It is considered by some researchers e. This contrasts with Baroody, Feil, and Johnson who define conceptual knowledge as being knowledge about facts, generalisations and principles, but claim that there is no requirement for the knowledge to be richly connected. Rather their research, and the research of others e. Again, it seems necessary to reflect on the qualities of knowledge rather than to just adopt a superficial and sometimes unconsidered, definition of this type of knowledge.
Pesek and Kirshner studied six fifth-grade U. Other mathematics education researchers have written about this issue. Anthony and Walshaw noted that it is important for teaching to move beyond the use of procedural rules and that, as students are encouraged to make sense of the underlying mathematics through realising the connections between ideas, they become less focused on purely finding the answer and are more inclined to think about why certain processes work.
Anthony and Walshaw , p. Similarly, Warren and English , p. Furthermore, Clements and Sarama recently discussed the importance of effective professional development for teachers based on the notion of learning trajectories. They appear to be alluding to, amongst other points, the importance of teachers seeing and understanding the mathematical connections in order for them to enable their students to do so.
Recently, Downton, Russo, and Hopkins reported on a study of 25 students aged 11 and 12 years and their understanding of handling zeros in multi-digit multiplication. First, there are legitimate mathematical procedures, and, as has been noted on several occasions, the learning of procedures is an important part of mathematical understanding.
Multiplicative thinking is a complex set of ideas that underpin most of the mathematics learned beyond the middle primary years. The following definition, Hurst, is based on the work of Siemon et al. The capacity to understand this concept is built upon the base-ten property of place value, that is, that the values of the places increase in powers of ten from right to left Ross, This gives rise to the understanding that if a digit of a number is moved one place from right to left, it becomes ten times bigger.
If a digit is moved one place from left to right, it becomes ten times smaller. The capacity to understand, generate, and use extended number facts is also linked to this idea. The theoretical framework for the study was developed from the connections between these ideas Figure 1. Theoretical framework for the study Table 1. Responses to calculations for the four quiz questions Question Frequency correct Percentage correct 1. A Multiplicative Thinking quiz was administered to the students as a whole-class exercise during the final stages of the school year for each jurisdiction.
The quiz contained 18 questions on different aspects of multiplicative thinking. Only data generated from four of the quiz questions are considered here because those four questions were relevant to the specific research question. The quiz questions concerned are as follows: 1. Work out the answer to 16 x Explain what happens when you multiply a number by Explain what happens when you divide a number by 10 3.
What is the answer to 1. Explain or show how you did it. With regard to the calculations of correct answers, the results are shown in Table 1. When considering the age of the students involved and the attention to multiplication that is stipulated in the curriculum documents, the results shown in Table 1 are lower than what could reasonably be expected.
When breaking down the data to year levels, the proportion of correct responses for Question 1 ranged from This pattern, that the Year 6 students performed better than the Year 5 students, was repeated for the other three questions with the difference being more accentuated with Questions 3 and 4. For Question 3, correct answers were given by Summary of responses to the explanations for the four quiz questions Question Frequency correct Percentage correct 1.
For Question 4, the results were While this is interesting, and a generally expected outcome that Year 6 students would do better than Year 5 students, it is not the main issue here. The main focus of the paper is not the capacity of the students to calculate though this in itself is not trivial , it is the extent to which students can explain what occurred when the numbers were multiplied or divided by a power of ten.
Responses to the second part of each question, regarding explaining what was happening in the completion of the calculation, are summarised in Table 2. No student gave an appropriate explanation while making an error in the calculation. Although only a small percentage of students could explain in a conceptual way what happened to the digits, the explanations of those who did so were strong. Elijah said that, [for 1.
Some students described it in different ways but still acknowledged that the digits moved to different places. Some students did a drawing to show a place value chart. Samples from Izzy, Cassie, and Freddie are included in Figure 2. Samples from Izzy 2a , Cassie 2b , and Freddie 2c Figure 3. Sample from Adam All of the samples above can be described as showing some conceptual understanding of the processes of multiplication and division. Of the participant students, 49 provided a correct or appropriate explanation for at least one of the four questions.
Only five of them did so for each of the four questions. It is interesting that most of the students who provided a strong conceptual explanation for one or more question reverted to a procedural response for another question. Two such students were Ryan and Andrew. I had to divide by 10 which means my answer had to be in tenths so 1. Andrew responded in this way - [For 1.
Some students demonstrated partial understanding or expressed their thinking in less elegant ways. Another student, William, also gave a procedural answer followed by a conceptual answer.
Samples from William Figure 5. This also shows partial or developing understanding. The earlier cited examples from Pete and Charlie are conceptually stronger because they talk about moving the digits to a different place and the zero being added as a place holder. Ollie was the only student to use a vertical algorithm to multiply 1. Sample from Ollie Of the students in the sample, only 49 9. The development of adaptive expertise and flexibility: the integration of conceptual and procedural knowledge.
Mahwah, NJ: Erlbaum. In article. An alternative reconceptualization of procedural and conceptual knowledge. Journal for Research in Mathematics Education , 38 , Baykul, Y. Ankara, Turkey, Ani Printing Press. Bryan, T. The conceptual knowledge of preservice secondary mathematics teachers: How well do they know the subject matter they will teach? Unpublished Doctoral Dissertation.
Byrnes, J. Role of conceptual knowledge in mathematical procedural learning. Developmental Psychology , 27 , View Article. Canobi, K. Cognitive Development , 19 , Developmental Psychology , 39 , Engelbrecht, J. Undergraduate students' performance and confidence in procedural and conceptual mathematics.
International journal of mathematical education in science and technology, 36 7 : Faulkenberry, E. Gelman, R. Enabling constraints for cognitive development and learning: domain specificity and epigenesis. Gilmore, C. British Journal of Educational Psychology, 76, - Can children construct inverse relations in arithmetic? Evidence for individual differences in the development of conceptual understanding and computational skill.
British Journal of Developmental Psychology, 26, - Haapasalo, L. Two types of mathematical knowledge and their relation. Sormunen eds. Towards Meaningful Mathematics and Science Education.
University of Joensuu. Bulletins of the Faculty of Education 86, pp. Halford, G. Hallett, D. Individual differences in conceptual and procedural knowledge when learning fractions.
Journal of Educational Psychology , , Individual differences in conceptual and procedural fraction understanding: The role of abilities and school experience. Journal of Experimental Child Psychology, - Hiebert, J. Conceptual and procedural learning in mathematics. Karmiloff-Smith, A. Kerslake, D. Khashan, K. Conceptual and procedural knowledge of rational numbers for Riyadh elementatry school teachers. Journal of Education and Human development, 3 4 , Kilpatrick, J.
Adding it up: Helping Children Learn Mathematics. Mabbott, D. McGehee, J. Unpublished Doctoral Dissertation, University of Texas. Nunes, T. The relative importance of two different mathematical abilities to mathematical achievement. British Journal of Educational Psychology, 82, Resnick, L. Learning to understand arithmetic. Glaser Ed. Syntax and Semantics in Learning to Subtract.
Carpenter, J. Romberg Eds. Rittle-Johnson, B. Conceptual and procedural knowledge of mathematics: does one lead to the other? Journal of Educational Psychology , 91 , Developing Conceptual and Procedural Knowledge of Mathematics. Dowker Eds. Oxford University Press. The relation between conceptual and procedural knowledge in learning mathematics: A review.
Donlan Ed. Schneider, M. The developmental relations between conceptual and procdural knowledge: A multimethod approach. Developmental Psychology, 46 , Siegler, R. Conscious and unconscious strategy discoveries: a micro genetic analysis. Journal of Experimental Psychology: General , , Star, J. Reconceptualizing procedural knowledge. Journal for Research in Mathematics Education , 36 , Teaching strategies for improving algebra knowledge in middle and high school students NCEE Department of Education.
However, no child was above chance on Picture—symbol 85 98 92 LCD items but below chance on any of the conceptual knowl- Simple morphism 88 90 89 edge items. Thus, it appears that conceptual knowledge was a Order 35 78 57 Total 69 88 necessary but not sufficient condition for procedural knowl- Procedural knowledge edge in the case of LCD items. Thus, it was possible to perform well on overextending whole number multiplication to fractions multiplication items when some knowledge of the category of yielded the correct answer.
In addi- tion to providing the preceding statement regarding the dis- Discussion tinction between whole numbers and fractions, we illustrated The results of Experiment 1 tend to favor the dynamic inter- why the LCD was correct and why simply adding numerators action view rather than the simultaneous activation view. First, and denominators was wrong using concrete manipulatives. If it was found that an impoverished conceptual knowledge base the simultaneous activation position is correct, the distinc- did not underlie erroneous computations on LCD items.
If the dynamic interaction view is correct, able conceptual knowledge of the category of fractions. Thus, no differences should emerge because the manipulatives should prior studies that used interviews may have underestimated not add anything to the distinction. We also added a third ap- competence. The poor performance of the fourth graders on LCD items We used a pretest-training-posttest design modeled after simply reflected lack of instruction.
The poor performance of Paris, Newman, and McVey's study of memory strate- the sixth graders can be attributed neither to low conceptual gies. Paris et al. These children had been a memory strategy in addition to simply demonstrating it in taught the LCD method in both the fourth and fifth grades.
These authors found According to the simultaneous activation view, knowledge of that a single instructional session was sufficient to promote the the relative size of fractions should have produced a critic that acquisition and maintenance of several strategies in children detects the error of adding numerators and denominators, but who were in the production deficiency phase of memory devel- this error was still common among the sixth graders.
We attempted to mirror this approach by using fifth In fairness, however, the sixth graders were not taught com- graders who had been taught fraction computations over a 2- putations with fractions by their teachers in a meaningful way, week period when they were fourth graders.
We used an imme- as specified by the simultaneous activation view. In addition, diate posttest to assess relearning and a delayed posttest to the lackofwithin-grade correlations between conceptual knowl- assess maintenance. Additionally, we broadened our assess- edge and LCD scores may have resulted because our measure of ment of conceptual knowledge to include items that required children's conceptual knowledge of the category of fractions comprehension of equivalent fractions and items that required may have been too narrow.
We conducted an instructional children to create a numerical fraction given a verbal descrip- study in Experiment 2 to address these issues.
Experiment 2 Method We devised several instructional interventions designed to Subjects remediate errors when adding fractions. Children were recruited from two crimination-generalization processes involved in learning classrooms each of two public elementary schools located in east-cen- when and where to apply a procedure.
We suggest that the error tral Maryland. Discussions with the teachers and consultation of their of simply adding numerators and denominators reflects an texts revealed that, as fourth graders, the children had been taught overgeneralization of the procedure for adding whole numbers. By extending whole number addition to anything reducing, addition, subtraction, multiplication, and division. Chil- that is a "number like entity," children imply that they consider dren were tested early in the spring semester several weeks prior to whole numbers and fractions to be part of a single, intuitive being retaught fractions.
Their errors on order items further support this claim. We attempted to invoke a differentiation of this category into Design two subclasses whole numbers and fractions to eliminate chil- dren's tendency to overgeneralize the whole number addition A pretest-training-posttest design was used. Subjects were randomly procedure to fractions. On the "fractions are not like ordinary numbers" because one cannot first testing day, children were given a modified fractions knowledge add fractions in the same way that whole numbers are added.
Of booklet as a pretest. Over the next several days, groups of 2 to 4 sub- note is the fact that we intended to affect conceptual knowledge, jects were given instruction on the LCD method in a min session. We add also that Five days after training, children were again given the fractions knowl- the high scores on multiplication items arose from children edge booklet as a posttest.
In addition to condition, a second indepen- getting the answer right for the wrong reasons. It turns out that dent variable was used. WASIK the LCD method applied to addition of fractions and did not apply to me show you why it's wrong to just add the numerators and denomina- multiplication.
The other group half were not told this. Pilot testing tors. Each equal section was in the overgeneralization phenomenon. Addition- ally, colored plastic pieces thatfitover one section of a region were also used to instantiate a fraction. Word-symbol items were included to assess children's ability to should also think of this orange piece.
For example. What fraction of the pie did Joe eat? Finally, they were told that "you have to fraction corresponding to part-part relations e. For example, in one item, the exemplar was a circle that had one of with thefiveinitial sentences and were then told its four equal parts shaded. The correct response was a configuration of eight triangles with two shaded triangles. If I add 1 overeat Children had to choose from among four alternatives for each item.
That's Given the subtlety of the response, a "not given" alternative was not right 2 over cat. Now what about this one, one over dog plus 2 over provided. That's right, 3 over dog. Now what would I get when we add 1 over dog plus I over cat?
In total, there were three tokens for each offivekinds of conceptual Would we get something that's half-dog, half-cat? We would knowledge items: picture-symbol, word-symbol, simple morphism, get 2. When the bottoms are the multiple morphism, and order items. Thus, subjects could receive a same, we can just add the tops. But when the bottoms are differ- total score of 15 for conceptual knowledge.
In addition, there were the ent, we have to come up with a bottom that they both go into, right? Thus, sub- A dog is a kind of animal and a cat is a kind of animal. Well the jects could receive a total score of 6 for procedural knowledge.
When the bottoms are the same, we can just add the tops. But when the bottoms are different, we Instruction conditions. In each of the three conditions the material need tofinda number that they both go into.
Let's look at lfi plus! All conditions again.. First, all scripts began with the follow- ingfivesentences: After this analogy, children were walked through the four example When you add fractions, you have to use the Least Common LCD problems and then asked to solve the six practice problems on Denominator method. Fractions are not like ordinary numbers their own. In summary, the conditions were identical except that the such as 1, 10 and They make a mistake by adding meaning through an analogy with a well-established knowledge do- the 1 and 1 on top and the 2 and 3 on the bottom.
During this time told, "Remember, this special method called the least common denom- the experimenter solicited help from students with statements such as, inator has to be used when you add fractions.
When you multiply "What's the smallest number that both 2 and 3 go into? Is it four? The last should not be used when multiplying fractions.
Subjects in the distinction only condition were presented with the Results five introductory statements followed by the further line, "Let me show you how to add them correctly. They received no other instruction from the experi- sented in four parts: preliminary analyses, analysis of concep- menter. Newman-Keuls post hoc tests were used to interpret sig- sented with thefiveintroductory statements. Then they were told, "Let nificant main effects and interactions. However, the average math grade of subjects in the distinction-only condition was Distinction only 2.
The maximum mean score is 3. Therefore, for all subsequent analyses of the average difference between each child's predicted and actual posttest effects of instruction, we computed residualized gain scores for scores. Each student's residualized gain score was computed by sub- tracting his or her predicted posttest scores from his or her dualized gain scores for LCD items immediate and delayed actual posttest scores.
Such gain scores reflected how much posttest as the repeated measure. This analysis can be used to instruction adds to what would have been predicted by math determine whether certain forms of instruction increased post- grades and pretest LCD scores alone. The average resi- dualized gain scores for each instructional condition are shown Conceptual Knowledge in Table 3.
Although inspection of Table 3 reveals that the average residua- As can be seen, the fifth graders in this experiment performed lized gain score for subjects in the distinction-only group was similarly to the fourth and sixth graders in Experiment 1 on higher than that for subjects in the other conditions, post hoc picture-symbol, simple morphism, and order items.
The nonsignificance of the condition effect implies that Post hoc tests demonstrated four levels of per- the use of the manipulatives and the analogy added little to formance. At the highest level was the nonsymbolic simple simply making the distinction between the category of frac- morphism item.
Children performed significantly better on tions and the category of whole numbers. At the second level was word- The foregoing analysis should not imply, however, that LCD symbol and picture-symbol items, for which performance was performance was not enhanced by instruction. A repeated- better on these than on order or multiple morphism items. At measures analysis of covariance ANCOVA using math grade the third level was order items, and at the fourth was multiple as a covariate revealed a large and highly significant effect of morphisms.
This four-level pattern was also found at the post- trial,. Here, children performed test. Although performance on the immediate posttest was significantly higher than that on the Effects of Instruction delayed posttest, there was still substantial maintenance as a Procedural knowledge. We performed a repeated-measures result of the significant difference between the pretest and de- ANOVA using condition as the independent variable and resi- layed posttest means.
Recall that only half of the subjects were told not to use the LCD method on multiplication. We labeled Table 2 the treatment of being told not to use the LCD method when Experiment 2: Percentage Correct for multiplying "constraint given" and the other group "constraint Conceptual Knowledge Items not given.
Thus, when told that it was necessary to use the LCD multiple morphisms at the posttest. Hence, although Table 4 method when adding fractions, a substantial number of chil- implies such children should first do well on multiple mor- dren spontaneously overgeneralized this algorithm to multipli- phisms before doing well on LCD items, these seven children cation of fractions. No subject who was given the constraint seemed to skip this item. Nevertheless, the majority of children made this error.
Table 5 presents the correlations for acquiring procedural knowledge. We first determined between individual conceptual knowledge items and scores on whether a given child performed at an above-chance level on the fraction addition items at the pretest and at the immediate LCD items and on each of thefiveconceptual knowledge items.
We omitted the correlations involving We then determined the number of children who fell into one of simple morphism items because the children performed ex- the seven patterns shown in Table 4. If the dynamic interaction tremely well on this item. This high level of performance pro- view is correct, the majority of children should fall into one of duced a restricted range of scores.
Also shown are the correla- the patterns at both the pretest and posttest. Moreover, chil- tions between total scores for pretest conceptual knowledge dren should either fall into the same pattern at the pretest and omitting simple morphism scores andfractionaddition scores posttest indicating stability or move from one pattern in Table at the immediate and delayed posttests. If the simulta- As can be seen, both individual conceptual knowledge items neous activation view was correct, few children would fit the and total conceptual knowledge scores correlated significantly patterns shown, and many would learn the LCD method with- with addition scores at the immediate posttest.
However, by the out high levels of conceptual knowledge. One way to test these hypotheses is to perform one scalogram This pattern suggests that conceptual knowledge appears to analysis for pretest patterns and a second for posttest patterns. To rectify this prob- lem, they developed the Longitudinal Guttman Simplex analy- General Discussion sis that effectively combines two Guttman scalograms for 2 points in time.
This analysis computes CL, a consistency index The present set of experiments were designed to consider for longitudinal Guttman scales developed by Collins, Cliff and whether the simultaneous activation or the dynamic interaction Dent Conceptually, CL indicates whether the propor- view offers a more adequate account of the relationship be- tion of stability and growth patterns exceeds what would be tween conceptual and procedural knowledge in the case of expected by chance.
Although CL typically ranges between 0 mathematics. The simultaneous activation view specifies that and 1. Adherents to this view as- consistency. Even children with high conceptual knowledge still Immediate Delayed committed common procedural errors. Finally, Davis and Pretest Pretest posttest posttest McKnight and Rosnick and Clement found that relating algebraic formulae to real-world content did not facili- Picture-symbol -.
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