Utilizing Student Mistakes for Effective Learning

It used to be common practice for teachers to announce errors in front of the class in order to motivate students to do better in the future. However, rarely was there further discussion on the types of common errors or the thought processes that led to the errors. More recently, and understandably, that practice has fallen out of favor and now teachers are hesitant to discuss mistakes in a public forum for fear of embarrassing students. This new approach limits the amount of learning that can result from making mistakes (Tulis, 2013).

During the first day of each of my math classes, I start my lesson with an Albert Einstein quote: “Anyone who has never made a mistake has never tried anything new” (Thorpe, 2000, p. 149). I tell my students that there is no learning without mistakes and that each time they share their mistakes with the class they support me in teaching. There are two main reasons for highlighting errors. First, we can avoid mistakes in the future by understanding why and where they arise. This is the essence of learning and teaching. And second, we can reduce the shame associated with making mistakes. It is normal to make mistakes, and the probability that someone else in the class is making the same mistake is quite high.

“We learn from failure, not from success!” (Stoker, 1897). It is important to make mistakes because they offer an opportunity for growth, improving how we face challenging situations and motivating us to find new approaches to solving problems. Most of all, recognizing and understanding our own mistakes clarifies our comprehension of mathematical concepts (Boaler, 2013; Metcalfe, 2017). When students understand what is wrong in their answer and they are able to fix it, there is a good chance they will not make that mistake again and will feel more confident in their abilities.

Teachers can use students’ mistakes to allow and promote learning without embarrassing students. Encouraging students to share their mistakes promotes constructive rather than transmissive teaching. It is not effective to simply point out a mistake. It is important for students to reflect on errors, understand why they made a mistake, recognize how making mistakes is common, and realize what they need to do to avoid making them. Thus, it is essential for the teacher to understand the error and guide students in analyzing it to prevent the error from being repeated and to clarify the mathematical concept underlining the mistake.

In math, we first need to distinguish between conceptual and computational errors. In the first case, the “intervention” (instruction to address misconceptions, errors, or gaps in learning) should focus on the mathematical concept while in the second case it should be centered on calculation rules. Let’s consider mistakes that can be made in completing the following sequence:

15        23        31        __        47        __

The missing numbers are 39 and 55. Finding the answer requires identifying the rule, adding 8, and correctly performing the calculations, 31 + 8 = 39 and 47 + 8 = 55.

A conceptual mistake may be present when the answers are inconsistent and/or far enough from the correct answers to make us believe that the rule has not been identified (e.g., 35 as the first answer and 53 as the second answer). The error may be identified as a calculation mistake when the given answers are close to the correct answers (e.g., 38 or 40 as the first answer and/or 54 or 56 as the second answer). We may suppose that the rule has been correctly identified because the answer is almost correct.

Sometimes the mistake could be due to distraction. For example, if the first answer is correct and the second answer is missing. If the mistake is due to inattention, the teacher can explain how to check the answer. We can confirm the cause of the error by asking students to explain how they arrived at the answer.

The response to a conceptual mistake should be oriented to revisiting the mathematical concept. If the rule of the sequence is not identified, the student may not see the regularity in the number sequence and the connections among numbers. The intervention should focus on numbers and their relationships.

A possible intervention to strengthen the number concept might be centered on using the number line. Number lines help students visualize the position of numbers and associate them with their corresponding magnitudes. On the number line, the unit defines the distance between two numbers. If students draw a number line and insert the numbers of the previous sequence on it, they can see that the distance between two consecutive numbers is maintained, and it is equal to 8. Through this work, students “create a new experience” around solving the problem and analyzing number relationships (Booth & Siegler, 2008).

In case of a computational mistake, the intervention should focus on calculation rules and techniques. The teacher can make explicit mental strategies used to add 8. The decomposition strategy can be applied to the first addend (31) for the first answer (e.g., decomposing 31 as 30 + 1, adding 8 to 30 to get 38, and adding 1 to 38 to get 39). For the second answer, this strategy can be used by decomposing the 8 and then adding it in parts to the second addend (47) to get 55 (e.g., decomposing 8 into 5 + 3, adding 3 to 47 to get 50, and then adding 5 to get 55).

Computation mistakes can also be differentiated. Let’s analyze three different erroneous answers for the subtraction problem 23 – 15, supposing that students use a written calculation strategy. Are we able to identify the nature of the mistake in the following three cases?

If the answer is 7, we might suppose that the calculation procedure is correct, but the computation is incorrect: the counting-on process from 15 to 23 may be the culprit. On the contrary, if the answer is 18 or 12, the standard calculation algorithm is likely attempted but executed incorrectly. The answer 18 may have been due to an incorrect application of the borrowing or regrouping rule (regrouping the 2 tens and 3 ones into 1 ten and 13 ones but forgetting to change the 2 tens to 1 ten). The answer 12 may have been obtained by inverting the order of the digits in the ones column while performing the operation: 5 – 3 = 2, possibly to subtract a smaller number from a larger number. The intervention should be designed based on the types of errors or misunderstandings that are uncovered.

In addition to conceptual and calculation mistakes, we might observe writing errors for numbers. Let’s analyze two possible mistakes that can be made in transcoding a number from verbal representation to digital representation: lexical and syntactical mistakes. Lexical mistakes can be observed when digits are incorrectly chosen but the order of magnitude is correct (e.g., to transcode thirty-nine, students write 36 instead of 39). Syntactical mistakes occur when digits are correctly chosen but the order of magnitude is incorrect (e.g., to transcode thirty-nine, students write 309 [“30”and “9”] instead of 39). Lexical mistakes might be due to dyslexia issues (e.g., 9 is confused with 6) or incomprehension about the number’s name. Syntactical mistakes are made when verbal aspects intervene in transcoding the number. In this case, the intervention should focus on the digit’s role in the number highlighting the difference among hundreds, tens, and ones.

Particular attention should also be given to the name of the number. In English, the number- words after 10 might be confusing for students. The numbers 11 and 12 have unique names compared to teen number-words (from 13 to 19). For 13 to 19, the number-words are constituted by a basic number (e.g., three, four, five, six, seven, eight, and nine) and the word “teen” (thirteen, fourteen, fifteen, etc.). However, 11 and 12, with number-words “eleven” and “twelve,” have no such consistency or common rule. Additionally, the teen number-words reverse the order of the one’s and ten’s digits, making the number concept unclear (e.g., from 20 through the 90s, the ten’s digit is said first, “eighty-one,” but the teens start with naming the one’s digit, “eight-teen”). After 100, the numbers are said with the number of the hundreds and a basic name (e.g., one hundred, two hundred, three hundred, and so on.) but transcribed in a condensed format (e.g., two hundred and one is written 201 rather than 2001 or twenty-one is written 21 instead of 201). It is important to make explicit these aspects of numbers and corresponding number-words with students who make lexical and syntactical mistakes with numbers.

It is interesting to observe that in the Chinese language this problem does not exist because the number-words clearly express the number magnitude. For example, 11 is expressed as “ten, one,” 12 as “ten, two,” and so on. That makes it easy for students to understand that the number system is based on units of 10 and to clarify the value of the position of each digit in a number.

It has been shown that inserting error analysis in teaching practices positively increases mathematical understanding (Boaler, 2013). However, it is the causes leading to a mistake that must be deeply analyzed. It is through this analysis that the teacher can implement an effective intervention utilizing student mistakes to increase learning.

References

Boaler, J. (2013). Ability and mathematics: The mindset revolution that is reshaping education. FORUM, 55(1), 143–152.

Booth, J. L., & Siegler, R. S. (2008). Numerical magnitude representations influence arithmetic learning. Child Development, 79(4), 1016–1031.

Metcalfe, J. (2017). Learning from errors. Annual Review of Psychology, 68(1), 465–489.

Stoker, B. (1897). Dracula. London, UK: Archibald Constable and Company.

Thorpe, S. (2000). How to think like Einstein: Simple ways to break the rules and discover your hidden genius. Naperville, IL: Sourcebooks, Inc.

Tulis, M. (2013). Error management behavior in classrooms: Teachers’ responses to student mistakes. Teaching and Teacher Education, 33, 56–68.