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## Reducing Mistake Anxiety

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.

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. The 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. This 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. This 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

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. The 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. The most common error students make when they add fractions results from adding the numerators and denominators without first changing the fractions so that they have common denominators. It would not be unusual for a student to see 2/3 + 4/5 then add the numerators (2 + 4 = 6) and denominators (3 + 5 = 8) and conclude that 2/3 + 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. The best way to eliminate this misconception is to allow students to work with math manipulatives when they first work with fractions. This 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.” The 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.

### Creating the Right Environment for Younger Students

What we know about the brain suggests that suitable learning environments for young students can differ 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.

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.

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 sufficient challenge, their math brains won’t grow. 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. Through 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). This process goes on without conscious awareness.

Age-Related Changes

In children up to eight or nine years old, the dopamine-modulating reward center in the nucleus accumbens reacts strongly to positive feedback (activating the prefrontal cortex) and minimally to negative feedback. In older children, increased activation still occurs in the PFC when dopamine is released in response to positive feedback (particularly in response to correct answers/ predictions). However, the greatest age-related change is the higher reactivity of the NAc to negative feedback and the accompanying drop in dopamine, decrease in pleasure, and reduced input through the amygdala filter to the PFC. The NAc increases in reactivity through the teen years, then settles down into the adult pattern of less sudden, profound emotional shifts (Crone et al., 2006).

The high response to positive feedback in younger children is neuro-logical
because their brains need motivation to keep exploring and making sense of the world. In upper elementary school, things begin to change. Because the prefrontal cortex is more reactive to the drop in dopamine release by the NAc that occurs with mistake recognition, students from about 6th grade through high school are affected more by negative feedback and less by positive feedback. Mistakes become high-stress experiences, and the risk of making mistakes, especially in front of classmates, limits their opportunities to learn.

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.

### Reducing Negative Attitudes Toward Mistakes

The following strategies apply not to errors made on tests, but to mistakes made in front of classmates. We begin with a couple of general strategies to encourage thoughtful participation and to increase attentive focus. Then we consider strategies to refine your reactions to students’ mistakes, strategies that encourage participation, and strategies to create low-stress ways of using mistakes for learning.

#### Strategy: Enforce Wait Time

When you plan to call on younger students, even those who don’t volunteer, an enforced wait time after the question is asked is necessary to keep overeager classmates from calling out the answer. Use of “engaging challenge” to enforce this wait time serves this purpose. For example, tell students that to be considered, they cannot call out an answer or raise their hands until you say a particular number (for example, a multiple of five, a number greater than today’s date, and so on). When you ignore their call-outs and raised hands, because these actions violate the rule, students will realize that they won’t be recognized unless they listen carefully and follow all the rules; they must wait for the clue they need to hear before they can answer.

#### Strategy: Call On Multiple Students

You will increase attentive focus among younger children by calling on multiple students to respond to the same question without saying if their answers are correct. Only the person who thinks, learns. When students hear a classmate give an answer and the teacher say it is correct, they have no investment in trying to evaluate the information themselves. However, if the class culture is one in which students know they are all responsible for trying to answer all questions (because you will ask several students for answers before acknowledging the correct answer), they continue to work the question in their minds or on their papers because they have not been “given” the answer by the first person to solve the problem.

After selecting several volunteers to offer their solutions, call for a vote among the entire class. If the remaining students know that they are still accountable for their own answers, they will continue to be invested because they made an active prediction and will want to know if they were right or wrong. This then motivates finding out how to do it correctly in the future.

You can also ask estimation questions in which each student must select an estimate higher or lower than the previous suggestion so they don’t just repeat what a classmate said and so they continue to follow along while other students answer.

#### Strategy: Intervene Immediately

It is important to immediately reduce students’ stress when they give an incorrect response, especially with students who typically don’t participate in class and whom you coax into responding. To provide quick intervention, it helps to be prepared. When students respond to a question in front of their peers, they put themselves at risk, so thank them for any response they volunteer. For example, say, “You took a tough question. Good for you.” If the response is incorrect, see if there is some part of their suggestion that is correct, and then restate the question to fit the response before posing the original question to the class in a different way.

You can also say that the student’s answer is very close and you want him or her to listen to the ideas of a few other students, think about how to revise the answer, and then you’ll come back to him or her. Return to this student quickly with a different question, with one that is easier for him or her to address, or with the original question, if you feel that he or she has had sufficient time and exposure to other students’ suggestions to think of the correct answer.

You’ll probably know by the incorrect answer whether the error was computational (e.g., incorrect addition) or procedural (e.g., addition when the problem called for subtraction). Your knowledge of the student’s general arithmetic foundation will inform you if he or she knows how to add and subtract, but the error was made because the student didn’t know which procedure to use. What happens next depends on the student and his or her reason for the error. If choosing the correct procedure is a problem for other students, immediate investigation of the cause of the mistakes will be valuable for all.

You can respond with a positive tone of voice and facial expression and say, “I’m so glad you gave that answer. It reminds me that I didn’t fully explain that different words are cues that tell us to subtract. In the question that I asked, I said, ‘What is the difference between 15 and 5?’ You said 20 because you added them, and that did give a different number than either 15 or 5. Let’s revise (or add to) our list of words that are cues for subtraction.” Follow this with similar practice questions, with students responding on personal whiteboards or other response devices.

If the student’s incorrect answer is indicative of a below-level mastery of rote memory facts, there is little value in using the error for teaching purposes. To reduce distress over making the mistake, give the correct answer and immediately ask a different question that is correctly answered by the student’s response. For example, “You said that 5 times 7 equals 30. Actually, 5 times 7 equals 35, but you are correct that 5 times a number close to 7 equals 30. Do you know what that number is, or would you like to call on a classmate?”

If the error is conceptual or procedural about a topic that the class has already mastered, you can say, “I like that answer because it answers another question that I plan to ask later in this unit.” Write down the response in your own words, but add something that will make it a “taking-off point” for something that will come up in the following days: “You said that 1/5 is more than 1/3, which fits with the topic of dividing fractions. I’ll write ‘fraction division’ and use your great example as part of our fraction division lesson later this week.”

In the next few days, you can give this student some additional coaching and practice work on denominators to bring him or her up to class level. When you move on to fraction division, remember to return to this point with the question, “Can dividing ever give you more than you start with?” This provides a bit of cognitive dissonance to promote curiosity in the lesson. After the discussion, direct the class to the fractions you wrote down when the student made his or her original mistake.

Write and ask, “What is 1/5 divided by 1/3?” Write the answer as 3/5, circle the denominators, return to the student, and say, “Now we know that larger numbers have different meanings when they are in the denominators of fractions. This brings us back to the question I asked the other day. Let’s make a list of what we learned about that question, which I kept up here on the board. When you divide a fraction by another fraction, why do you end up with a quotient that is more than either fraction?”

#### Strategy: Use Estimation and Prediction to Increase Participation

Because older children have more negative feelings about errors, a good way to promote participation is to ask “mistake-proof” questions. These questions are also opportunities for differentiation, because they include options for students who already know the procedure or facts to respond at their level of increasing conceptual understanding. You can ask open-ended questions about how to solve problems (rather than to actually calculate answers) or questions that require only estimation or prediction. Putting students in pairs and groups also reduces mistake negativity.

Opening a discussion with a variation of “How can we find out…” is a great way to engage students through their strengths. It also allows students to hear multiple approaches and to choose one they find most understandable. An example is, “If we want to give everyone at your table an equal number of raisins from this bag, what could we do?” Hold up a clear bag with too many raisins to count, so students know they are not expected to know a specific answer.

Your question puts the focus on concept and process. Students are reminded not to give specific numbers, because you are looking for approaches, or ways to start to evaluate a situation. As in most math discussions, ask students why they think their suggestion might work. Write down three suggestions and let the class evaluate which ones could work. Students can move to corners of the room that represent each of the three suggestions, plus one that represents “none of these.” They can work with other supporters of that technique to create evidence and examples, or relate the process they support to other similar procedures.

Estimate. Students often don’t take the time to estimate or check their answers, and when estimation is called for, they may first solve the problem, then write a close round number as their estimate. They likely do this because they have never experienced the “here-me-now” value of estimation.

Encourage estimates by valuing them in multiple aspects of math. Give partial credit for reasonable estimates labeled as such on homework or quizzes. Have students start homework in class, but just with estimates. Even if they know the answer without calculating, have them write a range of answers and a reason why their estimate range is logical. These can be shared with the class when estimates are reviewed before the students leave school to do homework independently. Students then have accurate estimates you already approved with which to compare their homework answers to see if they are on target or should rethink the problem.

Predict. Making predictions, like estimating, is a safe type of “risk-taking behavior” that can stimulate the dopamine-pleasure response and encourage fearful or perfectionist students to take chances without the anxiety (amygdala stress) of being wrong. Emphasize that predictions don’t have to be right, and that sometimes even the smartest math students make incorrect predictions.

As an example to demonstrate that everyone makes incorrect predictions (and estimations), explain that you will flip a coin and that students should write down whether it will land on heads or tails. Flip the coin and ask students to hold up their predictions if they were wrong. Students can then look around and see that even the “smart kids” made incorrect predictions.

Experiences such as these gradually help reluctant students recognize that incorrect predictions are not signs of ignorance, so they will be more confident in your explanation that predictions are opportunities for the brain to try something out, then take the actual outcome and use that information to make future predictions more accurate.

Students in upper grades, especially Explorers, enjoy building their estimation skills in real-world situations. Map Readers will enjoy the opportunity to see the goal and the sequential steps that will lead them to success. Offer choices whenever possible so students can participate through their strengths. Written instructions or a demonstration can be offered at the beginning, especially to Map Readers, and notes of the process used can be written at the conclusion by Explorers as they describe what they did. All students can select the manner in which they present their ultimate discoveries for assessment and class sharing.

The use of novelty, surprise, and discrepant events to demonstrate the importance of accurate estimations can help students remember concepts. Examples include the following:

• Overfill a glass of water so the water spills onto the floor. After the class has a good laugh, ask them what you did wrong. (You didn’t plan. You didn’t estimate or predict how much the glass would hold and when to stop pouring.)
• Arrive a minute late for class and tell students you didn’t estimate how much time it would take to walk to class from your new parking space (or other location).
• Bring a bag with about 10 small, undividable pieces of candy to class and hold it up. With a smile, say that you will be giving one piece to each student. They will surely notice that you did not bring enough and voice that sentiment. You can say, “Well, I didn’t count exactly, so how could I know? What could I have done to bring a more reasonable amount?” Guide them to suggest that you could have estimated approximately the same number of candies as students in the class. Just be sure to have the rest of the candy in another bag!

#### Strategy: Create an Estimating Center

Quantity. Create an area in the classroom where you keep containers of the same size that are filled with objects of different sizes. For an extended center activity, children can write down their estimates of how many objects the containers hold. Encourage students to periodically look at and revise their estimates, if necessary, and have volunteers read some of their estimates. When the actual number is revealed, have students jot a note in their journals if they were over or under (perhaps by how many) and why. Fill the containers with different objects and have students do the activity again, making and revising estimates over a certain number of days.

Weight. To build numeracy and estimation skills, an estimating center can have a scale with items to be weighed. The items can be math manipulatives of the same size and weight, or they can be other objects such as old golf balls. Students first lift and feel a one-pound weight and then put it on the scale to see the weight confirmed. They then estimate how many of the designated items would weigh one pound. They can do this by putting them into a bag and using any method they choose, such as holding the weight in one hand or just remembering the feel of the weight. They then weigh the bag and add or remove items until it weighs one pound. Students should keep notes and repeat the activity with different items. Extensions of the activity can have students work with two or three pounds or mix and match objects—tennis balls and golf balls, for example.

Comparisons. Select two boxes or cans of food that weigh 8 ounces and 16 ounces, respectively. Have students hold each as you tell them (or they read) the weights of the containers. Give students a box or can with the weight covered and have them compare the weight of the new package to the weight of the 8- and 16-ounce samples. They can then estimate whether the new item’s weight is closer to 8 or 16 ounces. As students become more successful, they may want to predict a more specific weight. Ask them to tell you why they think the new can weighs 10 ounces, for example, and encourage them to respond with, “It is a little heavier than the 8-ounce can” or “It is much lighter than the 16-ounce can, but not as light as the 8-ounce can.”

With this estimation-promoting activity (which can easily become an independent activity), students also build number sense by experiencing the relationships between numbers and real measurements and by developing concepts of more than and less than.

To further develop these concepts, or for challenge work at the center, you can ask students how much they think an item costs. The goal is not for them to know prices, but to develop the concept that larger objects don’t necessarily weigh or cost more. If a student predicts that a \$3 box of cereal costs \$1 and you say “more,” he or she may say \$2. You say “more” again, and the student will continue giving answers as you direct with “more” or “less” until the correct dollar amount is guessed. Continue this activity with a small can of a costly item, such as artichoke hearts.

Students doing this extension can keep records of the item size and price and repeat the activity with items they select, perhaps because they know they are more or less costly. When they think they know the concept that price is related to both size and value of the item, they can write their discovery in their journal. Let students know that they did something more important than get a correct answer; they discovered a concept—a key that will help them solve many future problems.

Size. Distribute plastic storage bags of the same size, and have students fill the bags at home with their choice of items (e.g., beans, marbles, lemons) in quantities that they are able to count. In class, students each weigh their bags and remove items so all the bags weigh the same amount. Each student then counts how many items in his or her bag are needed to equal that constant weight, such as one pound. Have them add their data (and attach their bags) to a class chart. Then cover the numbers with sticky notes and have students play a game where they guess how many of each item are in a pound. This game can later become an independent activity during center time. Students build or extend the very important concept that size and weight do not always have a direct relationship to quantity.

To adapt this activity for higher levels of achievable challenge, have students work in pairs or small groups with a scale and a variety of objects to put in bags. Options include predicting (estimating) how many objects would equal a given weight and how many objects should be removed or added after each weighing to get the bag as close as possible to the given weight.

This activity can be further extended as students develop the early concept of multiples. Students can make new predictions to reach a new weight, such as two, three, four, or five pounds. Extend the challenge further by asking students to estimate how much each individual item weighs. This activity can take place before students formally learn fractions, but they can communicate in their own words what they predict once they see, for example, that four objects weigh one pound.

#### Strategy: Estimate Weight with Familiar Objects

This is a great “here-me-now” activity. We know our students’ backpacks are getting heavier all the time. They already have prior multisensory knowledge about their own backpacks and those of their classmates. They pick up one another’s packs to get to their own, or they pass bags to their respective owners. They carry their own when it is heavy and light. This is an object that transcends language and culture in almost every classroom. This estimation activity therefore provides the comfort of familiarity plus personal interest because it is all about the students themselves and their classmates.

Select several backpacks from volunteers who predict that theirs are light, medium, and heavy. Use a scale and have the students weigh their bags, read the number on the scale (with your help, if necessary), and then write the weight on the board under one of the three categories: Heavy, Medium, and
Light. After each bag is weighed and the weight is announced and recorded (providing both auditory and visual input), pass the bag around so each student experiences the tactile sense of what a bag that weighs X pounds feels like.

After the first three examples (one in each category), have a volunteer predict if his or her bag is heavy, medium, or light. The bag is passed around and each student privately writes down an estimation of its weight before the owner weighs it and posts the weight on the list. You may want to label each bag with its weight written on a sticky note. Students can then return to these bags and lift them again to reinforce their sense of what a 5-, 10-, or 15-pound backpack feels like. This step is especially important when students repeatedly make incorrect estimates.

To keep participation up and fear of mistakes down, students’ estimates are kept private. The goal is for each student to improve prediction skills at his or her own achievable-challenge level. Keeping private records reinforces the idea that students are working to improve their own skills, rather than to be better than their classmates.

After students finish weighing and predicting the weights of all the bags, they now make charts or graphs comparing the actual weight with their predictions in the order in which the bags were weighed. Several options are available for students at different levels of mastery. Provide below-level students with appropriate comparison charts, while at- or above-level students can choose the style of chart they want to make. The charts should have spaces where students can fill in the weights of each bag. Next to the numbers representing the actual weights are the students’ predictions. If they make bar graphs of the data, they should see that the bars on the bar graphs become closer in height as more bags are weighed.

Advanced students can use subtraction to find the difference between their predictions and the actual weights and then plot those numbers on a line graph (from the first to the last bag), with the bag number on the X-axis and the difference between their estimate and the actual weight on the Y-axis. This will also illustrate the trend in accuracy.

When students evaluate their various graphs, they can discuss what trends they find and how they are represented in different graphs. Why do the bars on the bar graphs become closer in height with more predictions, and why does the line on the line graph drop downward with more predictions? Students can go on to discuss what they predict the next three additions to their graphs would look like and explain how they used the trend to predict this outcome.

For homework, students can choose household objects that they can lift and that can fit on a bathroom scale. They replicate the backpack experiment with 5 to 10 objects, graph their predictions with the actual weights, and describe what they noticed about their prediction accuracy as they gained more experience. (Note: Be sure all students have access to bathroom scales before assigning this homework.)

This activity is an opportunity for students to have a positive, fun experience that develops confidence in their estimation ability. They will be more willing to use estimation in future math lessons and activities, and they will be more comfortable participating in class because they see that estimates are “correct” when they are within a range of answers (rather than one specific number). Estimation begins to feel like a safe way to participate. In addition to increasing comfort with participation by recognizing that everyone makes mistakes (that is, their estimates are not always correct), students see that the more they practice, the better they become at the skill of estimating. Another goal is for students to recognize the value of estimation because it helps them see if their answers are reasonable. That “aha!” moment can promote use of estimation for more of their math homework and test questions.

#### Strategy: Estimate Circumference

Select classroom objects for circumference estimation and place them in random order. Students work alone or in small groups to arrange the objects in order, according to their prediction of smallest to largest circumference. They then select a small, medium, and large object and use string or a tracing of the base to determine circumference. From these observations, they estimate and then measure the remaining objects, keeping track of their estimates and actual values. Remind students to record the data in the order in which they make their estimates so they can see their prediction-accuracy trend.

As in the previous activities, students use their preferred method of comparing their estimates with the actual measured circumferences and plot a bar or line graph to find a pattern. As you see students progressing, suggest to more advanced students that they take the more challenging approach of making a graph or chart after gathering data from only half the objects. Each student in this challenge then considers how to use the collected data to more successfully estimate the next half. Suggest that students ask themselves the following questions:

• Am I over- or underestimating?
• How can I adjust my next estimates to be more accurate?
• How did the second half of my graph or chart differ from the first half?
• Can I use this same approach and apply it to other classroom objects for additional estimations?

This activity can also be used to estimate and measure the perimeter of squares, rectangles, and even objects with more than four sides; alternatively, students can advance to predicting surface areas to keep the challenge appropriately engaging. You will see the success of these estimation activities as students demonstrate more confidence in taking risks and participate in increasingly challenging discussions as their comfort and achievement levels increase. Your encouragement and formative feedback help students feel supported, safe, and engaged. Students recognize that your feedback is a valuable tool—not criticism—and they appreciate the benefits of using information about their mistakes to improve their future predictions (answers).

Students increasingly experience mistakes as learning tools, which helps them develop a more confident, positive attitude about math and life. Risk taking (estimation), error analysis (charting results), and perseverance in using the error analysis to make revised predictions reward students with the attainment of greater skill and success.

#### Strategy: Lower Risk with Small-Group Practice

When possible, to keep all students engaged in problem solving, you’ll want to have a number of students give responses before acknowledging if any are correct. To build confidence so all students actively participate, and to lower the stress if you call on students who don’t usually volunteer, peer and small-group practice is valuable scaffolding. Reluctant participants are more comfortable working with a peer as a way to gain confidence about the accuracy of the answers they offer.

Model peer work before you ask students to work together, first by playing both roles yourself, then with a prerehearsed pair of students or by demonstrating with a prepared student partner. This demonstration will show that the nature of pair work is to solve problems independently and then to describe one’s answer to the partner for validation or corrective feedback (from the partner or from you). Students need to understand this is not a time to work as partners to reach a solution together. Once the students have independent answers, they can compare. If their answers are the same, even if incorrect, the high-anxiety student will have the safety of knowing he or she was not the only one who made a mistake.

If students in the pair or small group disagree on their independent solutions, they take turns explaining their reasoning and follow the rule that no one interrupts until the person speaking is finished. Eventually, this practice can work up to a higher level of pair teaching, in which the partner doesn’t show or tell his or her way of reaching an answer, but rather asks leading questions to guide the partner to reach an accurate solution. Peer preparation before whole-class response time increases comfort, risk taking, and active learning from mistakes.

As an extension, have students do peer reviews of homework or class work before you call on participants, or use peer reviews as test review. Peer comparisons offer another opportunity to increase verbal math communication and confidence as evenly paired students compare answers and try to convince each other why they believe their answer is correct.

Students work best in this situation when they are at the same achievable-challenge level, which reduces the likelihood of one student telling the answer instead of listening and guiding the classmate to the correct answer. However, the ability to choose a partner raises dopamine levels and lowers stress, so you may sometimes allow students to select partners. In order for pairs of students at different achievable-challenge levels to work together, it is helpful to introduce the concept of accountability. In other words, both students are accountable to explain how the problem is solved, and they know that either of them could be called upon to do so. If the peer work is for test preparation and the student learner achieves significantly higher accuracy on the type of problems reviewed that day, both students (“learner” and “guide”) receive extra credit. This places a tangible value on successful peer work and encourages students to take the task seriously.

#### Strategy: Find Multiple Approaches

Multiple-approach problems strengthen risk taking and participation, increase options for achievable-challenge levels, and reveal math to be a creative process. To ensure that students experience the valuable tool of considering several options and using logical reasoning to select the best approach (for their learning strength or for the type of problem), explain that you are looking not for answers, but for ways to solve the problem and reasons why the student finds one of the approaches best.

The first instruction is to write with words or to show by examples at least two different ways to reach a solution. Depending on the level of achievable challenge, some students may have time for or awareness of only one approach. The strategy still works in their favor because, while others are writing about at least two approaches, those students won’t feel rushed. Let them know privately in advance that they are not under pressure to find multiple solutions this time because their current goal is mastering whatever procedure or concept they are at in your extended goal plan.

Tell the class, “This time I don’t want you to tell me what the actual answer is. Just tell me what you could do to solve the problem.” After students describe their approaches, ask for other approaches without indicating which are correct. To keep everyone engaged, let students discuss the approaches they understand or agree with, and when the different approaches are tested for accuracy, the conversation can continue as they talk about why one approach works better for them.

For example, if the problem was to find the answer to 8 x 6, students may suggest three options: memorizing the multiplication table for 6, knowing that 8 x 5 = 40 and adding another 8 to equal 48, or adding a column of six 8s. Allowing students to personally choose among approaches all confirmed as correct and to support their choice will increase their comfort levels. This process also builds math logic, intuition, and reasoning skills that extend into other academic subjects and real-life problem solving.

Another example might be to ask students how to find out which fraction is greater: 2/5 or 3/7? Encourage students to draw diagrams or to use any math tools in the room (e.g., manipulatives, rulers, graph paper). The answers are likely to match the learning strengths of the students. Explorers may use three manipulatives that are each 1/7 of the same whole and compare the size to two 1/5 pieces. Map Readers may draw two equally sized circles or rectangles on graph paper, divide one into five parts and the other into seven parts, color two and three sections of the respective shapes, and then compare the two colored regions.

Students who have mastered a higher conceptual level of equivalent fractions may find common denominators. Other students who understand that fractions represent division may divide the numerator by the denominator and find which quotient is larger. Other options include making two number lines so students can fairly accurately divide one number line into seven sections and another into five sections. Students who are comfortable with estimation may evaluate which of the two fractions is closer to one whole.

With the large number of options, and a problem in which an exact answer is not required, students come to realize that if they can’t remember a particular rule, they can create their own system of comparison. This approach also reinforces for students the benefit of knowing supporting concepts so they don’t get stuck because they can’t remember the algorithm—a memorized procedure they can reproduce but don’t necessarily understand. The important message with multiple-approach problems is that participation is not limited to the students who are faster or always correct, because you emphasize the value of the different ways of approaching the problem, not just the solution. If a student devises an appropriate method to reach a solution but makes an arithmetic error, he or she can still be recognized for the accuracy of the reasoning. You can take this method, demonstrate how it works perfectly when the subtraction or addition is corrected, and prove it by using the method to solve a similar problem with different numbers. The student who suggested this method will feel the dopamine reward of a correct approach because he or she realizes that that approach can generate lots of correct answers. The student has discovered a concept or creative idea that belongs to him or her and is a useful tool.

#### Strategy: Use Problems with Multiple Correct Answers

Problems with multiple correct answers are fun for students to work on individually or in pairs, then in small groups where they explain their different solutions to their classmates. This approach lowers mistake negativity because students know they can use their learning strengths, strategies they recall, and rote facts they remember, and then check with a partner before sharing with the class or group. These problems also provide for achievable challenge because students who find one way are instructed to keep going and find as many variations as they can. This gives students who need more time the comfort of knowing that they won’t be the last to finish.

For example, a game called “This Is Not a…” encourages multiple solutions and is played in a relaxed environment that encourages creativity. Students pass around an object—such as a toy telephone—and say, “This is not a…” Younger students name an object that is not a toy telephone (for example, “This is not a pencil.”). Older students continue and say, “This is not a toy telephone, it is a…” and they gesture or mime to suggest the object that they are pretending the toy telephone is—perhaps a computer mouse or a hair dryer. The gesturing students can eventually name what they are pretending the object is, or classmates can guess. Games such as this one help students become comfortable with a certain amount of ambiguity and gain the confidence to speak up when appropriate.

The most important learning activity in math, or any subject, is participation. This naturally leaves one open to making mistakes, but the brain learns by restructuring neural networks that make incorrect predictions. Playing this game lets students participate without the fear of making a mistake, which gradually builds confidence to participate, even if the response is wrong.

#### Strategy: Learn from Mistakes

One of the most effective ways to help reduce students’ fear of making mistakes is to model the process of how you learn from your mistakes. You can then progress to demonstrating how students can learn from errors you purposely generate, and when students are prepared to reflect on, not react to, mistake negativity, they can be guided to learn from their own mistakes.

Strategies in this category aim to lower the emotional overreaction the brain has to mistake negativity; to help students discover the motivating memories of perseverance, including perseverance through mistakes; to build students’ tolerance for mistakes; to reduce excessive anxiety-related errors; and to encourage students to strive for achievable challenge. These strategies, and your modeling, help students understand the value of mistakes.

Start a discussion with a statement or question such as, “Does a guitar player prefer playing songs with chords only after mastering dozens of chords?” Then promote discussion of personal memories of mistakes that led to success and gratification: “Can you describe a time when you kept trying even though you felt like giving up?” “How did you learn to play soccer so well?” “Do you still enjoy the beginner snowboarding runs now that you are advanced, or do they seem boring?”

This discussion will remind your students that once they built up their skills in playing sports, a musical instrument, or video games, it became boring to stay at the same level—but they made mistakes as they took on challenges to advance. Gradually, with effort and practice, they made fewer mistakes and enjoyed the pleasure of doing something with greater skill. When they make the connection to math challenges, students come to understand that mistakes are a natural part of new skill development in math just as they are in mastering a new video game or an athletic skill.

You will reduce your students’ fear of making mistakes when you help them understand that when they feel disappointment or embarrassment from mistakes, their brain is working well, and the rewiring that’s occurring will lead to a smarter brain for future answers. For example, an error recognized in homework or during class participation may be disappointing, but without that response, the brain would not be stimulated to reprogram the faulty information pathway, and the same mistake will be made again.

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Making mistakes in front of others is stressful for most students, yet correct understanding is constructed as much from recognizing mistakes as from rote rehearsal of procedures. The most brain-friendly environment is one that encourages participation and corrects the assumption that making errors means you are not smart. A positive, growth mind-set can become integrated into the class culture by using the strategies in this chapter to increase participation, reduce mistake anxiety, and build students’ confidence in the brain’s great power to grow smarter from mistakes.