?he 'ARITH3''Page %'
?fo 'Ann Lewis'- % -'2nd March 1977'
.mh
CHILDREN'S THINKING ABOUT ARITHMETIC
.br
====================================
.hm
.sp
Purpose
.br
-------
.sp
.in+5
.ti-5
i)\ \ \ Identifying the range of maths that children encounter
in primary schools (up to age 11).
.sp
.ti-5
ii)\ \ Finding cases (and topics) that seem to bother children a lot.
.sp
.ti-5
iii)\ Finding cases that bother teachers.
.in-5
.sp3
1. Introduction
.br
---------------
.sp
Initially, I spoke to a friend who teaches at a local primary
school (Hove/Portslade area). She obtained permission for
me to visit her class.
.sp
NB  I was advised by the Headmaster that the name of the school,
of individuals in the school, and the actual work of the children must
not be used (i.e. quoted) in any reports etc. Therefore all names
are changed, and examples used are similar to, but not copies of children's
work.
.sp2
The Class
.br
---------
.sp
- Consists of 32 4th year (10-11 year old) children. Their teacher is
Mrs Smith.
.sp2
Notes on arithmetic teaching in the class (general)
.br
----------------------------------------------------
.sp
.in+5
.ti-5
a)\ \ Children are divided into 4 groups according to ability (groups
have colour-names, not numbers, - this in theory avoids 
demoralising the less able).
.sp
.ti-5
b)\ \ This class has more children than last year's group having difficulty
with arithmetic. Four or five children are described as "poor",
(statistically, teachers expect one or two in a class of this size) and
the class as a whole needs a lot of guidance.
.br
Teaching through arithmetic topics, for example, which requires
a lot of self-learning on the part of the child, has proved impossible
this year.
.sp2
.ti-5
.tp5
c)  Progress
.br
.ti-5
------------
.sp
Progress in working and in teaching depends to a large extent on the
ability of the child. Those children who are "poor" may have
difficulty in getting beyond the simple problems of the four rules,
while those with greater ability and grasp of the subject work on
progressively more difficult problems, some moving onto new
subjects -> fractions etc.
.in-5
.sp2
2. Range of Arithmetic taught (Stage (i)).
.br
-----------------------------
.sp
(a) The four basic rules:	addition*, subtraction,
 				multiplication, division.

* Working both in columns(1), and in horizontal lines(2)
 	1. e.g. 3456		2. e.g. 3456+1234+987 = 5677
 		1234
 		 987 +
 		----
 		5677
 		====
.sp
b) Long multiplication, long division
.br
-------------------------------------
.sp
c) Fractions:
.br
-------------
.br
First year work - "practical" fractions - cutting up card shapes
etc. to gain a concept of fractions.
.br
By 4th year - application of the 4 rules to fractions and mixed
numbers.
.sp
d) Money, Decimal Fractions, Metric measurement
.br
-----------------------------------------------
.sp
e) Time
.br
-------
.in+5
.ti-5
.sp
i)   This class begins with a little information on the earth,
moon and sun, which introduces them to the concepts of time, seasons
and night and day.
.sp
.ti-5
ii)  Basic information on time (the clock) and the calendar (including
the "months" jingle) follows.
.sp
.ti-5
iii)\ Working on the calendar and on days of the week  -
.sp
.tp4
e.g. If July 3rd is a Thursday, what day do these dates fall on?
 			July 5th
 			June 30th
 			July 23rd  etc.
.sp
.tp3
.ti-5
iv)\ \ Work on the clock
.sp
 		e.g.	written time converted to figures
 			- quarter to eight - 7.45   etc.
.sp
.ti-5
v)\ \ \ \24-hour clock
.sp
.ti-5
vi)\ \ Brighter children may progress to applying the '4 rules'
to time.
.tp5
 		e.g.	days	hours
 			 2	 12
 				  4  x
 			------------ 
 			10	  0
 			------------
.sp
.in-5
f) Percentages
.sp
g) Symmetry
.sp2
The main text-book followed throughout the school is:
.sp
"Beta Mathematics",	Compiled by T.R. Goddard & A.W. 
 			Grattidge, in Association with J.W.
 			Adams & R.P. Beaumont.
.br
(Pub: Schofield & Sims, Huddersfield).
.sp
.ce
--------------------
.sp2
* At the beginning of the year, the following reference information
is given:
.sp
Meanings of terms
.br
-----------------
.sp
 	Words for addition:		Words for subtraction
 	-------------------		---------------------
 	plus				take away
 	add				minus
 	cost of				subtract
 	total				less than
 	greater than			decrease
 	increase			find the difference
 	find the sum of			reduce
.sp
 	Words for multiplication:	Words for Division
 	-------------------------	------------------
 	times				share
 	multiply			divide
 	multiply by			divide by
 	find the product of	
.sp
.ce
--------------------------------------
.sp2
.tp2
3. Finding Children's problems. (Stage (ii))
.br
-------------------------------
.sp
I approached this question basically in two ways:
.br
.in+5
.ti-5
.sp
a) by chatting to Mrs. Smith and looking at examples of children's
work - this gave an idea of common mistakes and/or problems.
.sp
.ti-5
b) by chatting to the children themselves. This wasn't at all
easy: they tend to express problems in terms of "like" and "don't like",
so I took to asking about likes and dislikes in arithmetic (and also
compared with other subjects) and got around to what they found
hard in this way.
.sp
.in-5
These two approaches and their results (which are far from satisfactory)
are discussed below.
.in+5
.sp2
.ti-5
a) While I was there, the class was given an arithmetic test covering
one example of each type of question that they had learned so far,
and a test on time questions.
.sp
The first of these tests was as follows:
.sp
.tp8
 	1. 367		2.		3. 1,486
 	    22 x	18)976		     999 -
 	   ---				   -----
 					   -----
 	   ---
 	   ---
.sp
4. 876 + 32 + 1780 + 15 =
.sp
.tp8
5.  L	p		6.  L	p		7.  L	p
    8	16 1/2		    36	31		    4	16 1/2
    0   99		     1  99 1/2 -	         5 x
    2	55 1/2 +	    ----------		    ----------
   -----------
              		    ----------		    -----------
   -----------
.sp2
Problems:
.br
---------
.sp
Qu.(1) - The school's policy for long multiplication is
to teach children to multiply by the units first, then tens,
then hundreds etc. This creates the problem that children often
forget noughts at the end of rows, and produce answers such as:
.tp8
 			 367
 			  22 x
 			----
 			 734
 			 734
 			----
 			1468
 			----
.sp
Qu.(2) - 30 was a common answer to this one! Children often
forget to bring down figures, and to carry on. Often the remainder
is given as the answer, instead of the top working.
.sp
Qu.(3) - Examples I and IX of Max's subtraction examples
often apply here. (See appendix I).
.br
(Mrs Smith says most of these are fairly common
but she doesn't recognise examples III and IV as ever occurring
in her experience).
.sp
Qu.(4) - Nothing specific, but people seem to get into all sorts
of muddles with sideways working: another example I saw was:
 		L5.93 - 38p = 65p
.sp
Qu.(5) - Halves in money problems often lead to confusion when they
add up to an even number: thus:
 		L	p
 		8	16 1/2
 		0	99
 		2	55 1/2 +
 		--------------
.br
May give	11	710
 		--------------
.sp
Qu(7) Multiplying by 1/2 again causes problems and often
leads to answers such as:-
 		L	p
 		4	16  1/2
 			 5      x
 	       ----------------
 			802 1/2
 	       ----------------
.sp
I saw several like this.
.sp2
Time questions - commonly, errors occur when converting
"spoken time" (e.g. "five-to-four", "quarter-to-ten") after the
half-hour into figures:-
 	e.g. "five-to-four" = 4.55
 	     "quarter-to-ten" = 10.15 (or 9.15)
.br
etc.
.sp2
General - children often don't seem to know what to do with '0' when
.tp6
it occurs, Thus:-
.sp
 	(i) 3		(ii) 3
 	    0 -		     0 -
 	    -		     -
 	    0		     7	(This one just seems
 	    -		     -	to be clutching at straws,
 				but I did see it!)

.sp
b) My talks with children were not very satisfactory. This may
have been partly due to my lack of knowledge of how to approach
them about
their problems, and partly through the very real difficulties that
children have in actually recognising that a 'problem' exists - they
have (or seem to have) very straightforward ways of looking at
things - "I can", "I can't", "I like", "I don't like", "s'alright,
I s'pose" etc.
.in-5
.sp
However, I did glean the following, which may be useful:-
.sp
i) Time questions were often popular - this seemed to be because of
their practicality (they're less abstract than other arithmetic).
.sp
ii) Fractions are disliked, particularly subtraction of fractions -
children seemed to find they got muddled doing them.
.sp
iii) Long division is the other main bug-bear in this class -
forgetting to carry down, working out how many xx's in xxx, etc. (I
can remember hating this, too). One girl showed me how they are
taught to approach these problems, i.e.:
.sp
 		13)1234	! 6 x 13 =  78
 		        ! 9 x 13 = 117	etc.
.sp
She went on to explain that she often forgets to carry the top
figure (pink) across from the right. (This would perhaps account
for some of the reproduction of remainders as answers?).
.sp
iv) There was a general feeling that arithmetic was an unpopular
subject (which is no change from my own school days), and I was amused
to contrast the obvious lack of enthusiasm for it with the forest
of arms which appeared when Mrs. Smith asked for ideas about
"snow" for a creative writing exercise (it was a snowy day.).
.sp2
4) Finding Problems that bother teachers (Stage (iii))
.br
---------------------------------------
.sp
I asked Mrs Smith two questions (formally) on this:
.br
.in+5
.ti-5
a) Do you think that arithmetic poses special problems in
teaching, either for children or teachers?
.sp
.ti-5
b) Are there topics which teachers find particular difficulty
in conveying? Or have mental blocks about?
.in-5
.sp
In relation to (a), she spoke generally of the limitations imposed
by lack of facilities. (Related to our subject, she
for example, had been unable to teach any geometry until recently
because the classroom has no protractors or compasses).
.br
Reading abilities pose further problems. Some children have
difficulty in grasping the range of vocabulary they need in order
to be able to read the questions. Arithmetic is not a popular
subject, and thus poses additional problems of holding the
children's interest sufficiently for information to be conveyed.
.sp
b) The main topics with which she experiences problems are
fractions and decimals and converting fractions to decimals and vice versa, and particularly fractions. The difficulty
is in the level of her own arithmetic - i.e. she feels a little
shaky about fractions herself. This implies that if a child does
not understand her explanation, she has difficulty in paraphrasing
it. She believed that this was a common problem with teachers.
.sp
.sp2
5) In Conclusion
.br
----------------
.sp
The main problems experienced commonly by Mrs. Smith and her
class are concerned with fractions.
.sp
In addition to this, children dislike long division, while Mrs.
Smith suggests difficulties with decimal fractions.
.sp
If it were possible to set up models for manipulating
fractions*, this might help, but it is a long way beyond
modelling addition.
.ce
-----------
.sp
* On the other hand, is the child's problem one of manipulation
alone, or of grasping the concept of portions of a whole?
.bp
Appendix I -	Examples of Arithmetic Errors (from
.br
------------	------------------------------------
 		Max Clowes, December 1976)(Subtraction)
 		---------------------------------------

.sp3
.sp3
I	 37		II	 48		III	 28
 	-19			-19			-15
 	 --			 --			 --
 	 22			 39			  3
 	 --			 --			 --
.sp5
IV	 72		V	 47		VI	 56
 	-57			-23			 27
 	 --			 --			 --
 	 20			 54			 11
 	 --			 --			 --
.sp5
VII	 52		VIII	 25		IX	 74
 	-37			-17			-49
 	 --			 --			 --
 	 85			 18			 36
 	 --			 --			 -- 
 						('counting' error)
.sp5
.ce
