Mar 10 2013

Are the common core math standards developmentally appropriate for 1st graders?

My sister’s friend is a first grade teacher in a ‘challenging’ school in New York.  People who know about education know that poor students often enter kindergarten, already a few years behind their more affluent peers.  Good teachers, when given the freedom to teach their students at an appropriate level for their incoming skills, are able to help students progress.

But when teachers are forced, from a top down mandate, to teach an untested curriculum based on the concept that by adding more ‘rigor,’ students will rise to meet the new expectations, well, that can be a problem.

I have a vivid memory of being in first grade and having to answer some questions like :  5 + blank = 7.  I would do these by counting up from 5 until I got to 7 and see how many I needed to count.  My teacher, seeing that I was pretty good at this suggested that I could get the answer even more quickly by just doing 7 minus 5.  I remember ignoring this suggestion.  Even now, as a math teacher and enthusiast, I have to think twice to be convinced that this method should work.  (I’m pretty comfortable with it now, but I still see it as a subtle thing.)

Something that ‘math people’ sometimes have is ‘number sense.’  There are actually some great mathematicians who truly have trouble with basic arithmetic since the math they study is in a completely different realm, but most, I’d say, have the number sense too.  So when I need to add 98 + 147, I might, instead add 100 + 145.  How and when I learned to do this, I’m not sure, but it certainly wasn’t in first grade.

The new first grade common core standards, apparently, were written by people who thought something like “Math people have this number sense so if we ‘teach’ kids to have it at an early age then they will become math people.”

So my sister’s friend has to teach students who would struggle to add 9 + 7 by just counting seven numbers beyond 9, to take one from the 7, give it to the 9, turning the problem into 10 + 6 which then is a ‘trivial’ problem.  My first issue with this is what if 10 + 6 is not trivial to the students?  Then this conversion is a complicated process that leads them to a question that is just as hard for them.  Also, this idea of getting students to ask:  “How far is 9 from 10?  Take that away from the other number and then add the result to 10″ is extremely complicated.

The students are also expected to do a question like 12 – 7, by noticing that 12 is 2 away from 10, so if you could do 12 – 2 to get 10, you’d just have to take away 5 more (since 7 – 2 = 5) so the problem becomes 10 – 5 which is ‘obviously’ 5.

Finally, students are taught the doubles of each number so they can add two number that are very close together, like 7 + 9 by doing 2*7 + 2 = 16 (since 9 is 2 more than 7).

Now I am not a specialist in early childhood education.  I certainly can’t be expected to know as much as the new Teach For America Early Childhood Education director Brittany Toll, who I believe taught lower elementary for two years.  But I have some experience as the father of a precocious five year old.

When my daughter Sarah was about 7 months old, I got the very stupid idea that I would train her to be the youngest ever solver of the Rubik’s cube.  So I got a cube and removed all the stickers except for two white stickers.  My plan was to have her ‘discover’ how to get the two white stickers together and then I’d slowly add more of the stickers until she could do the whole cube.  I didn’t force her to practice, but I would conveniently leave this custom cube around the apartment to see if it would interest her.  I learned pretty quickly that you can’t force someone to understand something that they are not ready for yet.  I gave up on the training and Sarah still has no interest in the cube (and she’s way past the world record age anyway.)

A private joke I have with my wife is “Seven … beven” since about six months ago I thought I’d work with Sarah on her number sense.  I said “Let’s play a game where I say a number and you say what number comes next.”  I found that when I’d say 5, she’d have to start counting from one until she got to five and then went one more, rather than just start at 5.  So I was quizzing her and she was getting a bit better at this.  I said 3 and she said “Three, four.”  I said 9 and she said “Nine, ten.”  Then I said 7, and she said “Seven … beven” indicating that the game was over.

Here are some videos illustrating what happens when you try to teach something developmentally inappropriate.  I hope nobody sees these as me exploiting my children.  I think of my blog readers as my friends and my kids are so cute I can’t resist.  But if people feel strongly that I should remove the videos, please say something in the comments, and if enough people do, I’ll take down the videos.

Here’s my daughter trying to teach my almost two year old son a complicated color matching game before he’s developmentally ready.

Here’s Sarah when she was about 2 1/2 trying to teach me how to properly drink from a sippy cup and properly say “ahhhh” afterwards, before I was developmentally ready.

27 Responses

  1. Neil James

    If a child is struggling with addition or subtraction conceptually, counting methods are a better fit than combination methods, and certainly no less rigorous than doubling or recombining. It is far more important that the student is convinced that the algorithm works and gives a consistent and true answer than to memorize a specific method.

    Take the 9+7 example. To make a ten is cognitively complex, involving a memorized concept, some understanding of betweenness, and remembering and manipulating several intermediate values. I’d have had to repeat these steps hundreds of times to be convinced that this algorithm is true (without the advantage of having variables).

    Given x,y in Naturals, where x+y>10, and y<x<10. Find the sum by "making a ten:"
    1. Recognize the sum is greater than ten.
    2. Recall the rule: the sum of 10 and some digit, d, is 1d.
    3. Apply an understanding of betweenness to determine that y<x<10.
    4. Subtract and hold a value in memory: 10-x.
    5. Subtract and hold a second value in memory: d=y-(10-x).
    6. Add the first intermediate value to x to make 10: x+(10-x)=10.
    7. Apply the rule: 10+d=1d.

    The "make a ten" algorithm for x+y is six or seven steps to obtain "x+(10-x)+y-(10-x)," with no guarantee that the student understands where the result comes from (because step 6 can be skipped and the result replaced with 10 every time, without affecting the answer). It does not sound like a strategy for a student struggling with addition concepts. It may be okay for an advanced first- or second-grader (and even then, it is perilous if she doesn't convince herself of it's truth on her own).

  2. meghank

    I was teaching first grade last year, with the horrible Envision Math curriculum (owned by Pearson).

    I quickly realized that the methods they used are useless to first-graders, and, besides that, impossible to teach. Most teachers I know tossed that curriculum (despite severe threats from the district) and taught math the way they knew from experience students need to be taught (when no one was watching, of course).

    (I’m teaching kindergarten this year and don’t even bother tearing out those worthless workbook pages anymore. I let them use their math workbooks as a coloring books. Here is a page from an Envision Math textbook: http://atthechalkface.com/2013/03/06/guess-the-grade-level-of-this-common-core-homework/)

    Please also do a post on how the Common Core State (sic) Standards in Reading are harmful to the lower grades. The requirement that half of all reading done at elementary school be non-fiction is particularly devastating in the lower grades, and for teachers who believe that children should choose what they read and read what interests them most.

    • I am also not a fan of the “with prompting and support” line in a bunch of the Kindergarten Common Core reading standards. Not only is this awfully broad, it seems like a pretty clear advisory that the skill to follow is well beyond Kindergarten students. If children need “prompting and support” to complete a task, is this task something to be mandated in a set of standards? Shouldn’t we set tasks that students will be able to master themselves at the end of the year?

      The Common Core math activities I have seen for Kindergarten just seem ridiculous. This one:

      http://shaunpjohnson.files.wordpress.com/2013/03/badkworksheet1.jpg

      was apparently designed by robots who have never encountered five year olds (or even high frequency word lists, for that matter). Even the size of the box for the illustration is badly thought out.

      The utter lack of educators involved in the creation of these standards and the lack of field testing are doing some horrible things in early primary classrooms.

    • Tee

      I have spent a lot of time thinking about the lower grades and the nonfiction requirement. I’m not an elementary grade teacher, but I did spend a few years volunteering in kindergarten and second-grade classrooms where students had a range of books. I have also spent about a decade of my life babysitting for babies, toddlers, and elementary-aged children.

      I always found that little kids seemed equally drawn to, say, books about Martin Luther King, dinosaurs, and dolphins as they were to fantasy or fiction books. Increasing the nonfiction requirement doesn’t mean that students won’t be able to choose what they read and read what interests them most, does it?

      Again, I could be completely wrong about this, as I am not an elementary-school teacher. This was just what I’ve noticed in my experience.

      • meghank

        If they happen to want to read mostly fiction, as many fantasy-lovers do, or mostly non-fiction, as many children, particularly boys, do, then it does conflict with that requirement.

        Just think about it. Would you like to be told now, as an adult, exactly what percentage fiction and what percentage non-fiction you can read?

  3. Monica

    I for one find your children adorable, and obviously of superior intelligence. I was especially charmed by your baby boy’s wiggling feet and toes in the first video. My own children are 22 and 16, and their little sausage feet are a distant memory. Sign me, “patiently awaiting grandmotherhood”.

    Regarding the larger point of early mathematics, I am a first grade teacher involved in a CCSS seminar at our county office of ed, and am both hopeful and concerned. I am worried mostly how it will all roll out, and that much will be lost in translation between the standards themselves, and how their implementation will likely be convoluted when translated into commercially published materials, with the specter of test proficiency hanging over the whole thing. Right now I’m predicting the CCSS may collapse under its own weight.

    That said, here is a short description of very early number sense development given by Prof. Doug Clements:
    http://www.youtube.com/watch?v=sbNFjHqzXF4

    There are lengthier talks by him on youtube. But this is a good start. (He does have something to sell, but I believe he is sincere and very knowledgeable!!)

    • meghank

      You say that much will be lost in translation between the standards themselves and the curriculum that will result, but I believe that the standards themselves are the problem.
      Don’t you see that this guy (Prof. Clements) is suggesting that children develop at different rates, and that it is working at cross-purposes to require that all children meet ALL of the standards by the last day of kindergarten?

  4. Cosmic Tinkerer

    Your kids are adorable! That was not exploitative and no harm was done, IMHO. Older sisters often enjoy playing teacher with younger siblings and you captured it. I’m sure there will be many more “lessons” to come. Thanks for sharing!

    • jaded serf

      I must agree with Cosmic Tinkerer’s comments. Your wee ones are lovely little learners!

      Your clip shows us that big sisters make great teachers. Maybe this is what our schools need for improved first grader test scores: Big sisters–one for each student! :)

  5. eminnm

    I don’t know very much about teaching the tiny ones, and I’m willing to get behind you on the sometimes-less-than-perfect rigor levels of CCSS (such as my fourth graders supposedly needing to learn what dimeter and pentameter are in poems, but not onomatopoeia or rhyme patterns, which are a whole lot more interesting and relevant to nine-year-olds in my opinion). I definitely think they’re a step up from our state standards in general though.

    My question is: When does it become developmentally appropriate to teach adding and subtracting like this? And how do you make that shift happen? My kiddos still count on their fingers much of the time for things like 9 + 4, except the ones who have that elusive number sense. They should be “ready” for this by fourth grade, but it’s still hard to convince many that this way of thinking will eventually be faster when, right now, tapping each finger is MUCH faster than working through all that thinking. Is it just repetition? Memorization? Maybe if we started teaching it when they’re little along with the traditional ways of adding, even if they don’t totally master the concept, at least they’ll have it in the back of their mind when they get older. But I see what people are saying about why bother making it a standard if we aren’t really expecting all kids to meet it, too. I don’t know. Tricky.

    • meghank

      “Maybe if we started teaching it when they’re little..”
      The key word here is maybe. Why haven’t these standards been field-tested, so that we KNOW what will result when we use them?

      • eminnm

        Are most standards field-tested? Is that how they usually develop standards? I honestly have no idea…It does seem like a field test for CCSS would take multiple years though, which is probably why they haven’t done it.

        • Why would it matter if a field test would take several years? It isn’t as if states are totally lacking in standards at present. It would be possible to complete a rigorous evaluation and field test (preferably with lots of public comment, reference to child development and neurology, etc.) before unleashing a new set of standards.

          The rush around the Common Core is artificial. They’re being sold as a solution to the problem of Our Failing Schools. Whether or not our schools are failing, it isn’t at all clear that what our schools lack is high expectations. Many states adopted these standards as part of applying for Race to the Top money. So our funding crisis plus a little competition = Common Core in most states. Are those good reasons for enormous changes?

          My school lacks pencils and copy paper at this point in the year. We need these more than we need a new expensive curriculum aligned to new standards or the computers to test those standards.

          I have a hard time believing that the proponents of these standards are so worried about our failing children that they cannot sleep until educators bring Rigor and High Expectations Common Core style, and that such is the crisis that our children can’t wait. If our children can’t wait for these standards, why must they wait for pencils, working heat, and safe drinking water?

    • emmay8

      As a second grade teacher, that is exactly my approach. The standards state that my students need to be fluent in number facts within 20 by June. The usual method of rote memorization doesn’t work for many, as is evidenced by thier performance on fluency activities. It also teaches them nothing about number relationships, which is neccesary for growth in Math. I taught several different strategies for addition and subtraction, with the hope that the one that made the most ssens to each of them would be the one thiey’d use. That is exactly what is happening. I approach the Common Core as a tool to make the studnets think. Memorization does not broaden you scope of thinking. And when a standard says that it needs to be met with support, I look at this as a sign that this is a foundational standard. They will eventually be doing it without support but they need the foundation and experience first. I may be coming from a differnt angle because I work in a Catholic school, where studet perfromance on Common Core aligned assessments, does not threaten my job.

  6. Hilary

    Your kids are so cute!

  7. E

    I teach in a Montessori school, and children as young as first or second grade use the “positive snake game” to practice addition, and even multiplication, using tangible objects at the same time developing their number sense and mental math. You can see some of the amazing Montessori math materials in the downloadable albums at freemontessori.org. Whether you teach kindergarten, middle, or high school, there is something here for you. I think the key is show, don’t tell…let the children see the formulas as a fun shortcut that has application to a task, not a task in itself. Montessori has many, many math materials that teach in this way, helping the children go from concrete to abstract as they are ready (so it may not work really well as a one-day lesson objective but for broader themes or more relaxed schedules–and for how children learn–it’s really great!)

  8. Allison

    Tell me if I’m a failure of a mother or if my kid needs serious tutoring (mind you I have two older daughters who were on task and now I have a boy – it’s all new to me): The problem was to find out how many kittens Katie has. Clues: Between 300-400, even number, sum of two of the five numbers listed below (215, 248, 137, 226, 182). I don’t recall doing triple digit addition with regrouping when I was in second grade much less this sort of problem. Believe me I want my kid challenged, but this was frustrating. For him and for me (I was so tempted to stab myself in the eye with a pencil).

    I’m curious if any teachers here worry that this “expose them to things they’ll deal with in a few years” gets in the way of the kids mastering the concepts they need to know now.

  9. Kathy

    Allison-I’m a parent worried that all the dilly dallying in future concepts is slowing the ability to master. Our educator calls it “the spiral” & insists it eventually “all comes together”. I disagree. Instead of focusing on mastering counting money, my first grader brings home worksheets to count :NQDNQQDP. I’d prefer he just learn to count money first, before introducing tricky concepts. Or 1/2 past 1/4 after clock to introduce fractions-wrapped in telling time. Just teach the :30 & :15! Geesh-I’m so frustrated.

  10. Poppy

    Wow, your blog post brought back grade school math memories! Forgive me for reminiscing…

    In first grade I had a system of “number friends”, and their associations were governed by “rules”. These “rules” bridged the gaps between what I had memorized, and operations I hadn’t. So, taking your example of 9 + 7, my rule was any time 9 was around, and adding needed to be done, the answer would be one less than the number with 9 (aka, 7 brings a 6). This was (obviously!) totally made-up and not taught.

    All the basic combinations of numbers had emotional and/or value judgements too: 8 + 6 was “annoying”; so was 7 + 4. But combinations of 7 and 5 were “good”–probably because they are harmonious, ie, subtracting you get 2 and adding you get 12.

    I had a full dose of old math, heavy on the memorization. I can still remember the day I discovered adding and multiplication were the same thing: in the middle of a test in 3rd grade, when I had forgotten part of the times table sequence.

    I wonder: can taught revelation be as lasting as something you discover yourself?

  11. Daniel

    I don’t think you’ve demonstrated in this post the the validity of ‘developmentally inappropriate learning’ at least not from an age perspective. What’s been demonstrated is that the challenges/lessons as you presented them, were either not fun enough, or not incremental enough to sustain your children’s motivation/attention.

    P.S. I don’t think you did anything wrong, and I’m not sure why it seems you’re ashamed of posting videos of them.

  12. drsandy

    My kid is a math wiz and he’s going to die from this. He’s 4 years ahead in arithmetic and this is not for him. He knows the calculations. complex ones too. I agree with EVERYTHING you wrote and let me tell you it gets worse because some of the questions make NO sense even in a universe far far away

  13. drsandy

    developmentally inappropriate varies by person but obviously there’s an average. If one kid can ride a bike at 4, almost all cannot. So I agree with the author though it’s besides the point. It’s poor pedagogy. And I agree that it should be taught later because its more arduous at an early age and not important at that time.

  14. Lisa

    Allison, I agree with you completely. I have a 3rd grader, 5th grader and 9 th grader. The 9th grader told me her math teacher tossed aside this spiral math homework telling the students that it would confuse them more than anything because they were a year away from learning that concept and that they needed to just work on homework dealing with the concept they had learned that day. Smart lady! On the other hand, my 3rd grader comes home with her columns of homework dealing with concepts she has yet to be taught all crammed together on one sheet of paper. She has one problem per concept. The problems are predominately not ate appropriate and I am spending time trying to teach a new concept for one problem only to move on to yet another new concept. Ugh. I have heard the same thing from the teachers. If they see a concept now they will understand it better when it is taught later on in their life. Excuse me but don’t they need to invest the time in learning and mastering the concepts they are being taught, oh I don’t know, today??

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By a somewhat frustrated 1991 alum

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