Teaching with and through Maps

Hic Sunt Dracones = Here Be Dragons

In working with maps, possibly for the first time, it is expected that students will feel and display feelings of discomfort. They may insist that they cannot draw or express fear of making a mistake. The Touchstone Atlas unit makes space for partial understandings; in fact, it necessitates risk-taking with partial understandings to promote sense making. When operating from a deficit mindset, teachers tend to focus only on the gaps in understanding instead of pinpointing the knowledge students possess that has led them to misinterpret a task. Encouraging students to show their work or explain their thinking assists teachers in analyzing student errors. Often, it is not that students don’t know anything about the content, it is that they are misapplying strategies or rules. In my experience with students, this happens more often when students try to memorize strategies instead of contextualizing problems.

Early mapmakers labeled hidden or unknown territories poles “hic sunt dracones,” implying that the sea monsters or dragons en route to the region had not yet been confronted as a result of exploration. On Abraham Ortelius’ map, Typus Orbis Terrarum (Antwerp, 1570/1587) the north and south poles are labeled “terra septentrionalis incognita” and “terra australis nondum cognita,” respectively. The mapmaker makes a distinction between the unknown northern lands and the southern land not yet known, implying that exploration of the southern land is expected to be upcoming.  In this labeling, the mapmaker acknowledges that land existed in areas that had not yet been explored, so, too will students be encouraged to map using what they know and recognize partial understandings as exploration opportunities. The teacher might likewise differentiate between concepts that need to be retaught (because students are close to discovery) and concepts that require explicit instruction or intervention. Tasks that extend beyond a student’s capacity to independently complete necessitate scaffolding. Appropriately scaffolding ensures students have an entry point to the learning resulting from a belief that all students, even those with comprehension gaps or “learners with emerging language – at any level – can engage deeply with central disciplinary ideas under specific instructional conditions”¹³ One way to adjust instructional conditions is to provide an item that students have familiarity with using.

Discussing role player game maps in this context can provide students with a more contemporary illustration of working with maps that acknowledge their blind spots. The map of the game world expands, and parts of the map that were once remote are revealed as players navigate and overcome challenges. In the game Horizon Zero Dawn, players must confront robotic beasts, called tall necks to expand their view of the game map (a scaled map of the southwestern United States, mainly depicting Colorado and Utah). Prior to confronting them, the player can only see symbols on the map that indicate a tall neck occupies the space. Once they climb the unsuspecting tall necks, they can override them to reveal features of the surrounding terrain and possible threats.  In effect, the game mapmaker’s tall neck symbology is the early mapmaker’s hic sunt dracones. Whether students relate to the hic sunt dracones designation or tall neck symbology, the goal is for students to normalize risk-taking and become more willing to exhibit what they know despite gaps they may not be fully able to articulate. Students develop mathematical language more organically when they first exhaust the language they already have and then discover its inadequacy or incompleteness. During this struggle for articulation as students discuss concepts, the teacher can “adapt and respond more effectively to what students are saying and doing as they develop disciplinary language and content understanding concurrently.”¹⁴

Mercator’s Atlas included three distinct sections before the maps consisting of biography, exegesis, poems, and letters. In like manner, student atlases will include their written reflections on their learning as they map. In these constructed responses, students will explain why they made certain choices, name the challenges they encountered, and acknowledge parts of mapping and math that still confuse them. The corresponding reflection for each map serves as a data source for the teacher. Not only can the teacher monitor for proficiency or gaps in certain math skills, but she can also monitor how well students are able to explain their work in writing and track growth in that regard.  Student explanation of their thinking is aligned to several of the Standards for Mathematical Practice, namely the following: MP2 - reason abstractly and quantitatively; MP3- construct viable arguments and critique the reasoning of others; and MP7- look for and make use of structure, discussed in a later section.

As the creation of an atlas typically happens organically, students will focus on creating one map at a time rather than becoming overwhelmed or focused on the scope of all the maps they will create to make an atlas. They are charting courses and confronting dragons as they encounter them, revising their work as necessary. Jacob describes the atlas as an open-ended form, making it suitable for changes, substitution, and addition.¹⁵ A goal of the revisitation and revision is to discover how student mapping changes over time, becomes more precise, and presents different information.  Just as a static map cannot fully capture a place because places undergo constant change,¹⁶ one work product cannot fully capture evolving student understanding. The Touchstone Atlas is not only a collection of various maps, but it also contains versions of understanding for students, with an emphasis on valuing the version in each stage. The atlas serves as a culmination of all the static maps which promote dynamic and shifting understanding.  Most, if not all created maps should make it into the atlas, emphasizing the value of partial understandings in sense-making. Making space for this development over time produces “mathematically proficient students who can apply what they know and are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later.”¹⁷ Notably, while early remarkable atlases are rife with errors and partial understandings of the world, the documentation of what mapmakers knew at the time was not only useful for contemporary navigators, but also sparked further curiosity and study which led to later maps containing more complete information. This legacy of atlas creation should put students at ease; in working despite and through partial understandings, they are in the good company of mapmakers and mathematicians who were necessarily mistake-makers by virtue of being sense-makers.