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!

 

OFF SUBJECT – OR MAYBE NOT?

Two great sites that show how mathematics and physics can be taught in a truly exciting way:

 http://www.numberphile.com/

English mathematicians  present mathematical anecdotes in short, well paced and visually styled videos.  Their enthusiasm for math is catching (as attested by my 14 year old son).

http://what-if.xkcd.com/

Physicist Randall Munroe gives serious and entertaining answers to impossible questions.  Fascinating!   I also recommend looking at his main site http://xkcd.com/

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.

Cognitive predictors at kindergarten of first grade math achievement

Passolunghi, Maria Chiara, and Silvia Lanfranchi. "Domain‐specific and domain‐general precursors of mathematical achievement: A longitudinal study from kindergarten to first grade." British Journal of Educational Psychology82.1 (2012): 42-63.‏

A wide scale research looking into the links between different cognitive abilities assessed at kindergarten and math achievement at the end of first grade.  Seventy children participated in this study.

 At the beginning of the last year of kindergarten (children aged 5.2 years) the children were  administered a battery of cognitive tests (Vocabulary and Block Design from the Wechsler , and tests assessing working memory, verbal and visuo-spatial short-term memory, phonological ability, verbal counting ability, and processing speed).   

At the end of the last year of kindergarten (children aged 5.8 years)    numerical competence was assessed (by tests assessing concepts of comparison, classification, one-to-one correspondence, seriation, the use of number words, counting and general understanding of numbers).

At the end of the first year of primary school (children aged 6.6 years)    mathematical achievement was assessed (by tests assessing spatio-temporal analysis, seriation, and classification (i.e., the ability to order spatio-temporal sequence and to order objects from smallest to largest), acquisition of the concept of natural numbers (i.e., to be able to position numbers on the ‘number line’), understanding of basic arithmetic operations (+, −), and the capacity to locate objects in space).

The paper adds some background on the assessed cognitive abilities:

Phonological awareness and mathematics  - one study found a significant relationship between phonological awareness (rhyme detection and initial consonant detection) assessed at 4 and 5 years of age and mathematical competencies as judged by teachers at the beginning of first grade.  Another study found that phonological awareness scores (rhyme tasks) assessed at 5 years of age predicted reading as well as arithmetic abilities 1 year later. Yet another study reported that phonological processing (measured by rapid digit naming, first sound matching, and last sound matching) was a unique determinant of fact fluency but did not predict other aspects of math performance (e.g., story problems).  There are also studies showing that phonological awareness does not predict mathematics ability and is only a predictor of reading ability.

Counting ability  - A verbal counting task (e.g., repeating as fast and accurately as possible a sequence of numbers) is a mixed measure that requires numerical knowledge of number sequence as well as speed of articulation and, therefore, the activity of phonological loop.  Counting is linked to phonological abilities, since children learn to count at 2 years of age without employing number words to describe the quantities.

 Processing speed - A number of studies showed that children with poor mathematical abilities have poor performance in processing speed.    Processing speed was found to be the best predictor of arithmetic competence among 7-year olds.  One study found that  fourth grade children with arithmetic learning disability suffered from a deficit in the speed of activating both numerical and non-numerical information from long-term memory.  Speed of retrieval of information from long-term memory  is a correlate of arithmetic skills in second to fifth graders.    In a group of third grade children, processing speed was a significant predictor of arithmetic ability when assessed by visual matching and retrieval fluency tasks.

How were the cognitive abilities assessed in this research?

 Processing speed was assessed with the WJ III visual matching test.   The child was asked to locate and circle two identical symbols that appear in a row of six symbols.  In speed pattern comparison task  60 pairs of patterns were presented, and the child had to decide as quickly as possible whether the two

patterns of each pair were identical.

Verbal Memory span was assessed with a forward digit and word recall tasks.  Visuo-spatial  memory span was assessed with a task in which   the child was shown a path taken by a frog on a 3 × 3 or 4 × 4 chessboard.    The child had to   reproduce the experimenter’s moves.  Another task was the Corsi block task that requires children to watch the experimenter pointing to a series of blocks that are arranged randomly on a board, and then to point to the blocks in the same order.

Working memory was assessed with a list of two to five words of which  the child had to  remember the first word and to concurrently perform a dual task.   Digit backwards recall task was also administered.

For the word fluency task  the child was asked to generate   as many words as possible in 1 min beginning with a particular letter.  For the semantic fluency task the child was   given a minute to name as many items as possible from particular semantic categories: animals, objects, and occupations.

Phonological awareness was assessed by word and pseudo-word repetition, recognition and repetition of the first or last sound of the word or non-word said aloud by the experimenter   in words and pseudowords, phonological analysis and synthesis of words.

For the verbal counting task, the child was asked to count as fast as possible from 1 to 10 three times.

Counting speed was assessed with   a task requiring the child to count objects as fast as possible.

Results:

Numerical competence as measures at the end of kindergarten was significantly correlated s 0.45 with processing speed, 0.42 with working memory , 0.42 with phonological skills and 0.46 with verbal counting and counting speed.

Mathematical achievement as measured at the end of first grade was significantly correlated 0.35 with vocabulary, 0.34 with processing speed, 0.32 with verbal counting and counting speed, and 0.42 with numerical competence as measured at the end of kindergarten. 

Path analysis model showed that the strongest predictors of numerical competence at the end of kindergarten are processing speed and working memory.  The strongest predictors of mathematical achievement  at the end of first grade are numerical competence at the end of kindergarten, processing speed and vocabulary.

   Processing speed  may facilitate the execution of several tasks, such as counting speed, as well as recall of partial results of problem solutions, preventing the decay of the information that must be processed. Phonological awareness and counting skills were highly correlated with processing speed in this research.  This might mean that the faster a child can process stimuli, the faster he/she can access  phonological information about numbers and counting.

Visual processing and mathematics

I continue to search for information about the relationship between cognitive abilities and math performance.  The more we know about it, the better we'll be able to link specific math difficulties to cognitive abilities that may be causing these specific difficulties.

Here are some of the things we know thus far about the relation between visual processing and math:

·         Visual processing is significantly and consistently linked with math in "older" children ("older" children probably refers to junior high school and high school students) (Flanagan et al, 2006).

·         Precision in placing numbers on a number line predicts future math achievement.  Some see . this ability as a measure of the number sense (Geary et al, 2012).

• Poor visuospatial skill is related with dyscalculia (Furlong et al, 2014)

• Visual processing as measured by the WJ3COG is not predictive of math achievement as measured by WJ3ACH . (McGrew& Wendling,  2010) .

Here is another study examining this relationship.

Tibber, M. S., Manasseh, G. S., Clarke, R. C., Gagin, G., Swanbeck, S. N., Butterworth, B. & Dakin, S. C. (2013). Sensitivity to numerosity is not a unique visuospatial psychophysical predictor of mathematical ability. Vision research, 89, 1-9.‏

http://www.sciencedirect.com/science/article/pii/S0042698913001491

More than 300 participants aged 6-71 were recruited to an experiment in the London science museum.  They completed a questionnaire about their age, sex, general and math education level, took a math test and performed a series of visuospatial matching tasks.

Mathematical ability was assessed using a computer-based multiple-choice test adapted from the Mathematics Calculation Subtest (WJ-Rcalc) of the Woodcock–Johnson III Tests of Cognitive Abilities.  Participants were presented with a series of problems that increased in difficulty from simple addition and subtraction through to multiplication and division of fractions and negative integers . Participants were given 30 s to respond to each problem.   The test was  terminated if the participant made a certain number of errors.  

Visuospatial processing was assessed by a visual matching test.  Two patches of oriented Gabor elements were presented to the left and right of screen centre (see figure below). Participants used a mouse to adjust the patch on the right (the test) so that it matched the appearance of the patch on the left (the reference) for a given parameter (orientation, size, numerosity or density).  As far as I know, this task does not fit any of the narrow CHC visual processing abilities.

The results showed, that older participants performed significantly better on all tasks ( visuospatial and math tasks).  This finding is surprising because I would expect a decline in performance above the age of 20-30.

It was also found that men significantly outperformed women on all tasks.

No correlation was found between the ability to match density and size and math achievement.

In participants older than 18, there was a significant correlation between the ability to match orientation and quantity and math achievement.  Such a tendency was also found in children but it was not significant.  The authors think that's because the children's sample size  was too small.

The correlation found between the ability to match quantities and math achievement is not surprising at all.  The ability to match quantities is essentially a math task (reflecting number sense) that is only performed visually.  The correlation found in this study and other studies between the ability to match quantities and math achievement is similar:  between 0.2 and 0.4.   Sensitivity to numerosity is a significant and reliable predictor of math scores.

How can we explain the correlation between the ability to match orientation and math achievement?  The authors ask this question but don’t answer it in a convincing way.  They say that there probably is a common system at the base of math and visuospatial tasks, and that people whose system is more sensitive (better) will tend to study high level math.

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.

Mathematics and cognitive abilities part 2: links between cognitive abilities and math (in light of chc theory).

The 2^nd of a three part presentation series about math and cognitive abilities is here. 

In this part we'll try to see the influence of cognitive abilities on math.  The presentation contains ideas for interventions when there are math difficulties resulting from various poor cognitive abilities.

This presentation will appear from now on in the right hand column of this blog, under "Cognitive abilities and math".

Part 1 of the series that deals with the development of arithmetic skills, is also found in the right hand column of this blog, under "Cognitive abilities and math".  I enclose it here too.

Part 3 is going to be about learning disabilities manifested in math and about dyscalculia.  This may take me a while to publish.

Dr. Philip A De Fina - the neuropsychology of mathematics

שווה להציץ:  מצגת מעניינת על מתמטיקה ויכולות קוגניטיביות מאת DR. PHILIP A DE FINA

Check this out: an interesting presentation about math and cognitive abilities by Dr. Philip A De Fina

[PPT]The Neuropsychology of Mathematics - Grasp-online.org

Cognitive - achievement relations: findings from a research done in 2010

McGrew and Wendling examined and synthesized studies that looked into the relations between CHC abilities and reading and math achievement.  Their research was published in this paper, available for free download:

CATTELL–HORN–CARROLL COGNITIVE-ACHIEVEMENT RELATIONS: WHAT WE HAVE LEARNED FROM THE PAST 20 YEARS OF RESEARCH. KEVIN S. McGREW AND BARBARA J. WENDLING. Psychology in the Schools, Vol. 47(7), 2010   

The research included 19 studies examining five or more CHC abilities in children.  Because most studies used the Woodcock Johnson (WJ) test, the findings presented here describe mainly the relations between cognitive abilities as measured in the WJ test and achievement as measured in the WJ test.  Most studies were done on the same sample – the national norm sample of the WJ test.

Basic reading skills were measured in this research by single word and non-word decoding, word recognition and linking sounds with symbols.  Reading comprehension was measured by multiple choice and open ended questions about texts and cloze procedures.  Basic math skills were measured by calculation tests ranging from basic math calculations to complex algorithmic computations.  Math reasoning was measured by word problems, number series, and application of math operations and concepts. 

The main findings, in my view (but not all findings) are presented here in an easy to understand visual way.

The results show that sometimes broad abilities are the best predictors, but sometimes narrow abilities are better predictors of achievement.  The results also show that if there is no specific need for an IQ score or a g score, it's not necessary to test the child with the whole intelligence test battery.  It's better to choose the broad and narrow abilities to be tested according to the referral question and to the child's age. 

The authors wonder why visual processing is not a predictor of math or reading achievement.  As for math, past research shows that skills like length estimation are core deficits in math learning disability.  As for reading, the ability to quickly map graphemes to phonemes is known as important for reading.  One possibility is that visual skills related to reading or math are not measured in intelligence tests that measure visual processing.  An argument the goes the other way around would be, that maybe the math tests used in this research don't measure math areas that draw heavily on visual processing (geometry or trigonometry, for example).

I have other questions regarding the fluid ability.  The authors write, that fluid ability tests that predict math achievement best, measure fluid ability through…reasoning with numbers and numerical relations (for example number series or number matrices tests).

Also, fluid ability (as measured by the WJ tests) was not a predictor of reading comprehension (as measured by the WJ tests).  This is odd.  How is it possible, that the ability for abstract reasoning, the ability to draw conclusions and understand consequences is not related to reading comprehension?  One possible explanation is that fluid ability is measured in the WJ test through non-linguistic tests that use visual or numerical stimuli.  It's possible that measuring fluid ability by tests that use verbal stimuli and require verbal abstraction would have helped to find the missing, obvious link between this ability and reading comprehension.

Learning disability definition steps, and math learning disability

A previous post discussed learning disability definition steps.  In order to implement step 3 we have to be familiar with findings about links between poor math functioning and cognitive abilities.

A few research studies looked into these links in the general population.  Some studies used the standardization sample of the WJ3 test.  This test has two parts:  a cognitive part (WJ3COG) which measures CHC abilities and an achievement part (WJ3ACH) which measures reading and math.

Studies which looked into the correlations between children's scores in math tests (in the WJ3ACH) and cognitive abilities, found fluid ability, crystallized knowledge, working memory (a narrow ability within short term ability in WJ3) and processing speed to be related to math achievement.  There are also studies that found that long term storage and retrieval is related to math achievement.

This paper (available for free in the net):

PROCTOR,  B, E., FLOYD R. G. AND SHAVER, R. B. CATTELL-HORN-CARROLL BROAD COGNITIVE ABILITY PROFILES OF LOW MATH ACHIEVERS. Psychology in the Schools, Vol. 42(1), 2005

assessed CHC abilities of children who have poor achievement in math.  Children who had low scores (one or more standard deviation below the mean) on the math tests of the WJ3ACH (but had an average or higher score on the reading tests of the WJ3ACH) participated in the study.  This group was compared to a control group of children that had average or higher scores on the math and reading tests of the WJ3ACH), 

The math tests included two test clusters.  The  math calculation skills cluster (problems in the four math operations, and application of math rules like commutative rule).  The math reasoning cluster included tests of verbal math problems, application of rules and axioms, relations, and math concepts.

Math calculation skills:  there was no difference in CHC abilities between the low math calculation skills group  and the average achieving group!  None of the CHC abilities of the low math calculation skills group was significantly low compared to the population mean.  

Math reasoning:  the low math reasoning group had significantly lower scores in cognitive abilities, in general, compared to the control group.  This might point to   group differences in g.

The fluid ability of the low math reasoning group was significantly lower that the population mean.  The crystallized knowledge of the low math reasoning group was lower than that of the control group but was not lower than the population mean.

Short term memory, processing speed and long term memory were not found in this study to be related to math calculation and to math reasoning skills.

It seems to me, that in order to receive more significant results, maybe it would have been better to choose children whose math performance is lower than the 7-10^th percentile   (not the 16^th percentile).  This is the group which is usually chosen in dyscalculia studies.  In addition, as the authors also write, it would have been better to include children who perform poorly both in math and reading.  Many children with math problems also have difficulties with reading.  Because this study included children with math problems only, it might have not covered the entire range of  poor math achievers.

This study found no pattern of low cognitive abilities among children with low math calculation skills.  It's possible that many children who are referred to us with poor math achievement are not learning disabled in math.  We know that poor math achievement can be a result of exclusionary factors like lack of experience, low motivation, anxiety, bad teaching or inappropriate math books.  I believe that exclusionary factors influence math more than reading.  Addressing exclusionary factors may solve the math problems of some children in this group.