Exploring Learner Errors and Misconceptions in Algebraic Expressions with Grade 9 Learners Through the use of Algebra Tiles

Figures & data

Figure 1. A representation of algebra tiles

Figure 2. The manipulation of algebra tiles (Hall, 1999)

Table 1. Categorisation and description of errors (adapted from Ncube, 2016)

Categories	Description for errors	Examples of common errors
Conjoin error	Lack of understanding of like terms and unlike terms.	$3c+4d=7\mathrm{cd}$ $8a+6=14a$
Misapplication of rules	Adding instead of multiplying, incorrect application of previously learned procedures.	$4m\times m=5\mathrm{m}$ ${\displaystyle \frac{x}{y}+{\displaystyle \frac{w}{z}=\mathrm{xz}+\mathrm{yw}}}$
Misinterpretation of symbolic notation	Partial misunderstanding of factorisation.	(1) ${\displaystyle \frac{\mathrm{ma}+\mathrm{mb}}{m+\mathrm{md}}={\displaystyle \frac{a+b}{d}}}$
Invalid distribution of brackets	Incomplete expansion of brackets and overgeneralised the distributive law.	$2(3a+4)=6a+4$ $(2m-n)+n=2\mathrm{mn}-n^2$
Sign errors	Problems working with integers and misunderstanding operation signs.	$(8x^2+3x+4)\text{–}(5x^2-7x+2)$ $=3x^2-4x+6$
Substituting letters with numeric values	Substituting letters with random numbers.	(1) if $b+d=6\mathrm{then}b+d+e=?$ Learner solution $b+d+e=9$ Learner assuming b, d, and e are identical

Table 2. Intervention lessons

Lesson	Focus of the lesson	Summary of each lesson
1	Examining pre-test errors	To make learners in this study aware of their errors and use them as a teaching point, the pre-test assessed learners’ work and revealed the errors.
2	Algebra tiles add and subtract like terms	Learners combine tiles to produce 0s by pairing like terms. Exercise 8.1 question (a) focuses on adding and subtracting like terms.
3–4	Multiplying binomial products and squares	Learners often multiply binomials by squaring the first and last term, e.g. $(x-2{)}^2\ne x^2+4$ This activity aimed to clarify for learners that all terms must be multiplied. Learners solved exercise 8.2 (a) and (c) by multiplying binomials with algebra tiles.
5–6	Polynomial division by integers and monomials	Modeling the divisor vertically. Learners utilised dividend tiles to form a rectangle whose height equalled the divisor’s length and whose quotient was the horizontal width. In addition, learners conducted exercise 8.1 question (d).
7	Factorisation: highest common factor	Learners built a rectangle with the tiles: rectangle length is the solution; width is the highest common factor. Exercise 8.5, part (a) was solved using the factorisation method.
8	Factorisation: common bracket	This lesson showed the learners that distributing a number outside the bracket is like finding the highest common factor, but in expanded form. Exercise 8.5 (b) required factoring, hence the common bracket must be removed. In certain instances, learners must alter the symbol to fit the bracket. Learners did not mind the obligation to change signs.
9	Factorisation: difference of two squares	Learners were told to first assemble the squares and then add the ‘missing’ pieces to complete the rectangle to find the difference of perfect squares. As part of their work, learners completed exercise 8.8 (a) and (b).
10	Factorisation-trinomial	The solutions for trinomial factorisation were illustrated with algebra tiles. Learners solved exercise 8.9 (a) and (b). Learners who use algebra tiles effectively appear to understand the relationship between multiplication, division and factorisation of algebraic expressions.

Figure 3. Learners’ performance on each test

Table 3. The average overall performance on the pre- and post-tests

22 Learners	Pre-test (%)	Post-test (%)	Gains (%)
Mean (%)	48	66	18

Table 4. A summary of the results from both the pre- and post-tests that demonstrate learner achievement

Algebraic expressions
	Pre-test		Post-test		Increase/decrease
Learner	40	%	40	%	%
1	8	20	19	47.5	27.5
2	14	35	27	67.5	32.5
3	15	37.5	27	67.5	30
4	27	67.5	31	77.5	10
5	19	47.5	22	55	7.5
6	6	15	30	75	60
7	24	60	23	57.5	−2.5
8	24	60	30	75	15
9	18	45	20	50	5
10	18	45	20	50	5
11	21	52.5	30	75	22.5
12	18	45	29	72.5	27.5
13	16	40	25	62.5	22.5
14	11	27.5	18	45	17.5
15	26	65	32	80	15
16	25	62.5	16	40	−22.5
17	27	67.5	27	67.5	0
18	12	30	24	60	30
19	15	37.5	25	62.5	25
20	26	65	35	87.5	22.5
21	28	70	31	77.5	7.5
22	27	67.5	38	95	27.5
Average	19.31818	48.29545	26.31818	65.79545	17.5

Table 5. The dependent sample test results

Pre-test	Post-test	Difference (post-test − pre-test)	Standard deviation (Diff-M)	Sq-Dev
20	47.5	28	10.55	111.21
35	67.5	33	15.55	241.66
37.5	67.5	30	12.55	157.39
67.5	77.5	10	−7.45	55.57
47.5	55	7	−10.45	109.3
15	75	60	42.55	1810.12
60	57.5	-2	−19.45	378.48
60	75	15	−2.45	6.02
45	50	5	−12.45	155.12
45	50	5	−12.45	155.12
52.5	75	22	4.55	20.66
45	72.5	28	10.55	111.21
40	62.5	23	5.55	30.75
27.5	45	17	−0.45	0.21
65	80	15	−2.45	6.02
62.5	40	-23	−40.45	1636.57
67.5	67.5	0	−17.45	304.66
30	60	30	12.55	157.39
37.5	62.5	25	7.55	56.93
65	87.5	23	5.55	30.75
70	77.5	6	−11.45	131.21
	95	27	9.55	91.12
		Mean: 17.45		S: 5757.45

Difference scores calculations
Mean: 17.45
μ = 0
S^2 = SS⁄d.f. = 5757.45/(22 − 1) = 274.16
S^2M = S^2/N = 274.16/22 = 12.46
SM = √S^2M = √12.46 = 3.53

t-Value calculation
t = (M − μ)/SM = (17.45 − 0)/3.53 = 4.94

Note: Column 4 is ( difference-mean)

t-statistic -hypothesis test statistic;

Sq-Dev-Squared deviations from the mean;

s, variance;

μ - Population mean;

M-sample mean;

S^2 sample standard deviation

Figure 4. Learners’ common errors on the tests

Figure 5. Some common slips on the tests

Table 6. The average percentage decrease between the pre- and post-tests for common errors

	Actual number of errors
Errors	Pre-test	Post-test	Percentage decrease
Slips	75	43	43%
Sign error	56	42	25%
Misconception	117	78	33%
Multiplication error	80	47	41%
Substitution error	38	20	47%
Total	366	230	37%

Figure 6. A representation of addition and subtraction of like terms using the algebra tiles

Figure 7. Some sign errors that were recorded

Figure 8. Squaring a binomial using algebra tiles: $(x-3{)}^2=x^2-3x-3x+9$

Figure 9. Typical misconceptions that were encountered on the pre-tests

Figure 10. Some of the most common multiplication errors discovered on the tests

Figure 11. Substitution using algebra tiles: $3(3)-2=7$

Figure 12. Example of substitution errors observed on the pre-tests

Hall, B.C. (1999). Using algebra tiles effectively: Tools for understanding. Prentice-Hall. https://www.bgsu.edu/content/dam/BGSU/nwo/documents/COMP-6-8/June20_2016/UsingAlgebraTilesEffectively.pdf

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Ncube, M. (2016). Analysis of errors made by learners in simplifying algebraic expressions at Grade 9 level. Unpublished doctoral dissertation, University of South Africa.

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