhaps).

Do you think there’s a benefit to learning different ways to get the same answer?

54 Responses to “Doing math multiple ways”

  1. Comradde PhysioProffe Says:

    I totally agree. This is exactly why learning the connection between algebra and geometry is so intellectually transformative.

  2. plantingourpennies Says:

    In general, I think learning more than one method makes someone understand the concepts and relationships better than just learning a single method. (There’s a difference between being able to use the quadratic formula and understanding how it’s derived…)

  3. Perpetua Says:

    Would you recommend the Singapore maths at home for a bright-but-probably-not-gifted 5 y.o.? Mine loves math & is frustrated by not getting enough at school. (Last year he started some basic multiplication, division, and fractions.)

    • nicoleandmaggie Says:


      I would recommend Singapore math to anybody. Though I would also recommend Saxon math to kids who are lower than average intelligence because it’s so good with the repetition.

      One thing that we do is skip some of the extra practice problems in the Singapore textbook if it seems like DC “gets it” without them. But those problems are still there if it takes a little longer.

  4. bogart Says:

    Not sure it counts as a pedagogical technique, but here’s yet more data to suggest that there are very smart/clever people with (too much?) time on their hands — for when you get ready to introduce string theory:

    A fan, here, of learning to do things multiple ways and also to think through how things can be done (i.e. “Here’s a problem, how can we solve it?”).

  5. Laura Vanderkam (@lvanderkam) Says:

    I love sneaking in more advanced math under the guise of simpler problems. If a kid has just learned that 5+3=8, there’s no reason she can’t then be asked to look at that problem in different ways: 5 plus what equals 8? 8 take away what equals 3? What is we used a letter in place of that “what”? What if we look at 5+3=8 and we take away 3 from both sides of the equals sign? There’s no reason algebra should blow people’s minds later on.

    • nicoleandmaggie Says:

      Yeah, I remember back in the old days we used to do things like 5 + ? = 8 way back in first grade when we were learning addition. (Back before Houghton Mifflin math was Chicago math.) It’s not so hard to move from 5+ ? to 5 + x.

      • Debbie M Says:

        So true. But my eighth-grade algebra teacher tried to get us to say that we knew the answer was 3 because we had subtracted 5 from both sides, when actually we just magically knew the answer because we had worked with these basic facts for so long. She should instead have said that for those cases where you don’t magically know the answer, you can subtract five from both sides. Then show us that it works on this easy problem. Then shown a hard problem. And check the work by doing the actual addition with the discovered number.

  6. First Gen American Says:

    We randomly do math problems all the time, but not as structured. I bought the brainquests but we haven’t really used them. It’s easier to slip math into normal conversation.

    • nicoleandmaggie Says:

      We do workbooks on weekends and in the summer, otherwise DC1 is insufferable because ze has too much energy.

      • Debbie M Says:

        Make sure everyone knows what the written symbols look like, too, though. My dad taught me square roots in the car once, but I had to do a million problems when I failed the pre-test because I didn’t recognize that symbol. I also had to do a million multiplication problems involving two- and three-digit numbers because I didn’t know that a dot could mean “times.”

      • nicoleandmaggie Says:

        haha, yes. I teach two courses in the same sequence and the textbook notation changes in the two courses… drives the students crazy. But it’s good for them and helps them to think about variables as variables.

  7. chacha1 Says:

    I feel like I never really learned to understand math at all. :-( I wish someone, somewhere, had presented different ways to approach it because I *know* my brain is capable of it; I just don’t have the right toolbox. It was One Way Till Eternity where I went through K-12, and if you didn’t get it, tough nuts. I remember crying over a set of long-division problems in 4th grade. Trauma!!

    • nicoleandmaggie Says:

      That’s so sad! If you got a masters degree in my program, I’d totally teach you math. :)

      • Debbie M Says:

        Doesn’t everyone cry over long-division problems in the 4th grade? I know I did!

        I think even the same-old methods can make sense later when your brain is more developed. I definitely found some incomprehensible books suddenly easy to read after college. (Not Camus, though.)

      • nicoleandmaggie Says:

        (#1 loved long division in 4th grade)

      • Debbie M Says:

        Yea #1!

      • Rented life Says:

        I faked being sick when we learned long division because it frustrated me so much. I struggled the same with subtraction and he teacher sent a note home saying I was lazy. In Algebra 1 I tried to get tutoring from the teacher and again heard I was lazy. I’m smart, and probably could have majored in math if it was presented differently. I loved stats class and math theory in grad school.

      • nicoleandmaggie Says:

        That is SO SAD!!! But I’m so glad you recovered in grad school!

      • rented life Says:

        I just wish I had learned more at those earlier ages, because when it’s presented right, math not only makes sense but is pretty interesting!

      • nicoleandmaggie Says:

        Oops! DH is teaching DC1 long division via Singapore math this morning. I meant to wait on that a bit and review time or money in the Brainquest book this morning instead, but I was on the phone with a coauthor and DH took over homeworkbook help. DC1 seems to be doing fine though.

    • Revanche Says:

      I feel the same way! My mom taught me arithmetic and multiplication/division when I was 5, but that was around the last time I truly felt comfortable with it. For some reason, the way teachers taught (no clue what style it was) algebra and up made zero sense to me. I always sensed that geometry could make sense but I never grasped it like I should have and just felt plain stupid in high school.

      • nicoleandmaggie Says:

        Your school failed you!

      • chacha1 Says:

        I got an A in geometry, Bs in algebra, a C in trig. Then went to college and got As in calculus and statistics. I have no idea if the teaching approach was the key but I suspect so. Now, I am just happy that the tax returns generally come out right. :-)

      • nicoleandmaggie Says:

        My calc 2 teacher was really really good at showing all steps, so we’d find ourselves finally understanding something from a much earlier class. (My calc 1 teacher was TERRIBLE, so I basically learned all my calc 1 while taking Calc 2, and he also did the same for people finally getting trig for the first time and many other things, I think in his calc 3 class). For that reason, in my math classes here, even though it makes my students giggle, I say things like, “Because these have the same denominator, we can just add across the top.” Because maybe I’ll catch someone who isn’t so good with fractions.

      • Revanche Says:

        I’m really replying to N&M here: can I sit in on DC1’s lessons please? I would learn so much :)

      • nicoleandmaggie Says:

        Hahaha, no probably not much out of DC1’s lessons, ze is only in third grade. But you could sit in on my office hours and learn a lot. :)

        You missed my best lecture though– that was this week. I do an amazing job teaching how to do normal distributions.

  8. CG Says:

    My dad loves loves loves math. And as a kid I HATED it. It’s still an area where I feel like I don’t have a firm foundation, even though I now need to use stats in my job all the time. So, when my dad was helping me with math homework, and he would “trick” me into doing algebra (or something), and then say, with a big happy smile on his face, “See, you just did algebra!” I would cry, because it was so hard and frustrating for me to do even the basics. I didn’t want to know multiple ways or more interesting ways of doing things, I just wanted to know the Single Simplest Way and get my homework done. Ugh. I want to cry 25 years later just remembering how demoralized I was. And by the way, my dad is a lovely lovely person and we have a great relationship, so it wasn’t that he was somehow being mean or anything, I think I just couldn’t take his enthusiasm for something that felt so oppressive, and I didn’t have the mental or emotional capacity for complicating it in any way. Not that this touched a nerve or anything.

    • nicoleandmaggie Says:

      I’m expecting to see several folks in my office this week with the same kind of stories about their relationship with math. But lightbulbs will come on and they’ll get it and do really well on the midterm. The first part is getting rid of the fear.

      Of course you had the mental capacity! You were just missing an important building block and emotions got in the way.

  9. Debbie M Says:

    Wait, people can understand math?

    Just kidding.

    Another advantage of multiple ways is that you can check your work.

  10. jlp Says:

    I could not agree more about providing multiple approaches. We have Singapore math sitting on our bookcase (it just got unpacked after our move), and I haven’t yet had a chance to break it out for our kids. Now I’m excited to get them started!

  11. Julie Reynolds Says:

    Life at our house is similar to CG’s where she is my 8 year old and her dad is my husband. Tear and screaming are part of math homework almost every night. K struggles to understand math and numbers, and using multiple methods to teach it just makes her frustrated. The result is that, instead of understanding multiple approaches to math, she doesn’t understand any of them.

  12. First Gen American Says:

    On perfectionism. Does dc ever relapse? We are finding the perfectionist tendencies coming back out again en force now that my son is in a new school. We linked it mostly to Spanish as he didn’t have that class at his last school and felt behind, then every little thing after that compounded his belief that suddenly he was dumb and he’d shut Down. We did a weekend of Spanish immersion and now he’s fine again in all classes.

    • nicoleandmaggie Says:

      Oh yes, ze relapses on a fairly predictable basis. Whenever ze goes too long without a challenge and is suddenly confronted with one. Whenever one of hir teachers praises hir for hir intelligence. Ze had a full melt-down requiring an email home earlier this year and we had an email exchange with hir teacher about it. Ze does it more for some teachers than for others, and the main thing seems to be just like the books say, ze needs challenge on a regular basis, ze needs to be constantly reminded that mistakes help learning, and ze needs to not be praised for being smart.

      But every time it gets a little easier. And whenever we have one of these experiences, like with the math flashcards (most recently) in our case, or Spanish immersion in yours, it’s something we can use to remind ze that ze once felt like this before, but ze put in the extra practice and ze triumphed.

      • First Gen American Says:

        You know, my son’s teacher immediately recommended therapy when I told her about his perfectionist quirks. I know psychology has it’s place, but I found it a little sad that this thirst for wanting to excel is somehow a “problem.” I assured her that it’s not a constant thing and told of her the coping strategies we use (many of the ones you cited already in your perfectionist article) and it worked out fine. His teacher is supposedly the best one in school and was super super responsive to my initial email so it all worked out fine. I just think it’s just an interesting observation that many people assume that if you’re not in the normal distribution of kids there’s something wrong with you that needs fixing.

  13. Matt Healy Says:

    When I see something like 16 * 23 in real life, I don’t think about the digits, I think “well 16 quarters would be 4 dollars, take 32 cents off that and I get $3.68 so its three hundred and sixty eight.”

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