The Building Blocks of Mathematical Competence

Prof. Daniel Ansari: Building Blocks of Mathematical 

Competence

https://www.youtube.com/watch?v=PtsGcT9ivLk

Prof. Daniel Ansari is one of the leading researchers in the study of the development of mathematical skills. 

Here he talks with an audience of educators.  Ansari speaks clearly and is a joy to listen to.  This talk is recommended despite the annoying fact that the photographer concentrated only on him and did not photo the slides he shows.

Here are several messages from the lecture:

Early math skills (in preschool and the early grades of elementary school) predict not only later math skills but also later reading skills.  Ansari deduces that working on early math skills helps not only later math but also later reading.  I think it might be possible that the same cognitive abilities lie both at the base of early math skills and at the base of later math skills and later reading skills (for example, working memory or fluid ability or even comprehension knowledge).  So I would think that it's a good idea to improve these skills as well as early math skills.

Ansari reviews (briefly) research that shows the ability of babies and animals to discriminate between quantities.

Ansari highlights the importance of young children's ability to process mathematical symbols (the digits 1,2,3…are mathematical symbols.  Each of them represents a specific quantity).  Children with difficulties in math find it hard to match math symbols with quantities (for example, to match the symbol "3" and three objects).  The ability to name digits and to match digits with quantities are very important for the progression from kindergarten to first grade – from formal to informal math education.  We need to make sure that kindergarten (five to six year old) children are able to name math symbols (to name digits) and to link digits with quantities.   It's important to practice counting, digit comparisons and quantity comparisons, ordering digits, matching digits and quantities and so on with kindergarten children,

The number line helps children to understand relations between numbers.  There is a very strong connection between number concepts and visuospatial concepts.  Playing games like "ladders and snakes" helps children understand this relation and relations between numbers. 

Ansari thinks that the difficulties of children with dyscalculia do not originate from the non-symbolic system (quantity perception) but from difficulties in linking symbolic and non-symbolic systems (linking quantity to digit).  This is an optimistic viewpoint because it's possible to work on these links, while it's difficult to improve quantity perception itself.

Ansari talks about developmental dyscalculia and different cognitive abilities that are tied with difficulties in math (working memory in general and especially visupspatial working memory, phonological awareness, executive functions and language).

As for math anxiety, Ansari says that children "inherit" their parent's math anxiety.  When a child is doing homework with his math anxious parent (even if they are working on basic math), tension rises and makes the child's math anxiety even worse.  A parent with math anxiety does not talk with his children about math concepts - even about simple, basic quantities or matching quantities to numbers.  This affects the development of his children's math skills.

Furthermore, teachers with math anxiety cause their students to "catch" math anxiety too.  Math anxiety of teachers, assessed at the beginning of first grade, predicted math anxiety of these teacher's students at the end of first grade. 

At the end of the lecture, Ansari talks about several myths in education that are not related to math, like the erroneous myth about learning styles.

A very interesting lecture!

 

Children with developmental dyscalculia have more difficulty with subtraction than addition. Why? And what is unique in their brain activity?

Rosenberg‐Lee, M., Ashkenazi, S., Chen, T., Young, C. B., Geary, D. C., & Menon, V. (2015). Brain hyper‐connectivity and operation‐specific deficits during arithmetic problem solving in children with developmental dyscalculia.Developmental science, 18(3), 351-372.‏http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4320038/

 In this research done with 7-9 year old children, the authors compared addition and subtraction abilities of children with developmental dyscalculia (DD) and typically developing children (TD).

Children diagnosed as DD scored at or below the 25^th percentile on the Numerical Operations subtest of the Wechsler Individual Achievement Test – Second Edition; WIAT-II.  Children diagnosed as TD scored at or above the 75^th percentile on this test.  Children in both groups had a FSIQ of 80 or above, and scored at or above the 25^th percentile  on  the Word Reading subtest of the WIAT-II.  Sixteen  DD and 20 TA children participated in the study.

The fMRI experiment consisted of addition and subtraction problems which were either simple or complex.   Each calculation trial lasted five seconds.  In the Complex addition task, participants were presented with an equation involving two addends and asked to indicate, via a button box, whether the answer shown was correct or incorrect (e.g. ‘3 + 4 = 8’). The first operand ranged from 2 to 9, the second from 2 to 5 The Simple addition task was identical except that one of the operands was always ‘1’ (e.g. ‘3 + 1 = 4’). In the Complex subtraction task, the first operand ranged from 3 to 14 and the second operand from 2 to 5. In the Simple subtraction task, the first operand ranged from 2 to 14 and the second operand was always ‘1’.

Here I'll focus on a few findings that are of interest for me, and not on all findings of this study.

·         DD children solved addition tasks with the same level of accuracy as TD children, but were slower.  DD children were significantly deficient with the subtraction tasks, in comparison with the TD children. Children with DD failed to respond in the allotted time in a large proportion of trials during the subtraction task. However, for trials in which they made a response, accuracy in the DD participants was relatively high at 75.4%, suggesting that DD participants were actively engaged in the task but were unable to solve many of the problems with the same fluency as their TD peers.

·         Timed trials exacerbate the difficulties children with DD have when solving subtraction problems consistent with their difficulties on timed number fact and story problems. The latter are typically due to use of slower and more effortful counting strategies to solve the problems, as contrasted with direct retrieval of the answer in children without mathematical difficulties. This pattern may be exacerbated with subtraction because, unlike addition, subtraction problems are not commutative (e.g. 4 − 3 ≠ 3 − 4), which makes memorization of answers more difficult and thus results in less fluent problem solving for all students.

·         Children with DD engage multiple fronto-parietal circuits differently from TD children. Children with DD may require greater engagement of these circuits, even while achieving only weaker levels of performance. Alternatively, greater engagement of these circuits may result in the activation of problem-irrelevant information that in turn disrupts problem solving. The latter view is consistent with behavioral studies that show the intrusion of problem-irrelevant information into working memory when children with DD attempt to retrieve arithmetic answers from long-term memory. 

·         Hyper-connectivity, rather than gross under-activation, is the primary neural source of problem solving difficulties in children with DD.  DD children showed hyper-activation on both addition and subtraction problems in multiple frontal, parietal and visual areas. Children with DD showed especially high levels of hyper-activation in parietal cortex for both correctly and incorrectly solved subtraction problems.

·         There is a network of brain regions that show aberrant responses during arithmetic problem solving.  Arithmetic deficits in DD are unlikely to be localized to a single brain region.  Rather, both localized processing deficits in multiple brain areas as well as the coordination between multiple brain circuits are impaired in DD. These conclusions are consistent with the proposal that most neurodevelopmental disorders and learning disabilities arise from diffuse disruptions and aberrant connectivity between regions rather than focal lesions.

Mathematics and cognitive abilities part 3: dyscalculia and learning disability manifested in arithmetic

I'm happy to present the third part of the presentation series "mathematics and cognitive abilities" dealing with dyscalculia and learning disability manifested in arithmetic.

The presentation is here.

The two previous presentations in this series, dealing with the development of arithmetic skills and links between cognitive abilities and math, are found in the right hand column of this blog, under the caption "cognitive abilities and math".

Enjoy!

Quantity and time, time processing in dyscalculia

Cappelletti, M., Freeman, E. D., & Butterworth, B. L.

(2011).Time processing in dyscalculia. Frontiers in psychology, 2.‏

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3243078/

How do we judge the length of time of events (without looking at our watch…)? Of events that last a few seconds? We probably conduct an inner counting of the number of "seconds" the event lasted.  This means that we use numbers to measure time.

This is obvious when we learn to tell time (especially with an analogical watch).  In order to be able to tell time we have to master a few arithmetic concepts ("half past four"; "a quarter to nine"; "a quarter past seven") and to know the "time system" (there are sixty seconds in a minute, sixty minutes in an hour, 24 hours in a day), that in some respects is similar to the base 10 number system.

This interesting study looked into aspects of these phenomena.  Twelve dyscalculic adults and 22 non-dyscalculic adults participated. 

It seems to me, that the assigning of participats to the dyscalculic and non-dyscalculic groups wasn't optimal.  This might have, in my opinion, weakened the results.

How were participants deemed dyscalculic?  They had to satisfy four criteria:

·         A score in the Dyscalculia Screener that is one standard deviation or more below average (more on this test here http://beyondiq.blogspot.co.il/2014/07/dyscalculia-screener-computerized-test.html  ).  They did satisfy this criterion.

·         An average IQ score (at least).  They satisfied this criterion as well.

·         A low score in an arithmetic achievement test (GAD, Graded Difficulty Arithmetic Task).   A look at the data reveals that eight of the twelve dyscalculic participants had a "dull average" score in this test.  A dull average score is not a score that is significantly below average.  The average score of all twelve participants in this test was dull average.

·         Deficient functioning in the arithmetic subtest of the WAIS-R.  A look at the data reveals that out of twelve participants, four scored between 8 and 9 and another had a score of 7.  Since the subtest's average is 10 and the standard deviation is 3, these five participants did not satisfy this criterion.

The authors write that the 22 participants in the control group were not given the dyscalculia screener.  They don't supply the control group's data on the three other criteria.

Under these limitations I will consider the results with caution.  The questions that were asked in this study are interesting in themselves.

The authors first asked the participants questions about everyday situations involving time estimation or knowledge about time:

An example of questions that require time estimation: How much time is needed to make a cup of tea?(they are English…)  How much time is needed to fly from London to New York? (this question is influenced by general information knowledge).

An example of a question that requires exact calculation:  If the time is now 10.35 p.m., what time will it be in 2 h and 50 min?

An example of a question that requires  knowledge about time facts: How many hours are in a day?

An example of a question that requires time comparison:  What time is the latest: 11:45 or 15:30?

There was no difference between the dyscalculics and the control group on questions about time estimation, time comparison and time facts.  Dyscalculics performed significantly worse than controls on questions requiring exact time calculations.

After this phase, the authors looked into the influence of numerical stimuli on the perception of time.  For this purpose the participants performed two tasks.  I'll refer here to one of them:

The participants saw the digit  5 projected on a computer screen for a certain length of time. Then a second digit was projected for a certain length of time.  The second digit could have been 1 or 9.  The participants had to decide whether the second digit was projected for a longer or a shorter period of time than the first digit.

We already know that  children who are not dyscalculic display a numerical stroop effect.  The numerical stroop task involves making a fast decision about the physical size of digits (which digit is physically larger?).  When there is congruence between the digits' value and physical size (5  3) performance of typically developing children is faster than when there is incongruence between the digits' value and physical size (5  3).  This effect does not happen with dyscalculic children.  The reason for that may be that dyscalculic children don't link numbers with their quantitative value.

The numerical stroop effect  indicates  that we link quantitative value with physical size.  Do we likewise link between quantitative value and time perception?

This leads us to the hypothesis that participants in the control group would think that  "1"  is projected for a shorter period of time than "5" was (disregarding the actual situation).  That's because the low value of  1  would affect the subjective perception of time.

We may also hypothesize that participants in the control group would think that "9" is projected for a longer period of time than "5" was (disregarding the actual situation).   That's because the higher value of "9", compared to "5", would affect the subjective perception of time.

We may also hypothesize that this effect will not appear with dyscalculic participants.  They will not perceive the digit 1 as projected for a shorter period of time relative to the digit 5, and will not perceive the digit 9 as projected for a longer period of time than the digit 5.  That's because dyscalculics don't link digits with their quantitative value.  When digits or numbers are not linked with their quantitative value, it's hard to take the next step and link the quantitative value with perceived time length.

The results indeed show that the perception of time of dyscalculic participants was not affected by the quantitative value of numbers.  The perception of time of control subjects was affected by the quantitative value of numbers.  The control group participants perceived the number 9 as projected for a longer period of time than the number 5.  They also perceived the number 1 as projected for a shorter period of time than the number 5, but as projected for a shorter period of time than the number 9.

The meaning of these findings may be that we link between quantitative value and subjective time perception.  People with dyscalculia apparently don’t make such a link.  More research is needed with larger groups and stricter group criteria in order to confirm these findings.

Dyscalculia: Characteristics, causes, and treatments

Price, Gavin R., and Daniel Ansari. "Dyscalculia: Characteristics, causes, and treatments." Numeracy 6.1 (2013): 2.‏

http://scholarcommons.usf.edu/cgi/viewcontent.cgi?article=1112&context=numeracy

While preparing the third presentation in the series "mathematics and cognitive abilities" I came upon this paper.  It is written very clearly, and I highly recommend it.  Here are some interesting findings from this paper:

 Dyscalculia characteristics:

·         Poor retrieval of arithmetic facts from long term memory.  By third grade, typically developing children have developed a store of arithmetic facts in memory, from which they can quickly recall the solution to a given problem.  Children with dyscalculia, on the other hand, typically fail to develop such fluent fact-retrieval mechanisms, continuing to employ procedural strategies long after their typically developing peers have progressed to memory-based  strategies.  One of the immature procedural strategies children with dyscalculia use is "count all", in which the child displays two addends on his fingers or by drawing lines, and then counts the fingers or lines from 1.  As an indicator of the severity of the fact-retrieval deficit in children with dyscalculia, typically developing children have been found to recall an average of three times as many arithmetic facts as those with dyscalculia.

·         Poor number sense.  This difficulty is proven by research finding such as:

o   Israely scholars Avishai Henik and Orly Rubinsten  reported a lack of facilitation from numerical information in  children with dyscalculia during a numerical stroop task.  In this task, the child is presented with two digits differing in physical size (e.g. 3 5 or 3 5).  The child determines as fast as he can which digit is physically larger.  When there is congruence between digit physical size and numerical value  (like this:   3 5), reaction time in typically developing children is faster than when digit size and value are incongruent.  Children with dyscalculia don't show this effect.  It's not clear whether the reason for this is that the underlying semantic representation of quantity is impaired in children with dyscalculia, or whether they have a deficit in the link between the semantic representations and their symbolic referents (i.e., Arabic digits).

·         Children with dyscalculia have slower reaction time to determine which of two digits (having the same physical size) has a larger numerical value.

·         Children with dyscalculia also have a qualitatively different “distance effect”.  The distance effect refers to the behavioral phenomenon that, as the distance between two numbers being compared decreases (e.g., 2 – 9 versus 7 – 9), reaction times and errors increase. In other words, numbers that are closer together are harder to compare than numbers that are further apart. The numerical distance effect (NDE) is taken by many researchers to reflect the integrity of the underlying representation of numerical magnitude along a “mental number line” with a larger NDE indicating a less-precise or more noisy representation. In support of this idea, the NDE decreases in size over the course of development, suggesting an ontogenetic increase in the precision of the number sense. Children with DD have been shown to have larger NDEs than typically developing children, in much the same way that typically developing children show a larger NDE relative to adults, suggesting that DD children may have a less-refined, immature representation of numerical magnitude compared to their typically developing peers. Recent evidence suggests that the magnitude of the developmental delay in the precision of this representation may be on the order of five years, with DD children showing numerical-representation precision equivalent to typically developing children five years their junior.

Numerosity – Genetic? Environmental? Predetermined and fixed? Changeable? There's room for optimism.

     

As I mentioned in an earlier post (about Dyscalculia screener), numerosity is the ability to represent, estimate and manipulate quantities.

This new article sheds more light on numerosity:

Why do we differ in number sense? Evidence from a genetically sensitive investigation.

  M.G. Tostoa, S.A. Petrill , J. Halberda , M. Trzaskowski , T.N. Tikhomirova , O.Y. Bogdanova ,R. Ly , J.B. Wilmer , D.Q. Naiman, L. Germine , R. Plomin, Y. Kovas. 

 Intelligence 43 (2014) 35–46

http://ac.els-cdn.com/S016028961300192X/1-s2.0-S016028961300192X-main.pdf?_tid=597f97e2-0d73-11e4-8029-00000aacb35e&acdnat=1405575160_5e8e1932bcc2f055dfab4594f8449661

Apparently, numerosity is a developing ability.  Six months old babies can distinguish between arrays of items of 4 from 8, and 8 from 16 (ratio 1:2).  Nine months old babies can discriminate between displays of 8 and 12 items (ratio 2:3).  Three year olds can distinguish between ratios of 3:4 and six year olds – of 5:6.  Adults can discriminate between arrays with ratios of 9:10 (!).  The ability to discriminate between quantities improves till the age of 30 and then begins to deteriorate.

The article describes a large scale research about numerosity with 16 year old twins.  The twins were presented with arrays of yellow and blue dots for a very brief time.  They had to decide whether there were more yellow or blue dots.  It was found, that numerosity correlates 0.3 with math achievement, 0.25 with processing speed, 0.22 with visual spatial working memory, 0.27 with nonverbal ability and 0.2 with language.

It was also found, that individual differences in numerosity among the 16 year olds were only modestly (32%) influenced by genetic factors.  Most of the variance was explained by environmental effects. 

Individual differences in the ability to distinguish between quantities appear as early as in 6 months old babies.  The authors suggest  that  in infancy, individual differences in numerosity are heavily influenced by genes, but in later development, numerosity is more affected by exposure to math stimuli, interest in activities with numbers and the amount of practice in math activities.  That means that numerosity can be trained and improved.  Even in infancy, 6 months old babies that received quantitative stimuli that were both visual and auditory, were able to distinguish between quantities with a ratio usually found in 9 months old babies.  It's possible that the multisensory information the babies received improved numerosity. 

It's also possible, that children born with poor numerosity and/or low general cognitive ability are less inclined to engage in activities with numbers and thus don't give their numerosity a chance to develop  to its full potential. 

That’s why it's especially important for children with poor numerosity not to give up and be exposed to math experiences and games.  This will help them train and improve their numerosity.