Learning Elementary Maths: Case Study of a Dutch Boy

Erik de Graaf and Marian de Graaf-Posthumus
Stichting Down's Syndroom
Bovenboerseweg 41
NL-7946 AL Wanneperveen
The Netherlands
+31 (0) 522-28 13 37
Fax: +31 (0) 522-28 17 99
  Reprinted with the permission of Erik de Graaf
Workshop presented at the Down Syndrome World Conference, Madrid, Spain
October 1997

That children with Down syndrome perhaps could learn to read became generally known in the field at the end of the eighties. However, the dogma that "they will never learn maths" remains alive to the present day. In literature, there is very little description about the process of learning maths for our children.
The authors of this paper did not accept the dogma and decided to attempt to teach their little boy with Down syndrome elementary maths.
Teaching maths was conducted simultaneously with teaching reading at a very early age and made effective use of the emerging reading proficiency. At the basis were the specific materials from the Macquarie Program. These were followed-up by a Dutch maths method for children with learning problems. Several innovative adaptations were made by the authors.
In the present paper, learning to do additions under 10 in a fully internalized way (i. e. without using fingers, an abacus, or otherwise) is described.
The authors see it as their goal to convince other parents and professionals that the attempt is very well worth the trouble.
A brief discussion of the method used will be given.
In De Graaf (1992) a description was given of how one boy learned to read from a very early age, even from before his third birthday up until the first months in the Dutch Group 3, which is the equivalent of the first year of the regular school elsewhere. This story has motivated very many, also because it has been published not only in Dutch, but also in English (De Graaf, 1993a) and in German (De Graaf, 1993b). The boy with Down syndrome concerned, David de Graaf, only had an average level of functioning from the outset. The important thing was the fact that with this case, for the first time, as far as is known, the entire trajectory was given in detail and - very importantly - from the point of view of parents. In addition to that, the entire process has also been documented on video.
Now, reading is one thing, maths is another. That children with Down syndrome could maybe learn to read became generally known in this field at the end of the eighties. However, the dogma "they will never learn maths" has remained very strongly alive until the present day, despite very remarkable achievements in individuals (Engels, 1995). In the literature, there is very little description about the process of learning maths for our children. For that reason, we will again describe the learning process of David. Here, we have to do that very briefly. Elsewhere, we have done this in much more detail (De Graaf & De Graaf-Posthumus, 1994). It is important to state again that David certainly is no "wonder boy". His starting position is described in detail in De Graaf (1992, 1993a and 1993b).
The authors are aware of the fact that there are children with very different characters (with parents with strongly different capacities) with whom, using different methods, even much better results have been achieved. This description is only meant to serve as an illustration of a development and to encourage other people, so that, hopefully, in the future, very many of these descriptions will follow.

Acoustical counting
The start of the process was when the Reading Program and the Number Program of the Macquarie University (Pieterse, 1981) arrived at David's home around his first birthday in the early spring of 1985. The essential element of the last mentioned programme in particular, is the decoupling of learning rote counting from learning number symbols and later the use of the same symbols as visual mnemonics during counting.
The first step on his way to learning maths was learning him to rote count. We started off with counting 1 to 3. This meant that all kinds of activities, like picking him up, getting him out of bed, etc., were preceded by counting, hoping that David would remember it in that way. More or less, we made that into a lifestyle. After a certain amount of time, we often counted only until 2, and watched David expectantly, hoping that he would say "three", but while doing so we were hindered very severely by the fact that David, in the first four years of his life hardly vocalised at all.

Figure 1 Matching
Learning the number symbols was conducted by means of the four basic concepts also used in the process of learning to read. In Buckley's terminology (1985) these were matching, selecting, naming, and comprehending. After David for the first time successfully had matched separate letters, around his third birthday, it was not very difficult to have him do the same thing shortly after with the figures 1 to 5, followed only a few months later with the figures 1 to 10 (Fig. 1).

More effective learning time
While the matching phase was fairly easy, in the course of the years we had to intensify our attempts to have David rote count. Basically, it was always about the increase of the total amount of effective learning time without David being bored, or we ourselves not being able to do it. In practice, this almost always meant dealing with all kinds of popular activities for short periods in order to achieve our goal. In all the applicable games we could think of, David first had to count from 1 to 10, later from 10 back to 1, (three, two, one - in particular remembering the correct position of the "two" was difficult). After some more time, he had to count from a random figure shown to him, first upward then downward.

Resultative counting
Simultaneously, David was often required to count a number of objects presented to him within the context of all kind of games: resultative counting. There we encountered another problem. David proved hardly able to combine the motor activity of pointing to the object he had to count with the counting itself, giving the impression that he wasn't able to count. However, directions like this were handled successfully by taking the motor part off him, or rather by moving his finger one by one alongside all cars, Duplo blocks etc. etc., or by pointing at them ourselves while David counted. Many years later, during real computations, especially by counting so-called "maths cubes", real three dimensional cubes as well as cubes on the computer screen, we would still be confronted with the same problem.

Dice configurations
Another very important game we have to mention here is throwing up a big, soft foam dice. The essence of all the variants of the game were, every time, throwing up, catching and then reading how many dots one saw. A game that matched this very closely was the well known German game, Ludo, an almost ideal environment to have a child very often and very consistently counting and exercising dice faces in a group situation. Due to all these exercises, after a certain amount of time, David was very good at recognising the dice faces.
After doubling Group 1 at the regular elementary school to which children go at age 4 in the Netherlands, David already sat in Group 2, when finally counting to 10, and from 10 downwards again, speeded up a little bit. During the entire process we had experiences as mentioned by Wishart (1993): Things that were very secure one day had completely vanished a few days later, to reappear shortly after again, especially with interesting reinforcement in sight. Giving much more emphasis to not learning something as to learning it, as mentioned by Wishart, seemed to have been written about our son.

Very bad pronunciation
When the counting rote finally came into existence, we had, unfortunately, not been able to deal with an old problem: David's pronunciation was so bad that he was hardly understandable. Even more strongly, even he himself was often confused by his own bad articulation. Fortunately, around his fifth birthday he was able to make the analysis (breaking up a word in the phonemes it is comprised of) and shortly thereafter the synthesis (reading by spelling) of consonant-vowel-consonant (CVC) words. By the age of 5 years 8 months, David knew all the letters and letter combinations of the Dutch language. He could name them spontaneously (1992, 1993a and 1993b). This fact put us in a position to start with presenting the words "one", "two" and "three" in letters from that time on, thereby enabling us to point out, via these letters, the correct pronunciation of these words.
However, during our efforts to emphasize that there is an "n" behind "ten", to distinguish it from "three" (in Dutch), we derailed and David began saying "treen" instead of "tree" for three. Although, in the meantime, we had arrived in a position to change his pronunciation by adding individual letters: e.g. the "r" in "tree", we then had to omit a phoneme that was redundant. We couldn't think of anything better than exercising on and on and on infinitely and correcting. Only during the summer vacation just preceding going to group 4, were we successful in getting "treen" back to "three", omitting the "n". Again, very often playing Ludo was very helpful in this respect. (But even so, a year or so later, especially with verbally offered computations, David would make mistakes from the type 3 + 1 = 11. On the basis of his former habit he then had mixed up the 3 and the 10 and had computed correctly 10 + 1 = 11 instead of 3 + 1.)

To Group 3
Finally, at the age of 7, David went to Group 3, the former first grade. At that moment, he had been able to count from a randomly chosen figure under 10 either upwards or downwards until zero for a few months. His pronunciation still led to problems, at least for his environment which often could not hear what was meant, three or ten.
While the initial reading in Group 3, owing to our own preparations (1992, 1993a and 1993b), posed no problems at all for David - at that moment he was quite a bit ahead of his class - as far as maths was concerned we chose following the class, seeing what he could do with the maths method available in the school. At that time, we couldn't think of anything better to do. However, even the first exercise sheets of this method, which was aimed at giving insight instead of offering structure, appeared to be far too difficult for David (at least that was our perception at that time). Good advice was hard to find. At this point our readers can appreciate that the last bit of the dream for our child with Down syndrome, being able just to catch up with computations amidst the weaker brothers in the class, went up in smoke forever. Only in the course of that October, by chance, we came across the brochure of a publishing house and discovered the maths method for children with learning problems by Boonstra, an employee of an organisation for the support of schools for children with special needs in the Greater Rotterdam area. It was called Low-Stress.

Low-Stress appeared to offer just what was needed at that time. Working in very little steps, with very strict criterion tests after every single step, exactly like in the Macquarie Program. The method offered an almost perfect fit to David's achievements so far, or rather a few months before. Had we been able to start considerably earlier, we then would have been able to have him work ahead, already in Group 2, just like we had done with so much success with reading. This would certainly have promoted his integration. The speed with which David took the first steps of Low-Stress (see the Table on the next page) showed us that we were right in that respect. After only a few days at the end of October he could add a certain number of cubes to an existing number to establish by counting how many there were in total. Due to all kinds of technical problems with finding the right handwriting method, David's writing education had begun by learning figures instead of letters. Due to this, at the time we started with Low-Stress, he was able to write a very, very weak, trembling, thin pencil figure in a box and by doing so he could state how many identical illustrations were present in his addition exercise. After this the introduction of the "+" sign and writing down the correct number to the total of two counted series didn't pose any more problems. Also in the next Steps (5 and 6), in principle, he could fall back on earlier skills that had been exercised, or rather that had been overlearned, until the end. In these, the identical illustrations in the first term had been replaced by the corresponding numbers, at first instance by David himself and later on rightaway in the exercise, after which he had to establish the total of those amounts. In essence this was no more than continuing to count from a certain shown figure. At the end of February 1992 we could check off this step.

Working with dice configurations
Description of the addition steps of Low-Stress (not the literal text) Age that David
reached criterion
(year; months)
 1 Adding a prescribed number of cubes to an existing number. Counting total number (maximum 9). 7; 08
 2 Counting the number of identical illustrations (maximum 9). Writing the figure in a box. 7; 08
 3 Two series of illustrations to be counted. Starting with the first and continuing the counting with the second one. Denoting the correct figure for the total number. 7; 08
 4 Ditto, with the introduction of the + and the = signs. 7; 09
 5 Ditto. Counting of the first series and writing the correct figure next to it. Determination of the total by counting upwards from the figure representing the first series (maximum 9). 7; 10
 6 Replacing the first series by a figure. Determination of the total by counting upwards from that figure (maximum 9). 8; 00
 7 Abstractly noted additions below 10 (e.g. 3 + 2 = _ ). Maximum of second term is 5. Dice configurations as mnemonic. 8; 01
 8 Dice configurations on the top of the worksheet. Student has to determine all by himself which one to use. 8; 07
 9 Abstractly noted additions below 10, with second term over 5 (e.g. 3 + 6 = _ ). Switching both terms and continue counting. 8; 07
10 Additions below 10. Continue counting with second term below 5. Switching terms and continue counting with second term over 5. 8; 07
11 Ditto. Continue counting immediately if second term is smaller than first. Switching terms and continue counting if second term is larger than first. 8; 07
12 Ditto. Continue counting by means of blind dice. 8; 10

An essential new element appeared in Step 7, still solving abstractly noted exercises with the aid of continuous counting, but now in using dice faces one to five as mnemonic. In doing so, Boonstra had in mind to rigorously avoid counting on the children's fingers and in addition to offer a means that could be faded in steps. We liked this approach very much because it matched our early intervention approach of the preceding years. Yet, it was the first point where we had to adapt the method a bit. The dice that were printed in the book had sides of 1 cm which were much too small for David's weak motor function. Right from the outset it was obvious that he would never be able to point with one index finger to the individual dots on the dice. However, if we drew a strongly enlarged corresponding dice with 4 cm sides above every row of exercises (in that step all of them still with the same second term!) this would work. In doing so, at the end of March David made his first dice additions. But in Step 8, several dice configurations stood next to each other at the top of the worksheet and the student had to choose all by himself which configuration he had to use. Drawing all these dice at the top of every exercise row meant much too much workload for the group teacher, who also had many more other children. Therefore, we made several strong cardboard sheets with all five dice on it (Fig. 2), initially with sides of 5 cm and later on a smaller version with sides of 3 cm. These were used by David at school as well at home. This principle worked fairly well with just another adaptation. One row of computations from the book was too much information per page. But if we used no more than two enlarged computations per page of an exercise book with very coarse (1 cm by 1 cm) squared paper cut in half, things went fairly well.

Real problems
Figure 2 The cardboard sheets with the dice did not red us of other difficulties. The last type of exercises gave us large problems, especially totally unexpectedly with exercises from the type 1 + _ = _ . Although realising that we had to deal here with a typical "Wishart-problem", (Wishart, 1993) still we never totally understood why things went wrong just here, when all other exercises went so well so often. David, both his teacher and ourselves exercised, exercised and exercised and had to deal very strongly with the large dilemma to continue exercising on the one hand and making things likeable enough for David by adding enough variation on the other hand. Therefore, we introduced a new type of exercising material: colorful magnet figures on a sheet of steel, which can be bought very cheaply in every toy shop. With this, we achieved a higher "production" per unit of time, because we needed to wait no longer for David's still very weak and very unclear writing. Finally, the material could easily be carried in a bag (together with the cardboard dice card), meaning that we could carry it on every car or train trip and he could do a few exercises on the road. Under the eyes of a few girls in the same compartment of the train, David almost always wanted to do his very, very best.

Figure 3 Furthermore, at this point the computer could start to play a role as exercise medium. Exercising so often meant that David knew more and more computations by heart. He would achieve them, as this is called, at a memorised level. This meant a test case for another idea of Boonstra, the author of the method: If a child doesn't know a computation, he can always fall back on a previous strategy. In David's case, this meant routinely working with the computer, using all figure keys, enter (to reinforce) and backspace (to correct a wrong answer by himself), while in addition from time to time using his little cardboard sheet with dice faces in between (Fig. 3).

Concise procedure
During the summer holidays of 1992, we kept ourselves busy very intensively with 1 + _ = _ type computations. After having introduced an even more procedural approach, gradually his achievements improved. In addition, we could see that he was merely making mistakes in the final step: continuing the counting. Shortly after the start of Group 4, many months after checking off the previous step, we decided that we only had to deal with real mistakes (only counting mistakes) and no longer with real procedural errors, an enormously important distinction. A formal criterion test with an 80% score, David, at that time, most probably would not have met. But yet we all had seen him do things right so very often.
During the previous months, at one point, we had sinned against Boonstra (and our own!) ideas. To achieve more variation during months after months of almost identical exercises, and also to adapt better to the computer programme we had, we also had exercised Step 9. In that step the second terms were bigger than 5 and David had to learn to change the terms, if necessary, before using his dice. David understood this very well. The same point hold for Step 10, where the second term no longer per definition was bigger than 5, but probably could be greater than 5. In other words: now David had to decide this all by himself. This left one extra instruction: "What is more?". Finally, in Step 11 it was no longer about second terms larger or smaller than 5, but only about learning to see whether changing terms rightaway would result in less counting, like in the case of 1 + 4. Also these three steps appeared to run without any problem. So finally steps 9, 10 and 11, were checked off together with Step 8 with general votes.

The "blank" dice
Figure 4 By means of Step 12, the last step of the chapter about additions under 10, Boonstra wanted to achieve that working with the dice was internalised. Children first had to choose a method of how to solve the problem. Then they had to actually solve it by bouncing with their fist on the first term and in addition by counting with their index finger on a separate loose square, little card, the "blank" dice. However, in this way, it was too abstract for David. Fortunately, Marian could think of a very effective in between step: a separate "blank" dice that looked very much like the dice on his cardboard sheet. We made a blank dice with sides of 5 cm on another piece of white cardboard. The way of working the procedure was as follows: First, David would establish whether the first and the second term had to be switched or not, and, if necessary, he rewrote the exercise accordingly. Successively, on his card with the five dice, he established which dice he was intending to use. Next, he would cover this with his blank dice to continue to count by heart on the blank dice (Fig. 4). This went perfectly well and on New Years Eve 1992, it also went without any problem in the way Boonstra had meant it: only using the blank dice. To make things easier, we drew this blank dice on the back of the piece of cardboard with the 5 other dice. In doing so, we always had all the attributes which were needed to fall back on an earlier strategy near at hand, at home, in the car, in the train, in short, everywhere where we could exercise.

In Boonstra's view, after additions under 10, first subtractions under 10 had to follow to be able to use these as a tool to cross multiples of tens later on. In that case these cross links can be computed from exercise to exercise without the necessity of having 36 splits from the type 7 = 3 + 4 available at a very high level.
Looking back, while we have seen the method work very well when it was needed for David, it is unclear how important this emphasis on not having to memorise splits is, especially after having seen with how much ease the same David remembers his tables of computation. Perhaps he would have been served just as well by a more innovative use of a regular method.

     Boonstra, H. H. (1985?), "Low-Stress; methode voor het aanvankelijk rekenen;" (Low-Stress, method for elementary maths, in Dutch), formerly Kok Educatief, Kampen, presently Stichting Down's Syndroom, Wanneperveen, Netherlands
     Pieterse, M. (1981), "Number Skills Program" (belonging to The Macquarie Program developementally delayed children), Macquarie University, Sydney, Australia
     Buckley, S. (1984), "Reading & language development in children with Down's syndrome", Down's Syndrome Project, Portsmouth Polytechnic, Portsmouth, UK
     Engels-Geurts, N. (1995), "Hoe Peetjie leerde rekenen" (How Peetjie learned maths, in Dutch), Down + Up, no. 30, pp. 17-20
     Graaf, E. A. B. de (1992), "Kinderen met Down's syndroom leren lezen en schrijven (Children with Down syndrome learn reading and writing)", Stichting Down's Syndroom SDS, Wanneperveen, the Netherlands, 95 pages
     Graaf, E. A. B. de (1993a), "Learning to read at an early age: case study of a Dutch boy", Down's Syndrome: Research and Practice, Vol.: 1, no.: 2, pp. 87-90
     Graaf, E. A. B. de (1993b), "Ein kleiner Junge lernt lesen; Fallstudie aus Holland", Down-Syndrom Aktuell, Sonderdruck "Down-Syndrom heute"
     Graaf, E. A. B. de & M. de Graaf-Posthumus (1994), "We hebben negen pannekoeken en we eten er zeven op; hoeveel blijven er dan nog over? Hoe David leerde rekenen" ("We have nine pancakes and two of them; how many do remain? How David learned maths") (in Dutch), Down + Up, no. 27, pp. 20-33
     Wishart, J. (1993), "Learning the hard way: Avoidance strategies in young children with Down's syndrome", Down's Syndrome: Research and Practice, Vol.: 1, no.: 2, pp. 47-55

  Revised: February 12, 1998.