Research on number sense tends to be invalid

The preceding weblog text considered the pronunciation of numbers in English, German, French, Dutch and Danish.

There better be a general warning about invalidity of current research on number sense.

Update Sept 3: There now is also this proposal on developing an international standard for the mathematical pronunciation of the natural numbers.

Warning 1. The object of study concerns a chaotic situation

Research on how children learn numbers, counting and arithmetic, is mostly done in the context of the current confusing pronunciations. This is like studying people walking a tightrope while saying the alphabet in reverse order. This will not allow conclusions on the separate abilities: (a) dealing with arithmetic, (b) dealing with a confusing dialect.

In methodological terms: common studies suffer from invalidity. (Wikipedia.) They aren’t targeted at their research objective: number sense. Perhaps they intend to, but they are shooting into a fog, and they cannot be on target.

A positive exception is this article by Lisser Rye Ejersbo and Morten Misfeldt (2015), “The relationship between number names and number concepts”. They provide pupils with the mathematical names of numbers and study how this improves their competence. This reduces the chaos that other studies leave intact.

It is insufficient to state that you want to study “number sense in the current situation”. When you grow aware that the current situation seriously hinders number sense, then you ought to see that your research objective is invalid, since the current situation confuses number sense. If you still want to study number sense in the current situation, hit yourself with a hammer, since apparently this is the only thing that will still stop you.

Warning 2. Results will be useless

Results of studies within the current chaos will tend to be useless: (a) They cannot be used w.r.t. mathematical pronunciation, since they don’t study this. (b) Once the mathematical pronunciation is implemented, results on number sense within the current chaotic situation are irrelevant.

Validity and reliability (source: wikimedia commons)

Validity and reliability (source: wikimedia commons)

Warning Sub 2. Don’t be confused by a possible exception

There seems to be one exception to warning 2: the comparison of English, which has low chaos in pronunciation, to other situations with higher chaos (Dutch, German, French, Danish). This presumes similar setup of studies, and would only be able to show that mathematical pronunciation indeed is better. Which we already know. It is like establishing over and over again that drinking affects driving. The usefulness of this kind of study thus must be doubted too. One should not be confused in thinking that it would be useful.

Indeed, we might imagine a diagram with a horizontal axis giving skill in addition with outcomes in the range 10-20 and a vertical axis giving skill in addition with outcomes in the range 20-50, both giving the ages when satisfactory skills have been attained, and then plot the results for English, German, French, Dutch and Danish. We would see that English has lower ages, and French might actually do better than German, since the strange French number names are for 70-99. It might make for a nice diagram, but the specific locations don’t really matter since we already know the main message.

For example, Xenidou-Dervou (2015:14) states:

“Increasingly more studies are suggesting that this inconsistency between spoken and written numbers can have negative effects on school-aged children’s symbolic processing (e.g., Helmreich et al., 2011).”

Compare this with our earlier observation that professor Fred Schuh of TU Delft already proposed  on these grounds a reform of pronunciation in Dutch in 1943, 1949 and 1952 … Parliament in Norway (their “Storting”) decided in July 1950 to rename the numbers above 20 in English fashion.

It is not only problematic that Xenidou-Dervou isn’t aware of this, but also that she doesn’t see that the current chaotic situation invalidates her own research setup.

She remarks (2015:14) that the logical clarity (Schuh’s insight) has not been subjected to statistical testing. This may be true. When you don’t understand that drinking affects driving, then you might require statistics. Doing such tests is as relevant as statistical research on verifying that drinking affects driving. She states (my emphasis):

“To the best of our knowledge, the effect that the language of numbers can have in the development of a core system of numerical cognition such as children’s symbolic approximation skills [using Arabic numbers], controlling for their nonsymbolic approximation skills [using representations like dots but apparently not fingers] has not been previously addressed.”

Thus, the statistics on drunk driving are corrected for the performance when drunk riding a bicycle. It might be suggested that nonsymbolic number sense would be independent from language, and we might readily accept this for numbers smaller than 10, but to properly test this for 11-99 we need a large sample of Kaspar Hausers who are unaffected by language. Xenidou-Dervou’s correction does not remove the contamination by language.

Statistical tests may indeed be used to establish that large males tend to have a higher tolerance for drinking than small females, and to test legal standards. But questions like these are not at issue in the topic of number sense.

The relevant points are:

  • It is already logically obvious that a change to mathematical pronunciation will be beneficial. There is no need for statistical confirmation, e.g. by comparing English with other language situations. To suggest that such research would be necessary is distractive w.r.t. the real scientific question (see next).
  • The study of number sense can only be done validly in a situation with mathematical pronunciation, without the noise of the current chaotic situation of the national language dialects.

(PM. This is inverse of the case that there was statistical information that smoking was highly correlated with lung cancer, but that the tabacco industry insisted upon biological evidence. This analogy might arise when researchers would have stacks of statistical results proving that weird pronunciation is highly correlated with slow acquisition of mathematical understanding and skill, while there would be a strong lobby for maintaining national pronunciation who insist upon biological evidence. Thus do not confuse these statistical situations.)

Curiously, the press-release on Xenidou-Dervou’s promotion event and publication of the thesis of January 7 2015 states that she ‘discovered’ something which was already well known to Fred Schuh in 1943, 1949, 1952, if not some present-day teachers and children themselves:

“From age 5 the influence of teaching is larger than of natural abilities. What hinders Dutch children is the way how numbers are pronounced in Dutch. These relations have been found by Iro Xenidou-Dervou (…)”

“One of the teachers in the researched schools could confirm this with an anecdote from practice. She had heard one pupil telling another pupil doing a calculation: “Do it in English, that is easier.””

“Xenidou-Dervou thus suggests to start in Holland with education in symbolic calculation [with Arabic numbers] already before First Grade [age 6].”

Perhaps we might already start with Arabic numbers before First Grade indeed. Some children already watch Sesame Street. It would be more advisable to do something about pronunciation however. It is perhaps difficult to maintain common sense when you are in a straight-jacket of thesis research.

Warning 3. Such studies will not discover the true cause for the current chaotic situation

The barrier against the use of mathematical pronunciation doesn’t lie with the competences of children but with the national decision making structure. Thus, most current studies on education and number sense will never discover, let alone resolve, the true problem.

That the mathematical pronunciation will be advantageous is crystal clear. Of course it helps when you are allowed to first walk the tightrope and only then say the alphabet in reverse. Thus we have to look at the national decision making structure to see why this isn’t done.

Of key importance are misconceptions about mathematicians. Policy makers and education researchers often think that mathematicians know what they are doing while they don’t. Education researchers may be psychologists with limited interest in mathematics per se. Few are critical of what children actually must learn.

We may accept that psychology is something else than mathematics education, but when a psychologist researches the education of mathematics then we ought to presume that they know about mathematics education. When they don’t understand mathematics education then they should not try to force it into their psychological mold, and go study something else.

Two relevant books of mine on this issue are:

Warning 4. Mathematics education research has breaches of scientific integrity

Current research on education and number sense assumes that there is an environment with integrity of science. However, there is a serious breach by Hans Freudenthal (1905-1990) w.r.t. the results of his Ph. D. student Pierre van Hiele (1909-2010). Van Hiele discovered the key educational relevance of the distinction between concrete versus abstract, with levels of insight, while Freudenthal interpreted that as the distinction between applied and pure mathematics, and henceforth used his elbows to get Van Hiele out of the way.  Freudenthal was an abstract thinking mathematician who invented his own reality. There now exists a Freudenthal “Head in the Clouds Realistic Mathematics” Institute in Utrecht. Its employees behave as a sect, reject criticism, will not look into Freudenthal’s breach of integrity of science, and will not undo the damage. See my letter to IMU / ICMI. Other researchers tend not to know about this, and tend to accept “findings” from Utrecht assuming that it has a “good reputation”.

This warning holds in general

Just to be sure: this warning on invalidity of research on number sense is general. We might for example think of issues discussed in the Oxford Handbook of Numerical Cognition (2015), edited by Ann Dowker. Or think about issues discussed by Korbinian Moeller et al. (2011), or E. Klein et al. (2013). But, this weblog is about a major problem in Holland, and thus it might help to make some remarks concerning the anatomy of Holland.

Comment w.r.t. the Dutch MathChild project

The Dutch MathChild project can be found here, with contacts in Belgium, UK and Canada. Its background is in psychology and not in mathematics education.

The Amsterdam thesis by Iro Xenidou-Dervou (2015) is not fully online and it should be.

There is the full thesis by Ilona Friso-van den Bos (2014). She did the thesis at the dept. of education & pedagogy in Utrecht, but now she is at the Freudenthal “Head in the Clouds Realistic Mathematics” Institute (FHCRMI). I looked at this thesis only diagonally. Issues quickly become technical and this is secondary to the first question about validity. At first glance the thesis does not show sect behaviour (allowing for contagion from FHCRMI to other places at Utrecht University). The names of Freudenthal and Van Hiele are not in the thesis. The thesis has a neuro-psychological setup with a focus on working memory, which suggests some distance from mathematics education.The scheme of the thesis is that you define a test for number sense, a test for working memory, and a test for mathematical proficiency (try to imagine this without number sense and working memory), and then use children to see what model parameters can be estimated. Criticism 1 is that “mathematics achievement” is in the title and used frequently (see also the picture on p282), and taken for Holland as the CITO score (p160), which has a high FHCRMI content (so we find contagion indeed). Criticism 2 is that working memory belongs to the current fashion in neuro-psychology but is less relevant for mathematics education. For ME it is important to get rid of Freudenthal’s misconceptions and to look at Van Hiele levels of insight. Thus, get proper use of working memory, rather than train it to become a bit larger to do crummy FHCRMI math.

Criticism 3 concerns our present issue: the handling of the pronunciation of numbers. The thesis gives:

“(…) a difference between participants from linguistic backgrounds in which number words are inverted (e.g., saying six-and-twenty instead of twenty-six), because these inversions have been suggested to be a source of difficulty in number processing (Klein et al., 2013), and that errors related to inversion can be associated with central executive performance (Zuber, Pixner, Moeller, & Nuerk, 2009).” (p82)

“Publication year and inversion of number words did not play a role in the prediction of effect sizes.” (p97)

On p197-198 we find, my emphasis:

“An alternative explanation for the deviation in findings between previous studies (e.g., Barth & Paladino, 2011) and the current study is that in all previous studies, children were taught in English, in which the number system is more uniform than the Dutch number system. Dutch number words include the ones before the tens, instead of tens before ones (e.g., instead of saying thirty-five, one would say five-and-thirty), which is inconsistent with the order of written numerals. This may make it more difficult for young children to gain insight into the number system, and might explain the large number of children being placed in the random group during kindergarten, leading children to prevail in using less mature placement strategies and skipping the strategy with three reference points to inform number line placements in favour of the most advanced strategy, which is making linear placements. This hypothesis, however, rests under the assumption that children make placements through interpretation of verbal number words, either by transcoding the written number or by listening closely to the experimenter reading the numbers out loud. A study by Helmreich et al. (2011) indeed suggested that inversion errors may be of influence on number line placements in primary school children, although an important difference with the current study was that no numbers were read out loud by the experimenter, making the chance of inversion errors larger. More experimental studies are needed to investigate similar differences in findings and manipulate strategy use through variations in instruction in various groups.”

Criticism 3 thus generates the sub-criticisms:

  1. It is not only problematic that Friso-Van den Bos doesn’t give the earlier reference to professor Fred Schuh of TU Delft in 1943, 1949 and 1952, but also that she doesn’t see that the current chaotic situation invalidates her own research setup. Yes, we do see that she makes a correction at times, but the point is that the proper correction is that the thesis as a whole is shelved, since the situation that she studies cannot render the data that she needs.
  2. It is curious that she states that “more experimental studies are needed”. Compare this with a study of drunken driving in London, Paris, Oslo, Athens, … to test whether there are differences … I cannot understand how an educator can observe the crooked pronunciation of numbers, and not see immediately how important it is to remove the bottleneck rather than further research it. This is like finding a cancer and not remove it but argue that it needs more study. One might say that it is “only a Ph. D. study”, but the idea of a dissertation is that it shows that one can do scientific research by oneself individually. A researcher should be able to spot issues on validity. (Perhaps most Ph. D. students are too young or perhaps standards are too low given current academic culture.)
Concluding on the responsibility of educators of mathematics

As in the earlier weblog text, the main responsibility lies with Parliament: to investigate the issue.

It will still be the educators of mathematics who have the responsibility to re-engineer the mathematical pronunciation of numbers, to be used in education, and subsequently also in society and courts of justice. As a teacher of mathematics, I have presented my suggestions in the earlier weblog text, see here.


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