We begin by presenting a description of the difficulties Dylan encountered as a student and an overview of the physical tools and compensatory strategies she used to make mathematics accessible. Second, we provide a detailed illustration of each of the compensatory strategies. As we discuss each strategy, we highlight Dylan’s issues of access, the specific ways she compensated, and the additional demands implementing each compensatory strategy placed on her. Third, we report on the structural barriers that Dylan encountered as she completed her degree in statistics. Finally, we present Dylan’s recommendations for educators working with students who may have dyscalculia.
3.1. Overview of Dylan’s Issues of Access, Tools, and Compensatory Strategies
Dylan experienced significant, persistent, and pervasive issues with mathematics. Dylan explained that she had difficulty inverting numbers (e.g., writing 18 as 81), distinguishing mathematical symbols, understanding dense mathematical notation (e.g., fractions), and understanding the magnitude of numbers and how operations might affect numerical values. To address these difficulties, Dylan developed a repository of tools and compensatory strategies that enabled her to access mathematics. Dylan used a collection of physical tools that she explained was essential for her to do mathematics, which included
Graph paper (4 squares per inch) to help her align and space her notation;
Mechanical pencils to ensure a consistently sized lead thickness;
A retractable eraser to enable her to erase precisely; and
Colored pens to color code her notes and help her track various symbols and elements.
She used these tools along with the following compensatory strategies when learning and doing mathematics:
Rewriting in words—she translated mathematical symbols into words (e.g., 3 as “three”);
Creating a dictionary—she created an internally consistent and differentiated way of writing each mathematical symbol so it had only one meaning and was not perceptually confused with other symbols;
Rewriting for consistency—she rewrote problems in a consistent form to make use of space to maintain clarity about her solution process;
Consistent solution process—she developed and used consistent solution processes to solve particular kinds of problems; and
Grounding the abstract—she grounded her understanding of values and mathematics concepts in real-world examples.
We illustrate each of these compensatory strategies below using examples and excerpts drawn from the video recordings. Note that during the sessions with Cindy, although they discussed tool use, Dylan used felt-tipped markers for visibility within the video and did not have access to her standard complement of tools.
3.1.1. Rewriting in Words
Dylan had difficulty understanding and tracking the meaning of mathematical symbols. She explained one of her strategies to make sense of notation was to rewrite the mathematical symbols in words. This helped her understand the meaning of the symbols and helped her solve problems. She gave an example of how she would write out a problem like 5 + 3 = to help her make sense of the symbols, and how this was a strategy she still used with higher-level mathematics such as calculus.
Dylan: “I am stronger with language, so this is a thing that I did when I was a kid. And then I actually started doing it—it came up more in my higher level math classes. If I have like five plus three equals eight [writes 5 + 3 = 8; see Figure 2
], I actually do really well with language. [writes out “five plus three equals eight”
By translating the symbols into words, she would more readily be able to connect the symbols to their underlying meaning. She explained that mathematical notation was a condensed notational form of writing words; “math is shorthand for this [points to “five plus three equals eight”]”. It was this “shorthand” for quantities that Dylan found hard to access, which was why she routinely translated these kinds of symbols into words. When explaining this approach to Cindy, she went on to give an example for calculus and how she would regularly translate function notation (i.e., f(x)) into words to make it more understandable.
Dylan: “This is calculus, but you can see it illustrated with this notation [writes f(x) = x + 4x3; see Figure 2
] this notation, the way people say this is “f of x” which is also terrible. It’s the function
equals this [as writing “function of x = x + 4x3; see Figure 2
]. This little notation here [points to f(x)
] would throw me off really badly in my classes, because f times x? No, it’s a notation that is basically applying this function to the variable x. I would sometimes write out something like this [writes bracket underneath “function of x”
] right underneath whatever it was and again, this is really verbose, [but] it might be helpful.”
Dylan noted that she often used this kind of translation in her notes to help her decode the meaning of the symbols, and described it as her creation of metadata for the notation. For example, she would translate the notation P(A|B) into “probability of A given B” (see Figure 3
). This rewriting into words was essential when she was learning new concepts or new notation.
This translation of symbols into words took Dylan extra time both when writing notes and when solving problems. Because this kind of translation was not available in her classes or textbooks, she paid tutors to provide this kind of support. She explained that “I would force tutors to give me the English words for the symbols, which was always funny because these are grad students who haven’t thought about this stuff in years. ‘How would you use this in a sentence? I haven’t thought about it that way’”. The kinds of supports that Dylan needed were not something that the tutors were skilled at providing. Although this compensatory strategy provided her with a way of understanding the mathematical symbols, it placed additional demands upon her requiring that she spend more time and money than her peers to have access to mathematics.
3.1.2. Creating a Dictionary
Not only did Dylan experience difficulty understanding and decoding symbols, but she experienced issues of access when mathematical symbols were perceptually similar. She perceived several numerals to be perceptually similar and explained that, “I have a hard time with ones and sevens; and threes and eights”, she also had difficulty differentiating her fours from her nines and her sixes from her fives. To address these issues, Dylan created her own system of distinct and consistent ways of writing each symbol, which she referred to as her dictionary. “That is one of the techniques I rely upon really heavily: [I’m] super consistent in the ways I write things and the notation I decide on. Trying my best to make it really distinct, so my ones don’t look like my sevens and my threes don’t look like my eights”. The perceived perceptual similarity was not limited to the 10 numeral symbols. For example, she explained her difficulties differentiating the symbol x from upper case X and the plus sign.
Dylan: “There is the lower-case x and the upper-case x, and God forbid you rotate it and it looks like a plus. Coming up with some distinct way so that I know what my variable means and then the language that I associate with the way that looks right.”
One of her primary goals was to “make [mathematical symbols] very distinct from one another”, so that when she saw the symbol she was able to tell what it referred to. “I [have] come up with a really consistent form of notation, so it’s unambiguous”. Because she had difficulty in differentiating both number symbols and other mathematical notation, she explained: “I was dogmatic about how clean and crisp I wrote my notation to differentiate it as much as possible”. This was one of the reasons she relied so heavily upon mechanical pencils that could provide lines of uniform weight and a retractable eraser to erase precisely.
Dylan also had difficulty if one symbol was used for two different meanings (e.g., x standing for both the multiplication operation and an unknown). She explained how sometimes “one symbol has multiple meanings. And math is really lazy and they tend to reuse [notation] to mean different kinds of things”. She explained that she would differentiate the different meanings by writing each symbol in distinct ways depending upon the context.
Dylan: “So I would be taking two math courses in the same semester and one would use an “r” for one meaning and one would use an “r” for another meaning. I would have to do a lower-case r for that class and an upper-case r for this class.”
Although this often enabled her to differentiate the meanings of the same symbols, “I had to sometimes go back and reassess”. One of the challenges of maintaining this system was that as she moved through higher-level mathematics courses she would need to revise her dictionary. She explained: “Then, the next semester I would take a different class and all of a sudden the r would have a third meaning and I would have to reassess my dictionary”. Although this dictionary enabled her to access and understand the meaning of symbols consistently, it was challenging to maintain, particularly as the mathematics notation got more complex and symbols were reused for new purposes.
3.1.3. Rewriting with Consistency
Dylan also experienced issues of access with the density of mathematical symbols. She often would rewrite problems to space her notation out in a uniform way. For example, when given a problem in a book or on a test, she would rewrite the problem in her own handwriting often using graph paper, in which each symbol occupied one square of the graph paper (see Figure 4
Dylan: “Another thing I do is, although I could have started working off your problem, I won’t. I’ll rewrite it in my own handwriting. One, because I’m familiar with my handwriting. And two, because it forces me to rewrite the problem again, so I know what I’m solving.”
Dylan’s rewriting would enable her to use her dictionary of consistent and distinct symbols to rewrite the problems into an accessible format. “When you see the symbol, the first thing I do is, depending upon what type of problem I’m doing, I rewrite it in my handwriting that I have identified as unambiguous.” When there were multiple canonical or acceptable ways to write a problem, she would rewrite the problem using the form that she preferred. “I come up with whatever operator I’m most comfortable with and before I even attempt a math problem, I rewrite it.” She gave examples of how she handled both multiplication and division, which had multiple notations that were not all equally accessible for her. For example, multiplication can be represented with an x, a dot (i.e., •), or with parentheses. Although she had used a dot for multiplication throughout high school, she found that this was problematic in her advanced mathematics courses. “It becomes problematic in higher level math beyond this, in linear algebra it means something else.” She also avoided using the dot as multiplication because it could be confused with a decimal: “I’m also careful with it because of decimals—it floats up and down [gestures up and down with pointer finger].” Because the dot had a different meaning in linear algebra and because the dot could be mistaken for a decimal point, Dylan would rewrite any problem that used the dot symbol for multiplication. She gave an example of the problem: 4x • 3 = 5.
Dylan: “So let’s say we have the problem 4x
times 3 equals 5 [writes 4x • 3 = 5; see Figure 5
]. If I saw this written down, I will actually rewrite it like this [writes 4x(3) = 5; see Figure 5
], so parentheses are used as multiplication. I come up with a really consistent form of notation, so it’s unambiguous.”
Similarly, notation for division was also problematic; Dylan found the standard division symbol inaccessible. For the problem 6 3 = she explained, “This is terrible [pointing to the division sign in 6 3 =] is it the 3 divided by the 6 or the 6 divided by the 3?” She also found the perceptual similarity of the long division sign and the square root sign to be problematic, so she tended to rewrite division problems using the fraction bar symbol.
Dylan: “Division [writes , see Figure 6
] is typically taught like this to younger students. I despise this symbol, because I’m not sure what is acting on what. Long division is a little easier [writes long division sign to right of ; see Figure 6
]; however, long division, once you start doing square roots [writes square root sign below long division sign; see Figure 6
], so I actually use the fraction symbol [draws fraction bar; see Figure 6
Although rewriting problems to use space and her preferred notation enabled Dylan to access the meaning of the mathematics symbols, she acknowledged that—like many of her compensatory strategies—it took more time and effort. For her, the tradeoff was worth it. She explained: “rewriting is another technique that I suggest for kids that adds more time, but increases my odds of accuracy.”
3.1.4. Consistent Solution Process
In addition to her difficulties with notation, Dylan described herself as having poor number sense [2
]. She often had difficulty understanding the magnitude of quantities (e.g., fractions, negative numbers) and how operations might affect their values. To address this, Dylan relied upon a set of processes that helped her solve problems. The consistency of these processes enabled her to know and trust her answer, even if she had difficulty understanding the quantities involved. In our previous work [2
] we identified six consistent processes that Dylan used across a range of mathematics topics. The consistent processes identified in our prior analysis were
Converting all fractions to a common base or language (i.e., denominator);
Showing all steps of her solution process;
Pulling out like terms (e.g., negatives or variables);
Distributing first to get rid of parentheses to simplify the problem;
Using her “abcd” method for solving fraction division problems (i.e., ); and
Solving problems as inverse operations.
We illustrate Dylan’s explanation of two of these consistent processes (converting all fractions to a common base and showing all steps) and how these consistent processes enabled her to make sense of and solve problems. Then, we present an exemplar problem Dylan solved in the interview that illustrated many of her consistent processes in one problem.
Dylan had difficulty conceptualizing the relative magnitude of fractions, which made comparisons and operations with fractions challenging. “It’s a very dense notation, when you think about the concept it’s trying to portray.” She often referred to fractions with different denominators as speaking different languages or having different bases. To address her difficulty in conceptualizing fractional magnitude, she used a consistent process of converting all fractions to a common denominator by multiplying the numerator and denominator by the denominator of the other fraction. For example, to compare the fractions 1/3 and 1/4, she explained that she would first determine a common base or language.
Dylan: “So this is the classic thing, that I see these two numbers and they aren’t in the same language. I would immediately try to get these into the same common denominator so I can try to understand them. And that’s where I have my cross multiplying thing that goes on [draws arrow between the denominator of each fraction and the numerator and denominator of the other fraction; see Figure 7
], one of those little rules. So three times four, to get them into the same base is 12. So now I will have twelve on the bottom for both of them [writes /12 and /12
], and to get this 4 times 3 is 12, 4 times 1 is 4 [writes 4
]. And 3 times 4 is 12 and 3 times 1 is 3 [writes 3
]. And now I can actually compare these two numbers [writes >
]. The three is smaller and now I can rewrite them [writes 1/3 > 1/4
In order to compare fractions, Dylan would convert them to a common denominator so that she could operate on the values as if they were whole numbers (e.g., comparing 3 and 4 and ignoring the denominator). After she determined the relationship, she then rewrote the fractions in the simplified and original form (1/3 and 1/4). Her use of the “language” analogy helped her understand how to operate on and conceptualize fractions, and she sometimes also used this language analogy to refer to like and unlike terms (as is seen in the next example).
The second consistent process involved Dylan showing every step of her solution process. Dylan sometimes had difficulty losing terms or symbols, and tracing a problem from one step to the next if multiple operations were performed simultaneously. To address these issues, as she solved problems, she represented each step of her solution process. This often involved more writing but it enabled her to understand the transformation from one step to the next and enabled her to catch her mistakes more easily. She gave an example of how she would write out every step when talking through her solution to the problem 12x + 5 − 3 = 6.
Dylan: “I’m a big proponent of writing out every single step including anything that is intermediate even if it seems redundant. If I have [writes 12x + 5 - 3 = 6; see Figure 8
] and I wanted to solve for x. This is actually a good one. I’ll just go through it with my steps of how I would do it, so you can see the way I think through problems. So, the first thing that I immediately see is that I want to start combining things that are similar. I use the word language a lot so these guys [points to 5 and 3
] speak in the same language, because they are just numbers, this one [points to 12x
] has a variable attached to it so it’s speaking in a different language, there is one that is just a number [points to 6
]. So, I’ll start with the language that these guys are speaking. So, I’ll rewrite twelve x [writes 12x
] because I’m not going to do anything to it. Plus [writes +
] I’m not doing anything to. Um, five minus three, now I’ve got 2 [writes 2 =
]. I’m not doing anything to the 6 [writes 6
]. Now I want to get rid of this two because it speaks the same language as the 6. So I need to, and this is where my colors come in. So, I’m going to minus 2 from this side [writes −2
] and minus 2 from this side [writes -2
]. And sometimes I’ll rewrite this out [writes 12x + 2 − 2 = 6 − 2
]. So, now I’m leaving the 12x alone [writes 12x
], two minus two is zero [writes +0
]. This is one of the steps I may or may not do depending upon how much I have going on. Six minus two is 4. [writes 4
]. Now I can get rid of my plus 0, since it doesn’t do anything 12x = 4 [writes 12x = 4
]. I’ve got to get this 12 somehow over here, so I can isolate this x by itself. I know I can divide by 12 [writes /12
] and divide by 12 [writes /12
]. 12 divided by 12 is 1. One x equals four-twelfths [writes 1x = 4/12
]. This [pointing to 1
] is redundant because 1 times x is just x. [writes x = 4/12
] I can go further and simplify that, but I don’t think I’ll do that right now.”
As Dylan solved the problem she was careful to copy over each part of the problem that remained unchanged and made sure to only do one operation each step, often rewriting the equation so the operations were embedded within it rather than notated on a separate line. Dylan illustrated in this example how she used color to help her differentiate between the equation (blue) and the operations applied to the equation (green). Her deliberate effort to show all steps helped her track her progress through the problems and avoid mistakes.
To provide an example of how Dylan’s consistent processes might make a problem lengthier, she explained how she would solve the problem 3(x + 2) = ½(3 − x). Although most statistics majors might solve this in four or fewer steps (e.g., 6(x + 2) = 3 − x; 6x + 12 = 3 − x; 7x = −9; x = −9/7), Dylan solved this problem in 14 steps. Her written work (with annotations) illustrates how she used several of her consistent processes as she solved the problem (see Figure 9
In this problem, Dylan made sure to show every step of the solution process, and this included copying the original problem Katie had written into her own handwriting and copying the current equation verbatim when moving to a second sheet of paper. Both these reproductions were considered part of her step-by-step process because they were necessary for her to make sense of the symbols and to help her track the problem as she switched to a new sheet of paper. As she copied over the initial problem, she stated “Oh—that’s one-half, I was reading that as y2.” Katie’s representation with a slanted fraction bar was perceptually similar to a y2 and so Dylan took the deliberate step of differentiating this notation. When she copied the equation verbatim onto her second page, she explained: “I’d recopy that over, this is why grad students hated grading my homework.” Throughout this problem, she tended to operate on only one side of the equation at a time, copying the other side without any changes. She explained: “I try not to do multiple operations at once. [It’s] also a good way to catch errors.” As was often the case, Dylan’s consistent processes (distributing first and always converting to a common denominator) resulted in her solving a more lengthy and complex problem.
As was evident in each of the examples, Dylan’s consistent processes enabled her to make sense of hard to conceptualize values, and provided a consistent approach to solving problems to enable her to track her progress. This approach, however, often resulted in more time and effort because of the demands of each of the consistent processes and because implementing these processes sometimes led to her solving a problem in a less optimized way and therefore solving a more difficult problem.
3.1.5. Grounding the Abstract
Dylan had difficulty understanding abstract or decontextualized numbers, operations and concepts. She explained that numbers and concepts were often not sufficiently connected to meaningful contexts: “You learn this really abstract concept for 5 + 3 = 8. What is the 5? What is the 3? What is the 8?” To try to make sense of numbers and concepts, she would try to connect these symbols and ideas to the real world. She recommended helping students develop a meaning of number by starting with manipulatives: “Now we have these objects and we can start to label them.” “I found visual things to click with me a little bit better, if someone uses manipulatives.” Dylan found that manipulatives helped her connect to the underlying meanings of the symbols, and having an opportunity to move, cut, combine and interact with these manipulatives was helpful for Dylan in conceptualizing quantities and operations on quantities. “When I’m struggling with a concept, if I can give it a really concrete foundation, it will stick with me much better.”
In addition to concretely representing quantities with manipulatives, Dylan found it helpful if she could connect the concept to a real world application. Because she found mathematics connected to application easier to understand, she found statistics more accessible because it had an applied component where patterns in data were quantified.
Dylan: “The reason why it’s funny that I did statistics, but it makes a lot of sense, is that if I could take one of these concepts and apply it to something that was actually tangible to me, so like the percentage of people that are over the age of 55 that have a landline phone, if I come up with these stories or narratives, then I could actually recall how to do an operation, but I had to have some sort of narrative that was actually based in the real world.”
She found that the internet was a good resource for helping her understand the purpose behind particular concepts or ideas. “If you are approaching a subject, before you even dive into it just go out into the world of the internet and try to figure out how people even use this and then go back into the abstract.” She explained,
Dylan: “Sometimes I would google “real life examples of blah blah blah” because even if it wasn’t that exact problem, all the sudden if I could connect to that—Oh! That’s why I need to know this. I really struggled with math when I was like “why are we learning this?”
Dylan found herself often turning to the internet for these kinds of real world connection because her tutors were often ill-equipped to provide this kind of grounding, which she described as “another big problem in the math community.”
3.2. Structural Barriers
In her discussions with Cindy and her interview with Katie, Dylan identified several structural barriers she encountered. These were policies and practices which marginalized her participation in upper division mathematics classes. Dylan explained “in my first semester, I failed my two weeder classes.” The “weeder” courses (e.g., calculus and linear algebra) are predicated on the assumption that if a student cannot pass the class, they are not enough of a “math person” to succeed. After being diagnosed with a disability, she was told that the primary accommodation for students with dyscalculia at UC Berkeley was to waive the university’s mathematics requirement. We view this as an exclusionary practice, which, rather than accommodating for disabilities, excludes those individuals from participation in mathematics—and by extension all other STEM fields. Not only were the policies exclusionary, but she was actively discouraged from pursuing and completing her degree in statistics.
Dylan: “The math department is very unaccommodating. People in math tend to have an attitude of, if you can do it then great, if you can’t, then you’re just not meant to be. Which is ridiculous, because it’s not acceptable to not be able to read, so I don’t know how it’s acceptable to not be able to do math. I got told that I should probably drop out of Berkeley and reapply the next semester into a different degree program. At this point I was a junior in college. Which is not really possible, it’s like, no I’m committed.”
The institutional policies and advising practices were exclusionary and Dylan was left to figure out how to navigate this system on her own.
Dylan: “The way I was able to get through was brute force. I had this professor, and she said there are two kinds of mathematicians: those that are smarter than everyone else, and those that work harder. And so I worked harder. I took classes multiple times, I created this incredible library of stuff, I have this really specific system of how I write and rewrite… In order to learn a basic concept, I would attack it from every angle until something stuck. So it was just brute force, and working harder than everyone else. Which is frustrating, right? Because you are being asked to do a phenomenal amount more work than anyone else in this class has to do and it would be really lovely if we just lived in a world where we were given lots of options of all these different ways of learning this one particular concept. And unfortunately, right now you have to cultivate all these different ways yourself.”
Not only did she have to spend considerably more effort than her peers just to transform the mathematics into a form that was accessible, but she was given no support to do this.
Although she was able to succeed by developing compensatory strategies, these all required more time and effort, which was not easily accounted for in the standard instructional or assessment approaches. Dylan explained how her compensatory strategies resulted in her having to take classes multiple times:
Dylan: “When I’m learning something, I try to remove myself really quick, and go learn why would I ever do this, what are the definitions of all these terms, then come up with my note taking system for how I’m going to do it, then I can dive into the subject. Needless to say, this is why I had to take so many classes in college a couple times.”
Dylan’s demanding compensatory strategies did enable her to access mathematics, but not always at the pace that she needed to pass her classes the first time through. Dylan remarked that taking classes multiple times was “very expensive,” and paying for tutors placed an additional financial strain upon her.
In these sessions, Dylan identified time as the most significant structural barrier: “Time is always the hugest block.” Implementing her compensatory strategies took more time, and this was time she was not afforded on assessments. She explained how this time constraint was arbitrary and did not match her experiences in her career as a data analyst.
Dylan: “It’s really really unfortunate to me, I think that, in mathematics, it’s not just that you understand the concept. It’s not just that you can get a correct answer, you also have to do it in a certain amount of time, which inevitably is like this really bizarre artificial constraint. Especially now that I’ve worked in industry at as a data analyst. And yes, I had time constraints, I had to get a report done in this amount of time or whatnot. It’s nothing like sitting down for a test and you have 60 min to do this insane page of stuff and this huge mental dump, where these kinds of rewriting things would really become problematic, I would just flat out run out of time. I would have to either not rewrite things and inevitably make mistakes or rewrite things and not finish exams. I would have to make calculated decisions at the beginning of exams.”
Not only were the structures of assessment (time constraints) and the policies (weeder classes and disability waivers) sending messages to Dylan that she did not belong in upper division mathematics classes, but she often had graduate students devaluing her legitimate approaches to solving problems. In implementing her compensatory strategies, she often encountered resistance because she did show all her steps and methodically wrote and rewrote notation so that she could understand its meaning. “I would also frequently get the professor’s assistants that would grade our work; they would gripe that my solutions were too long”. Graduate students, prioritizing the economy of notation, identified Dylan’s processes as problematic.
Dylan: “I had grad students—one in particular, I think it was multivariate calculus—he would write on the top of my assignments “please shorten steps” or whatever because he has to grade x numbers of assignments and he hated how verbose mine were.”
Dylan experienced significant structural barriers in completing her statistics degree. These existed within the policies and practices at the institutional level, and also in her individual interactions with people tasked with evaluating her competency.