An Essay into ArithmeticAuthor(s): Tom MillerSource: Mathematics in School, Vol. 8, No. 2 (Mar., 1979), pp. 31-33Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30213452 .

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AN ESSAY INTO

AIRITHiMETIC

by Tom Miller

This article outlines one teacher's attempt to analyse his pupils' difficulties with arithmetic. He would welcome any suggestions or advice from readers and to hear of other teachers' attempts to deal with the problem. Send comments to the Editors. We will forward them to Mr Miller and publish any of general interest - Editors.

I decided to break down the arithmetic I was teaching into it's component groups and see if by doing this I could discover where the main stumbling blocks were and what - if anything - I could do td avoid them; or if they are unavoidable, to see how my approach to them could be modified or even drastically altered.

I teach girls, aged approximately 11-18, and as they come to me at what one would normally class as fairly late in the arithmetic stage (at this point I must restrict my definition to the four rules and at a later stage I will restrict it even further, the reasons will become obvious), I have been tempted to make certain assumptions of knowledge. I thought hard about this, trying to discover how a child of 11 could slip through the net and eventually come to me being unable, for example, to set out and add a simple addition sum with acarryoverfrom each column, e.g. 27+ 103+ 89+ 196.

I eventually decided that it may be unfair to make any assumption, so I was inclined to break addition down to what I personally would class as remedial mathematics and go as far as having addition sums with no carry on any columns. After long thought I have abandoned this idea and my starting point remains as addition with a carry from each column. I am fairly certain that this particular starting point will be difficult to justify under fierce criticism. My only defence is that, in practice, I have found it to be adequate and I

have been able to deal quietly with any child who has found difficulty with this section.

I have had massive difficulty with the breakdown. It is a matter, I feel, of personal preference as to which particular section should be broken down minutely and which can have broad area titles. I have tried to strike a "middle of the road" balance, but I am always hopeful for a better arrangement!

I gave two sums in each section of my breakdown feeling that it was unlikely that any child who was capable of completing the section would get both sums wrong. A similar case can be put forward for having three or more sums in each section but, as my purpose is to find out something and not bore my pupils rigid, I think two is sufficient.

My sections are listed here:

Addition I have already stated my case for the starting point and I feel that most mathematics teachers will accept the "shortness" of this section.

A) Adding without Decimals E.g.

7 984+387+89 604, 3 557+2 679+305+22 This section is self explanatory.

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B) Adding with Decimals

E.g. 16.7+305+8.75, 3.773+265+0.04 I had a lot of difficulty in deciding whether or not to break this section down any further. It is obvious that there are many sub-divisions and my defence for leaving it as one section is sheer weight of work. Subtraction A) All Numbers Smaller

E.g. 794- 632, 877- 766 I decided that this section should be kept in although I had serious doubts.

B) With One End Zero

E.g. 790- 597, 460- 273

C) With Two End Zeros

E.g. 3 600- 577, 2 400- 597 Both Sections B and C refer of course to the minuend not to the subtrahend.

D) One Decimal Place to be Subtracted from Integer E.g. 227-138.5, 846- 789.8

E) Two Decimal Places to be Subtracted from Integer E.g. 36- 27.85, 133- 117.89 With Sections D and E there are many sub-divisions, but these two cover the main points and I feel that to go further at this stage may be pedantic. In fact I am inclined to think that the only justification for Section E is a serious check on the validity of Section D.

Multiplication A) Integer by Units

E.g. 796x 9, 467x7 This is a basic start to the multiplication section. It gives most children a sense of achievement to feel that they can cope with the first part and gives them the confidence to carry on.

B) Integer by 10, 100, 1 000, etc.

E.g. 36x 100, 865x 1 000

C) Integer by Tens (i.e. 20, 30, 40, etc.) E.g. 36 x 70, 865 x 40

D) Integer by Tens and Units

E.g. 36x 47, 865x 86 I have deliberately made each column a "carry" as I feel that there is little point in not doing so.

E) Decimal by Units

E.g. 467.83x 7, 89.755x 6

Again, each column has a "carry".

F) Decimal by 10, 100, 1 000, etc.

E.g. 46.3x 10, 8.975x 100 It is difficult to judge whether a child is capable of multiplying a decimal by a large number like 10 000 000 merely by giving it two fairly simple sums. I am afraid that I have not done enough here.

G) Decimal by Tens (20, 30, etc.)

E.g. 46.85x 70, 894.73x 50

H) Decimal by Tens and Units

E.g. 46.85 x 49, 894.73x 86

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Division This is a massive section which threatened to grow out of all reasonable control. I have pruned it mercilessly to what I hope is an almost manageable size. I could not make it any smaller without glaring omissions. I have named the sections Al . .

. 11 and A2 . . . 12. This was

my original method of recording. I have changed the recording method but not yet the titles.

Al. Integer by Unit with no Remainder

E.g. 1 888+8, 45 801+7

B1. Integer by 10, 100, 1 000, etc. with no Decimal Answer

E.g. 400+ 10, 8 600+ 100

Cl. Integer by Unit with Remainder

E.g. 66+4, 7 708+8 I am happy to accept a decimal answer to this as I think it implies a greater knowledge of division than the simple leaving of a remainder. I have offered guidance on the question "Shall I leave a remainder or carry on?" by saying that a decimal answer in this case is preferable.

D1. Integer by 10, 100, etc., with a Decimal Answer

E.g. 672+ 10, 859+ 1 000

El. Integer by Tens (i.e. 20, 30, 40, etc.) with no Remainder

E.g. 27 280+40, 86 380+ 70 This was a difficult one to make decisions on and I am sure that this section could take up several pages.

Fl. Integer by Tens (20, 30, 40, etc.) with a Remainder

E.g. 21 444+ 60, 36 536+80 Again, here a decimal answer is acceptable.

G1. Integer by Unit with Decimal Answer

E.g. 12+8, 327+6

H1. Integer by Tens (20, 30, 40, etc.) with Decimal Answer

E.g. 31 925+70, 12 849+40 When I am given a remainder rather than a decimal answer I would consider it a wrong answer unless the child had given decimal answers correctly in another section. Even then I would set two or three sums to the child with specific instructions to give decimal answers to them.

11. Integer by Tens and Units with no Remainder E.g. 31 968+48, 78 960+ 56

A2. Integer by Tens and Units with a Remainder

E.g. 20 677+ 56, 30 684+ 36 A decimal answer is acceptable.

B2. Integer by Tens and Units with a Decimal Answer

E.g. 456+ 16, 369+36 A remainder is not acceptable here.

C2. Decimal by a Unit with no Carry to the Next Decimal Place

E.g. 148.5+ 9, 258.86+ 9

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D2. Decimal by Unit with a Carry to the Next Decimal Place

E.g. 17.2+6, 18.73+9

E2. Decimal by 10, 100, 1 000, etc.

E.g. 18.62+ 10, 182.5+ 100

F2. Decimal by Tens (20, 30, 40, etc.) with no Carry to the Next Decimal Place

E.g. 18.62+ 20, 182.5+ 50

G2. Decimal by Tens with a Carry to the Next Decimal Place

E.g. 182.5+ 60, 3.796+ 80

H2. Decimal by Tens and Units with no Carry to the Next Decimal Place

E.g. 620+ 17, 2 363+ 65

12. Decimal by Tens and Units with a Carry to the Next Decimal Place

E.g. 17.5+18, 35.91+36

Reflections Addition This section is in general satisfactory. The first section was dealt with adequately.

The second section caused some difficulty with the setting out of the sums. All of the children in the experiment actually added up correctly, but 35% of them set the sums out incorrectly and so produced wrong answers. The mistake made by all 35% was to set down all of the numbers in columns, e.g.

3.773 265

0.04

I have wondered whether this is in fact a fair test of addition, because I am not actually testing whether or not a child can add; but whether or not the child has a good enough knowledge of place value to be able to set out a sum correctly and then add it up. This, as every teacher of mathematics is only too well aware, is not the same thing.

Assuming this to be so, perhaps the children should be taught place value if they have difficulty with this section, rather than addition. This is an aspect which had not occurred to me before. I have not yet taken it further but I think it will be a fascinating study.

Subtraction I am happy with the sums and the breakdown in this area. The only mistakes made were on the last two sections, D and E, both of these involve subtracting a decimal from an integer. The answers given varied from the totally nonsensical, where a child had written absolute rubbish or at least I could not find out how on earth she had arrived at her answer, to the wrongly set out but computationally correct subtraction.

Forty-five per cent of the children tested got Sections D and E wrong. I have no figures yet on what proportion of these actually subtracted properly, although I suspect that it is rather high; I base this on the evidence that all of them completed the other three sections with no apparent difficulty. This again may suggest that place value is in need of massive revision with those children.

(As an aside at this point: it may well prove worthwhile to investigate the reasons why some of these children think that there is a "uniths" place and therefore cannot see any difference between the values of a digit before and after the decimal point.)

Multiplication I am dissatisfied with this section. I consider the breakdown and the actual sums to be satisfactory but the number of sums is too great and the paper should be made into two or three separate, and as far as the children are concerned, totally unconnected sheets.

The failure rate was rather interesting in that the same children were not consistent - no one failed everything (although one child nearly managed it).

Section A 5% failed Section B 10% failed Section C 10% failed Section D 15% failed Section E 40% failed Section F 30% failed Section G 65% failed Section H 70% failed

The really significant failure rate began on Section E. I wonder whether it is mere coincidence that every time a decimal rears it's ugly head, the failure rate zooms off into the blue? Perhaps one should be extra careful when teaching decimals. Division I am very unhappy with the setting out of this section. There are far too many sums and I am afraid that this is unavoidable. I am unable to see how I can justify reducing the sheer size of this section. I intend before taking the investigation further to split the papers up into much smaller units. The results are very mixed and I have so far not drawn any conclusion other than the rather worrying and increasingly obvious problem with decimals. It has shown that any sum involving decimals creates massive difficulty.

Conclusions I think that I made a bad mistake on the timing and the setting out of all of the papers. I originally decided that I would try to get them completed as quickly as possible and so have the whole thing out of the way within a week. The first two or three papers were welcomed by the children but they rapidly became bored and dis- interested, and many were inclined to give up when faced with the massive division section.

I intend to re-set the whole series and I have come to the conclusion that it is not necessary to have a complete section on any aspect. I intend to reduce the size of each sheet to six or eight sums (it may be that less would be even better), to spread the investigation out over a term or longer and to have different kinds of sum on each sheet. This, I hope, will give more accurate results.

I have also been working on decimals with the forms who took part in the investigation and it may be that on the second attempt there will be some improvement. (I only wish that I had a foolproof method of teaching decimals - I do not like any method that I have seen.) I intend to rewrite the methods of teaching place value that I know of and have used, and it may be that a method based on the best parts of all of the other methods is one which would suit me. I have found my method of recording results satisfactory and I intend to keep it.

My general conclusion is that this has been a worth- while exercise though somewhat unrefined. I feel that if I can smooth off the rough edges I will get some valuable information from a repeat - not necessarily with the same forms. My next "project" is a similar investigation with fractions. I have not yet completed the breakdown but I look forward to even more frightening results with them!

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