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Crisis-Ready Educational Design: The Case of Mathematics

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Title: Crisis-Ready Educational Design: The Case of Mathematics
Language: English
Authors: Foster, Colin (ORCID 0000-0003-1648-7485), Burkhardt, Hugh (ORCID 0000-0001-9635-0520), Schoenfeld, Alan (ORCID 0000-0003-1858-959X)
Source: Curriculum Journal. Nov 2022 33(4):519-535.
Availability: Wiley. Available from: John Wiley & Sons, Inc. 111 River Street, Hoboken, NJ 07030. Tel: 800-835-6770; e-mail: cs-journals@wiley.com; Web site: https://www.wiley.com/en-us
Peer Reviewed: Y
Page Count: 17
Publication Date: 2022
Document Type: Journal Articles
Reports - Descriptive
Descriptors: Curriculum Design, Mathematics Education, Distance Education, COVID-19, Pandemics, Electronic Learning, Mathematics Curriculum, Equal Education, Formative Evaluation, Feedback (Response)
DOI: 10.1002/curj.159
ISSN: 0958-5176
1469-3704
Abstract: The COVID-19 pandemic has made abundantly clear how far our school systems are from being crisis-ready. The lockdowns seen across many parts of the world left schools and teachers scrambling to provide parents with whatever teaching materials they could find to enable some semblance of distance learning to take place. Despite heroic efforts, the immediate solutions found were far from optimal. This should not be surprising, since no curriculum or school system was ever designed with crisis-readiness in mind. In this article, we look back at the experience of school education during the pandemic, but mainly forward to what educational design can learn to make school curricula and systems more robust and crisis-ready. Taking the mathematics curriculum as our focus, we set out design strategies and tactics devised to ensure that all students are equitably engaged in productive struggle with important content and processes, feel that they have agency over their mathematics, and receive actionable formative feedback on their learning. Through a fully-worked out example, we illustrate the sorts of approaches that we envisage, and we conclude by discussing how we might transition towards such a curriculum.
Abstractor: As Provided
Entry Date: 2022
Accession Number: EJ1351165
Database: ERIC
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  Value: <anid>AN0159609416;5az01nov.22;2022Oct13.08:13;v2.2.500</anid> <title id="AN0159609416-1">Crisis‐ready educational design: The case of mathematics </title> <p>The COVID‐19 pandemic has made abundantly clear how far our school systems are from being crisis‐ready. The lockdowns seen across many parts of the world left schools and teachers scrambling to provide parents with whatever teaching materials they could find to enable some semblance of distance learning to take place. Despite heroic efforts, the immediate solutions found were far from optimal. This should not be surprising, since no curriculum or school system was ever designed with crisis‐readiness in mind. In this article, we look back at the experience of school education during the pandemic, but mainly forward to what educational design can learn to make school curricula and systems more robust and crisis‐ready. Taking the mathematics curriculum as our focus, we set out design strategies and tactics devised to ensure that all students are equitably engaged in productive struggle with important content and processes, feel that they have agency over their mathematics, and receive actionable formative feedback on their learning. Through a fully‐worked out example, we illustrate the sorts of approaches that we envisage, and we conclude by discussing how we might transition towards such a curriculum.</p> <p>Keywords: COVID‐19; crisis‐readiness; curriculum design; home learning; mathematics; virtual learning</p> <hd id="AN0159609416-2">INTRODUCTION</hd> <p>Scientists had long warned that it was only a matter of time before a global pandemic caused widespread chaos (e.g., Bostrom & Cirkovic, 2011). Indeed, some have since described COVID‐19 as merely a 'dress rehearsal' for a more contagious, even civilisation‐threatening future global catastrophe (Greger, 2020). Clearly, no education system can be designed to be pandemic‐proof; however, in these uncertain times, with further waves of COVID‐19 and its variants seeming likely, it is timely to consider what it might take to build a greater degree of robustness and crisis‐readiness into the school curriculum. What might the design challenges be? Could it be practicable? Might there even be low‐cost adjustments to curriculum design that could lead to curricula far more robust to short‐notice school closures, whether global, national or restricted to particular local areas? In an uncertain world, where pandemics and other global emergencies, such as those relating to climate emergency, are an increasing possibility, it is essential to take crisis‐readiness seriously. At the same time, perhaps this moment in history could be leveraged as an opportunity to reconsider what a school curriculum might look like, so as to enhance aspects of student learning and 'build back better'. The overarching question we wish to explore in this article is: How might we design curricula that are much more robust in the face of short‐notice school closures and that have the potential to improve students' experiences of learning? We use the focus of mathematics as a particular case to enable us to explore issues of relevance to mathematics, as well as more broadly.</p> <p>We will address the school curriculum broadly, but take our own area of mathematics as a test case to explore what kinds of strategic re‐engineering of a curriculum might be worth considering in the context of global or national crises (Skovsmose, 2021). Mathematics has an important role in society, and children get one chance at their education; in particular, it is clear that disadvantaged children miss out the most from any interruption to their schooling (CEPEO, 2020; EEF, 2020; Hodgen et al., 2020). The cumulative and hierarchical nature of mathematics makes it particularly vulnerable to interruptions to study, yet it is a critical subject to students' overall academic success and a gatekeeper qualification to subsequent education, especially within areas of science, medicine, technology and engineering. Leaving school without a good mathematics qualification negatively impacts a young person's future prospects in employment, economic and social stability, and health and wellbeing (Moses & Cobb, 2002; Royal Society, 2016; Schoenfeld, 2002). So, while we will use mathematics as a test case, the principles we outline will generalise, at least to science subjects, and perhaps more widely.</p> <p>Fundamentally, our starting point is a pedagogic approach based on agentic interaction (Schoenfeld, 2014) above content delivery. While there are many arguments for such an approach, in this paper we focus on ways in which this is particularly helpful during a period of extended educational disruption. Indeed, we see a synergy between features of these pedagogical principles and the pandemic‐imposed constraints of online learning, and we seek to exploit this in our elaboration of possible forms of practical implementation. Drawing on our extensive experience in educational design, we here propose some practical approaches to curriculum design that could allow future mathematics curricula to be considerably more crisis‐ready, so that the detrimental educational impact on students of an unanticipated period of home learning is lessened. We set out what may be possible and what would be needed to put this in place, as well as what the potential benefits could be, both from a crisis‐readiness perspective and more broadly. We argue that the current crisis affords valuable opportunities to 'think big' about what we want our curricula to do for our young people.</p> <hd id="AN0159609416-3">RESPONSES TO THE CRISIS</hd> <p>The closure of schools due to the COVID‐19 pandemic caused unprecedented challenges (see NCTM, 2020). Such a dramatic change to the social setting placed enormous demands on schools, teachers and family units. In the context of a stressful and rapidly changing international health crisis, teachers acted heroically by swiftly adapting their materials to digital learning (Hodgen et al., 2020). Distance learning in mathematics during the school closures took a variety of forms, from synchronous teaching via platforms such as <emph>Google Classroom</emph>, through assigning questions on subscription websites (e.g., HegartyMaths.com), to teachers creating bespoke narrated <emph>PowerPoint</emph> presentations (Dawes, 2020; Hodgen et al., 2020). However, despite enormous efforts, with hindsight many of the approaches taken were far from ideal. Trying to conduct business‐as‐close‐to‐usual‐as‐possible in our schools—whether virtual or in‐person—was far from optimal (see Hodgen et al., 2020).</p> <p>It is now clear that school closures are detrimental, both through lost instructional time and the forgetting of previous material, and have a negative effect on achievement. An early report (DELVE Initiative, 2020) found that closing schools leads to "loss of learning and deterioration in children's mental and physical health ... [and] increases inequalities, in both children's education achievement and their long‐term prospects". A report published by the Royal Society suggested that school time lost due to the COVID‐19 pandemic could damage the UK economy for as long as 65 years (BBC News, 2020), and students across the world face losing USD $10 trillion in labour earnings over their working lives (Azevedo et al., 2020; Dorn et al., 2020). In The Netherlands, where broadband access is extremely good, even the short lockdown that took place there led to students making very little progress while learning from home (Engzell et al., 2021). Kuhfeld et al. (2020) predicted that returning students were likely to begin the 2020 academic year with around two‐thirds of the typical learning gains in reading and less than half of the learning gains in mathematics that would be expected in a typical school year.</p> <p>Multiple studies make clear that students from low‐income households are especially disadvantaged by school closures (Bonal, & González, 2020; CEPEO, 2020; EEF, 2020; Hodgen et al., 2020; Kuhfeld et al., 2020; van Lancker & Parolin, 2020). A report published only a few months after school closures began suggested that students were already substantially behind, especially in mathematics (Soland, 2020). During school closures, inadequate provision for home learning widens the socioeconomic achievement gap, with wealthier parents more likely to have the cultural capital, computers and internet connections—and even just the physical space necessary to support education at home—and, of course, this includes the ability to afford one‐to‐one private tuition (Hodgen et al., 2020).</p> <p>The tendency of virtual learning designs to further exacerbate inequity has been broadly noted (e.g., Goodnough, 2020; Greenlining, 2020; Hobbs & Hawkins, 2020; Walravens, 2020), widening the digital divide through differences in both technological provision and home circumstances. This is expected to have both a short‐term and a long‐term impact on social mobility, and the magnitude of this effect will depend on the alternative forms of education accessed during the closures (CEPEO, 2020).</p> <p>The right kind of support for effective remote learning could mitigate the extent to which the gap widens (EEF, 2020). In the UK, with £300,000 of initial funding from the Department for Education, the Oak National Academy assembled a set of teaching materials over a 2‐week period (Diamond, 2020). These initially consisted of hundreds of 1‐h video lessons from teachers across different subjects, from early years up to grade 9 (see https://www.thenational.academy/). These free resources initially had a mixed reception on social media. As even the designers themselves commented (Thomas, 2020, np): "Oak won't change the world. It's not supposed to revolutionise teaching. We just want to make life a little bit easier during one of the most difficult periods in our lifetimes. If we can do that, then it's mission accomplished." While the materials were widely adopted, Paul Whiteman, general secretary of the National Association of Head Teachers, was not alone in feeling that the "resources have a shelf‐life that should not go beyond the coronavirus lockdown in their current form" (NAHT, 2020, np).</p> <p>In some ways, the difficulties in resource design during the COVID‐19 school closures highlighted, in an intensified form, wider systemic problems with educational design. Even during normal times, a commercial textbook series is typically assembled under tight deadlines that are driven by powerful economic imperatives (Foster et al., 2021). This normally leaves little time for careful consideration of alternative approaches or detailed analysis of curricular coherence across the entire product, let alone trialling of lessons with real students and teachers in real classrooms to discover what actually happens. 'New' textbook schemes are often assembled as a patchwork, with text and exercises copied and pasted from previous series. Economic pressures tend to favour a demonstrate‐and‐practice imitative pedagogy, still dominant in mathematics classrooms worldwide, that fails to develop extended reasoning or non‐routine problem solving (Burkhardt, 2006a). Alternative approaches, used in some high‐performing countries, have been shown to be effective in developing these higher‐level skills (see Burkhardt, 2014).</p> <p>The competencies that would enable young people to understand the world they live in and its unfolding events, of which COVID is a particularly vivid example, are not well developed by the current focus on narrow, granular standards, curricula and testing. We see part of the problem as how 'progress' is commonly conceived. Concerns about learning 'losses' mirror, on a larger scale, those expressed about the "summer slump", during which students appear to forget much of what they learned in the previous academic year (Alexander et al., 2007). It is likely that one explanation for this is that the material was not learned or understood deeply, but crammed superficially for examinations. Isolated procedural skills and facts are only likely to be retained if regularly practised (Roediger & Butler, 2011), and this may be impossible to maintain as the number of disparate facts and skills taught in school accumulates. More importantly, most school‐related learning vanishes with time, a fact made all too clear when parents try to help their children with homework and realise that their own knowledge is no longer available to them.</p> <p>We advocate for "less is more", by which we mean a slimmed down set of content, providing the opportunity for students to master the ideas by teaching for <emph>depth</emph>, thereby increasing the possibility that they will retain a large fraction of what is taught (Foster, 2013). A reduction in the quantity of content should not be perceived as 'dumbing down', and is in fact intended as the opposite: an overcrowded curriculum can only ever be experienced in a superficial way; by consciously tackling less, we allow the possibility of deep mastery. Such a curriculum would be centred on a relatively small number of <emph>big ideas</emph>—essential concepts that are fundamental to the structure of mathematics (Gardiner, 2014). For example, an appreciation of 'proportion' is perhaps the central concept within lower secondary mathematics, offering the means to make sense of vast areas within the subject—it is far more than 'a topic' which can be covered across a short series of lessons. By isolating such essential 'big ideas', a coherent and powerful curriculum might be created (see Martin, 2011). Such ideas, once encountered at sufficient depth, will be remembered—and, even if forgotten, can be reconstructed for themselves when needed.</p> <p>This approach is sensible not only for mathematics and the sciences, but also to some degree within the humanities. To make sense of the problems that surround us, such as how to deal with a situation like a global pandemic, or how to address social inequities, students need an <emph>integrated</emph> understanding of science and humanities. Students should emerge from school as sophisticated reasoners, with the disposition to think through challenging situations using the information that they know and can find. That includes checking the reliability of what they find, and not accepting things at face value. Such predilections are the basis of true critical literacy in <emph>any</emph> curriculum area.</p> <hd id="AN0159609416-4">DESIGN STRATEGIES AND TACTICS</hd> <p>Over the last 40 years, an international collaborative team has developed a set of principles for the design of teaching materials whose power is widely recognised (e.g., https://<ulink href="http://www.mathunion.org/icmi/awards/past‐receipients/2016‐icmi‐award‐winners">www.mathunion.org/icmi/awards/past‐receipients/2016‐icmi‐award‐winners</ulink>). These principles reflect the five dimensions of the <emph>Teaching for Robust Understanding</emph> (TRU) framework (Schoenfeld, 2014; Schoenfeld et al., 2019; see https://truframework.org/). In brief, the dimensions of classrooms that lead to powerful student understanding are:</p> <p></p> <ulist> <item> <emph>Content</emph> : Are students engaged with important content <emph>and</emph> processes, bringing out the "big ideas" that underpin and link concepts?</item> <p></p> <item> <emph>Cognitive demand</emph> : Do activities involve students in <emph>productive struggle</emph> ?</item> <p></p> <item> <emph>Equitable access</emph> : Are <emph>all</emph> students actively involved in each phase of the learning activity sequence?</item> <p></p> <item> <emph>Agency, Ownership and Identity</emph> : Does each student feel that they can contribute their reasoning, and that it is recognised as "belonging to them" and their fellow students?</item> <p></p> <item> <emph>Formative assessment</emph> : Are lessons structured to consistently reveal student thinking and provide formative feedback?</item> </ulist> <p>These student‐focused dimensions clearly require a different "classroom contract" (Brousseau, 1984)—the set of mutual expectations of students and the teacher as to the roles each will play—than the more typical, instructional "demonstrate and practice" mode. Students take more responsibility for their own learning, explaining their reasoning, not just giving answers, while teachers move from directive to facilitative roles (Sapona et al., 1989). We note that this role‐shifting is very much in harmony with the constraints of remote learning, where the teacher is a scarce resource but some support from parents may be available. This provides a potential opportunity, although it does not, of course, make the design challenges straightforward.</p> <p>Out of this framework, a collection of design strategies and tactics have been developed to help designers to create lesson sequences consistent with the TRU dimensions (see Burkhardt & Pead, 2020, for an overview, with examples). Here, we consider how these approaches might be modified to enable student‐centred home learning.</p> <hd id="AN0159609416-5">Make the tasks carry the load</hd> <p>The choice of problems that students are asked to tackle is central to curriculum (re)design (Watson & Ohtani, 2015). High quality mathematical tasks are authentic, intricate, interesting and powerful (Steen & Forman, 2001). The design goal is to choose <emph>substantial</emph> tasks that involve important mathematics (TRU Dimension 1 above) and provide challenges that result in productive struggle (Dim. 2) for all students (Dim. 3), at their own level, so that they emerge with a solution that they feel is their own (Dim. 4).</p> <p>To this end:</p> <p></p> <ulist> <item> Use tasks that students can engage with <emph>substantially</emph> , for a significant length of time (e.g., up to an hour), without recourse to adults.</item> <p></p> <item> Make the entry phase of each task accessible—e.g., based on familiar, real‐life situations or an easy‐to‐comprehend game.</item> <p></p> <item> Most tasks should be relatively straightforward technically, so as to compensate for the strategic and tactical demands that complex, non‐standard tasks present (Foster, 2019).</item> <p></p> <item> To introduce more challenging content, design tasks in which students can start in "understand, then critique" mode by using provided examples of "student work" (carefully‐designed fictional student responses, see Evans & Swan, 2013).</item> <p></p> <item> Following this mode, the task can then ask the student to extend the ideas and, for example, "Classify and define mathematical objects and structures," "Represent and translate between mathematical concepts and their representations," "Justify and/or prove mathematical conjectures, procedures and connections" and "Identify and analyze structure within situations" (Swan & Foster, 2018).</item> <p></p> <item> Such understandings can be built on, and fluency acquired through, <emph>études</emph> (Foster, 2018), which allow essential practice of important skills to take place within a rich, stimulating context, rather than through repetitive, routine exercises.</item> <p></p> <item> Connections can be consolidated through the use of modelling problems, such as "Which of the following real‐life scenarios can be described by this function? Identify the variables and the relationships."</item> </ulist> <p>Central to this approach, there is a <emph>product</emph> of the work of each student or pair of students that is <emph>owned</emph> by its creators (Dim. 4), which they build on through discussion and review (Dim. 5). This might be a report with recommendations, a design, or simply an analysis well explained.</p> <p>We now turn to the design challenge of crafting these kinds of tasks that can function well in a home environment.</p> <hd id="AN0159609416-6">Develop a student‐centred pedagogy</hd> <p>In home learning the teacher becomes a scarce and valuable resource, serving in concert with other available resources—notably technology, peers from the same class, siblings and parents. This shift complements the central need to develop and support student agency (see Calleja et al., 2021; Swan, 2006).</p> <p>The three main modes of interaction for learning—with a peer, a parent and the teacher—may be expanded on as follows.</p> <hd id="AN0159609416-7">With a peer</hd> <p>Here, students initially all work on the same problem individually, before sharing and critiquing each other's reasoning. Peer interactions providing formative feedback can be a powerful resource for supporting learning (see Hodgen et al., 2018; Mercer, 2000), but they are underused, sometimes even disapproved of (a relic from notions of "cheating" on simple exercises). Fullilove and Treisman (1990) found that when college students worked together on hard problems, they performed much better overall.</p> <p>We highlight two modes of peer working:</p> <p></p> <ulist> <item> <emph>Team mode</emph> : Two students work on the same task in their homes, knowing that they will need to explain their approach to a friend, and roughly "scripting" how they are going to do this. At a prearranged time, each explains their approach and what they have found, with an eye towards producing an improved solution. Following their discussions they each write individual improved versions and exchange them. They know that the teacher may call on either of them to explain, thus providing a means of formative assessment. This structure provides accountability, which is an important ingredient of successful collaborative learning (Slavin et al., 2008).</item> <p></p> <item> <emph>Complementary mode</emph> : Here, the two students play <emph>different</emph> roles, typically one "making a case" and the other "interrogating the witness" to find holes in the argument. Initially, each student in the pair prepares the case for a different problem and sends an outline to the other; then, they take turns at questioning the other. (Interpretive tasks, such as finding meaning in a dataset, are natural tasks for this mode.) This mode can be an effective way of building dialogic classrooms (Ruthven et al., 2017).</item> </ulist> <p>Supportive materials built around rich tasks, with a fellow student providing feedback, are powerful resources for learning. They can be combined with e‐presentations to the whole class, which are an adaptation of "gallery walks" from in‐person classrooms. All of these peer interactions are designed to contribute to productive discussions which provide opportunities to clarify and deepen learning (see Swan, 2006).</p> <hd id="AN0159609416-8">With the parent</hd> <p>Parents will have other pressures, sometimes extreme, such as work or other children or family members to care for, so it is important to acknowledge the practical limits of parental involvement. Where parents can participate, it is helpful to set out a suggested "pedagogical posture", namely:</p> <p></p> <ulist> <item> Parents are <emph>not expected to be subject</emph> ‐ <emph>matter teachers</emph> : their primary job is to be interested listeners and questioners. It is fine—and may even be advantageous—if they do not know the relevant content. Attentive listening can be an extraordinarily powerful tool (see Coles, 2002).</item> <p></p> <item> <emph>Learning is the child's responsibility</emph> —parents should give them plenty of time to work on the tasks before intervening. They should not be alarmed if their child initially appears to be 'stuck'.</item> <p></p> <item> Parents should <emph>avoid instructing but mainly ask questions</emph> to provoke <emph>the child</emph> to explain their thinking about the task: "Tell me what this is about," "What do you think you might try?"—later on they might ask, "Tell me more about your thinking," "What have you tried?", "What did you find out?" (see Foster, 2014).</item> <p></p> <item> Parents should <emph>consistently ask for more and better explanation</emph> : "I don't quite understand. Tell me a bit more." Doing this prompts students to think more deeply and to refine their communication of their ideas (see Stein et al., 2008).</item> <p></p> <item> A <emph>Common Issues Table</emph> (see Wake et al., 2016) is provided with each task, stating some specific things that students tend to find difficult with the task—each issue accompanied by some questions that the parent might ask to support them in their learning. The purpose of this is to help the adult to provide targeted scaffolding support (Holton & Clarke, 2006) in response to the difficulties diagnosed, but generally in the form of questions, rather than explanation (Foster, 2014).</item> </ulist> <p>The features of this pedagogical posture seek to capitalise on the presence of a caring and interested adult, but without assuming the availability of specific subject content or pedagogical content knowledge (Shulman, 2004).</p> <hd id="AN0159609416-9">The roles of the teacher</hd> <p>Teachers are the most valuable—and limited—resource for learning (Mercer, 2000; Watson et al., 2003). In a typical lesson of up to 60 min, with around 30 students, a teacher can spend 1–2 min at most individually with each student. Making effective use of the teacher is a key element in any curriculum design—and a new challenge for a lockdown curriculum with home‐based learning. Standing at the front to explain a new skill, working an example, and then watching the students do exercises, though still prevalent, is rarely optimal use of an expert teacher's time or pedagogical skills (Swain & Swan, 2007). Teachers' expertise is, first of all, diagnostic—being able to perceive more deeply than a peer or parent what a student's approach might reveal about their understandings and misunderstandings (Swan & Burkhardt, 2014). The next step is choosing an intervention (perhaps a question) that will move forward a student's thinking about the problem, without taking it over by being too directive about the next phase of the work (Foster, 2014). Many teachers are not strong in this <emph>adaptive expertise</emph> (Hatano & Inagaki, 1986), and one of the roles of high‐quality teaching materials is to help them to develop it. Common issues tables, as mentioned above, can be a useful way to do this (Wake et al., 2016; Wiliam, 2017).</p> <p>The following modes illustrate different ways to utilise teachers' expertise during online learning:</p> <p></p> <ulist> <item> <emph>Sampling mode</emph> : The teacher can sample the progress of each student's/pair's work at various points in the thinking process. This sampling will be teacher‐controlled, though <emph>informed</emph> by the end‐of‐task notes from students and calls for guidance from students or parents. This makes effective use of the teacher's time (see Swan & Burkhardt, 2014).</item> <p></p> <item> <emph>Coaching mode</emph> : From the sampling and prior knowledge, the teacher will judge where significant interventions are needed. These need time and a structure that uses that time most effectively. Consequently, any given student or pair will rarely have an extended discussion with the teacher. When they do, they will be expected to discuss the task as "fellow mathematicians."</item> <p></p> <item> <emph>"Whole‐class" mode</emph> : There can be a real gain in bringing together a larger group of students (perhaps the whole class) to present their solutions. The teacher may choose an order that showcases approaches of increasing sophistication, such as is common within the <emph>neriage</emph> phase of the Japanese problem‐solving lesson (see Baldry et al., 2022 ; Takahashi, 2006). At the end of each presentation, students may be asked to further develop their own solutions or to tackle a related problem in the light of the discussion.</item> <p></p> <item> <emph>Changeover points</emph> : At some point, the teacher will judge that the activity sequence for this task has gone on long enough. If the task has been chosen well, this will normally be after several lessons or hours of student work. The teacher will then launch the next task. This may be done for each pair, but there is considerable advantage in keeping the class as a whole working on the same rich task at the same time, so that the potential for fruitful interactions is high.</item> </ulist> <p>We do not underestimate the challenges of making the suggested changes in the typical didactical contract (Brousseau, 1984). We do note, however, that with carefully designed instructional materials, such as the formative assessment lessons available at https://<ulink href="http://www.map.mathshell.org/lessons.php,">www.map.mathshell.org/lessons.php,</ulink> teachers can and do make such changes—and learning is enhanced (Herman et al., 2014; Research for Action, 2015).</p> <hd id="AN0159609416-10">Roles for technology that stimulate thinking</hd> <p>We have deliberately left discussion of technology until now, not because it is unimportant, but because it can easily dominate thinking about pedagogy in an unhelpful way (see Drijvers, 2015). Communication is key to learning, and technology must now play a central role in this. Currently, educational software is often used to support the elements that <emph>least</emph> need support, such as generating repetitive practice. Here, we have deliberately avoided specifying what technology is minimal. Smartphones are now nearly universal in developed countries, and much can be achieved using their camera, send and voice features alone. With more power and wise usage, more can be achieved. However, the technology should be the <emph>servant</emph> of the learning; it can easily become the focus instead. In our experience, simple uses of a computer's strengths have proved most powerful: presenting tasks to the student; offering mathematical tools for standard procedures, from calculators and spreadsheets upwards; providing "mathematical microworlds," small and large, to investigate. Attempts to move the computer into analysing student thinking through their responses have proven effective for simple tasks but not yet for tasks involving extended chains of autonomous reasoning (ISDDE, 2012; Pead, 2010), and the gains do not yet appear to justify the costs. However, sophisticated computer algebra systems, such as STACK (see https://stack‐assessment.org/, Sangwin, 2013), allow answers to be evaluated mathematically and permit open questions, such as example‐generation prompts. And <emph>comparative judgment</emph> offers teachers the facility to monitor students' development in fuzzy, hard‐to‐define constructs, such as problem solving and conceptual understanding (e.g., see Jones et al., 2015).</p> <hd id="AN0159609416-11">An example learning activity</hd> <p>We now illustrate how the simple technology of smartphones and video‐conferencing software such as <emph>Zoom</emph>, might be used with a Grade 6 lesson, 'Consecutive Sums' from the Mathematics Assessment Project website,[<reflink idref="bib1" id="ref1">1</reflink>] adapted (to 3 sessions of about an hour) for the environment and pedagogy discussed above.</p> <p>The sequence begins by scheduling the activity. The teacher may choose to introduce the task on screen to make sure that the students understand it, or leave it to the students in the first instance. It can be difficult enough for students to listen patiently to the teacher in person; it is even more difficult through the screen. So, shifting the focus to orchestrating student actions, getting students to work individually, then with each other, and providing feedback, becomes that much more important.</p> <p>Each student begins by tackling the entry task:</p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/5AZ/01nov22/curj159-fig-0001.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="curj159-fig-0001.jpg" title="." /> </p> <p></p> <p>This problem is deceptively simple, in that it is easily approachable, but can lead to very rich mathematical explorations—which can build agency and provide opportunities for conjecture and proof (a <emph>low‐entry‐high‐ceiling</emph> task, see Kiddle, 2020).</p> <p>For this first session, students are asked to experiment with some numbers, then to try to write down a conjecture about which numbers can be made in this way. An online "help sheet" provides access to the meanings of words such as "consecutive", "conjecture" and "prove," but it deliberately does <emph>not</emph> give answers. After about 20 min of working individually, students are permitted to call their partner, exchange pictures of their work, and talk about any differences in what they have done. The teacher can be copied in on these exchanges when the students think that they have either a product or a need, but the teacher only intervenes when they judge this to be necessary. The teacher chooses when to move on, perhaps announcing that each student should try to substantiate their conjectures, writing down their explanation of why each conjecture is true, first individually and then comparing wordings. For the rest of the session, if there is time, they each try another task: to write all of the other numbers from 1 to 20 as sums of consecutive whole numbers, and/or to state and prove another conjecture—such as that the sum of two consecutive whole numbers is always odd. The teacher may choose to guide that choice for some students.</p> <p>The second session, in <emph>coaching mode</emph>, will benefit from a parent or older sibling, to whom each student explains the problem, and what they have found out. (If such adults are not available, students can be grouped together, or students from more advanced grades can be recruited to play such roles: see Hodgen et al., 2018, for a summary of the evidence for the benefits of cross‐age tutoring.) As justified above, the adult functions as an interested and supportive person, but they do not require any specialized mathematical knowledge. Student and adult together review the <emph>Common Issues Table</emph> provided, which sets out likely misunderstandings, and questions that will help the student to resolve them—in this activity, student and adult <emph>together</emph> play a teacher role. (Moving students into teacher roles is a powerful way of raising the learning to a higher level; see e.g., Sapona et al., 1989.) Students then go on to evaluate the truth of a conjecture, perhaps that "sums of 4 consecutive whole numbers are always even," by looking at 5 + 6 + 7 + 8, creating other examples, and analysing their structure. They might also begin to explain why sums of 4 consecutive whole numbers are <emph>not</emph> multiples of 4 in the same way that sums of 3 consecutive whole numbers <emph>are</emph> multiples of 3.</p> <p>The next stage is to download and analyse four carefully constructed fictitious student responses to this conjecture (see Evans & Swan, 2013). These responses reflect both correct answers and common errors. The students' task is to understand and explain the reasoning in each case and to compare it with their own versions, asking, for example, which they find easiest to understand, simplest mathematically, or most convincing, and why. Each student then explains all this to an adult—later on, if need be—live or on video.</p> <p>The teacher samples these explanations and, for opening the third session, may choose to organise a class discussion (<emph>whole‐class mode</emph>), selecting a sequence of students to present their thinking to the whole class. Students are offered eight conjectures about consecutive sums and asked to decide if each conjecture is <emph>always true</emph>, <emph>never true</emph> or <emph>sometimes true</emph>—and, if the latter, under what conditions. Again, what the students produce are <emph>explanations</emph>, using some combination of numbers, words, symbols and/or drawings. Some of the eight conjectures are easy to classify and some are more challenging.</p> <p>The final "products" of each student or pair are sent to the teacher, who decides if and where intervention is needed (<emph>sampling mode</emph>). This must not be onerous for the teacher, who might decide that enough has been accomplished for the time being, pointing students to another quite different type of problem‐solving lesson—perhaps, for example, estimating how many matchsticks can be made from a tree.[<reflink idref="bib2" id="ref2">2</reflink>]</p> <p>This kind of problem‐solving activity typically involves challenging mathematical reasoning using relatively simple mathematical techniques—the balance that is most often required in everyday life and work. (Steen, 2001 contrasts mathematical literacy—i.e., the sophisticated use of (relatively) elementary mathematics—with the reverse situation that is typical of traditional school mathematics.) This, in itself, is a productive re‐orientation of the curriculum. Specifically, the lesson requires students to think in depth about the properties of integer arithmetic, including factors, odd and even numbers, and arithmetic progressions. The conjectures in the third session go deeply into these issues. The processes in which students engage—conjecture, explanation and proof—are at the heart of <emph>doing</emph> mathematics (Cuoco et al., 1996). These are all examples of the "big ideas" that are our focus here (Martin, 2011).</p> <hd id="AN0159609416-13">TOWARDS IMPLEMENTATION</hd> <p>Turning this model framework into a set of high‐quality learning materials for a strategically focused, distance‐usable curriculum is a challenging research and development project. The product must be attractive to students, acceptable to teachers and parents, and robust in use (see Clark, 2021; Foster et al., 2021; Kirschner & Van Merriënboer, 2018). <emph>We envision it as a collective endeavour</emph>. There is a lot to build on: creative designers around the world have produced a wide range of tasks that fit in with the principles set out above (see, e.g., numerous examples at https://<ulink href="http://www.bowlandmaths.org.uk/,">www.bowlandmaths.org.uk/,</ulink> https://<ulink href="http://www.desmos.com/,">www.desmos.com/,</ulink> https://<ulink href="http://www.map.mathshell.org/,">www.map.mathshell.org/,</ulink> https://<ulink href="http://www.mathshell.org/,">www.mathshell.org/,</ulink> and https://nrich.maths.org/).</p> <p>It takes time to build a comprehensive set of materials, but that may not even be necessary, since it is unlikely that spells of home learning would last for extended periods. Instead, we suggest that design could start with simple things that could be easily done at home and would enhance the curriculum by improving the balance of types of learning in the way that we have outlined. What is advocated here for learning at home will also enhance <emph>in</emph>‐<emph>person</emph> instruction, moving any mathematics curriculum towards more sensemaking and growing student mathematical expertise.</p> <p>The design and development process would be an iterative spiral, with two phases (Swan & Burkhardt, 2014). This is standard across fields for product development but, as it is so often short‐circuited in education, we review it here. The design phase of any project has two aspects: Designing an effective learning activity sequence that will move students towards specific learning goals, and communicating the design to users so they can realise the design in their own classrooms. There are well‐developed methods for the design of such materials (see, e.g., Burkhardt, 2006b; Burkhardt & Swan, 2017; Swan, 2006).</p> <p>These include:</p> <p></p> <ulist> <item> <emph>finding and developing tasks</emph> that: offer easy and inviting student and parent access; lead to important mathematics; are accessible at different levels of sophistication; and lead students to develop reasoning that is directly explicable to parents, as well as to teachers.</item> <p></p> <item> <emph>selecting relevant genres and learning activity sequences suitable for the tasks</emph> ; trialling each task on a small scale to identify common issues that students have in tackling the task; and devising non‐directive questions for each issue that parents and teachers can use to move students' reasoning forward, without "taking the solution away from them" (see Foster, 2014).</item> <p></p> <item> <emph>trying these materials out</emph> , again on a small scale, producing three linked guides—for students, for parents and for teachers—the communicating element, which can only be improved with "real users".</item> </ulist> <p>The resulting draft prototype unit is then ready for systemic development. An effective development cycle includes having independent observers take detailed notes on lesson implementation in a small number of classrooms, with their feedback informing a revision process by a team that includes new designers who are not "invested" in the original design. After two or three such cycles, with a diverse range of student populations, the materials can be considered "use‐tested" and ready for wide distribution (see, e.g., Swan & Burkhardt, 2014, for detail on this process).</p> <p>It is clear that this process differs fundamentally from the typical "farm things out and piece them together" mode that dominates mathematics curriculum development (Foster et al., 2021). It is, however, typical of design in fields in which product engineering is the norm. Unlike typically designed materials, materials designed in this way have warrants for their validity (Burkhardt & Schoenfeld, 2003, 2020) and effectiveness (<emph>cf</emph>. Herman et al., 2015; Research for Action, 2015).</p> <p>To develop a full curriculum to these standards would not be inexpensive, and yet it would comprise a tiny fraction—much less than 1%—of the running costs of a large education system over the decade or so that the design and development process would take (see Burkhardt, 2006b). In contrast, as Schoenfeld (1999) points out, the Mattel corporation spends 5% of its income on research and development, and pharmaceutical companies spend nearly 20% of their income on their research and development. The question is, how serious are we about obtaining instructional materials of high quality and in being prepared for the next crisis before it happens?</p> <hd id="AN0159609416-14">CONCLUSION</hd> <p>The COVID‐19 crisis has cast a number of issues into sharp relief. First, it is clear that, as a field, education systems have not been ready to take meaningful instruction online. At this point in the pandemic, it appears that attempts have favoured replicating as closely as possible a 'normal' classroom, and the more traditional the classroom the more difficult this is to reproduce meaningfully in an online context. Instead, we advocate for a different pedagogic approach, based on agentic interaction (Schoenfeld, 2014), rather than mere content delivery.</p> <p>We have attempted to show in this paper that there are approaches to design that can help to do this: not only to craft instruction that can withstand the rigours of virtual communications, but that can do what should be done in face‐to‐face instruction as well: focus on key ideas in ways that students will remember, and that will in addition offer the possibility of agency in their own learning. We recognise that there are considerable challenges in doing this and the limitations are apparent. For example, synchronous teaching into homes is not always realistic, given many families' limited internet access, limited numbers of devices and limited physical space for parallel interactions to proceed. We also do not underestimate the teacher professional development challenges of implementing such approaches as these. We also recognise the challenges on parents and other adults to support children in the ways outlined here. All of this requires a shift in thinking about what learning mathematics is about.</p> <p>Another limitation is that we advocate a synchronous approach, which is not always practicable, given the constraints of limited wifi and access to technology. However, aspects of our approach could be adapted for asynchronous (i.e., on‐demand) lessons. For example, student explanations and questions could be audio recorded using a phone and uploaded onto a secure school server, and then other students could asynchronously respond to and build on these in their own mathematics. Such an approach would not threaten the essential ingredients of the TRU framework (Schoenfeld, 2014), and would enable students to equitably (dim. 3) experience important content and processes through the tasks and on‐demand video (dim. 1), and would be engaged in productive struggle individually and in pairs (dim. 2). In some ways, having on‐demand access could be seen to heighten agency (dim. 4), and opportunities for formative assessment (dim. 5) would be equally strong.</p> <p>In this paper, our "proof of concept" examples have come entirely from mathematics, and that is our domain of specialism. However, we believe that the argument is more general than that, since none of the principles of TRU is necessarily mathematics‐specific, and the TRU framework has indeed been adapted successfully into other school subjects (see Schoenfeld, 2014). We leave to others with the necessary expertise the details of how this could be worked out. But there is now adequate evidence to show that the path we propose is feasible—and not prohibitively expensive. If we want our education system to be crisis‐ready, we need to learn from what has happened over the period of the pandemic and ensure that we are better placed for the future. Whether you attribute the quote to Machiavelli or Churchill, there is still wisdom in the saying, "Never let a good crisis go to waste."</p> <hd id="AN0159609416-15">ACKNOWLEDGEMENT</hd> <p>We would like to thank the Editor and the anonymous Reviewers for their very helpful comments on earlier versions of this paper, which have allowed us to improve it considerably.</p> <hd id="AN0159609416-16">CONFLICT OF INTEREST</hd> <p>The authors have declared no conflicts of interest for this article.</p> <hd id="AN0159609416-17">ETHICS STATEMENT</hd> <p>Ethical approval was not required for this work.</p> <hd id="AN0159609416-18">DATA AVAILABILITY STATEMENT</hd> <p>There are no data associated with this paper.</p> <ref id="AN0159609416-19"> <title> Footnotes </title> <blist> <bibl id="bib1" idref="ref1" type="bt">1</bibl> <bibtext> Downloadable free from https://<ulink href="http://www.map.mathshell.org/download.php?fileid=1602">www.map.mathshell.org/download.php?fileid=1602</ulink>.</bibtext> </blist> <blist> <bibl id="bib2" idref="ref2" type="bt">2</bibl> <bibtext> Downloadable free from https://<ulink href="http://www.map.mathshell.org/download.php?fileid=1691">www.map.mathshell.org/download.php?fileid=1691</ulink>.</bibtext> </blist> <blist> <bibl id="bib3" type="bt">3</bibl> <bibtext> Funding information None</bibtext> </blist> </ref> <ref id="AN0159609416-20"> <title> REFERENCES </title> <blist> <bibtext> Alexander, K. 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  Data: <searchLink fieldCode="AR" term="%22Foster%2C+Colin%22">Foster, Colin</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0003-1648-7485">0000-0003-1648-7485</externalLink>)<br /><searchLink fieldCode="AR" term="%22Burkhardt%2C+Hugh%22">Burkhardt, Hugh</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0001-9635-0520">0000-0001-9635-0520</externalLink>)<br /><searchLink fieldCode="AR" term="%22Schoenfeld%2C+Alan%22">Schoenfeld, Alan</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0003-1858-959X">0000-0003-1858-959X</externalLink>)
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  Data: <searchLink fieldCode="SO" term="%22Curriculum+Journal%22"><i>Curriculum Journal</i></searchLink>. Nov 2022 33(4):519-535.
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  Data: Wiley. Available from: John Wiley & Sons, Inc. 111 River Street, Hoboken, NJ 07030. Tel: 800-835-6770; e-mail: cs-journals@wiley.com; Web site: https://www.wiley.com/en-us
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  Data: 0958-5176<br />1469-3704
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  Data: The COVID-19 pandemic has made abundantly clear how far our school systems are from being crisis-ready. The lockdowns seen across many parts of the world left schools and teachers scrambling to provide parents with whatever teaching materials they could find to enable some semblance of distance learning to take place. Despite heroic efforts, the immediate solutions found were far from optimal. This should not be surprising, since no curriculum or school system was ever designed with crisis-readiness in mind. In this article, we look back at the experience of school education during the pandemic, but mainly forward to what educational design can learn to make school curricula and systems more robust and crisis-ready. Taking the mathematics curriculum as our focus, we set out design strategies and tactics devised to ensure that all students are equitably engaged in productive struggle with important content and processes, feel that they have agency over their mathematics, and receive actionable formative feedback on their learning. Through a fully-worked out example, we illustrate the sorts of approaches that we envisage, and we conclude by discussing how we might transition towards such a curriculum.
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