Kindergarten demands ‘algebraic thinking’

Kindergarten is too tough for little kids these days, New York City teachers complain to the Post.

Way beyond the ABCs, crayons and building blocks, the city Department of Education now wants 4- and 5-year-olds to write “informative/explanatory reports” and demonstrate “algebraic thinking.”

Children who barely know how to write the alphabet or add 2 and 2 are expected to write topic sentences and use diagrams to illustrate math equations.

Under newly adopted Common Core State Standards, kindergarten teachers read aloud “informational texts,” such as “Garden Helpers,” a National Geographic tale about useful pests.

After three weeks, kids have to “write a book about what they’ve learned,” with a drawing and sentences explaining the topic.

In math, kindergarteners learn about the “commutative property.”  (I recall learning that in middle school.)

 The big test: “Miguel has two shelves. Miguel has six books . . . How many different ways can Miguel put books on the two shelves? Show and tell how you know.”

Teachers rate students’ performance as “novice,” “apprentice,” “practitioner” or “expert.”

An “expert” would draw a diagram with a key, show all five combinations, write number sentences for each equation, and explain his or her conclusions using math terms, the DOE says.

Cathleen Vecchione, a kindergarten teacher at PS 257 in Williamsburg, Brooklyn, has taught her students to count by 10s, but hasn’t started teaching addition.

Her students are expected to write simple sentences, such as “I have a pet.”

I tutor first graders in reading and I once volunteered in my daughter’s kindergarten class. Writing is very challenging for little kids. Some can’t form letters. Most can’t spell. It’s especially tough for boys. And I haven’t met many five- or six-year-olds who are ready to write equations.

In fact, I’m 60 and I’m a little puzzled by Miguel’s book options. The Post suggests there are five combinations. I get 14 ways if it’s just about how many books go on each shelf. (Zero books on Shelf A and six on Shelf B and so on, then zero books on Shelf B and six on Shelf A and so on.) But what if Miguel is putting some books on their side, and other backwards and . . . Is he organizing by subject matter? Perhaps he’s got his physics books on Shelf A and his philosophy books on Shelf B.

In Developing the Habits of Mind for Algebraic Thinking, Barry Garelick implies that fifth graders aren’t ready to write algebraic equations. “Giving students problems to solve for which they have little or no prior knowledge or mastery of algebraic skills is not likely to develop the habit of mind of algebraic thinking,” he writes.

About Joanne


  1. Kindergarten should be a happy time of poems, stories, listening, good manners, sharing, coloring, learning basic days of the week, months of the year, colors, etc.and being encouraged to make connections. Without these simple little skills and schema, actual academic school is more difficult. Average and above children are not usually ready for calculus, and expecting children from an un/undereducated home to already know these things and be ready to begin algebraic equations at age 5 is the dreamchild of a worse-than-stupid mind. No wonder so many adults in this country don’t know how to work, or concentrate, or even read; they were expected to KNOW before they had a chance to LEARN, and this turned their educational experience into a nightmare. Let our children learn to play and learn together in a children’s garden, which is what kindergarten is supposed to be. Academics come later, when the planted seeds are mature enough to put down roots and grow.

  2. Michael E. Lopez says:

    I suspect, based on my limited knowledge of the subject, that the human brain isn’t really ready for this sort of abstraction this early.

    Kindergarteners would be better off playing tag. Seriously.

    • Your comment reminds me of something I heard about recently – starting kids in kindergarten at an older age because they’re “not ready” for the curriculum at the standard kindergarten starting age, either as a voluntary move by a parent or by changing enrollment standards. So we as a society pat ourselves on the back for putting first grade material into kindergartens, while raising the age for school enrollment because kids who are ready for traditional kindergarten aren’t ready for the academic version.

      I personally favor introducing a broad range of math and language concepts at the kindergarten level. I think it’s possible to do so in an age-appropriate manner, and that if done well you lay a foundation for academic preparedness as the kids encounter more complex material in future grades. But it seems like we’re presently… doing it wrong.

  3. The number of ways to put n books on two shelves is 2^n.

    • Foobarista says:

      The problem here is the general problem of what is “unsaid”, but assumed, in math word problems. Sure, if you use the unstated assumption that you put all the books in the two bookshelves, you’re right, but that isn’t stated in the problem. I got into a lot of trouble in math classes because I was always finding logic holes or unstated assumptions in the text of the problems instead of just going for the “obvious” answers (which weren’t all that obvious).

  4. The number of waysto put n books in k shelves is k^n.

  5. I can arrange 6 books on one shelf 720 ways

  6. Roger Sweeny says:

    I’m sure the answer they are looking for is that Miguel could put 1 book on the first shelf and 5 on the second, 2 on the first shelf and 4 on the second, 3 on the first shelf and 3 on the second, 4 on the first shelf and 2 on the second, or 5 on the first shelf and 1 on the second.

    That’s a total of 5 different ways. It does show commutativity in a concrete way: 1 on the first shelf and 5 on the second shelf uses up all 6 books just like 5 on the first shelf and 1 on the second shelf does.

    This could be a useful exercise if the class had done things like it before, and if the instructions were really clear. For example, that there has to be at least one book on each shelf and that order on the shelf doesn’t matter (both of which tripped up our hostess).

  7. The shelf question is unbelievably inappropriate for that age group, and poorly worded to boot. Is Miguel expected to use all the books for this exercise, or can it be anything up to and including six? Is order important, or not? Two basic questions which play a significant role in solving this problem.

    For crying out loud, I’ve taught college freshmen of presumably average intelligence who would be flummoxed by this problem – and Common Core standards anticipate kindergarteners being able to derive a correct answer? Seriously?

  8. Mark Roulo says:

    In Developing the Habits of Mind for Algebraic Thinking, Barry Garelick implies that fifth graders aren’t ready to write algebraic equations. “Giving students problems to solve for which they have little or no prior knowledge or mastery of algebraic skills is not likely to develop the habit of mind of algebraic thinking,” he writes.

    I don’t think his objection is to 5th graders doing this because they are only in 5th grade (although he might object). His objection is that they are being asked to do something that they haven’t been taught how to do.

    I’ll observe that a commonly used Russian 3rd grade math textbook contains the following problem:

    Solve the following equations:
    1,111 – x * 207 = 490

    [Note: There are more equations to solve]

    It is pretty clear that the Russians did not (and probably do not) consider this something that 5th graders cannot do [but note that Russian kids start school one year later than ours, so their 3rd grade kids would be old enough to be in 4th grade here. Still …]

    • GEORGE LARSON says:

      When you say commonly used how common is common? I read long ago that Russian education is not uniformly excellent. Non Russian speaking minorities, and non urban elites are not included in these advanced curriculums you describe. The Soviet system used early testing and tracking of its students. Those who did not get into the fast tracks did not get much. A large part of the Red Army could not easily speak or understand Russian.

      • Mark Roulo says:

        I am not Russian, so I can’t say for sure.

        I know that they tracked (and probably still track) earlier than the US does, but prior to 3rd grade seems pretty extreme. A 1958 Life article series comparing the US and the USSR education systems claimed that “only” 1/3 of Russian kids completed all ten years of pre-college education. But all of those 1/3 seem to take calculus, so I don’t think you can track too many out by 2nd grade. I could be wrong.

        Anecdotally, my wife had it at a karate class my son was taking and one of the other moms recognized it.

        The textbooks are referenced here:

        I had the impression that the Soviet Union kinda centralized things like textbook selection 🙂

      • George,
        commonly used would mean REQUIRED (usually nicely put as Recommended) for use by all schools by the Department of ED.
        And I graduated from school in Ukraine (former Soviet Union) in’93 – we had no tracking. Neither had my friends (who we met in the US) from Russia (Moscow, Novosibirsk, St.Peterburg) and other former soviet republics. There were no Honors classes, no special ed, and no electives – everybody marched through the same things in the same grades (centralized curriculum). The only thing different was that students were allowed to finish school after grade 8 with a certificate of incomplete general education. Those who were planning to go for higher education were staying for grades 9 and 10.

        I use Russian Math textbooks (as well as Physics and Chemistry books) with my son. Equations (simple ones) start in 2-3 grade… But so is “language of mathematics” – definitions of terms, introduction of variables, translation of word problems into mathematical terms, setting up the problems in format (useful for physics/chem). However, statistics, probability, and problems on “how many ways” are not introduced until grade upper grades. (From my schooling, I don’t remember them at all…)

      • Mark Roulo says:

        Another followup …

        A Japanese 3rd grade math textbook also shows very simple algebra problems, but with a box instead of a letter value for the variable.

        Example problem:
            [_] – 17 = 25

        Interestingly, the examples here involve iterative guess-and-check …

    • Barry Garelick says:

      Yes, I did not mean to imply it was inappropriate for 5th graders, but that the 5th graders were given a problem without the tools/skills to solve it efficiently. In addition, the problem was stated in such a way that it involved inductive assumptions rather than deductive reasoning.

  9. If the number of ways to put n books into k shelves includes the order on which they are arranged in each
    shelf then the number ofways is n! Cn+k-1,n which is
    also the product of the consecutive integers from k to

  10. My brain is boggled by this. My homeschooled first grader is very good at math, intuitively understanding decimals, negative numbers, and simple algebra. He’ll be finishing up 4th grade singapore math soon. He does multistep word problems, but I can’t imagine him doing that one. I also don’t understand how they teach reading and composition at the same time. He writes, but he learned to read first.

  11. to Foobarista – Your comment reminds me of a preparation course I once took for one of the actuarial exams. Our instructor told us that often the hardest thing in doing word problems was not figuring out what is the answer but rather figuring out what is the question.

  12. To George Larsen – Only about 50% of the population of the old Soviet Union were Russians. So probably a lot of the Red Army spoke Russian poorly. Hell, Stalin, who was Georgian, spoke Russian with a strong accent. The Athenians thought that Alexander the Great’s Macedonian Greek was hilarious and ridiculed it behind his back. In Stalin’s case I doubt that anybody said anything about his accent.

    • Well, somebody surely said something … once. Then everybody else wised up.

      This is yet another iteration of raise all boats … yet I doubt we’ll see any jumps in the exit exams (state, NAEP, ACT, SAT, whatever). The problems we face are likely socio-economic rather than limits of human ability.

      Plus, seriously, are our grade school teachers prepared to teach this? I mean, other than reading it from a script; but to answer deep and fundamental questions.

  13. This is yet another example of not taking developmental readiness into consideration. When we push kids to learn something before they are ready, we are wasting their time and ours. It’s more effective to teach them what they’re able to learn when they are able to learn it. By jumping into algebraic thinking like this, we are missing an opportunity to teach them basic math skills.

    I teach middle school math, and I see kids all the time who can’t figure out what a simple question is asking them to do. They are whizzes with manipulatives, but they tend to use them to build things rather than solve problems. If they’d had a solid math foundation before they got to middle school, I could more easily take them to the abstractions they need to go to at that age. But instead, I’m teaching them match facts and how to multiply fractions.

    There are, no doubt, children who are ready in Kindergarten to learn such things. They should get that knowledge if they’re ready. But everyone else should get what they are ready to learn. Pushing too fast slows everything down.

  14. GoogleMaster says:

    I get 1812.
    For arranging 6 and 0, 6! x 0! = 720.
    For arranging 5 and 1, 5! x 1! = 120.
    For arranging 4 and 2, 4! x 2! = 48.
    For arranging 3 and 3, 3! x 3! = 36.
    For arranging 2 and 4, 2! x 4! = 48.
    For arranging 1 and 5, 1! x 5! = 120.
    For arranging 0 and 6, 0! x 6! = 720.
    Sum = 1812.

    If you allow one or more books to be dropped on the floor, then I get 2212 “ways to put books on two shelves”.

    Even if you assume that “ways to put books on two shelves” refers only to using all six books and the order of the books on the shelves doesn’t matter, I still get 7, including the 0-and-6 and 6-and-0 cases.

    This assignment would have made my head explode. Correction, my questions about this assignment would have made my teacher’s head explode.

    • Roger Sweeny says:

      Googlemaster, if you ask a kindergartener to put 6 books on two shelves and you then put all 6 on one shelf, she will tell you that you haven’t put them on two shelves. The second shelf doesn’t have any books on it. You have simply put 6 books on one shelf.

      But, of course, you are smarter than a kindergartener and know that since zero is a number, it makes perfect sense to say, “I did too put the books on two shelves. I put zero books on the second.”

  15. To Googlemaster – Your calculation is incorrect. In the second line for example there are siix ways to select 5 books from a set of six books for the first shelf (or equivalently to select 1 book for the second shelf) so 120 must be multiplied by 6 to get 720. In fact all lines come out to 720 so the correct answer is 5040 = in accordance with the general formula – the product of the consecutive integers from k to n+k-1 where n is the number of books and k is the number of shelves.

  16. The different ways of interpreting the books in shelves problems illustrates the difference between Bose-Einstein and Fermi-Dirac statistics.

  17. Books follow Bose-Einstein statistics.

  18. Or is it that books follow Fermi-Dirac statistics? I’m an actuary not a physicist. I forgot which is which.

    • Foobarista says:

      Einstein would have likely gotten the question “wrong”, and dropped out of kindergarten.

  19. There is a simple inductive proof for the formula for the number of ways of placing n books in k shelves. The proof inducts on n for each fixed value of k. First note that for n=1 the formula gives k ways which is obviously correct. Assume for a fixed value of k that the formula is true for n-1. Then consider any way of putting n-1 books in k shelves. for such a way if the number of books on the j_th shelf is i_j then we must have the sum of the i_j’s from 1 to k equal to n-1. Then the final book may be placed on the j_th shelf in i_j + 1 different ways. So the total number of ways of adding the final book is the sume of the i_j + 1 from 1 to k which is n-1+k. so the answer for n books can be obtained from the answer for n-1 books by multipying by n-1+k which inductively establishes the formula.

  20. I don’t like the Miguel problem. It sounds like combinatorics. The kindergarten standards don’t include combinatorics.

    The specific standard being addressed here is presumably K.OA.3, “Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).”

    Here is a problem that would meet this standard: “Grandma has 5 flowers and two vases to put them in. Draw a picture of how she can arrange the flowers. Tell how many flowers are in each vase in your picture.”

    The teacher could summarize all the students’ different answers on the board by writing equations, 5 = 2 + 3, 5 = 4 + 1, etc.

    Being able to decompose a number in more than one way is helpful for “making 10” (see 1.OA.6), which in turn is an important strategy leading to the addition algorithm.

    In any event, this is not about combinatorics; it’s about understanding numbers and number relations.

    And by the way, nothing in the kindergarten standards requires the students to write equations either. Students could write equations if they want, but the standards don’t require it. It could be the teacher summarizing things that way.

    RTFS, please.

    • “It could be the teacher summarizing things that way.” Exactly right; this is the crux of the problem with Common Core as I indicated in an article about the standards in The Atlantic. William McCallum commented that there is nothing in the standards that dictates how the standards are to be taught, which I assume means that teachers could use whole class/direct instruction if they wished. But the Standards of Mathematical Pratice are being interpreted to require student-centered, inquiry-based approaches, along with attempting to instill algebraic “habits of mind” well in advance of a pre-algebra or algebra class which lend themselves more naturally to teaching such habits, as I discuss in the article that Joanne linked to.