1. The inverse relation between
addition and subtraction
This project investigates ways to teach pupils about the inverse
relation between addition and subtraction. The project is
currently being implemented in a school. You can find out
about how the investigation is carried out and what the results
so far are. You can also choose to carry out a similar project
in your school.
Background
It
has been known for sometime that pupils may be able solve
a problem correctly using their fingers to find the answer
without knowing what operation they would have to do in a
calculator to solve the problem. This is often the case when
the problem tells a story where a quantity increases
but the arithmetic operation that should be entered in a calculator
in order to find the answer is a subtraction or, vice
versa, when the quantity in the problem decreases but the
operation to be entered in the calculator to solve the problem
would be an addition. Consider, for example, the problem:
Jim had some stamps in his collection; his Grandmother came
to visit and gave him 19 stamps; now he has 43 stamps; how
many stamps did he have before his Grandmother came to visit?
Notice that Jim's stamp collection now has more stamps than
before but the operation to be entered in the calculator to
find the answer would be 43 - 19. Problems such as this are
known as inverse problems, in contrast to those where
there is a match between the story and the operation, which
are called direct problems.
Primary
school pupils often find it difficult to indicate the operation
that should be entered in the calculator to solve the problem
when problems are of the inverse type. This suggests that
they are not completely aware of the inverse relation between
addition and subtraction.
It
is important that pupils become aware of this inverse relation
between addition and subtraction for at least two reasons.
First, it is useful to know which operation to enter
in a calculator to solve a problem, particularly if the numbers
are large. But the second reason is even more important:
pupils' conceptual knowledge of addition and subtraction will
be incomplete without this understanding of the inverse relation
between the two operations. Think about some of the ways in
which mental arithmetic is taught in the Numeracy Hour and
you will see that pupils often need to use inversion to find
an easier solution to a sum.
Our
investigations with pupils in Years 2 and 3 in schools in
the Oxford area started by looking at their success level
in indicating the correct sum to solve a problem when they
were asked to enter the sum in the calculator to find out
the answer. The percentage of correct responses given by Year
2 and 3 pupils to direct problems was equal to 88%. For inverse
problems, the percentage correct was 37%. You can see that
there is a very large difference between the correct choice
of operation when the problems are direct and when they are
inverse, and that Year 2 and 3 pupils could benefit from instruction
on this concept.
The Initial and
Final Assessments
In
order to know how well your pupils perform in tasks that assess
this aspect of their understanding of the inverse relation
between addition and subtraction, you can administer a task
to your class. This disk contains assessment files that will
allow you to find out how well the pupils in your class perform
on such tasks. We call the assessment 'initial' and 'final'
because we use the assessments before and after the pupils
have participated in a teaching experiment. We strongly encourage
you to do the same in order to monitor how much your pupils
know before you start and how much they have learned through
the tasks.
In
order to carry out the assessment, you will need to use a
PowerPoint file and the file with booklets for the pupils,
which are in Word 2000.
Starting
the assessment
Before
you start the assessment session, you will need to print
the PowerPoint file for your use and the pupils' booklets,
one for each pupil.
To
print the PowerPoint file for your use: PowerPoint has
an option to 'print notes' in the print menu (when you are
in PowerPoint, you will need to click on 'file', then 'print',
and then choose the option 'print notes' from the box that
asks 'print what'). When you choose this option, you will
print the pictures and the text that goes with the problem.
This will allow you to read the instructions for each problem.
At the assessment session, you will display the PowerPoint
file in the slide show mode. This will display only the picture,
not the story. The assessment was designed with no need for
pupils to read the instructions in order to avoid disadvantaging
pupils who have reading problems.
As
the pupils booklets were composed in Word; you just print
these as any Word file.
In
order to give this assessment, you will need to use
a computer coupled with an appropriate projector. The images
will be presented to the pupils through the programme Powerpoint.
The pupils should have in front of them the pupils' booklets,
which can be printed from the Word file identified as 'pupils'
booklets'. Each page contains four rectangles and in each
rectangle the pupils should write the sum required to solve
the problem, as if they were entering the sum in a calculator.
Each item is identified by a picture, which is the same as
the first picture on that problem display.
When
you open the Powerpoint programme (labelled 'initial and final
assessment'), there is a first screen that welcomes the pupils
to the maths challenge. After this, each screen will present
a new problem. The screen only contains pictures, you will
need to read the problem for the pupils. Their task is to
write on their page the operation that they would enter in
the calculator in order to solve the problem. Ensure that
the pupils understand that you don't want the answer, you
want them to imagine that they are using the calculator and
write down the operation they would enter in the calculator
to find the answer.
To
start the test, put the programme into 'slide show'. When
you are in this mode, the slides will change when you push
'page down'. You can control the pace of the task so that
all the pupils have the chance to respond.
The
first problem is an example. After the pupils have written
the operation that they think is the right one to enter in
the calculator in order to solve the problem, click the mouse
on the screen and a little computer will appear. When you
click it again, the sum will appear on the screen. You should
check whether the pupils understood the instructions and wrote
an operation, not the answer. After this initial example,
there is no more feedback. Ensure that the pupils understand
that there will be no more feedback during this session.
You
will switch to the next problem by pushing 'page down'. In
the middle of the assessment, there is a screen that gives
a warning about a change from small to large numbers. At the
end of the assessment, there is a final screen that says that
the session has come to the end.
You
will notice that the assessment contains both direct and inverse
problems and that these appear with small and large numbers.
You should try to keep the problem instructions close to those
in the file. Should a pupil have difficulty in understanding
or remembering a problem, you can repeat the problem or rephrase
it slightly but ensure you do not give the pupils extra information
or new clues.
The Teaching Experiment
There
are different ways of designing tasks to give pupils practice
in skills that need to be relatively automatic even if they
are based on understanding. The tasks in this teaching experiment
were designed to give the pupils an opportunity to discuss
problem solving and to practice the skill of using the calculator
to solve direct and inverse problems.
We
have investigated how effective different task designs are
by comparing three groups of pupils before and after the teaching
experiment. Group 1 solved a series of direct problems and
then a series of inverse problems. Some teachers who were
working with us thought this would be the best way to design
this task because the pupils would be taught one thing at
a time. Group 2 solved the same direct and the inverse problems
as those in Group 1 but the pupils had the direct and inverse
problems mixed in the same task. Some of the teachers who
were working with us thought this was the best way of designing
the task because the pupils were taught that there are different
types of problems. Group 3 was a control group. The pupils
in Group 3 practised solving the operations of addition and
subtraction that those in the other groups had entered in
the calculator.
We
analysed the pupils' performance in the initial assessment,
before the teaching, and in two later assessments, after the
teaching, one immediately after and one eight to nine weeks
after. There was little difference between the groups in the
ability to select the right sum when the problems were of
the direct type. Their performance was similar both before
and after the teaching session and was close to 100% correct,
just like the performance of other pupils in previous studies.
At the immediate post-test, Groups 1 and 2 performed significantly
better than Group 3. It did not make a difference whether
they had learned one thing at a time, by having first blocks
of direct and then inverse problems, or whether they worked
with both problem types mixed and learned that there are different
types of problems. However, eight to nine weeks later, only
the pupils in Group 2 were still performing significantly
better than Group 3. The performance of pupils in Group 1
had decreased in level, possibly through forgetting, and that
of pupils in Group 3 had increased, possibly as a result of
instruction in the classroom. Nevertheless, the pupils in
Group 2 showed no signs of forgetting and still performed
significantly better than those in Group 3.
These
results led to the conclusion that it is better to design
this teaching task using a mixture of direct and inverse problems
in the teaching session than to teach them one thing at a
time. So you will see that the tasks contain four blocks of
problems in order to give the pupils some breaks and to keep
them motivated throughout the task but the problems are not
separated into direct or inverse within these blocks.
Teaching materials
You
will find the teaching materials in two Powerpoint files,
identified as Session 1 and Session 2. In Session 1 the pupils
will work with smaller numbers and in Session 2 with larger
numbers. The files for the sessions operate in the same way
as the one for the assessments. The screens that you present
to the pupils contain no instructions so you need to print
the 'notes pages' in order to obtain for yourself a booklet
with all the instructions. There are also pupils booklets
for Session 1 and Session 2 composed in Word.
When
you start the teaching session, the first screen in the Powerpoint
file contains an invitation to join the maths challenge, then
each problem follows. In contrast to the assessment phase,
during the learning phase the pupils will receive feedback
for every problem. You start the feedback process by clicking
the mouse two times. The first click makes a little computer
appear, the second one makes the sum that leads to the problem
solution appear.
It
is not sufficient to show the feedback in order to promote
an understanding of the inverse relation between addition
and subtraction in this situation. Some explanation should
be provided. You are the best judge of how this can be done
in your class. For each problem, you may want to ask one child
to provide his/her answer and an explanation for why that
answer was chosen. Often the children find it easy to chose
the right operation and explain why it is right when the problems
are direct. They say things like 'you add because you want
to know how much x has after he got some more'. The
difficulty is when they make a mistake in inverse problems.
We used explanations such as 'you want to know how much x
had before he got some from his friend; what sum do you do
to find out the number x had before he got some more?;
should he have more or less before he got the new ones from
his friend?' You may want to provide the explanation after
the initial two or three inverse problems if no pupil has
been able to produce the correct sum with an explanation but
after that it should be possible to obtain correct explanations
from different pupils.
Another
possibility is to let the children discuss which sum is required
in small groups and why, and then have each group offer their
answer to the class. After the groups have presented their
answers, you could summarise why the problem is solved with
that particular sum.
You
might find that discussing all the problems makes the task
too long. One option is to ask the pupils to discuss the problems
in the first half of the session, and then use only the feedback
from the computer for the second half of the same session,
with a final discussion of the last few problems at the end
of the session.
We
used the sessions on consecutive school days. The initial
assessment was given on one day, each of the teaching sessions
on a subsequent day, and the immediate post-test was given
on the first school day after the training. The delayed post-test
was given about eight or nine weeks later. The delayed post-test
is as important as the immediate post-test. You will want
to know not only whether the pupils learned but also whether
after some time they still know what they learned. You might
decide that they need a revision exercise.
Sharing
results
In
our study, the pupils worked on a one-to-one basis with the
experimenter. You can look at the results of our study by
accessing the webpage
http://www.brookes.ac.uk/schools/social/psych/childlearn
and
looking at the talk named 'Why children sometimes learn something
and later forget it'.
We
would like very much to know what you find with your own class.
If you want us to do an analysis, all you need to do is to
post the pages on which the pupils wrote their answers to
us. We will input the information into a statistical package,
do an analysis, and send the results and the pupils' answers
back to you. If you are mailing the information to us, in
order to be able to analyse it and to comply with the data
protection act, we ask you to assign numbers to the pupils
so we can identify the same pupils' responses on the different
occasions without knowing who the pupil is. We do not need
to know the pupils' names; we only have to identify which
answer sheets came from the same pupils.
If
you plan to do your own analysis, we would very much like
to hear from you. You can contact us through the Keeping
in touch section on the webpage or
mail one of the project directors, Dr. Ursula Pretzlik, at
upretzlik@brookes.ac.uk.
How
to use this CD:
If
you are planning to run the computer sessions using more than
one computer: i.e. one or two children per computer in an
IT suite then the following procedure must be carried out
on each computer.
- Start-up the
computer in the usual way, and log on if necessary.
- Insert the CD
and click on 'My computer' and then onto the CD (Maths 1).
- You will then
need to drag and drop the folder 'Initial and Final Assessment'
and the folders 'Session 1' and 'Session 2' onto the computer(s)
you wish to use for the sessions.
- Now if you double
click on the Initial and Final Assessments folder you will
see two files. Double click on the file Initial and Final
Assessemt.ppt and start the PowerPoint slide show (F5).
This will begin the presentation of the initial assessment.
- To start session
1, double click on the folder called 'Session 1' and you
will see two files, one for the child booklet for session
1 and one for the presentation. To start the presentation,
proceed the same way as you did for the assessment.
- To access session
2, repeat step 5 using the folder called 'Session 2'.
If you have any
problems stetting up or running the session, contact Daniel
Bell at Oxford Brookes University bdbell@brookes.ac.uk
If you plan to do your own analysis, we would very much like
to hear from you. You can contact us through the Keeping
in touch page. |
