You may have seen this video making the edu-blog rounds:

## There’s a Place for Everything…

Manipulatives are a great tool for teaching elementary math concepts. Converting the blocks to pictures makes it easy for kids to sketch their ideas. But blocks and pictures should be steps along the way to abstraction, and no student should be forced to use them any longer than necessary. Fourth grade addition is MUCH longer than necessary!

Our goal as teachers is to help students progress from concrete (manipulative blocks) to pictorial (like this student’s sketches or the Singapore math models I call bar diagrams) and from there to abstract work (numbers and variables).

If a student understands place value — that the order we write numbers indicates the size of “block” they represent — then drawing pictures like these is mere busywork. It distracts the student from the important goal of solving problems.

**[Hat tip:** I first saw this video at Out In Left Field.]

My middle school children learned with Investigations in elementary school, and my younger children are using it now. I have not seen this kind of thing happening at any time over the past 9 years that they have been working in this curriculum. The curriculum is not perfect, but if done correctly, it provides students with a strong understanding and many tools for mathematical thinking. My children all have excellent number sense and are able to use standard algorithms as well as other strategies to effectively solve a wide variety of mathematical problems.

I think this video is mind-blowing: thank you for continuing to share such great stuff.

I agree with fwteacher that the problem is not with Investigations: I think it can be a valid learning approach. The problem, however, is that elementary school teachers do not receive nearly enough preparation in teaching math–see Liping Ma’s “Knowing and Teaching Elementary Mathematics”. As a result, there isn’t always the understanding that manipulatives can serve as a crutch to further learning. This video was that lack of understanding taken to the extreme and it ties into the “let’s draw stuff in math because it’s fun” mindsight that I blogged about last month:

“Too many math teachers want to be elementary school teachers”

http://challenge-of-teaching-math.blogspot.com/2011/02/too-many-math-teachers-want-to-be.html

Paul Hawking

Blog:

The Challenge of Teaching Math

Latest post:

Pulled from the comments feeds (2-28-11)

http://challenge-of-teaching-math.blogspot.com/2011/02/pulled-from-comments-feeds-2-28-11.html

Hey, Denise

I couldn’t agree more – pictures and physical representations are stepping-stones on the way to abstraction for children learning the complexities of base-ten numbers.

Comments on the YouTube page are really negative, but I believe there is a tendency for folks who are not familiar with math pedagogy to quickly reach the conclusion that the method this child has learned under is really bad, seeing what a mess she made of this problem. But the child understands place value and multidigit addition – perhaps the pictures helped her a while back when she was learning it, but now she has no more need for them.

Thanks for introducing the video to a new pool of viewers – it is a great conversation starter.

@fwteacher: I’m glad to hear that some teachers are doing it right!

Whenever I hear a student say, “We aren’t allowed to do [insert topic] at school,” I figure that one of two things has happened:

(1) The student misunderstood what the teacher said.

(2) The teacher, like those in the Liping Ma study Paul mentioned, doesn’t understand what he or she is teaching.

It reminds me of a quote I read about the New Math in the 60’s, but which I can’t quite remember. The gist of it was: Given a teacher who really knew what she was doing, the New Math might help students learn and understand deep and interesting ideas — but if the teacher didn’t know math, the students were ruined. With the traditional curriculum, on the other hand, the clueless teacher couldn’t do much harm.

Update:Found the quote.Our school system just recently starting using Investigations at the middle school level. I am at the high school, and have not yet had any students who have been through it…however, there has been much controversy surrounding the adoption of this program. The head of the math department at the middle school pretty much said what you wrote in your comment, Denise – the program is a great conceptual tool IF a teacher understands the underlying concepts. I’m interested to see what happens when I get a batch of kids who used Investigations.

I cringed when the little girl (who was super-cute, by the way!) said, “We’re not allowed to stack in school.”

As I watched her cumbersome practice of drawing out the “cubes”, “sheets”, and “rods”, I was reminded of studying Egyptian numerals in my History of Mathematics course. Those numerals are easier to draw. 🙂

10^1 was represented by a one-dimensional “rod”, 10^2 by a 2-D “sheet”, and 10^3 by a 3-D cube…I’m curious as to how Investigations represents “10,000”.

You’re right, Heather!

Egyptian Math in Hieroglyphs

and Alex’s Puzzling Papyrus

The way I’ve seen base-ten blocks used, 10,000 would be a “super-rod” made of 10 thousand-cubes in a line. And the 100,000 is a “super-flat” of 100 thousand-cubes. But these are only mental images — I don’t know that anyone actually builds them.

Thank you for posting this video! I am in school right now going to be a high school math teacher and I find this fascinating. It is good to know before I even get into the trade where the pictures and physical representations belong. Though they make great stepping stones and help the student understand what is going on, these should not be the only tools that mathematicians use. I am sure that the Investigations are wonderful at first but I can see the confusion from students when they try problems like this. Thanks for the insight!

Hi, Nick! I’m glad you stopped by.

Even in high school math, pictures and physical representations are helpful. Base ten blocks can be used, or special blocks made for algebra (there are several varieties), to demonstrate addition, multiplication, and division of polynomials, for instance. These translate easily to a pictorial model, as in Euclid’s Geometric Algebra.

But in the same way that a 4th grade student should be way past the pictorial stage for addition, you don’t want algebra 2 students having to draw a box to multiply two polynomials.

What also gets lost in the shuffle is that those pictorial representations are ALSO abstractions, in this case of stickers. So, if they’re used as a step up to just having a “brain sense” of 1000, great. In Everyday Math they have the kids stack as well, but they also teach adding from left to right. As in, how many thousands (in this case 2000), how many hundreds (in this case 15 hundreds or 1500), tens, ones and then the children add those stacks again.

But all of these should be in the service of making sure that kids can add some numbers! What I see over and over is that the kids who “get” math and like math learn well no matter how convoluted the system, no matter how many ways they learn, etc. They are even enriched by it. This little girl clearly has a mom who likes math and who works with her on it at home. I am certain that’s why she can describe what she did so clearly — she’s been trying to explain it to her mom (and a very smart mom for making her do that, too!) and has really gotten her conceptual understanding as much from home as school!

However, at the other end of the spectrum, there are kids whose take-away message is that you can solve math problems a whole bunch of ways and it doesn’t really matter which one you use. They don’t have people at home willing to sit and figure out what they’re doing and showing them alternatives and explaining why they both work.

That mindset of “try anything” is how you get kids faced with word problems like this just calling out answers, or adding instead of multiplying or subtracting because it’s easier than dividing.

I used to ask 6th and 7th graders questions before starting problems like this and some would struggle with the question “Will our answer be bigger or smaller than 680? than 1500?” That is, the main abstraction of having three different size piles that we’re putting into one big pile was STILL missing at that late stage!

As an elementary school teacher who loves math and teaching math, I see this girl as one who could master the “traditional” algorithm (or partial sums or Everyday Math’s “opposite change”) in a single 10 minute lesson. I know that when I build a really solid foundational understanding of a concept like multiplication, children see the relationship among all of the algorithms for that operation. I once wrote up the same multiplication problem using 5 different algorithms (including one I had made up). The students saw right away that “it’s all the same! You’re just keeping track of the numbers in different ways.”

Without a solid foundation, children spend hours trying to master what to them is an arbitrary set of steps. Then, a week after the unit is over, they can’t do it anymore. While we can’t stop with manipulatives, we can’t start with an abstract algorithm, either.

No, we certainly can’t start with the algorithm. That’s not the endpoint, either, though some math textbooks seem to give that impression. The goal is for students to understand well enough to see the connections — “It’s all the same!” — and be able to use that insight to solve problems.

I’ve run into students like Jen described, who can’t see that addition is “putting the piles together.” I worry about them.

A few years ago, Keith Devlin threw math education bloggers into a tizzy (a kerfuffle?) by arguing that we needed to revise our understanding of multiplication (here, here, and here, with a coda here). I found the discussion fascinating, though some of it degenerated into entrenched positions with occasional name-calling. But the question that interested me was never fully answered:

Is there a way to teach addition and multiplication so that when students read a story problem, they can recognize which operation makes sense?

I have mixed feelings.

I came into this (math curriculum, policy, standards, etc) adamantly opposed to Investigations, and I remain so. Skills development gets short shrift, needlessly.

But I have come to like manipulatives. In fact, I like the abstracted drawings of manipulatives a whole lot.

In my own teaching, I find that even advanced students benefit by being challenged to move back and forth, by being asked why a physical model works, or fails to work.

Toys, I call them. And good students can do neat things both with and without them.

I am most impressed by this kid being able to add both ways, and understand what she is doing. I would be most unimpressed by her adding with algorithm (good, but not impressive) and I would be furious if she were using the model and did not know the standard algorithm.

In high school I like looking back and finding why algorithms work, and abstracting from place value to reconstruct similar algorithms in algebra. But then we can look back at the manipulative models, and analyze them as well.

Jonathan

Denise, I laughed when I read this:

“you don’t want algebra 2 students having to draw a box to multiply two polynomials”

because I have seen students in higher-level math courses doing just that!

Back when I was in high school, I learned to factor trinomials in standard from (“a” not equal to 1) by using trial & error (experiment with factors of “a” and “c” and the appropriate signs to find a combination that works). When I started teaching high school math a few years ago, a teacher introduced me to a lengthy algorithm that eliminates most of the guesswork and results in the two binomial factors every time (IF done correctly). I thought it was the coolest thing, and I showed my students the technique. After a bit of practice, they got into the rhythm of the algorithm, but over time, they did not retain the skill.

The following year, I taught my trial and error method, and walked them through the mental process. There was a fair bit of griping, but after a few days, the students were really getting it! I showed them the other technique, and they told me it was absurd. 🙂 Using the trial & error technique helped them make connections to FOIL that the other way could not, and the students demonstrated retention of the material months later.

@jd2718: There seems to be a constant changing tide from teaching algorithms effectively to only focusing on conceptual understanding to teaching pure mathematics to making authentic learning connections with everything we teach. I think it’s all important, but the right mix depends on the course and the ability level of the students.

@Heather: It must be a synchronicity thing, because I posted on my blog today a lesson on factoring quadratics that sounds similar to what your colleague uses. Maybe it was in the way s/he sold it that made it work. All I know is that for students who were taught the “four square” method of multiplying binomials, this approach seems to provide them with something familiar as well as a more organized approach than guess and check, erase if it doesn’t work, guess something else and check, erase if it doesn’t work, guess again a prior choice because they haven’t kept track of what they’ve tried, and then give up because they think they’ve tried everything. Having said that, I always find it interesting that there is not only more than one way to solve a math problem, but there is always more than one way to effectively teach mathematics 🙂

Paul Hawking

Blog:

The Challenge of Teaching Math

Latest post:

Teaching factoring quadratics when a>1

http://challenge-of-teaching-math.blogspot.com/2011/03/teaching-factoring-quadratics-when.html

Paul,

I probably get labeled most easily as a traditional teacher who emphasizes understanding. But really, I treat the whole thing like one big game. So I do all four of the things you discuss, and have fun. It’s better than doing 3 of them, and much better than only doing one of them, which would be dreary and boring for the teacher, and dull and mind-numbing for the kiddies.

Jonathan

Paul,

I just checked out the slide show on your blog – it is similar to the technique I had described, but your method makes a LOT more sense and is easier to execute than the other one. When I introduce students to the “trial & error” approach, I do teach strategies so that they’re (hopefully) not throwing in random combinations willy-nilly. Still, I like to provide them with a range of techniques from which to choose, because what may work for one student may not work for another. Thank you for sharing!

Heather

BTW, for those who don’t know what the

Investigationsreform math program is, Katharine Beals has written a series of enlightening posts comparing it with more “traditional” math. In the 4th grade school year:*first month of school*mid-November*late November*early December*mid-December*January (This could be an interesting lesson on mental math, IF the students have already mastered subtraction by the standard algorithm. Showing multiple methods is a good way to deepen understanding. But one solid, reliable way of doing the calculation should be firmly established first, or the different options are merely confusing.)*late January#1: Decide what we want to teach the children. The standard algorithm, or even typing on a calculator might be good enough. I suspect that we want children to really understand what addition and multiplication are. If so then:

#2: If we want kids to understand then that involves creating a consistent internal map. The child has to do this, not the teacher. The teacher helps though! Teachers need feedback from the children to understand how all the processes fit together, and they fit together very very simply, that is: Addition is a shortcut to counting (the girl proved it by using her complicated counting method) and multiplication is a shortcut to addition (ignoring the rants of Mr Devlin).

Students get counting by one fairly easily. You move them along from counting by ones to counting by something larger like twos, threes, fives and tens. Then suddenly you realize that counting by tens is easy. This leads you into the addition algorithm, which is just counting by base ten units.

To get the student to multiplication, you do the same as addition. Add up numbers a few times then realize that if you add a number ten time it’s just adding a zero. Show how the standard multiplication algorithm works by proving that you can chop up numbers, multiply the pieces, then add them — it works because multiplication IS repeated addition.

However you want to get those connections into the students’ head is between the teacher and the student. Specific methods are difficult to suggest because the needs are specific to each student, not just a generalized silver-bullet technique.

If the teacher doesn’t know anything, then the student is at a huge disadvantage obviously and the teacher might be doing more harm than good. But if the teacher understands, then it’s just a matter of sensitivity on the teacher’s part as to where the misconceptions are and how to break them.

One on one, I don’t think there is a problem with any student or teacher — the method almost doesn’t matter — addition will be learned. The problem lies in teaching a class of diverse levels under time constraints. Such is the struggle of a teacher.

The problem lies in teaching a class of diverse levels under time constraints.True, and that’s one reason I love homeschooling. But even in a classroom situation, using a coherent, well-designed curriculum can make a world of difference.

I’m not sure how many of you saw this article, but it was a favorite of the parents at my school: “How Einstein Started Solving its Math Problem”

http://www.voiceofsandiego.org/education/schooled/article_00c1eeda-ee09-11df-b7fc-001cc4c03286.html

In it, the students have a mnemonic for dividing by fractions that goes, “Don’t ask why, invert and multiply!” Coincidentally, I was teaching dividing by fractions to my fourth and fifth graders and I shared the article with them before we started. “Let me guess,” one said, “We’re going to ask why, aren’t we?” We broke out the playdough and proved to ourselves that three divided by one half is six by making three wholes and counting how many halves we had. Then we “made” a series of more and more complex division problems involving fractions — each time they solved the problem with playdough, I drew pictures and provided the number sentence on the board. Without prompting, they tried out the “shortcut” of multiplying by the reciprocal and were excited to “discover” (see Eleanor Duckworth) that each way provided the same answer.

The next day, we followed Marilyn Burns’ excellent lesson on dividing with fractions in which you list six descriptions of division and apply them to dividing with fractions. The students felt this really helped them solidify their understanding and we had a lot of “ah ha” moments. The next day, we made up a game in which you tried to get the biggest quotient possible by rolling a die and fitting the number into either the numerator or denominator of the dividend and the divisor and as a wrap up to that work, we went back to the shortcut algorithm and the students explained to me in several different ways why multiplying by the reciprocal provided the same result as dividing by a fraction.

It illustrates what so many of the comments have said — teaching mathematics (arithmetic?) requires a balance of manipulatives and more efficient processes. We have to have a variety of representations ready to help students develop the deepest understanding possible.

…teaching mathematics (arithmetic?) requires a balance of manipulatives and more efficient processes.And it takes time: time to talk about things, time for hands-on experiments, time to work problems, time to practice (“drill”, though I prefer to do it through games, like the dice fractions game Michelle describes), and time to consolidate everything mentally so it sticks.

Part of the craft of teaching is to know where to focus whatever limited time you have — and here is where a curriculum can make or break an inexperienced teacher. If the curriculum makes wise choices of what to emphasize and spend time on, the students will learn. But if the curriculum is scattered and incoherent, only the best teachers will be able to lead their classes through it.