COMPREHENSION & VISUALIZATION - Teaching Students to Solve Word Problems

ABSTRACT In this feasibility study the authors describe and evaluate a word problem solving instruction, based on the principles underlying instructional programs like Solve it! and schema-based instruction. This instruction is executed during a five-week intervention period in a group of four less successful second grade word problem solvers. The effectiveness of the word problem solving instruction is reported by means of students' performances on combine, change and compare problems before and after the intervention period, as well as by examining whether they executed the solution steps of the instruction correctly. This feasibility study provides important insights with regard to varying ways in which a word problem solving instruction can influence the solution strategies and performances of students who perform poorly on word problems. WORD PROBLEM SOLVING PROCESS Look at the following example of a word problem [Word problem example] "Mary has 9 marbles. She has 4 marbles more than John. How many marbles does John have?" Tim, a seven-year-old boy who is in the second grade of elementary school, has difficulties with solving word problems like the one that is given in the example above. While solving these word problems, Tim often uses an impulsive, superficial solution strategy. Significantly, he only focuses on selecting the presented numbers (9 and 4) and identifying the relational keywords (more than), which subsequently form the basis for his mathematical calculations. Tim's strategy often leads to an incorrect answer to the word problem. In this situation, Tim performed an addition operation where a subtraction operation was required, that is 9 + 4 = 13 instead of 9 -4 = 5. The incorrect answer is not the result of a lack of calculation ability, but a result of a problem with deeply and correctly understanding the word problem text. Mathematical word problem solving plays a prominent role in the curriculum of contemporary approaches to teaching mathematics [1

Model method drawing acts as a double-edged sword for solving inconsistent word problems

Drawing bar diagrams has been shown to improve performance on mathematical word problems wherein the relational keyword is consistent with the required arithmetic operation. This study extends this by testing the effectiveness of bar diagram drawing for word problems with an inconsistent keyword-arithmetic operation mapping. Seventy-five fifth graders solved consistent and inconsistent word problems while encouraged to draw bar diagrams. For each word problem, we assessed problem type (consistent/inconsistent), performance (correct/incorrect), and bar diagrams (accurate/inaccurate/no drawing). Overall, bar diagram drawing was associated with increased performance on both consistent and inconsistent word problems, but the strongest benefits of drawing were found for inconsistent word problems. For inconsistent word problems, bar diagram accuracy was more clearly related to performance (accurate bar diagrams related to correct answers, but inaccurate ones to incorrect answers) than for consistent word problems. We conclude that bar diagram drawing provides an effective graphical support for solving inconsistent word problems

The consistency effect in word problem solving is effectively reduced through verbal instruction

In mathematical word problem solving, a relatively well-established finding is that more errors are made on word problems in which the relational keyword is inconsistent instead of consistent with the required arithmetic operation. This study aimed at reducing this consistency effect. Children solved a set of compare word problems before and after receiving a verbal instruction focusing on the consistency effect (or a control verbal instruction). Additionally, we explored potential transfer of the verbal instruction to word problems containing other relational keywords (e.g., larger/smaller than) than those in the verbal instruction (e.g., more/less than). Results showed a significant pretest-to posttest reduction of the consistency effect (but also an unexpected decrement on marked consistent problems) after the experimental verbal instruction but not after the control verbal instruction. No significant effects were found regarding transfer. It is concluded that our verbal instruction was useful for reducing the consistency effect, but future research should address how this benefit can be maintained without hampering performance on marked consistent problems

Model method drawing acts as a double-edged sword for solving inconsistent word problems

Drawing bar diagrams has been shown to improve performance on mathematical word problems wherein the relational keyword is consistent with the required arithmetic operation. This study extends this by testing the effectiveness of bar diagram drawing for word problems with an inconsistent keyword-arithmetic operation mapping. Seventy-five fifth graders solved consistent and inconsistent word problems while encouraged to draw bar diagrams. For each word problem, we assessed problem type (consistent/inconsistent), performance (correct/incorrect), and bar diagrams (accurate/inaccurate/no drawing). Overall, bar diagram drawing was associated with increased performance on both consistent and inconsistent word problems, but the strongest benefits of drawing were found for inconsistent word problems. For inconsistent word problems, bar diagram accuracy was more clearly related to performance (accurate bar diagrams related to correct answers, but inaccurate ones to incorrect answers) than for consistent word problems. We conclude that bar diagram drawing provides an effective graphical support for solving inconsistent word problems

Proliferative and anti-apoptotic fractions in maturing hematopoietic cell lineages and their role in homeostasis of normal bone marrow

Recent developments in clinical flow cytometry allow the simultaneous assessment of proliferative and anti‐apoptotic activity in the different hematopoietic cell lineages and during their maturation process. This can further advance the flow cytometric diagnosis of myeloid malignancies. In this study we established indicative reference values for the Ki‐67 proliferation index and Bcl‐2 anti‐apoptotic index in blast cells, as well as maturing erythroid, myeloid, and monocytic cells from normal bone marrow (BM). Furthermore, the cell fractions co‐expressing both proliferation and anti‐apoptotic markers were quantified. Fifty BM aspirates from femoral heads of patients undergoing hip replacement were included in this study. Ten‐color/twelve‐parameter flow cytometry in combination with a software‐based maturation tool was used for immunophenotypic analysis of Ki‐67 and Bcl‐2 positive fractions during the erythro‐, myelo‐, and monopoiesis. Indicative reference values for the Ki‐67 and Bcl‐2 positive fractions were established for different relevant hematopoietic cell populations in healthy BM. Ki‐67 and Bcl‐2 were equally expressed in the total CD34 positive blast cell compartment and 30% of Ki‐67 positive blast cells also showed Bcl‐2 positivity. The Ki‐67 and Bcl‐2 positive fractions were highest in the more immature erythroid, myeloid and monocytic cells. Both fractions then gradually declined during the subsequent maturation phases of these cell lineages. We present a novel application of an earlier developed assay that allows the simultaneous determination of the Ki‐67 proliferative and Bcl‐2 anti‐apoptotic indices in maturing hematopoietic cell populations of the BM. Their differential expression levels during the maturation process were in accordance with the demand and lifespan of these cell populations. The indicative reference values established in this study can act as a baseline for further cell biological and biomedical studies involving hematological malignancies