During my investigation into why students dislike math I discovered that one of the main reasons for the hatred is that they feel incompetent. Understandable – no one likes to do something for which they lack the skills. The reasons underlying the incompetence are vast: one possible explanation is that much of math is not intuitive due to the lack of evolutionary need for it, and yet we have evolved in our acquired math capacity exponentially (pun intended). Another possibility involves the way math is taught. There seems to have been two opposing approaches to math education that put more effort into warring each other for the victory crown than attempting to understand what learning development and research says about the worth of their strategies. Turns out that the hands-on conceptual approach and the procedural approach that includes rote memorization are equally important and work together to effectively teach math skills. Imagine that?!? All facetiousness aside, I unearthed a further factor contributing to math competency. Since mathematics knowledge depends upon cumulative gains, it turns out that children who are exposed to “more numeracy-related activities at home show greater proficiency at [school].” In a study by LeFevre et al., parental “reports of numeracy activities were correlated with their child’s math performance.” The study looked at the correlations between activities like card games that directly involved skills such as counting and recognizing digits and activities such as baking that indirectly involve numeracy skills. The findings reinforced the hypothesis that the prevalence of direct and indirect numeracy activities at home is related to children’s “fluency with basic numerical skills, such as addition or number-line knowledge.” An interesting finding of the study is that a child’s “involvement in games predicted unique variability of the addition fluency measure,” that is, these children showed “substantial gains in their knowledge of number and magnitude.” While I don’t mean to pretend that certain math knowledge (such as LCM) is not important, the implications of this study with regards to what I witnessed in the elementary school classroom, are twofold. First, the students that I worked with must not have had the opportunity to engage in games and other numeracy activities at home for their unfamiliarity with numbers and basic math skills was obvious. (I blame it on television, but that's another blog.) Secondly, if games are a valuable tool for imparting math knowledge, teaching the class how to play cribbage and involving everyone in a round robin tournament may be a much more effective way to spend math class with a group of beginners than asking the same students to answer what is the lowest common multiple between two numbers on the chalk board. It is important to remember that as their teachers we must make the best choice for the direction of their learning. If they are two or more steps behind, it makes no sense to try to go forward. Playing a card game such as cribbage is a fun and sneaky way to slip math concepts into a child’s frame of reference! Something to think about anyway! References LeFevre, J., Skwarchuk, S., Smith-Chant, B. L., Fast, L., & Kamawar, D. (2009). Home numeracy experiences and children's math performance in the early school years. Canadian Journal Of Behavioural Science, 41(2), 55-66. doi:10.1037/a0014532
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According to Dr. Daniel Ansari, there has been a fearsome ongoing debate about the most effective way to teach math—rote learning tactics versus discovery-based strategies—that has failed to take into account the vast research findings about how children learn math.
The side that advocates “procedural knowledge” emphasizes explicit teaching of strategies and encourages students to memorize facts while the side advocating “conceptual knowledge” focuses on student construction of knowledge through hands-on materials, strategic-invention, and solving of open-ended questions that do not involve memorization. Neither side, in this polarized and emotional debate, has any use for the other. Keeping the previous blog post in mind (that some mathematical procedures are not intuitive and the subsequent symbols are not mental representations), it would seem that conceptual knowledge could only carry us so far. It seems we require the environment (in this case the math symbols as objects and procedures) to carry some of the cognitive weight. But any observer in today’s classroom can witness the limitations of rote memorization as well. Indeed, Daniel Ansari’s investigation found that children learn best when procedural and conceptual approaches are combined. “Researchers [like Bethany Rittle-Johnson] have demonstrated that an effective use of instructional time in math education involves alternation of lessons focused on concepts with those concentrated on instructing students on procedures.” It turns out that different parts of the brain are used in the various methods such that both approaches are interrelated and mutually determine successful outcomes in math acquisition. One of the specific arguments by the discovery-based camp targets time limits in math education: they oppose tactics such as the commonly known as “mad-minutes” wherein students must calculate multiplication products quickly. Ansari points out that there is no evidence that “speeded instruction necessarily has negative consequences.” In a study to which Ansari contributed, he found that young adults who performed better on high school math tests had more areas of the brain active: specifically, the fact retrieval area located in the left hemisphere was active whereas the students who achieved lower scores “recruited brain regions associated with less efficient strategies, such as counting and decomposition in areas of the right parietal cortex.” The “data suggests that math fluency and its neural correlates contribute to higher-level math abilities.” Since speeded practice in combination with other approaches results in “larger gains” it is thus useful for helping “low-achieving students in overcoming their math reasoning difficulties.” With his emphasis on research-based evidence, Ansari advises us to consider what is appropriate for the developmental level of the student when considering the sequence and content of learning. He leaves us with this important reminder: “[l]earning math is a cumulative process—early skills build on the foundations for later abilities.” If students enter the classroom without the necessary background in mathematics, if numbers and patterns are largely unfamiliar to students, then we must go back to square one with them. The next blog takes a peak at what that might look like in the classroom. References Ansari, D. (2015). No more math wars: An evidence-based, developmental perspective on math education. Education Canada, 55 (3), 16-19. Are mathematical symbols merely symbolic notations that exist separately from the abstract mathematical truths that underlie their actions? Some would say yes, that mathematical cognition occurs after we have converted the symbols into meaning, a meaning that exists outside of space-time, and a meaning that is as equally contained in an ancient mathematical manuscript as it is in a recent explication. Others think that perhaps mathematical symbols do something more than simply express mathematical concepts: that they “enable us to perform mathematical operations that we would not be able to do in the mind alone.” By assuming part of the cognitive load, mathematical symbols become “epistemic actions.” From the perspective of a mathematics philosophy, a causal account of how we obtain math knowledge is complex. Elementary numerical knowledge represents our ability to perceive natural numbers. From a young age we can see when there is more or less of something. Likewise we can “match the number of voices [to] the number of people speaking,” or the number of items in a hand to the number of items on a screen. However, many mathematical truths don’t commonly originate from direct sensory observation. Rather, they are attained from cultural sources. The mechanisms taught require cognitive reconstruction in order to understand the abstract structures underlying the concept. In many instances, mathematical mechanisms are not closely matched to our intuitions. For this reason we must build an entire mental framework around math concepts in order to make sense of them. An obvious example can be found in the number zero which equals both nothing and something at the same time. On its own it equals nothing. As a placeholder it means much more. All the properties of zero--for instance that it is the neutral element in addition (1 + 0 = 1), or that any number elevated to zero potency equals one, or 0! = 1! = 1--are not inferable. One of the ways we support mathematical cognition is through external devices such as calculators and slide rulers. Any device outside of the human that assumes some of the cognitive load is an epistemic action in that its “primary goal is to obtain information about the world.” When coins are sorted by value and stacked in tens, for instance, it is much easier to count them than if they are in a disorganized pile. By designating part of the task to the environment in this way, performance improves. In comparison, a notebook serves as an external memory device for someone with short-term memory loss. The extended mind thesis presents the notion of active externalism wherein objects within the environment function as part of the mind. The mind and the environment are a coupled system that function together with the same purpose. According to De Cruz and De Smedt, “[a] way to interpret the extended mind without contributing cognition and agency to artifacts is to argue that not all concepts are mental representations. One must then suppose that not all concepts can be entertained by human minds, due to intrinsic limitations of human cognition.” Using the example of ultraviolet light, which humans cannot perceive due to our lack of receptor cells for ultraviolet light, De Cruz and De Smedt show that we can entertain the concept of ultraviolet light without actually having a mental representation of it because of the special instruments we’ve developed to capture it. Similarly, “mathematical symbols can represent objects that are not representable with our internal cognitive resources alone [and this implies] that not all mathematical concepts are mental representations.” Through their research De Cruz and De Smedt found that the use of symbols is cognitively demanding on students despite their familiarity with symbolic notation systems. They prefer to use lengthy descriptions that imagine particular situations to solve problems such as “how to obtain the number of girls in a class when the number of boys is known, and you know that boys outnumber girls by four.” They also discovered difficulty for students shifting from one symbolic representation to another. Students consistently failed to represent 1+7 in blocks; rather than laying them out in two separate groups one of one block and the other with seven blocks, they arrange the blocks such that they spell 1+7 by their configuration. Students require explicit instruction to use the symbolic objects successfully. To understand symbols requires that we “decouple meaning from materiality.” While this is a process that occurs early on in development—by the age of two most children can discriminate between representational objects (e.g. a photo of a bottle of milk) and the actual object—the parts of the brain occupied with this endeavor differs from task to task and culture to culture based on the way the symbolic systems were learned to begin with. Chinese-speaking students who learned math with an abacus rely more on motor-related brain areas when performing mathematical tasks than English natives whose language-related brain areas are more active during the same task. Likewise, there are differences in cognitive organization between literate and illiterate people. Brains of illiterate people show more white matter, the function of which is “to connect different functional areas of the nervous system [indicating] that literate people have less connections in their brain.” De Cruz and De Smedt explain that literate people can afford to forget since they can use daybooks, notes and the like to store memory. Mathematical symbols refer both to procedures (the value of pi is the ratio of a circle’s circumference to its diameter) and to objects (structurally π represents 3.14159265…). Studies indicate that the process of moving from procedure to mathematical object occurs in students as they learn new math skills. In a study of adolescents’ working with negative numbers, many mistakes were made due to “a studious application of rules, rather than an intrinsic understanding of negative numerosities; [thus] the best predictor of success in individual students was their ability to use the minus sign correctly, not their conceptual understanding. Adolescents aren’t the only ones struggling with negative numbers. As it happens, negative numbers stumped mathematicians for eons and were largely rejected until well into the 1800s. De Cruz suggests that negative numbers are non-intuitive because they lack evolutionary import; “there seems to be no compelling reason why natural selection should have equipped [us] to deal with negative quantities.” Thoughts such as these just wouldn’t endure without mathematical symbols to carry them, for the demand on our cognitive resources and attention is great in the course of the problem solving and cultural conditioning that occurs moment to moment in life. Could it be that students find math difficult for the simple reason that we weren't designed for it? Perhaps we didn't evolve to intuit higher mathematics equations, but when we look how far we've come along in our understanding of the universe through mathematics we cannot deny that something has evolved: perhaps it is our capacity for evoking mental representations from symbols. One thing is for certain, we didn't come to this kind of understanding overnight. Understanding mathematics requires skillful integration of a variety of innate and acquired numerical processes beginning at the simplest level. Therefore, the next step in understanding why students perceive math as difficult involves looking at the ways in which math is taught. References De Cruz, H. H., & De Smedt, J. (2013). Mathematical symbols as epistemic actions. Synthese, 190(1), 3-19. During a recent shared learning experience in a grade 6/7 class, I was discouraged by the number of students who moaned and groaned when asked to take out their math books in preparation for a lesson. My position in the class gave me the flexibility of working one-on-one with several students and I took advantage of this opportunity to try to discover what exactly was underlying the contempt. I was surprised by the certainty with which students asserted their math incompetence. They hated math because they “couldn’t do it.” Since I relate to the feeling of not wanting to work on something at which I am consistently unsuccessful, the explanation made sense to me. But why can't they do it? What is so difficult? Taken in small steps, math logically unfolds. Or does it? During the first lesson, the students were supposed to be working on lowest common multiples (LCM). The student with whom I worked closely wasn’t very familiar with the concept of multiplication, so the idea that some numbers shared something in common to do with multiplication confused her even more. I ended up spending the entire lesson one-on-one with her, first reviewing the idea of multiplication through arrays, and then working on a multiplication chart in order to show her how she might use the chart as a tool to help with LCM. Later that week, in considering why math is difficult for students, I read several articles which shed light on the topic in different ways. For the next few entries of my math blog I would like to share these ideas. The first article suggested that the trouble might lie in the fact that many math concepts aren’t easily translated into mental representations (this idea will be fleshed out further on). The article didn’t dig into the depth of brain development research, so there wasn’t a focus on the relationship between formative development and math readiness. However, due to my own experience with mathematics, both personally and throughout my daughter’s course of learning, I am convinced that math readiness is an important consideration in the elementary classroom where learners possess a wide range of differences in learning styles, numeracy familiarity (previous knowledge) and levels of cognitive development. Perhaps this is an obvious reflection, yet the observations I have made in recent classrooms confirm my suspicions that math readiness is often unaccounted for—that is to say, it appeared that all students were expected to be able to do the work presented in the lesson, and yet they were not all able to complete the tasks. I realize the issue is more complex than student readiness; there are also considerations such as how well the teacher knows the subject, and how thoroughly it is taught. The second article speaks to the so-called “math wars,” a debate between “procedural knowledge” and “conceptual knowledge” as the most effective approach for math instruction, offering an evidence-based, developmental perspective on math education. The third article presents evidence that early home experiences that expose children to quantitative activities and indirect numeracy activities (such as card games) provide a foundation for better acquisition of mathematics in school. The study thus implies that students who may have missed out on home numeracy experiences could benefit from similar types of experiences in school. Perhaps, playing cards instead of learning about lowest common multiples will be a better use of time for the development of numeracy skills. Before we come to any definitive conclusions, we should look a little closer; please follow my math blog to see how the evidence unfolds. |
AuthorNatalie Nickerson; that's me. Archives
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