In the trenches with enVisionMATH

It’s been back-to-school time for everybody in our household (hence an excuse for the light posting). We started classes at the college today, and last week the 4.5-year old went back to preschool full-time and the 6.5-year old started first grade. (The 1.5-year old is rocking the local daycare.) One of the biggest changes for the kids is for our first-grader, Lucy, since she has real homework for the first time. It’s not much; the expectation is about 20 minutes a night, Monday through Thursday. Some of that homework is math, which I was very excited about — but then that excitement turned to alert caution when I learned my daughter’s class was using enVisionMATH.

I wrote this post on enVisionMATH almost three years ago, basically laughing it off the blogosphere for its happy-clappy, uncritical acceptance of unproven digital nativist frameworks and for going way over the top with smartboards. Little did I know that my own offspring would be in the middle of it just three years later. So, in an effort to process what she’s doing (for me, for her, and for anybody else who cares), this is the first of what might be many posts about the specifics of enVisionMATH, as viewed by a parent whose kid happens to be learning from that curriculum, and who also happens to be a mathematician and math teacher.

I’ll start with the worksheet Lucy brought home this evening, called “Making 8”:

I’ve never had a kid in first grade before, nor can I remember how I did this stuff in first grade, nor have I recently worked with a kid in first grade. So I’m going to share my thoughts, but realize I have no reference for what’s “normal” pedagogy for 6-year olds and what’s not.

This worksheet is really about subtraction, although it never comes out and says so. The first two exercises are attempting to build a sense about subtraction by getting kids to think about how parts fit together to form a particular quantity. enVisionMATH appears to be really big on getting kids to recognize numbers visually rather than by counting. I’ll need to blog about this in a later post, but Lucy’s had some other exercises that, for example, stress the ability to recognize this:

…as the number 6, just by looking at it and without counting the dots, almost to the point of telling kids that they shouldn’t be counting anything but rather arranging things into patterns. Again, that’s for another post.

So, back to the worksheet, kids are supposed to look at the first collection of balloons and, knowing that there are eight of them, see — and only “see” — that 8 splits into 2 plus 6, and then 4 plus 4. I did a few more of these with Lucy using coins (no balloons on hand, sadly). Biggest challenge here: Keeping Lucy from just counting the black balloons and then counting the white balloons. And the only reason this was a challenge was because, as a math person, I knew what the worksheet was getting at: recognizing quantities through visual patterns rather than counting, so the unwritten rule is for kids not to count the balloons. But other parents probably didn’t know this, and their kids just counted. I don’t think this is necessarily wrong, but it doesn’t necessarily help in the next sections either.

The next section is rather startlingly labelled “Algebra”. Remember: This is a worksheet for a first grade class. Why we are bringing up the word “algebra” at this point is anybody’s guess. I suspect this is more to make parents, school boards, and accreditors happy than it is to start getting kids to feel comfortable with the word “algebra”.  But anyway, as you can see, the two problems are just the first two problems in reverse.

Lucy had a hard time with this. First of all, she didn’t understand what “the whole” meant. This is not the first time Lucy’s struggled not with the math but with relatively esoteric vocabulary in her math lessons. Last week she had a worksheet where she was to arrange three integers “in order from greatest to least” and “from least to greatest” and we had to take a moment to figure out what all of that meant. Maybe other people’s kids don’t struggle with that, but on the other hand it’s been verified that Lucy is reading at a third or fourth grade level right now, so I wonder if it’s just her.

We had to work these out using manipulatives. We started with fingers because that’s the first thing I thought of. So, I said, if the whole is 8:

…and one part is 3:

…then what was the other part?

Lucy was able to get the answer of “5” with no problem. But… I don’t think she got it the right way. Because when we moved to the next problem and the “one part” was 1, for her, the other part was still 5! This was because when I held up one finger on my left hand this time, there were of course five fingers on the right hand. I tried holding up eight and wiggling one finger instead of putting the “one part” on one hand, but that just confused her. So, we went back to coins and built a “balloon diagram” like in the first two problems, and she got them just fine (and without counting).

I don’t think exercises 3 and 4 are bad problems necessarily, but I do think they came in here way too early. Perhaps I’m missing the context of the actual classroom interaction between Lucy and her teacher, but it would seem like a better idea to do as many exercises like 1 and 2 as possible before moving on to the “algebra”. After all, if you stick to positive integers, there are only seven ways to fill in the blanks __ + __ = 8. (And doing all seven might help kids discover the commutative property early on, which seems like a much more important thing to bring up than “algebra” in first grade.)

And then, it’s not clear to me that doing “algebra” is a better idea here than just doing straight-up subtraction.  What’s to be gained by saying “the whole is 8; one part is 3; the other part is ____” versus “What is 8 minus 3?” Again, maybe I’m out of touch, but subtraction is a fundamental skill that algebra builds upon; doing algebra before subtraction seems a little backwards to say the least. A kid who is comfortable with subtraction will be able to do these whole/part problems in a snap by using subtraction. A kid doing these “algebra” problems basically has to invent subtraction in order to do them, or else draw pictures of balloons and start counting. It feels like the curriculum is trying to be intentionally nontraditional here, just for the sake of doing things differently rather than because it works better.

Then we come to the “Journal” question, which is downright sophisticated: “The whole is 8. One part is 8. What is the other part?” Here we reach serious abstraction: You can’t draw balloons like in exercises 1 and 2, and in fact resorting to physical props is tricky.As Derek Bruff mentioned in a tweet about this earlier this evening, the use of the word “part” in conjunction with the quantity 0 is already sort of questionable. What does it even mean to say the “part” is 0? What “part”? I don’t see a “part”. The natural way of interpreting what a “part” is, is as a bunch of objects. If there are no objects present, then there really isn’t a “part”.

We had to resort to thinking not about objects but containers that hold the objects. I took two books sitting nearby. I took my eight coins and said: The whole is 8. One part is 2 — and put 2 coins on one of the books. What is the other part? — and put the remaining coins on the other book. Lucy got the right answer quickly, and she did so by looking back at exercise 1 with the balloons and noticing it was the same problem with different objects, which I thought was pretty smart. I’ll make an algebraist out of her yet! Then I repeated with one part being 1. Then I did it with one part being 6; then 7. Then I said, “The whole is 8; one part is eight.” — putting all eight coins on one book. “What’s the other part?” — showing her my empty hands and an empty book. “Zero,” she said right away.

For her, and maybe not just her, “zero” represents not a size of a part but a state of emptiness of a container — or perhaps the size of a set. It’s how much you see when nothing is there. To map the “zero” concept onto a concept of “part” that presupposes something is there just doesn’t make sense. If this sounds like the New Math, I think we’re barking up the right tree.

The “Tell how you know” was especially tough because it involves getting Lucy to talk about what she did, even though she’s doing it at a sort of visceral level, and then spell the words she needs to use — which is the other type of homework she has. I got her to say out loud what she was thinking, and then I had her say it back to me and then helped her spell the words.

So we made it through the worksheet, but there are a lot of questions in my mind about the pedagogical design of this stuff. And how in the world does this sort of thing work in a household where the parents don’t have the time, patience, interest, fluency, or comfort level in mathematics to sit down and work all this out with the kid?

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Filed under Early education, Education, Math

17 responses to “In the trenches with enVisionMATH

  1. My younger daughter has brought home worksheets similar to Lucy’s worksheets. I think it must be the same curriculum at her school. I was so baffled by the instructions on one worksheet that I had to call my dad, who has more math background than I have. Even he had trouble deciphering the task. It’s crazy when a math professor can look at a first grade math worksheet and see these issues and how confusing they must be for kids, but it also makes me feel better. I was never a bad math student, but I was decent, and even though I took four years of math when my school required three at the time, I never took a calculus course. I felt pretty embarrassed calling my dad for help on third grade math.

  2. Robert,

    I also started to blog.

    Your friend from Wittenberg Trail.


  3. So the reason I asked about Singapore was the part-part-whole relationship that you tweeted. One major problem with U.S. curricula is the tendency to jump from the concrete to abstract without a stage in between, like from the balloons to the multiple guess question on your daughter’s worksheet. Understanding a part-whole relationship precedes learning addition & subtraction.

    That jump to understand and explain zero seems a little abrupt, too. Isn’t the point of 1st grade home enjoyment to PRACTICE? You can’t practice if you haven’t been taught. Why would they introduce a new concept to a student, at home, where parents have to explain it?

    You should have questions about the “pedagogical design of this stuff”.

  4. I was lucky, in that the public school my son went to allowed me to substitute Singapore math books one grade up for the worksheets they did at the school (it saved them from figuring out how to handle a student a year or more ahead of the rest of the class).

    Incidentally, on the digital-native thing, people are still using the concept, despite its being rather threadbare now.
    I blogged on it recently:

  5. What a fantastic post!

    re: Why we are bringing up the word “algebra” at this point is anybody’s guess. I suspect this is more to make parents, school boards, and accreditors happy than it is to start getting kids to feel comfortable with the word “algebra”.

    It’s not about keeping any of those folks happy.

    Public schools aren’t accredited, parent opinion doesn’t matter, and school boards are typically compliant to administrations, not the other way around.

    If you want to know why algebra is being brought up in 1st grade, the answer lies in education schools and the National Council of Teachers of Mathematics.

    FANTASTIC blog!

  6. I must respectfully disagree with my colleague Catherine (she started Kitchen Table Math and I am a guest blogger there).

    1. Why we are bringing up the word “algebra” at this point is anybody’s guess.

    Check your state’s math standards. California has a rubric, “Algebra and Functions” beginning in Kindergarten. First grade standard:

    Algebra and Functions
    1.0 Students use number sentences with operational symbols and expressions to
    solve problems:
    1.1 Write and solve number sentences from problem situations that express relation-
    ships involving addition and subtraction.
    1.2 Understand the meaning of the symbols +, −, =.
    1.3 Create problem situations that might lead to given number sentences involving
    addition and subtraction.

    2. Public schools aren’t accredited

    Catherine is making a common mistake, which is to confound accredited with routinely evaluated for efficacy.

    As a board member of a private school, I’ve become quite familiar with the accreditation process for our region, the Western Association of Schools and Colleges (WASC), which accredits both public and private schools. It’s a quite complicated process, but the short version is:

    1. Does your school have expected outcome measures for your students?
    2. Does your school have a plan for producing the expected outcome measures?
    3. Does your school have the resources to execute the plan?

    Note that the validity of the outcome measures aren’t evaluated.

    • Thanks for your comment, Liz. I don’t so much have a problem with teaching 1st graders the concepts of algebra, but I do have issues with:

      (1) Throwing algebra-like problems at them before they’ve masted basic addition and subtraction. Algebra is just a lot easier once one understands how the operations themselves work, before we try to invert the operation to solve algebra problems. I can’t figure out what this curriculum is trying to do with either algebra or subtraction. Are they trying to substitute a sort of intuitive subtraction for the real thing — kids look at 8 things and split them into parts, etc.? Are they trying to have kids discover subtraction on their own? Are they just trying to circumvent subtraction altogether and go for something like a set difference operation (more New Math again) first? Or what? I’m curious about the instructional design here because it’s so different from the usual learn addition > learn subtraction > do algebra sequence that seems to have worked pretty well for the last century or two.

      (2) Using the word “algebra” in the first place. I’m reminded of a DVD my girls liked to watch when they were really little, “Big Bird Goes to China”. In one scene, Big Bird attends the school of a new Chinese friend, and learns how to write “Big Bird” in Chinese characters. Having learned that one word, Big Bird looks at the camera and says, “Hey, I can speak Chinese!” Here in this curriculum, students work with a problem that involves knowing the whole and one part and deducing the other part — or not, since all they have to do is count — and suddenly we say they are “doing algebra”. They don’t know algebra yet even if they get that one right, so I object to the nomenclature.

      • I agree with you, Robert. What I was trying to say is that using the word “algebra” in primary grades is probably a function of state standards — not that those standards are good, or lead to good instructional design.

        And the enVisionMath curriculum seems particularly….heinous.

  7. Catherine T.

    I have a first grade daughter, too. I part-time homeschool (she goes to school 3 afternoons a week mainly for the social benefits), and I really appreciate your post. I have a B.S. in Math and have been following kitchentablemath for a couple of years now because I want to give my daughter a solid foundation in math and keep her from thinking math is confusing. For my daughter’s age (turning six soon) and maturity, I have found that just two pages a day in a Bob Jones University Press math worktext has been ideal. I don’t homeschool for religious reasons, but they really have the best text for our situation. There are no expensive, complicated manipulatives (OK, sometimes I have to borrow my husband’s loose change for the money chapters 😉 ), and it builds gently (she loves the pictures), logically, and efficiently to teach her what she needs to know. She doesn’t realize that story problems are supposedly scary!
    I have strong feelings–negative ones–about math curricula that expect children to make jumps in math logic that they are not ready for. It just sets them up for defeat and makes them hate math, when math is a truly lovely thing! It sounds like you’re homeschooling math, too, only you’re having to do it after she spends the day in class. Good luck to you!

  8. LSquared

    “it’s not clear to me that doing “algebra” is a better idea here than just doing straight-up subtraction.”

    When you’re teaching subtraction in grades 1 and 2, you find that children have a lot more trouble with it than with addition. Indeed, a very experienced teacher of grades 1-3 assures me that teaching multiplication is much easier than teaching subtraction. The winning strategy in some way for learning subtraction is to realize that it’s just backwards addition: so if you know your addition facts really well, then you automatically know your subtraction facts. If children know their addition facts pretty well (unlikely at the beginning of first grade), and if the phrasing is clear (it seems that it wasn’t), using a missing number addition problem is an advantage because it prompts children to think about the addition fact they know to help them solve the problem. What I see here is a good idea implemented poorly (not by you, of course, you implemented it very well–the textbook company , not so much).

    And how in the world does this sort of thing work in a household where the parents don’t have the time, patience, interest, fluency, or comfort level in mathematics to sit down and work all this out with the kid?

    Ahh–that’s the real problem. This sort of thing works brilliantly for some teachers–the ones who are in touch with their students and the math, and they manage their classes so that children get enough instruction and enough assessment so that the children understand (and the teacher is sure that the children understand) exactly what the question is, and some ways to effectively figure out the answers. Eventually this way of teaching gets shared with other people, but it’s a bit like a game of telephone, and details get lost along the way. Eventually some textbook company comes along and puts bits of it into a textbook, and it shares enough characteristics with the original strategy that they can point to the papers on the original strategy and say “our textbook is proven by research to be good”, but they miss out enough things that it doesn’t work very well. This is sad. Ah well–you will have many great opportunities to do math with your daughter, and I predict she will end up good at math (good conversations about math with someone who’s really interested in listening are great motivators).

  9. You bring up some very interesting points. My daughter must have used a similar math curriculum when she was in school. She always had to explain HOW she got answers, even when the problems were extremely easy–it was very frustrating for her. (And I remember that her second grade teacher routinely terrified the class by using intimidating terms like “algorithm” and “algebraic thinking.”) Interestingly, her third grade teacher complained that the children didn’t have their subtraction facts memorized and asked the parents to make flashcards…

  10. I’ve nothing to add here, other than to say that I have a first-grader for the first time, too, this year, and I appreciate you taking the time to write this post. I’ll be taking a closer look at my daughter’s math homework this week…

  11. Veronica

    Though it might be better for your daughter and you both if you didn’t have to grapple with enVision Math firsthand, I am grateful that you are blogging this experience, and I hope that you will continue… followed a link here from Kitchen Table Math, which I found while Googling for knowledgeable, thoughtful critiques of the enVision curriculum, which I am convinced has been designed more to meet the marketing targets and perpetuate a proprietary multimedia system than to meet the needs of any living, breathing student.

    My son is in 2nd grade and attends a school for kids with high-functioning autism. To my mind, mathematics ought to be the subject MOST suited to these children, allowing them rich and frequent experiences of academic mastery and intellectual pleasure. Rule-based, consistent — and with clear connections between concrete, visual explanations and abstract concepts. No confusing, upsetting social context to interpret! No subjective literary/verbal content to untangle! In K-6 math, there are lots & lots of things to memorize (fantastic for many kids on the spectrum), rules to learn & follow (ditto), and in the end, a single correct answer to each question (SUPERfantastic for ASD kids in general).

    enVision takes ALL OF THAT AWAY by depriving students of the rules (or getting them wrong), failing to provide the material the students would happily learn by rote (because it’s better to let them “discover” it over & over again, or not to learn it at all? I am stumped), and introducing concepts in a completely mystifying sequence, dropping them abruptly to introduce something new in an unrelated way with no overlap, then eventually returning at some point as if there were no interruption… From what I have picked up lurking on teacher forums, because the curriculum is so balkanized, unless all components of the curriculum are implemented (CD-ROM, overhead transparencies, classroom lesson guide, in-class activities, manipulatives, God knows what else), big chunks of teaching will go missing, which teachers will not notice if they don’t understand math (as many elementary general ed teachers do not — and I am discovering that goes double for special ed teachers).

    My mother is a mathematician, and I grew up around educators. I have no problem with New Math — my mother was a proponent, actually, and worked on preK-6 math curriculum and teacher education for many years. I was a test subject pretty much from my infancy for how to teach math concepts, create math games, and use manipulatives. I understand what New Math was, at its best, and was supposed to be — the integration of MORE intellectual rigor and depth into primary math, allowing for greater challenge and exploration earlier on. It just never materialized — what came to be called and popularized as New Math was indeed watered-down, dumbed-down — and now, with monstrosities like enVision, tarted-up and pimped out at thousands of dollars a pop. If you’ll pardon the figure of speech. And the long, ranty comment.

    enVision purports to introduce “solid geometry” at the outset of 2nd grade, setting off several days of counting faces, vertices, and edges in a limited range of solid forms. I was perplexed by some of the assumptions that seemed to underly the answers marked right and wrong in my son’s worksheets, like: cylinders and cones have no edges. I asked if “edge” had been given a definition (the only definitions that would justify such an exclusion — like an edge being limited to a straight line or line segment at the intersection of two planes or two polygons — seemed a bit beyond the scope of the lessons, not to mention beyond the ability of the kids to follow, when a more straightforward, common-sense definition would serve better as well as still be mathematically correct, like, an edge is the boundary of a geometric shape). My son replied that anything with a curve has no edges, “like a sphere.” Seriously? These are 2nd graders, not topologists.

    He loves rules, and he’s bright, and he likes to generalize. An edge is an edge is an edge is an edge. So I asked him if my desk, or our dining table, both of which have rounded tops, have edges. “Nope.” Thanks, enVision Math. I can’t wait to see what I have to help him unlearn NEXT week.

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  13. Stefanie

    Great post. A few years ago I worked successfully with other parents in our school district to get rid of Investigations math (a totally fuzzy approach to math – no importance placed on math facts, no calculations practice) and it was replaced with enVision Math. enVision has a semi-fuzzy approach and sometimes you luck out with a teacher who ignores the fuzzy stuff and focuses on the real math. Sometimes you get a teacher who follows it page by miserable page.

    My suggestion as someone who’s “been there, done that” and has had both kinds of teachers, is for you to give the enVision math homework as little attention as you can, unless you deem it to be appropriate and helpful in education.

    Instead, spend time doing math workbooks at home on a regular basis or sign your chid up for a program like Kumon, a reasonably-priced math tutoring franchise which focuses on mastery and proficiency of math basics by doinjg a little every day.

    You obviously know what math your child needs to know. Go over the state curriculum guidelines and focus on what you believe to be important. It’s amazing how focusing on that will very quickly bring your child to head of the class and provide them with lifelong knowledge, confidence and success in math.

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