Picking up where I left off in my last post on Everyday Math students, *Ah, but surely their "conceptual understanding" is deeper*.

Yes, Everyday Math has eradicated most of those "boring," "tedious" calculation drills of traditional math, substituting the rote-calculating pre-21st Century mind with 21st Century technology like calculators, supposedly freeing the 21st Century mind for 21st Century pursuits like "conceptual understanding" and "higher-level thinking."

So what is the state of conceptual understanding among those whose entire experience with math consists of Everyday Math? The students in our 3rd-5th grade after school program, selected for good behavior, interest in learning, and parental interest, and who otherwise show no signs of learning disabilities, include a number of students who probably now meet the diagnostic criteria for dyscalculia. Not only are they still counting on their fingers and struggling with the standard algorithms, but also (*even many of the 5th grade cohort*):

- Lack a basic number sense: for example, are stumped, *totally stumped*, when asked to give a number between 4000 and 5000.

- Lack a basic understanding of fractions: for example, that 1/1 = 2/2 = 3/3 = 4/4 = 1, or that 1/2 is smaller than 3/4

- Lack a basic understanding of decimals, and how to convert simple fractions to decimals and vice versa--e.g. 1/4 to .25 or .5 to 1/2

Number sense, and conceptual understanding more generally, are supposed to be the pride and glory of Reform Math. They are supposed, somehow, to emerge from finding multiple solutions to a small number of basic problems, inventing your own methods, using lengthy procedures that "break things down" like the partial-sums method and the partial quotients algorithm, and explaining your answers in words, numbers and pictures.

But there's growing anecdotal evidence that average students who are educated exclusively through Reform Math are *more deficient in number sense than their traditionally educated peers*.

For those educational researchers who are both intellectually honest and ethically minded (as opposed to those who are not), these anecdotes cry out for well-designed studies exploring Wu's hypothesis that basic skills *versus* conceptual understanding is a bogus dichotomy.

## Tuesday, January 15, 2013

### Conceptual understanding under Everyday Math

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## 12 comments:

According to my experience at a local bakery (old comment), they not only lack number sense and understanding of the relationship between decimals and percentages (as well as fraction), they are also unable to translate a problem (in my example, calculation of sales tax)into math terms, so it can be entered into a calculator (let alone solve it without one).

My 2nd grader's school supposedly uses a "balanced" approach. ThinkMath and Harcourt materials for math. I think they use ThinkMath's instruction, but send home Harcourt pages for homework.

His math homework for this week is a single worksheet on 2-digit addition with regrouping, but the teacher attached a note with a copy of a hundreds chart, with instructions to use the hundreds chart or partial sums to solve the 14 problems.

From the weekly newsletter:

"We will continue to add 2 digit numbers. We will work with base ten blocks in small groups for adding. For their homework, I have attached a hundreds board for them to use. Please do not teach them the algorithim. We haven't gotten there yet. Right now, they have learned to use the number line to skip count by tens and ones, to use the hundreds board, and to add using the break apart strategy with place value (example: 56 + 23 they would do 50 + 20, then 6 + 3, then 70+9= 79)"

So I asked my son to show me the partial sums method she wants them to use. I never learned it, so I had no idea what it was...and yet I can easily add 2 digit numbers (imagine that). He didn't seem to know how to perform partial sums either. So he defaulted to the hundreds chart to add. 46+38 37+39 etc.

Bad parent that I am, I showed him how to solve the problems with regrouping. It clicked. What would have taken him forever to do on a hundreds chart, took him 5 minutes.

I am so sick of the word "strategies." They spend lots of time discussing strategies and very little time on anything else.

It is how they teach 2-digit addition with Singapore math - how to do it mentally. In first grade. Add the tens and then the ones. Add ones by making a ten if needed. They don't name the strategies though, and once they show them they let the student use any strategy they want. Which would include renaming or regrouping. 46 + 38 = 76 + 8 = 80 + 4 or 70 + 14.

Julie, partial sums means you break apart the numbers. So if you are adding 17 and 15 you'll first add 5 and 7, then 10 and 10, and finally you'll add the two sums together.

You can use the same method to introduce two by two digit multiplication. Effectively, this IS what you're doing with the traditional methods, you just aren't writing out every step.

As a very short term bridge into a more efficient algorithm, there is nothing wrong with it, but it is absurd to not tell parents what it is or what the reasoning is for using it, and to ask them to refrain from moving ahead if their kid is ready.

What I don't understand about many of these methods, is that they require the students to write many more things down and do many more calculations. Each of those extra pieces is a place where errors can creep in. At an age when kids' handwriting can often be illegible, having them write down more numbers is just opening up the door to error.

It's my understanding that EM also requires significant "writing about math/your answer" as opposed to the traditional "show your work". The former discriminates against kids whose writing ability lags behind their math ability.

With the Primary Mathematics, it is a mental math process. They are not asked to write down the intermediate steps. They learn how to do it with manipulatives, and they can write a simple number bond if they need, to help, but the goal is mental math. They may be asked to explain their thinking orally or at least show what they did with the blocks, initially. They are not asked to explain in writing. There are not really any more calculations than with the standard algorithm: add tens, add ones, rename if needed. But that depends on what the teacher requires, a teacher can always add that they have to show the steps, even if it not in the curriculum. And, in the US, it seems everyone wants kids to verbalize their thinking process over and over again... Getting the right answer does not seem to be enough. My son had all kinds of ways to do mental computation, and he always got the right answer, and had this way of multiplying by 12 before he learned how to formally multiply because of a computer game he was playing where he had to, and he figured it out and always got it right, but to have to explain it in words is a whole different skill and I think unfair to ask of a young child whose verbal skills may not have caught up to math skills.

Reading these stories brings me almost physical pain. I see no value in wasting time on retrograde "strategies". I think those waste the children's time and create potential for confusion. Instead of

masteringa single algorithm that willalways work,we're going to show them a half-dozen different ways and then let them decide. So now they have 6 processes to learn, hopefully not confuse or transpose any of the details among them, and then have to make one extra decision before they even get down to solving the problem!?! What's the saying, "jack of all trades, master of none."Here's the thing, there have been studies that show every decision we make over the course of a day, no matter how small or trivial, uses up a certain amount of "juice" and adds to our largely fixed cognitive load for that day. So, do you want your fixed amount of decision-making power spent on choosing a strategy, or actually running the algorithm and being ready for the next problem? I know what I and other smart people do.

That last post was getting long... :-)

Related to this thread, I think, Michael Goldstein had a good post discussing recent research showing the value of instant recall for math facts even when solving complex math problems. The relation is there's only one sure-fire way to get instant recall of math facts: drill.

Lastly, I hope people don't mind me bringing this up, and maybe everyone here has already has seen it at the KTM blog, but a friend and I created an Android app to help children practice arithmetic drill. Unlike a lot of the apps that are out there, it's not just mental math and recall app. It's useful for practicing the algorithms for solving multi-digit addition, subtraction, multiplication, and division. It's been out for a little over a month and has been doing quite well. I'm of the opinion 5 or 10 problems a day 4-6 days a week can make a world of difference in kids' math skills. Maybe not everyone is cut-out to be an engineer, but I think everyone can have some confidence towards math (instead of being intimidated and thinking, "Oh, I can't do math; I'm not good at it"), and that only comes from practicing.

The web page for it is http://www.edisongauss.com/blackboard-math-app/ Thanks.

Here's another article relating fluency of math abilities to chemistry ed:

http://pubs.acs.org/doi/abs/10.1021/ed085p724

>> In addition, we argue that these results indicate an inadequate degree of mathematics fluency for the majority of the students tested, which can seriously impede their abilities to develop a firm conceptual understanding of quantitative introductory chemistry. <<

My children (using Primary Mathematics and other math) gained math fluency by understanding and applying at least a few mental math strategies with smaller numbers. They also learned the standard algorithm. Some of the early mental math strategies for 2-digit numbers directly relate to the standard algorithms. It is when students have to learn a lot of different complicated paper and pencil algorithms other than the standard one that I think the difficulty comes in. My kids did not have to learn lattice method or area model and all that kind of stuff. But they did learn and use mental math strategies. One of my kids liked playing with numbers. He was the math whiz, even though he had a hard time memorizing math facts. I would hate to have had him in a class that required 100 problems in 5 minutes or only one way to do the math. And he definitely "could do math" and enjoyed it, even though he would not have tolerated drill sheets. So it varies with students what they need and what is pushed at what age. On the other hand, for some students mental math strategies might be too much.

Everyday Math and other new math does nothing but ruin learning for children. As a teacher, I refused to teach it and my students always scored higher using traditional math.

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