I had problems with [my] math teacher. When I asked a question, she wouldn't answer and [would] say I should have been listening, even though I was listening, just her explanation wasn't so great.
— A 7th grade student
With the exception of errors that result from carelessness or incomplete basic arithmetic facts, the errors that students make in math tend to be consistent. Th e most common involve incorrectly applying a procedure or
an algorithm learned by rote memorization. Such errors occur when students have not developed the mathematical reasoning that accompanies constructing the mental patterns of concepts; procedures and facts learned only by rote memory are not available for successful transfer to new situations.
As in other subjects, students have misconceptions about mathematics. These misconceptions hinder the learning process because they are strongly embedded into neural networks that have been activated again and again. Students need tangible experiences to break these misconceptions. Eliminating mathematical misconceptions is difficult, and merely repeating a lesson or providing extra time for practice will not help. A better approach is to show students common errors and help them examine completed sample problems that demonstrate these common errors. Th is method also gives you an opportunity to reinforce critical foundational skills.
Making Common Errors
Multidigit Addition. An example of a common error is 54 + 37 = 811. Th is error occurs when students line up the numbers 54 and 37 in columns and write the sums of each column: 4 + 7 = 11 and 5 + 3 = 8. Multidigit Subtraction. A common error occurs when students subtract whichever digit is smaller from the larger digit. For example, 42 − 29 = 27 because 9 − 2 = 7 and 4 − 2 = 2. Later, this error is repeated with negative integers, so that students write 45 − 55 = 10.
Combining Like Terms. Another concept that needs to be constructed with a framework of experiential learning is that one can add and subtract only like terms (i.e., objects of the same category, same units of measurements). Unless this concept is learned with complete understanding in elementary school, it will continue to confuse students when they move on to common denominators and the simplification of algebraic equations. An example of a common error in this category is 2a + 2 = 4a. Adding and Subtracting Decimals. Applying the rule they memorized for adding whole numbers, students may line up the numbers on the right side, instead of lining them up based on decimal points. For example, they may write
123.4
−4.593
instead of
123.4
− 4.593
Zero as a Placeholder. Unless students learn about place value early on, they confuse 0 as a placeholder with a 0 that doesn’t change the value of the number. Th e respective errors would be 3.04 = 3.4 and 3.40 < 3.400. This same confusion leads to the mistaken conception that in order to multiply decimals by 10, you just add a 0. Students learn to “add a zero” from working with positive and negative whole numbers, but this solution does not work with decimals and fractions.
Adding and Multiplying Fractions. Th e most common error students make when they add fractions results from adding the numerators and denominators without fi rst changing the fractions so that they have
common denominators. It would not be unusual for a student to see ⅔ + ⁴⁄₅, then add the numerators (2 + 4 = 6) and denominators (3 + 5 = 8) and conclude that ⅔ + ⁴⁄₅ = ⁶⁄₈. Similarly, students are confused when they are told, without conceptual understanding, why they need to multiply numerators and denominators across when multiplying fractions, especially since they are told they cannot add numerators and denominators across when adding fractions. Th e best way to eliminate this misconception is to allow students to work with math manipulatives when they first work with fractions. Th is approach allows students to visualize denominators and numerators broken down into their basic parts. Further along, confusion about the nature of addition and multiplication will result in the common errors of applying the distributive, associative, and commutative rules to subtraction and division.
“Multiplication Always Results in a Larger Number.” This statement is true for positive whole numbers. However, it is not true for fractions and negative numbers. Students latch onto the misconception that this statement is true in all cases because of initial experiences with positive whole numbers. Instead of saying, “one-half times eight,” try saying, “one-half of eight.” Th e use of the word of in this problem (i.e., when a fraction and a whole number are multiplied together) informs students that the answer will be less than eight.
Rates and Ratios Written as Whole Numbers or Fractions. Students need to understand that ratios and rates are about relationships between numbers, not the numbers themselves. For example, they may write “2 : 2” or “2 to 2” as 1. If they do, they are missing the concept of rates as a comparison of two different factors (such as miles in relation to hours), so they don’t understand why a single number or a mixed number does not represent a comparison and cannot be a rate.
What we know about the brain suggests that suitable learning environments for young students can diff er in some respects from what is suitable for older students. This is due to two important characteristics evident in young children: tolerance for mistakes and innate curiosity. Research suggests that young children are usually comfortable making mistakes. In children younger than eight, the areas of the brain involved in cognitive control show strong activation following positive feedback, and stress-reactive regions are not activated by negative feedback (Crone, Donohue, Honomichl, Wendelken, & Bunge, 2006; Van Duijvenvoorde, Zanolie,
Rombouts, Raijmakers, & Crone, 2008). If you are a teacher of younger students, you are the caretaker of their precious creative potential. Challenge builds skills, and without suffi cient challenge, their math brains won’t grow.
Trial and Error
Much of what we do or say is based on the brain’s interpretation of information stored in memory from prior experiences. Most of our decisions are predictions made on an unconscious level, guided by these memories.
Memories of decisions are embedded with the pleasure or displeasure that resulted from previous predictions. As previous experiences build, so does the brain’s stored network of data; as a result, our response to new input becomes more accurate.
Through curiosity, trial and error, and the dopamine-mediated pleasure from correct responses and the negative feelings from erroneous responses, our brains are better able to interpret the environment. The brain becomes more and more accurate in anticipating (predicting) what action (answer) is correct (will bring pleasure). These predictions send out signals to the parts of the brain that control our actions, words, or answers to questions. The older children get and the more experiences they have, the more their thinking, reflective prefrontal cortex can modulate the emotional (involuntary, reactive) response of the lower brain. Through trial and error, mistakes, and correct choices, the brain builds neural tracks to preserve and repeat the rewarding behavior. For students and others, this means that after making an incorrect prediction (answer), the next time the question comes up, prediction accuracy is better because the faulty information in the circuit has changed.
You want your students to remain comfortable making some mistakes so they will be willing to challenge themselves in the years to come. Innate curiosity is something we are born with, and young children retain much of this quality. From infancy, young brains need to make sense of their world in order to survive. Innate curiosity is critical to promote this exploration, and it unconsciously drives behavior. Th rough exploration, children gradually construct neural networks of categories (e.g., patterns, schema), and as exploration and experience continue, the networks expand to accommodate more detail. Networks are modified in response to mistakes (i.e., incorrect predictions based on existing information) as students make more accurate connections between what was predicted and what was experienced (i.e., sensory input). Th is process goes on without conscious awareness.
Because young students’ brains are driven more by curiosity than by sensitivity to error embarrassment, you can be more direct and call on them to answer questions even if they don’t volunteer. This approach is often necessary for younger children, because their brains haven’t developed much attention control, and they need you to pull them into the lesson by direct methods, such as saying their names and asking for responses.
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