Cosets, Lagrange's theorem, order and index of subgroups. Quotient groups. Problem Set 2 , Solutions.

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Homomorphism theorems. Partitions and equivalence relations. Direct products. Problem Set 3 , Solutions. Semidirect products. Smith normal form and structure of finitely generated abelian groups. Symmetries of regular polyhera. Group actions, orbits, stabilizers; Burnside's lemma. Automorphisms, class equation, solvability of p-groups.

Problem Set 4 , Solutions. Rings and fields. The participants in the present study included students from five primary and four secondary state schools in Zagreb. With respect to primary school students, all state primary schools in Croatia have the same curriculum, so their students have comparable experiences with algebra education. With respect to secondary schools, we tested students from two gymnasiums general education and foreign language type schools and two technical secondary schools.

These schools were chosen to represent the average secondary school population in Zagreb mostly preparing for university studies. Specifically, graduates from the two gymnasiums included in the present study typically continue their education at university, typically studying non-mathematics or science related majors. In comparison, graduates from the tested technical schools often continue their education majoring in technical fields.

Students from gymnasiums that specialize in natural sciences and mathematics were not included in this study. The participants in the present study included students from the seventh grade of primary school age 13—14 years to the second grade of secondary school age 16—17 years. Hence, our sample included the students of four age groups, i. The d2 Test of Attention Brickenkamp, , was also administered, but the data were not analyzed in the present study. In each trial, simple equations consisting of three elements numbers or letters were presented in the centre of the visual field.

The presented numbers and letters were black, displayed in 24 pt size Ariel font on the white background. Simultaneously with the equation, a potentially correct or incorrect answer was presented below the equation.

- Algebra: Abstract and Concrete.
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The participants were asked to decide if the offered answer was correct or incorrect. Within all presented equations, a and b stand for different letters and numbers which all appeared with the same probability during the experiment. The participants were tested during two school periods 45 min long.

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## Math 113: Introduction to Abstract Algebra—Summer 2015

On the same or on another day, students solved the computerized test of equation rearrangement and completed a post-measurement questionnaire in the computer lab. Before administering the equation rearrangement test, participants were familiarized with the task. They were instructed to respond as quickly as possible by pressing one of the two mouse buttons with their index and middle fingers, corresponding to correct and incorrect answers, respectively.

Prior to experimentation, the participants performed a training block consisting of 6 equations equivalent to those used in subsequent experimental trials. During both practice and experimental trials, each equation was presented until the participant responded, up to a maximum of 30 s. If the participant did not respond within 30 s, the equation disappeared from the screen and another 30 s were available to give an answer. After each response, the next equation was presented after a delay of 1 s.

No feedback was given to the participants. During the experiment, the participants were presented with the three previously described types of equations, which were randomized across four blocks. Each block consisted of 15 equations of each equation type, amounting to an overall of 45 presented equations per block. Two blocks contained equations with numbers, while the other two blocks contained equations with letters symbols. Equations in the first and third blocks contained numbers while those in the second and fourth blocks consisted of letters.

The participants could take a break between blocks if needed. After having finished the computerized test, the participants completed a questionnaire designed for assessing their strategies during equation solving. While responding to these questionnaires, the participants described how they solved each equation type and ranked them by difficulty. In addition, they indicated whether their response depended on the type of the offered answers, and whether they changed their problem solving strategies during the time course of the experiment.

For each participant and each condition, reaction time and accuracy were evaluated. Only correct responses were included in the analysis of RTs. Inverse efficiency was also calculated as the ratio of reaction time and accuracy Townsend and Ashby, Lower values on this measure indicate higher efficiency on a particular task. Inverse efficiency is used to account for the speed—accuracy tradeoffs, and we used it as a measure of task difficulty. To determine the effects of age, gender, level of abstraction, repetitions and equation type, a two-way repeated measures analysis of variance ANOVA on accuracy and RTs was conducted.

Repeated-measures post hoc tests using Bonferroni adjustment were used to further assess the differences between different conditions. Hence, different categories reflect different student equation solving strategies, some of which were correct, and some incorrect. Some participants used more than one strategy, and were accordingly assigned to two or more categories. Each participant was assigned to concrete, rule-based or mixed concrete and rule-based group.

Female were conducted to compare the mean accuracy and RTs. On average, girls were more accurate than boys, and the participants in the 7th grade of primary school were less accurate than those in the 1st and 2nd grade of secondary school, while those in the 8th grade were less accurate than the students in the 2nd grade of secondary school.

A Accuracy percentage of correct responses and B RTs for the participants in the 7th and 8th grade of primary school, and the 1st and 2nd grade of secondary school, separated for male and female participants. Boys were faster in equation solving in the first grade of secondary school, whereas girls were faster in the second grade. In the 2nd grade of secondary school there was no statistically significant difference in the accuracy of solving equations with numbers and letters. A Accuracy percentage of correct responses and B RTs for the participants in the 7th and 8th grade of primary school, and the 1st and 2nd grade of secondary school, separated for the equations with numbers and equations with letters.

RTs deceased with age, the participants in the 2nd grade of secondary school were the fastest, and the students in the 1st grade of secondary school were faster than those in the 8th grade. A Accuracy percentage of correct responses and B RTs for the participants in the 7th and 8th grade of primary school, and the 1st and 2nd grade of secondary school, separated for the different equation types the A, B and C equations.

Primary school participants solved the A equations faster than the B and C equations, while the secondary school participants were the slowest in solving the C equations. A Accuracy percentage of correct responses and B RTs for the participants in the 7th and 8th grade of primary school, and the 1st and 2nd grade of secondary school, separated for the first and second block.

We categorized their answers and divided them into two groups — concrete strategies and rule-based strategies. Thus, b is smallest and a is biggest. Then we get x by dividing a by b. This strategy gave correct responses for the A and B, but not for the C equations. The majority of younger participants from primary school used concrete strategies, whereas participants from secondary school mostly used more abstract, rule-based strategies. Some participants used both concrete and rule-based strategies.

Participants who used both concrete and rule-based strategies typically used a concrete strategy to solve the C equations. C equation was used.

### Journal of Symbolic Logic

If we adopt inverse efficiency as a measure of task difficulty Townsend and Ashby, , the results suggest that the C equations were the most difficult. There was no statistically significant difference between the A and the B equations. A Inverse efficiency on equations with letters for the participants in the 7th and 8th grade of primary school, and the 1st and 2nd grade of secondary school, separated for the different equation types the A, B, and C equations.

B Proportion of equation types ranked as the least difficult and the most difficult by the participants in each grade. Participants ranked different equation types by difficulty in the questionnaires. Three participants thought that equations with multiplications A type are easier than equations with division B and C. Eight participants did not provide an answer to this question.

Most participants reported that the A equations were the easiest. Most participants agreed that the C equations were the most difficult. However, our data indicate that students become more efficient, i. With respect to gender differences, the girls in our sample were on average more accurate in equation rearrangement than boys, while no significant differences in their speed were revealed.

This finding is in disagreement with a common belief that boys are better in mathematics than girls which is based on reports that boys outperform girls on standardized tests like SAT e. However, most studies report no differences between boys and girls on algebra assessments e. Edit comment for material Algebra: Abstract and Concrete.

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Become a Partner. Material Detail. Algebra: Abstract and Concrete The book provides a thorough introduction to "modern'' or "abstract'' algebra at a level suitable for upper-level undergraduates and beginning graduate students.