In my previous post, I shared some of David Sousa’s findings related to elementary students struggling in math. These kids find themselves in a tangle of partially and incorrectly learned skills and procedures. They are uncertain about how to begin problem solving and which operations are appropriate. They are not fluent in math facts. These kids often become highly anxious about math, describing themselves as incapable of learning. They may act out in school or withdraw; either way, they define their situation as hopeless. When they attempt homework, they valiantly try to recall the procedures, but are mostly unsuccessful. These are the kids who cannot tell me what topic they are learning in math.
As in all areas of special education, the first step is gradually leading kids to an understanding of where they are, while building confidence and hope that relearning is within their ability. This is a difficult undertaking for both me and my student. Struggling math learners have a hodgepodge of knowledge. If they are identified in upper elementary grades, that tangle is enormous. I have found that the smartest students have the greatest tangles. Some of these students are twice exceptional, gifted with a disability. They have been able to stuff many partial facts and procedures into their memory and may have camouflaged their disability for a few years. Students with weaker memory may be easier to “retrain” because they have less to unlearn.
I described this process as “tricky” because of the emotional aspects associated with relearning. These smart kiddos have been told, “You can do it!” because their working memory gives them the appearance of deeper understanding. These bright kids often have strong metacognitive skills: they are aware that something is wrong and may have interpreted that as “stupidity.” Anxiety has now begun to seriously impact every math lesson. They become hyper-vigilant about their performance, expecting to make a mistake at every juncture and dreading tests.
Depending upon the student, I disclose enough information about their weaknesses to provide motivation but not so much that they want to run away! My goal is to infuse hope by demonstrating how much they HAVE learned. Systematic assessment is crucial to this process. Providing external motivation is often necessary, especially if they are phobic about math. They have not yet experienced the value and joy of truly learning math skills, so they need something to get them started.
Brain-based research gives us a clearer picture of the optimal times for learning new content. According to Sousa in his book, How The Brain Learns Mathematics, there are two “best” times for learning: at the beginning of a lesson and and the end. Using a 40 minute lesson as a model, he explains that the brain’s capacity to download and retain new information declines in the middle of that lesson. This model of learning also makes plain sense. Kids’ brains (like that of adults) have a limited capacity to maintain attention and absorb and apply new information. After a “high” point of acquiring information and a relative period of reduced retention, there is another maximum learning opportunity in the last portion of the lesson (these times are approximate, of course).
How does this affect special needs kids who are struggling in math? The initial explanation of skills and procedures was not clear to them. It may not have meshed with previous learning (often because the previous topic was not not learned adequately or correctly). It may have triggered anxiety about past math failures. It may not have included visual cues or manipulatives. It may have had too much information presented too quickly. Guided practice may be completely embedded into the initial instruction, so that student were overwhelmed by both new vocabulary and new procedures. When these kids are “released” into independent practice at this point in a lesson, the teacher may not be available to provide corrective feedback, so the kids practice incorrectly. And practice makes permanent. Kids are also hitting that learning slump in the lesson, along with increased anxiety and perhaps task avoidance. Special needs kids may be heading down a dead-end road. When the teacher concludes the lesson with opportunities for students to apply this newly learned information to real problems, our special needs kids have partially memorized procedures, partial understanding of underlying concepts, and inadequate practice without corrective feedback. Yikes. Then they have homework on the topic, where they continue to practice incorrectly.
As Sousa points out, “unlearning and relearning that process correctly is very difficult…. (B)oth teacher and student have a difficult road ahead to unlearn the incorrect method and relearn it correctly.” (page 63) Obviously, the earlier the math intervention, the better the outcome. Younger learners relearn more easily and have had less time to practice incorrectly. Motivation to relearn is also a big factor.
There is hope. Stay tuned for how to navigate that “difficult road.”
In David Sousa’s book “How The Brain Learns Mathematics,” he cites researchers who have found that number sense is not intrinsic but can be shaped though both formal and informal activities. Sousa lists a number of effective strategies for developing number sense. Here are a few of them:
Meaningful estimates. Helping students practice meaningful estimates goes far beyond, “Guess how many goldfish crackers are in the jar.” In my experience, those kinds of estimates require more than number sense; they tap visual spatial understandings, as well. Estimation jars are often used in lower grades where many kids are unlikely to have sufficient practice with this skill. Early number estimates should involve items and quantities which can easily be counted (certainly less than a hundred) so that kids can improve their ability to estimate. Kids should also be exposed to multiple opportunities for making reasonable estimates. Teachers can create realistic and necessary opportunities for estimation. For instance, kids might estimate how many books are in a reading bin and how many pencils are in a caddy. It’s important for kids to learn early on that estimation is not just another tedious step on worksheets, that estimation is not a hurdle to make math more burdensome. One student recently blew off the estimation portion of his classroom assignment, saying, “It doesn’t matter what I get. Anything is right.” Clearly, estimation was taught in isolation and as a meaningless activity.
Solve problems and consider the reasonableness of the solutions. This strategy sounds reasonable, but for kids with math difficulties, an unreasonable answer can be hard to recognize. The kind of problem where I typically see younger kids struggling is comparing values or quantities. A classic question is: If Kevin is 8 and his sister is 5, how much older is Kevin? The majority of K-2 special needs kids I teach will add those two numbers and feel their answer is reasonable. They have learned to subtract when items are missing, eaten, or given away, but using subtraction to compare is another beast altogether. Some kids learn faster when they start solving this type of problem with pieces of cereal or other food. Using cubes that interlock or stacking blocks is another way to visualize the number comparison problem, helping them “see” how much more one quantity is than another. It can take a LOT of experience for kids with math difficulties to master this process. Simply asking students if their answer is reasonable is ineffective if not preceded by plenty of experience with manipulatives and real-life math problems.
Model the enjoyment of numbers and number patterns. Research studies conclude that the teacher’s attitude “is the most critical factor in establishing a climate for curiosity and enjoyment of mathematics.” For me personally, this statement means that I must watch myself for subtle (and not-so-subtle) signs that math has not been my favorite subject. As I set a goal of making math enjoyable for my students, I am also enjoying it more. I ask myself: If I am short on time, what subject is going to be curtailed? Do I look for opportunities to create and solve math problems? Do I encourage kids to talk to each other during math instruction? Is my math instruction engaging and meaningful? Integrating math into other content areas is a terrific way to make problem solving meaningful. Free Math is another classroom strategy to provide time and resources for those “random” math questions that arise during the course of the day. Dedicate a space where teacher and kids can jot/dictate questions on sticky notes, to be addressed during Free Math time.
Do you have favorite strategies for developing number sense?