#### 3.1. Myth One: Students with LD Cannot Benefit from Inquiry Instruction in Mathematics

Most often, I hear some variation on this myth, formulated multiple ways: students with LD cannot benefit from inquiry-based mathematics, or that students with LD must be taught using only explicit, direct instruction. This myth widely circulates, both in academic literature and in schools. I hear this myth from both special educators and mathematics educators.

This myth about students with disabilities has developed around the already existing parameters of larger debates about pedagogy in mathematics. When we compare research in mathematics education to research in special education that includes mathematics, we find two different research traditions who use different theoretical approaches to learning, as well as quite different methods [

30]. Mathematics educational research has long been dominated by constructivist and sociocultural theories of learning, which tend to advocate for an inquiry-based pedagogy at both individual and collective levels [

31]. Rather than breaking down topics into sub skills and teaching those directly, inquiry-based pedagogy first engages learners in problem-solving and discussion of complex problems, asking students to develop strategies without being explicitly taught them. Mathematics educational research has documented the effectiveness of inquiry approaches to learning mathematics [

8,

9,

10,

11], including for low achieving students [

32,

33]. Mathematics education has typically not included students with disabilities in research [

4,

30].

Advocates for direct instruction have long been particularly influential in special education research, which developed from behaviorism and experimental psychology [

34,

35]. Recommended pedagogy within the academic fields of special education tends to be “explicit, systemic instruction,” which “typically encompasses a step-by-step teacher demonstration for a specific type of problem along with teacher-guided and independent practice using the step-by-step procedure.” ([

36], p. 41). Special education mathematics has also been influenced by strategic instruction and, more recently, aspects of constructivism [

4]. Yet historically, special education has long been suspicious of constructivist theories of learning [

34]. Within the field, inquiry instruction was often framed as “discovery learning,” and presented as if there is no teacher guidance, an inaccurate oversimplification of the approach. When Jones and colleagues [

5] noted that allowing students with LD to construct their own ideas in mathematics is “illogical, indefensible, and unsupported by empirical investigation,” they are making a pointed critique of constructivist mathematics, which they described as an “ideology”.

There is a significant research base of quantitative studies that demonstrate the efficacy of explicit, direct instruction in teaching mathematics skills and procedures for students with Learning Disabilities [

37]. The myth emerges from the assumption that there is sufficient evidence that inquiry mathematics is

not effective for students with LD, or that explicit instruction is the

only method that is evidence-based. As the National Mathematics Advisory Panel states, “it is important to note that there is no evidence supporting explicit instruction as the only mode of instruction for students [with LD]” ([

38] p. 1229).

Another source for this myth is research in special education that has argued theoretically that students with LD cannot benefit from inquiry instruction because of their deficits. From one academic piece within special education: “To expect students who have a history of problems with automaticity, metacognitive strategies, memory, attention, generalization, proactive learning, and motivation to engage in efficient self-discovery learning…is not plausible.” ([

39], p. 296). Some use the argument that the cognitive load of inquiry instruction will be problematic for students with LD because of working memory issues [

40]. Others assume that students with MLD will be “confused” by inquiry-based mathematics because such instruction “places substantial demands on metacognitive skills, which may not be adequately developed in children with MLD” ([

41], p. 52). This returns to the quote that opens the paper [

5], in which the authors stated that it is illogical to assume that students with LD can construct their own knowledge in mathematics. These arguments frame LD with a medical model, which stresses the deficits associated with the disability. It is important to note that these quotes are taken from papers written between 1994 and 2004. I argue not that these exact statements would be made today, but that a legacy of these arguments is the pervasive myth that inquiry instruction cannot possibly work for students with LD.

However, there is evidence that students with LD can benefit from inquiry instruction in mathematics. In a series of rigorous mixed methods studies spanning over a decade, Bottge and colleagues have developed and assessed a curriculum specifically designed for students with LD to participate in inquiry mathematics, called Enhanced Anchored Instruction [

42,

43,

44,

45]. Drawing from both sociocultural theory, problem-solving in mathematics, and special education, this approach uses highly motivating, authentic contexts to scaffold the engagement of students with LD in inquiry-based mathematics. Through the use of video, researchers deeply immersed students with LD in contexts such as building skateboard ramps or hovercrafts to learn rational number concepts. By removing textual barriers, their work allows students to deeply engage in rich mathematical contexts. Their work has demonstrated the efficacy of inquiry-based mathematics instruction for students with LD in both resource rooms and inclusive general education classrooms. They found that some students with LD benefited from scaffolds such as computer representations of objects that could be rotated, additional instruction in underlying concepts such as fractions, and hands-on projects related to the context, such as building a ramp.

Qualitative research that explores student thinking provides additional empirical evidence that students with LD can learn in inquiry classrooms. Moscardini found that students with learning disabilities were able to engage in open-ended problem solving, constructing their own strategies for story problems without prior instruction in procedures [

46]. Chick and colleagues report on the learning of students with LD within a unit on fractions, using a reform-based curriculum [

47]. Behrend has written about how a group of students with LD learned through engaging in open-ended problem-solving, again without prior instruction in procedures [

48]. Foote and Lambert described a classroom in which third graders with IEPs were able to develop understandings of relational thinking and equivalency through engagement in a weekly open-ended problem-solving routine [

49]. These studies provide evidence that students with LD can indeed learn challenging mathematics through inquiry-based mathematics. Similar findings have emerged in science, where inquiry-based curriculums have been effective for students with LD [

50,

51,

52].

Increasingly, researchers in special education mathematics have integrated aspects of constructivism into their research [

4]. For example, Hord and Newton, moving beyond a binary between explicit and inquiry, analyzed mathematics textbooks for both pedagogical features [

53]. While recognizing the value in each pedagogy, they question if explicit instruction introduced too early can actually negatively impact conceptual understanding:

“In the short term, explicit instruction is potentially effective to help students solve problems more quickly; however, this earlier introduction of explicit instruction may slow the progress of students with LD in becoming resilient, persistent problem solvers and developing deep conceptual understanding of topics.”.

They suggest that we need to consider the long-term time scale: inquiry mathematics allows students to take on agentic roles in relationship to mathematics and offers potential for long-term engagement with the subject [

12]. When instructional decisions are focused on narrow time scales, rather than the long-term goals of developing agency and empowerment, we lose the forest for the trees. At risk are our students’ identities as mathematical learners, which can suffer in a procedural curriculum [

14].

My own research with Luis echoes these findings [

14]. In the first half of his seventh-grade year, Luis’s wonderful teacher, Ms. Marquez, engaged her students in inquiry-based mathematics. One day, she had her students working on the classic problem of buying hot dogs in packs of eight, and hot dog buns in packs of six. The class was discussing how many of each would need to be bought so that there were exactly enough buns for hot dogs, leading into the concept of Least Common Multiple. The discussion seemed over, and Ms. Marquez was about to move on when Luis raised his hand. She called on him with a smile, and he brought up the concept of infinity, asking how they could predict how the numbers would behave into infinity. The discussion started right back up again. When I asked Ms. Marquez who her top students were, she listed Luis right away, noting that he excelled at “conceptual” thinking.

However, later in his seventh-grade year, when classroom instruction was focused on procedural learning, Luis lost focus and interest. Ms. Marquez, Luis’s seventh-grade-math teacher, faced considerable pressure to raise student test scores. In the second half of the school year, she focused heavily on direct instruction of procedures. While Ms. Marquez strongly disagreed with this shift away from inquiry-based instruction, she felt she must do so for the sake of the students, whose scores mattered in their high school applications. One day she taught the students a method for combining like coefficients, clearly outlining the steps. Luis raised his hand, curious about the 2x written on the board. “But when,” he asked, “WHEN will we multiply it?” His question suggested that he wondered how an expression, made of two parts that will somehow eventually multiply, would ever multiply. Faced with limited time before the test, his teacher moved on, providing more examples for the students to practice using the procedure. Luis put his head down on his desk, and did not engage in the rest of the lesson. In interviews, Ms. Marquez was deeply troubled by Luis’s shift in engagement, and suggested that the shift in pedagogy was the culprit.

#### 3.2. Myth Two: Students with LD Cannot Create Their Own Strategies and Should Not Be Taught Using Multiple Strategies

I hear often that students with LD can’t invent mathematical strategies, or deal with multiple mathematical strategies. Not only is this myth common in the US, but Peltenburg, Heuvel-Panhuizen, and Robitzsch [

54] describe this as a commonly held belief in the Netherlands. There are two closely connected questions here: (1) should students with LD be directly taught strategies, or should they develop their own? and (2) should students with LD be taught one strategy or multiple strategies? Implicit in the assumption that students with LD need direct instruction of strategies is an assumption that students with LD cannot construct their own strategies. This myth is closely related to Myth 1, and can be understood as a justification for that myth. In this argument, students with LD become overwhelmed with multiple strategies during inquiry instruction.

Recent studies have provided evidence dispelling this myth, demonstrating that students with disabilities and low achievers in mathematics

can independently construct effective computational strategies. Peltenburg, Heuvel-Panhuizen, and Robitzsch [

54] explored whether students in special education schools in the Netherlands would spontaneously use the adding-on strategy for subtraction, and if so, what elements of the task or problem mattered to elicit that strategy. They found that in general students in special education were able to invent the adding-on strategy successfully and use it effectively–more successfully, in fact, than removal. Students were more likely to do so if the problem had a context that suggested adding on, and if the numbers were close together and crossed a ten. Contrary to the commonly held myth, these special education students did not need to be directly taught a strategy in order to invent it. Students could move between strategies without difficulty, without prior direct instruction. Similarly, Peters, De Smedt, Torbeyns, Verschaffel, and Ghesquière [

55] found that children with MLD were generally able to flexibly solve multi-digit subtraction problems, and like typically developing peers, these students switched between subtraction by removal and adding on. Other research has demonstrated that students with LD tend to develop the same strategies as typically developing peers, although the students with LD tended to use less sophisticated strategies longer than their peers [

56].

In the area of fractions, Hunt and colleagues have demonstrated that students with LD can and do construct their own strategies to understand fractions [

57,

58,

59]. These studies documented growth for students with LD when instruction was designed to closely understand student thinking in the area of fractions, and build tasks that develop those understandings. In short, we have strong evidence that students with LD can develop their own strategies for computation and problem solving, including sophisticated strategies.

Additional evidence debunking this myth comes from qualitative studies of students with LD in inquiry classrooms. Moscardini [

46] notes that without prior instruction, the special education students in his study invented all the strategies outlined in previous general-education research studies [

10]. Foote and Lambert [

49] document that third-grade students in special education were able to independently construct strategies to solve relational thinking equations, when provided time and access to manipulatives. In the study by Chick and colleagues [

47], two girls in special education constructed unique strategies for understanding fractions by using a clock as a model. Behrend [

48] found that the two students in her study with LD were able to construct their own strategies when presented with open-ended mathematical tasks.

Another set of studies more specifically explored whether or not students with special education could benefit from instruction that included multiple strategies. Kroesbergen and van Luit [

60] studied the effectiveness of Structured Instruction (SI) versus Guided Instruction (GI) in small-group interventions with both general education and special education, in multiplication. In the SI group, the teacher chose the strategy, modeled it, and students practiced. If students brought up a different strategy, the teacher led them back to the teacher's strategy. In the GI group, the teacher did not model any strategies after presenting the problems. Strategies emerged from the students, and the teacher led the discussion. Students then had a chance to practice the new strategies. Both small groups were given 4 months of instruction on solving multiplication problems. The study found that both groups were more effective than the control group (typical classroom instruction) in solving story problems in multiplication. Guided Instruction was slightly more effective for low-achieving students, but also proved effective for students in special education. This study suggests that assumptions that students with LD cannot learn from a multiple-strategies approach appear to be unfounded.

I know from years of experience teaching mathematics to students with learning disabilities that this myth is incorrect. My students with LD were able, just like other students, to develop their own strategies, given the opportunity. Each time he was given a problem, Luis sought to understand it, often drawing a picture or using a representation to find a way into the problem [

14]. When learning how to add and subtract integers, for example, he modeled a number line and used the model to solve the problems. Ever the creative and critical thinker, he told me that he liked to think of the 0 in the number line as the border between Mexico and the US. Later in the school year, the teacher presented a rule for subtracting a negative number, by combining the signs into a positive sign. I watched for a few classes while Luis struggled to memorize this way of solving the problems. About a month later, Luis came up to me grinning, and told me in a whisper that the rule did not make sense to him, so now he just uses “the giant number line in my brain” to add and subtract integers. His own strategy stuck with him, when the strategies of others did not make sense to him.

As a reminder about the diversity within the category of LD, another student with LD in that same classroom, Desi, bemoaned being asked to create her own strategies, and wanted to be told what to do. She also told me that her previous teachers had always given her procedures. In contrast, Luis had been expected to develop strategies since kindergarten, as he went to an elementary school that used inquiry mathematics. I wonder then how much this difference between Desi and Luis, both students with LD, related to their prior experience learning mathematics? Moscardini [

46] notes that while the younger students in his study were quick to use manipulatives to solve problems, older students were more hesitant to engage in independent problem-solving. He notes that these older students had spent significant time in classrooms in which they experienced only explicit instruction, and they were used to being told how to solve problems. This suggests another possible source for this myth: teachers who have experienced students who resist multiple strategies. When students who have only had experience with explicit, procedural mathematics are asked to develop their own strategies, they often initially hesitate. This is an issue for students with and without disabilities who have experienced primarily procedural mathematics. The issue here is not a deficit within the student, but learned habits that can be unlearned. As Hord and Newton [

53] suggest, when students are provided with explicit instruction before they understand the concepts at play, these students may have difficulty engaging in those concepts through exploration.