Download Lessons from the Classroom Learning to Record Math Meaningfully

Lessons from the Classroom
Learning to Record Math Meaningfully
The intent of this essay is to try and open people’s minds in terms of ways to record
calculations to math problems. For too long, many people have felt there is only one
“best” way to perform and record calculations. The traditional North American vertical
recordings are not the only, nor are they necessarily the best methods for recording
computations. In fact they may actually work against our goal of developing
computational fluency for students in the elementary grades. Constance Kamii has
written many books and articles on this topic and she has concluded, “the traditional
algorithms un-teach place value.”
If you listen to a person doing the traditional algorithms you get a sense of what Kamii
is talking about. When doing the calculations you use shortcuts that reduce the
numbers to digits. Listen to the thinking when doing the Traditional Column Method:
11
178
+145
323
8 + 5 = 13, but can’t record that so carry the 1 and record the 3
1 + 7 + 4 = 12, but can’t record that so carry the 1 and record the 2
1 + 1 + 1 = 3 record the 3
This may not seem like a big deal to you as an adult with full understanding of our base
ten place value system, but to a person just learning to work with numbers it is similar
to taking all the vowels out of the sight words they are trying to learn and asking them
to learn to read and write.
How can we teach students to record their thinking in methods that demonstrate the
qualities of computational fluency and retain the concept of place value? For the
purpose of this essay we will define computational fluency as having the qualities of
accuracy, efficiency, and transparency (Suzanne Jo Russell). In essence we want
students to be able to solve arithmetic problems using number based methods that are
relatively quick, accurate and at the same time the user understands how they work.
ADDITION
An interesting thing happens when you watch early elementary students solve problems
with multi-digit numbers. They almost always start with the biggest number and move
to the smallest. For example, if you gave students the task of figuring out how many
students were on the playground if the all the second grade (178) and the third grade
(145) students went out for a fire drill and they were using Base 10 blocks to model this
situation, they would likely start with the hundreds and move to the tens and then to
the ones. Isn’t it ironic then that when we ask them to record the very same problem,
we ask them to start with the ones? A much better method would be to teach them to
record in a way that matches their natural problem solving. The concern has been what
to do when the sum of a certain column has been more than two digits. This recording
method deals with that issue but also retains place value. It is often called left to
right or partial sums.
178
+145
200 (100 + 100)
110
(70 + 40)
+ 13
(8 + 5)
323
Notice this method starts on the left in a way that mimics the thinking of the students’
natural problem solving methods.
For those of you that think starting on the left is just too strange, think about driving.
We drive on the right side of the road and think of that as the right way, but people in
Europe drive on the left side of the road. Normal is defined by what you get use to.
Starting on the left is actually normal for students before they are trained to abandon
those natural tendencies. For adults (and even for many of our upper elementary
students) starting on the left now feels strange. Luckily this recording method is
versatile. You can start on the left or the right and you will still get the same answer as
long as your arithmetic is correct.
178
+145
13
110
+200
323
(8 + 5)
(70 + 40)
(100 + 100)
Students can usually combine the partial
sums mentally because the numbers are
multiples of ten (friendly), but this again is
reliant on the quick and accurate recall of
basic facts.
If the goal is to develop computational fluency, which method is better? Let’s compare
the two methods of recording addition based on the criteria listed above:
 Efficient: I would say they are both about the same in terms of time.
 Accurate: That is dependent on the user. Both require the accurate calculations of
basic addition facts. Some might argue that 2 + 3 is easier than 20 + 30 or 200 +
300, but I don’t agree.
 Transparent: I find the methods that have the students record the actual value of
the addition to be much more transparent. For instance, 2 tens + 3 tens = 5 tens or
50 instead of saying 2 + 3 = 5 when you really mean 50. Even with the best
intentions, we don’t always remember to say the place values when we are doing
the calculations, and just think how confusing that must be to a child who is just
learning to work with numbers. No wonder Kamii says the typical vertical recordings
“un-teach place value.” They reduce the calculation to digits to make use of basic
facts.
SUBTRACTION
Recording subtraction is not quite so clear-cut. Many times, students will think about
problems differently depending on the wording of the problem.
In a problem like this, many students will solve the problem by thinking addition and use
the equation 78 +  = 249.
“John was saving up to buy a new bike. The bike coast $249. If John has
saved $78 so far, how much more does he need?”
Many students will solve a problem worded like this using the equation, 249 – 78 = .
“Fred collects baseball cards. He was looking through his 249 cards and
realized he had a lot of doubles. He gave all 78 of his doubles to his little
brother. How many cards does he have left?”
This is idea that you can use addition or subtraction to calculate the answer to a missing
part or compare problem is actually quite helpful. Students are often taught about
inverse operations as a good problem solving technique once they get to algebra or prealgebra classes. These students are figuring that big idea out all by themselves.
In either case, students will often not record vertically, but instead will either add up or
subtract back in pieces.
249 – 40 = 209
209 – 30 = 179
179 – 8 = 171
These first two steps can be combined if you can calculate 249 – 70
mentally, crossing over a hundred can be hard for many people however
so I showed the way most kids tend to do this work.
249 – (40 + 30 + 8) = 171
Fred has 171 cards left.
Many people would argue that this method fails in a comparison with the traditional
column method in terms of efficiency, and I would have to agree that this method is
probably a little slower. The question then is whether the increased transparency and
fostering the idea that the student decides to add or subtract is worth the sacrifice in
time?
Another Method of Recording
There is another method of recording subtraction that I think is as efficient as the
traditional method, but I have not seen it develop naturally in the work of the students.
This method would need to be explicitly taught and practiced much the same way the
traditional method is taught and practiced.
For this example, I chose numbers that typically present challenges for many students.
1002 – 475 = 
This problem has what are commonly referred to as middle zeros. When regrouping
(borrowing, carrying, etc), in the Traditional Method, many students make errors in
this situation.
9 9 12
1002
- 475
527
2 – 7: can’t do, so borrow, but it is a zero so borrow from the
hundreds, that is a zero too, so borrow from the thousands. Cross off
the one, make the zero a ten and cross it off and make it a 9. Then
make the next zero a ten and cross it off making it a 9 as well. Finally,
make the 2 a 12 and then subtract 12 – 5 = 7 and record, 9 – 7 = 2,
record, and 9 – 4 = 5, record.
When you see all the steps involved with this method, it is easy to understand why so
many students make errors. To be fair, not all subtraction problems involve this level of
regrouping, but it does illustrate the point that this method of recording can and does
cause problems for many students as they are learning about numbers.
An alternative method is often called, left to right subtraction.
1002
- 475
600 = (1000 – 400)
-70 = (0 – 70)
-3 = (2 – 5)
527
When you combine the parts, this is usually
work that can be done using mental math,
especially if you do it in two or three steps.
(600 – 70 = 530)
(530 – 3 = 527)
This sectioning technique is very similar to the
adding up or subtracting back methods.
A lot of adults, teachers and parents have expressed concern about the use of negative
numbers. It has been my experience that this does not bother the students much at all.
I usually use temperature as an example. If it is 2 degrees out and it gets 5 degrees
colder, what temperature is it? Another good example is a checking account. If I have
$2 and I spend $5 how much overdrawn am I? A good model for teaching about
numbers is the number line. We use the number line to help students learn about
whole numbers, large numbers, fractions and decimals, so it should also be a good
teaching tool for negative numbers as well.
I first learned about this method from a third grader. I had never seen anything like it,
but sure enough it is a method that is taught in some countries in Europe.
Let’s compare these two methods:



Efficient: In this particular problem, the left to right is actually quicker because you
do not need to do all the regrouping.
Accurate: That is dependent on the user. As long as a student is able to do
calculations like 2 – 5 or 600 – 70 the left to right method is accurate. In the
traditional method, they must know their basic facts (that is true for all methods),
but the biggest challenge is actually the regrouping process for most students.
Transparent: I have the same issue with the traditional method in subtraction as I
do with addition. In order to make use of the basic facts, the thinking is reduced to
digits rather than the true value of the numbers. As long as students have a
conceptual understanding of negative numbers, I think the left to right is more
transparent.
Subtraction has traditionally been one of the most difficult operations for students to
calculate accurately. Maybe more place value based recordings would help eliminate
some of those traditional problems such as subtracting up or incorrectly regrouping.
Multiplication
Since addition and multiplication are so related, it is not surprising that the methods for
recording are also similar. Multiplication is often referred to as repeated addition. The
alternative method for recording multiplication is very similar to partial sums and it is
often called partial products.
If you were going to multiply 37 x 9 using the traditional algorithm, you would write
them in the vertical format and then follow the procedure.
6
37
X9
333
7 X 9 = 63, but cannot record so record the 3 and carry the 6.
3 X 9 = 27, add the 6 and record the 33.
Notice again the numbers are reduced to digits to make the calculations
easier to do, essentially reducing them to basic facts.
If you were to do this same problem using the partial products method, you would do
everything the same (although you could do the start on the left if you wanted), except
you would record the partial products as we did with the left to right partial sums
method.
37
X 9
63 = (9 x 7 = 63)
+ 270 = (9 x 30 = 270)
333 = (63 + 270)
You would use the same recording method for multi-digit numbers.
42 X 28 would be recorded like this:
42
X 28
16
320
40
+ 800
1176



=
=
=
=
=
(8 X 2)
(8 X 40)
(20 X 2)
(20 X 40)
(800 + 320 + 40 + 16)
Efficient: I think they are both about the same when doing up to 2 digits by 2
digits, but when you get bigger than that, the long column of partial products can
become challenging to combine. For that reason, I believe the traditional method of
recording should be the preferred approach once a child has developed full
understanding of place value concepts. Students can be shown the partial products
and array methods in the elementary grades as teaching tools to help them
understand how the algorithm works, but they should also be exposed to the
traditional method so that they can transition to it when they demonstrate full
understanding of place value.
This brings up an interesting philosophical question. What role should technology
such as calculators and spreadsheets play in computation and problem solving? I
am not a big fan of calculators at the early elementary level, but when you are doing
multi-digit multiplication and division I would no longer consider that early
elementary type work. When does the process of actually doing the calculation
become so tedious that using technology becomes the preferred method?
The Connecticut Mastery Test (CMT) sets the following standards for paper and
pencil calculations for March of Fifth Grade:
A. Add and subtract 2-, 3- and 4-digit whole numbers and money amounts less
than $100.
B. Multiply and divide multiples of 10 and 100 by 10 and 100.
C. Multiply and divide 2- and 3-digit whole numbers and money amounts less
than $10 by 1-digit numbers.
We teach beyond these standards, but these are the standards the State
Department feels should be mastery level for students at this grade and time.
Accurate: The most errors in multiplication come from forgetting where you are in
the process or making a mistake in the carrying and adding in steps. In general, I
think using the area model (open array) is the most accurate, but it is also not very
efficient.
Transparent: I find the methods that have the students record the actual value of
the numbers to be much more transparent so I think partial products is much better
in this regard
There is a very similar recording method called Arrays, which work sort of like a
spreadsheet. This method is not purely number based, but is a combination of numbers
and a rectangle graphic. This method is based on the area model of multiplication.
28 x 46:
20
40
6
8
800 320
120 48
Now add the 4 subtotals together (this can be done in many ways, I have shown one):
800 + 120 + 320 + 48 = (800 + 300 + 100) + (20 + 20 + 48) = 1200 + 88 = 1288
536 x 49
500
40
30
6
20,000 1,200 240
4,500
270 54
9
Combine the subtotals: 24,500 + 1470 + 294
11
= 2 4, 5 0 0
1, 4 7 0
+ 294
2 6, 2 6 4
Notice I used the traditional column method to add the parts this time to show
you can use any method to combine the subtotals.
Division
There are two approaches I am aware of. The standard long division algorithm and a
second method sometimes referred to as flexible factors.
Long Division Method:
37 r 2
9
)
335
- 27
65
- 63
2
Step 1: Divide the hundreds.
How many 9’s in 3? None
Step 2: Divide the tens.
How many 9’s are in 33? (3) Record the 3 on top and multiply 3 X 9
= 27 Record 27 under the 33 and subtract. (33 – 27 =6)
Record and bring down the ones.
Step 3: Divide the ones.
How many 9’s are in 65? (7) Record the 7 on top and multiply 7 X 9
= 63 and record under the 65 and subtract. (65 – 63 = 2) Record as
a remainder, a fraction, or a decimal.
Flexible Factors Method:
30 + 7 = 37 r 2
9
)
335
- 270
65
- 63
2
30 X 9 = 270
7 X 9 = 63
37 X 9 = 333
There is 2 left
Thinking involved to get started:
I knew that it couldn’t be 100 times
because that would be way too big.
50 is still too big and 40 is still too big but
it is close, so I am going to start with 30,
because 30 X 9 = 270 and that is pretty
close without going over. Subtract 270.
How many 9’s in 65? That is a basic fact,
over. 9 x 7 is 63 so I will use that.
Recording the multiplication on the right helps students look at the problem
with place value intact instead of the more traditional digit based approach.
Compare the two recording methods:
 Efficient: I would say they are both about the same if the student knows their
multiplication facts (and extended facts). If they don’t, the flexible factors method
can become very tedious because they have to use facts they know. Knowing your
facts is vital for both, but long division uses basic facts only. That is both the benefit
and the problem with that method.
 Accurate: That is dependent on the user. Both require the use of multiplication
and subtraction. Often I think the subtraction in the flexible factors is easier
because when you use friendly factors (multiples of tens) you often are subtracting
numbers that end in zeros (example: 4 X 30 = 120).
 Transparent: I find the flexible factors method to be much more transparent.
Even though I now understand that when I say, “ How many 9’s are in 33 I really
mean how many 9’s are in 33 tens or 330, I seriously doubt many students who are
just learning about multiplication and division are seeing the underlying place values
that are inherent in this procedure. The language is very digit oriented. The other
benefit of this method is it reinforces the importance of knowing your basic
multiplication facts (and the related extended facts).
Conclusion:
I hope this essay helped you realize if you did not already, that there are many ways to
record your thinking when performing arithmetic calculations. The methods most of us
learned growing up are not the only ways to solve problems and if you hold yourself
rigidly to those procedures you may be making the process of performing those
calculations more difficult and far less creative than they need to be.
Our working definition of computational fluency is that students can figure out what
math they need to do to solve a problem and then carry out that calculation in an
efficient and accurate way that they understand. We do not assess kids on what
procedure they use as long as it meets those above criteria.
This is not true with all math programs. Some programs test whether or not you can
carry out a certain procedure. We don’t because we realize there are many ways to find
the answer. So if you are not convinced that this is a better way to record addition, you
can teach your child the traditional carrying method, but just be aware that there is a
growing amount of research that indicates that the carrying method actually works
against students development of place value and understanding of our base ten number
system so it may be better to wait until that understanding is firmly in place before
teaching the traditional recording methods that use such digit-based language.
I know change is hard, but whenever I think about that, I look to the medical model.
Just think if medicine never progressed? Many surgeons in the Civil War did not wash
their instruments because they were afraid they would get rusty. They did not know
about bacteria yet, so that was acceptable behavior in that time. If a doctor today
behaved in that manner it would be considered criminal. As we learn more about the
ways students learn we need to change our teaching methods to incorporate those new
understandings. Since we now know, that teaching the traditional methods by rote
actually interferes with some children’s understanding of place value, shouldn’t we teach
children at these formative years how to record in a different manner? If we don’t,
wouldn’t that be criminal?
Hope this gives you some food for thought. Please feel free to contact me at
[email protected] with any feedback or questions.