The value of manipulatives

(Note: I'm "resurrecting" an old post, with added information and a video. The topic is still very much valid.)

Manipulatives are IN, in math education. But do they TRULY facilitate learning to such an extent as people promoting them claim?

The entry at Text Savvy, Hands-on, Brains-Off, explains some of the pitfalls in manipulative use. Very enlightening!

Quoting from an article at Education Week (Studies Find That Use of Learning Toys Can Backfire):

In a similar series of experiments at the elementary-school level, the researchers found that children taught to do two-digit subtraction by the traditional written method performed just as well as children who used a commercially available set of manipulatives made up of individual blocks that could be interlocked to form units of 10.

Later on, though, the children who used the toys had trouble transferring their knowledge to paper-and-pencil representations. Mr. Uttal and his colleagues also found that the hands-on lessons took three times as long as the traditional teaching methods did.

The video below also illustrates how manipulative use can lead to problems.

The girl solves an addition problem involving thousands by drawing thousand-blocks, hundred-sheets, etc. on the board, taking 8 minutes. Then she says that at home she has been taught to stack the numbers and add. She solves it that way, too, taking 1 minute. But she gets two different answers.

(Hat tip to Denise)



Clearly, manipulatives aren't the best way to solve such problems, and children shouldn't be led to believe so. Not that this girl believed that way... she seems to understand which way is easier and quicker.

My take is that manipulatives aren't the ultimate answer to math teaching.

They're a reasonable starting point. But children should NOT be taught to rely on them. Children need to be taught and shown how to transfer all of that concrete play into the abstract.

In my books, I often instruct concepts with pictures, which essentially take the place of manipulatives. Then the student does a bunch of problems that have the same pictorial representation in them, including having to draw  those same pictorial models to illustrate the math problems. THEN after that, the student goes on to totally abstract representation.

Here's one example from my book Add & Subtract 2-B: Adding with Whole Tens. The student first adds using the pictorial model, and then with numbers only.

I don't know if that approach suffers from the drawbacks mentioned in the article above. I hope not; from the feedback I get, it seems to work well.

The value of manipulatives

(Note: I'm "resurrecting" an old post, with added information and a video. The topic is still very much valid.)

Manipulatives are IN, in math education. But do they TRULY facilitate learning to such an extent as people promoting them claim?

The entry at Text Savvy, Hands-on, Brains-Off, explains some of the pitfalls in manipulative use. Very enlightening!

Quoting from an article at Education Week (Studies Find That Use of Learning Toys Can Backfire):

In a similar series of experiments at the elementary-school level, the researchers found that children taught to do two-digit subtraction by the traditional written method performed just as well as children who used a commercially available set of manipulatives made up of individual blocks that could be interlocked to form units of 10.

Later on, though, the children who used the toys had trouble transferring their knowledge to paper-and-pencil representations. Mr. Uttal and his colleagues also found that the hands-on lessons took three times as long as the traditional teaching methods did.

The video below also illustrates how manipulative use can lead to problems.

The girl solves an addition problem involving thousands by drawing thousand-blocks, hundred-sheets, etc. on the board, taking 8 minutes. Then she says that at home she has been taught to stack the numbers and add. She solves it that way, too, taking 1 minute. But she gets two different answers.

(Hat tip to Denise)



Clearly, manipulatives aren't the best way to solve such problems, and children shouldn't be led to believe so. Not that this girl believed that way... she seems to understand which way is easier and quicker.

My take is that manipulatives aren't the ultimate answer to math teaching.

They're a reasonable starting point. But children should NOT be taught to rely on them. Children need to be taught and shown how to transfer all of that concrete play into the abstract.

In my books, I often instruct concepts with pictures, which essentially take the place of manipulatives. Then the student does a bunch of problems that have the same pictorial representation in them, including having to draw  those same pictorial models to illustrate the math problems. THEN after that, the student goes on to totally abstract representation.

Here's one example from my book Add & Subtract 2-B: Adding with Whole Tens. The student first adds using the pictorial model, and then with numbers only.

I don't know if that approach suffers from the drawbacks mentioned in the article above. I hope not; from the feedback I get, it seems to work well.

Math Manipulatives at my Amazon Store

I don't get around doing much editing of my Amazon Store, but today I was prompted because someone asked me to write a guide for manipulative use. So I'm doing a writeup on them, but also went to Amazon to see what they had.

So I added various ones that I thought might be useful to my Amazon store:

I was disappointed that there aren't any really cheap abaci there. The one abacus with 5 and 5 alternate color beads was quite pricey. Most of them had 10 beads the same color.

Other ones I included were:
- base ten blocks
- cuisenaire rods
- fraction circles or tiles
- scales, thermometers, measuring cups

Math Manipulatives at Amazon store

Math Manipulatives at my Amazon Store

I don't get around doing much editing of my Amazon Store, but today I was prompted because someone asked me to write a guide for manipulative use. So I'm doing a writeup on them, but also went to Amazon to see what they had.

So I added various ones that I thought might be useful to my Amazon store:

I was disappointed that there aren't any really cheap abaci there. The one abacus with 5 and 5 alternate color beads was quite pricey. Most of them had 10 beads the same color.

Other ones I included were:
- base ten blocks
- cuisenaire rods
- fraction circles or tiles
- scales, thermometers, measuring cups

Math Manipulatives at Amazon store

Basic abacus as a manipulative

I didn't want to leave out one of the best manipulatives there is for grades 1-2: just a simple "school" abacus that has 10 wires and 10 beads on each wire.

I'm not talking about a Chinese or Japanese abacus with a special counting system.

I'm talking about just using this simple abacus for counting, and treating each bead as 1. You don't have to learn any of these sophisticated systems that have been in use with various abacuses. Just consider each bead being 1, period. Then you have essentially 10 tens, or a hundred, in your abacus.

And that goes a long way explaining tens and ones or 2-digit place value on 1st grade.

You can also show the child things such as similarities in
10 − 5
20 − 5
60 − 5

or let the student find sums of 2-digit numbers: 23 + 45. He can move 2 tens and 4 tens, then 3 and 5 individual pieces - so the abacus can model adding the tens and ones separately.

You can let the child explore what happens with 28 + 9.

It's better if the abacus has the first five beads colored differently from the next five, in each row, like in this silly picture.

Then the child will easily recognize 6, 7, and 8 beads without counting. Also, let's say you'd choose 6 beads on one wire and 8 on the next one. You can show how the five and five on those two wires makes ten, and some are left over.

You can also model multiplication: move for example 4 beads on each of the 5 neighboring wires and you have 5 times 4.

So this is not rocket science; it is very easy. No need to learn any new systems.

Here's a picture of an old school abacus.

Wikipedia has info on all different kinds of abaci, including this kind of usage of the school abacus.

You can browse Amazon's abacus selection here.

Basic abacus as a manipulative

I didn't want to leave out one of the best manipulatives there is for grades 1-2: just a simple "school" abacus that has 10 wires and 10 beads on each wire.

I'm not talking about a Chinese or Japanese abacus with a special counting system.

I'm talking about just using this simple abacus for counting, and treating each bead as 1. You don't have to learn any of these sophisticated systems that have been in use with various abacuses. Just consider each bead being 1, period. Then you have essentially 10 tens, or a hundred, in your abacus.

And that goes a long way explaining tens and ones or 2-digit place value on 1st grade.

You can also show the child things such as similarities in
10 − 5
20 − 5
60 − 5

or let the student find sums of 2-digit numbers: 23 + 45. He can move 2 tens and 4 tens, then 3 and 5 individual pieces - so the abacus can model adding the tens and ones separately.

You can let the child explore what happens with 28 + 9.

It's better if the abacus has the first five beads colored differently from the next five, in each row, like in this silly picture.

Then the child will easily recognize 6, 7, and 8 beads without counting. Also, let's say you'd choose 6 beads on one wire and 8 on the next one. You can show how the five and five on those two wires makes ten, and some are left over.

You can also model multiplication: move for example 4 beads on each of the 5 neighboring wires and you have 5 times 4.

So this is not rocket science; it is very easy. No need to learn any new systems.

Here's a picture of an old school abacus.

Wikipedia has info on all different kinds of abaci, including this kind of usage of the school abacus.

You can browse Amazon's abacus selection here.

The good side of manipulatives

Update: read also Mama Squirrel's excellent blogpost on the issue; basically there is probably a balance in this, as in everything.

---

If you haven't, go pay a visit to the National Library of Virtual Manipulatives.

Lots of good stuff there!

So, why would I suddenly turn around and praise manipulatives, if I just mentioned how they're not the "ultimate thing" in math education?

It's just one of those things that can be good, if used right, in its rightful place.

Manipulatives do help students to understand concepts, initially, on a concrete level. But they shouldn't stop there. Kids need to learn to make generalizations. That is where the power of mathematics is.

For example, maybe you'd use The Base Blocks Addition to illustrate the concept of "carrying" to tens and hundreds (or "trading") in addition.

Once the student understands that, they should be able to translate their knowledge into bigger and smaller place values (including decimal addition).

So it's the age-old truth; there is balance in everything.

Or, what do you think?

The good side of manipulatives

Update: read also Mama Squirrel's excellent blogpost on the issue; basically there is probably a balance in this, as in everything.

---

If you haven't, go pay a visit to the National Library of Virtual Manipulatives.

Lots of good stuff there!

So, why would I suddenly turn around and praise manipulatives, if I just mentioned how they're not the "ultimate thing" in math education?

It's just one of those things that can be good, if used right, in its rightful place.

Manipulatives do help students to understand concepts, initially, on a concrete level. But they shouldn't stop there. Kids need to learn to make generalizations. That is where the power of mathematics is.

For example, maybe you'd use The Base Blocks Addition to illustrate the concept of "carrying" to tens and hundreds (or "trading") in addition.

Once the student understands that, they should be able to translate their knowledge into bigger and smaller place values (including decimal addition).

So it's the age-old truth; there is balance in everything.

Or, what do you think?

Know Your Tools

Math teacher's tools are quite numerous nowadays.

First of all of course comes a black or white board, or paper - something to write on, pencil, compass, protractor, ruler, eraser.
And the book you're using.
Then we also have computer software, animations and activities online, animated lessons and such.
There are workbooks, fun books, worktexts, online texts.
Then all the manipulatives, abacus, measuring cups, scales, algebra tiles, and so on.
And then there are games, games, games.

The choices are so numerous it's daunting. What's a teacher to do?

Well, you just have to get started somewhere, probably with the basics, and then add to your "toolbox" little by little as you have opportunity.

There is no need to try 'hog' it all at once. It's important to learn how to use any tool you might acquire. Quantity won't equal quality. Knowing a few "math tools" inside out is more beneficial than a mindless dashing to find the newest activity to spice up your math lessons.

---

Basic tools

1) The board and/or paper to write on. Essential. Easy to use.

2) The book or curriculum. Choosing a math curriculum is often difficult for homeschoolers. Check my curriculum pages for some help. Keep these two things in mind:

i) Now matter what book you're using, YOU as the teacher have the control. Don't be a slave to the curriculum. You can skip pages, rearrange the order in which to teach the material, supplement it, and so on.
ii) Don't despair if the book you're using doesn't seem to be the perfect choice for your student. You can quite likely sell it on homeschool swap boards, and buy some other one.

3) Manipulatives. I once saw a question asked by a homeschooling parent, on the lines, "What manipulatives must I use and when?" The person was under the impression that manipulatives are a "must".

Manipulatives are definitely emphasized in these days. They are usually very good, but they're not the end goal of math education, and there is no need to "go hog wild" over them.

Manipulatives are something the student manipulates with his hands to get a better grasp of something. But the goal is to learn to do math without them.

Some of the most helpful manipulatives are:

• 
  A 100-bead abacus



• Something to illustrate hundreds/tens/ones place value. I made my daughter ten-bags by putting marbles into little plastic bags. You can buy base ten math blocks. These are sets of tiny blocks for ones, sticks for tens, flats for hundreds, and large cubes for thousands. See an image on the right.



• Some sort of fraction manipulatives. You can just make pie models out of cardboard, even. Stores sell ready-made fraction models made of foam or plastic. The one below sells for less than $10 at LearningThings.com:
  

For many kids, drawing pictures can take the place of manipulatives. That is especially true in the middle grades and on.

Check out also some virtual manipulatives.


4) Geometry and measuring tools. These are pretty essential However, dynamic software can these days replace compass and ruler and easily be far better.

---

The extras

These are, obviously, too many to even start listing.

*Some game or games are good for drilling basic facts. Games are nice for about any topic. Here's one that I played with playing cards with my dd; and now she seems to have learned the sums that add to 10. And here's a game that's worth 1000 worksheets. Of course the internet is full of online math games.

*I would definitely use some math software if teaching graphing, algebra, or calculus. Check MathProf for example, or Math Mechanixs. I've listed a few more here.

---

If you're ready to add something new to your toolbox from the online world, try The Math Forum's MathTools - a library of technology tools, lessons, activities, and support materials. Check also my pages listing interactive math activities online (there's a menu on the right).

Let me know if I forgot something.

Know Your Tools

Math teacher's tools are quite numerous nowadays.

First of all of course comes a black or white board, or paper - something to write on, pencil, compass, protractor, ruler, eraser.
And the book you're using.
Then we also have computer software, animations and activities online, animated lessons and such.
There are workbooks, fun books, worktexts, online texts.
Then all the manipulatives, abacus, measuring cups, scales, algebra tiles, and so on.
And then there are games, games, games.

The choices are so numerous it's daunting. What's a teacher to do?

Well, you just have to get started somewhere, probably with the basics, and then add to your "toolbox" little by little as you have opportunity.

There is no need to try 'hog' it all at once. It's important to learn how to use any tool you might acquire. Quantity won't equal quality. Knowing a few "math tools" inside out is more beneficial than a mindless dashing to find the newest activity to spice up your math lessons.

---

Basic tools

1) The board and/or paper to write on. Essential. Easy to use.

2) The book or curriculum. Choosing a math curriculum is often difficult for homeschoolers. Check my curriculum pages for some help. Keep these two things in mind:

i) Now matter what book you're using, YOU as the teacher have the control. Don't be a slave to the curriculum. You can skip pages, rearrange the order in which to teach the material, supplement it, and so on.
ii) Don't despair if the book you're using doesn't seem to be the perfect choice for your student. You can quite likely sell it on homeschool swap boards, and buy some other one.

3) Manipulatives. I once saw a question asked by a homeschooling parent, on the lines, "What manipulatives must I use and when?" The person was under the impression that manipulatives are a "must".

Manipulatives are definitely emphasized in these days. They are usually very good, but they're not the end goal of math education, and there is no need to "go hog wild" over them.

Manipulatives are something the student manipulates with his hands to get a better grasp of something. But the goal is to learn to do math without them.

Some of the most helpful manipulatives are:

• 
  A 100-bead abacus



• Something to illustrate hundreds/tens/ones place value. I made my daughter ten-bags by putting marbles into little plastic bags. You can buy base ten math blocks. These are sets of tiny blocks for ones, sticks for tens, flats for hundreds, and large cubes for thousands. See an image on the right.



• Some sort of fraction manipulatives. You can just make pie models out of cardboard, even. Stores sell ready-made fraction models made of foam or plastic. The one below sells for less than $10 at LearningThings.com:
  

For many kids, drawing pictures can take the place of manipulatives. That is especially true in the middle grades and on.

Check out also some virtual manipulatives.


4) Geometry and measuring tools. These are pretty essential However, dynamic software can these days replace compass and ruler and easily be far better.

---

The extras

These are, obviously, too many to even start listing.

*Some game or games are good for drilling basic facts. Games are nice for about any topic. Here's one that I played with playing cards with my dd; and now she seems to have learned the sums that add to 10. And here's a game that's worth 1000 worksheets. Of course the internet is full of online math games.

*I would definitely use some math software if teaching graphing, algebra, or calculus. Check MathProf for example, or Math Mechanixs. I've listed a few more here.

---

If you're ready to add something new to your toolbox from the online world, try The Math Forum's MathTools - a library of technology tools, lessons, activities, and support materials. Check also my pages listing interactive math activities online (there's a menu on the right).

Let me know if I forgot something.