Author Archives: amanda ghegin

Mental Math T-GEM

My posting for this week involves the T-GEM cycle and the interactive “Make a Ten” activity from PhET. I believe this would be particularly helpful in the lower elementary classroom, as the way in which the visualization is organized, and the speed in which one can manipulate numbers makes it ideal and quicker to use than traditional manipulative tools. This simulation shows an example of how “…computers might be the best educational option for our students” (Finkelstein et al., 2005), using layering of numbers and essentially makes it an online and more improved/technology integrated version of the Numicon tools. Essentially, it provides a model-based inquiry for math manipulatives, but it’s responsiveness works much faster than using the physical pieces. While the articles I read closely this week were more geared towards the sciences, I found that much of the content could be applied to this mathematics simulation as well, as it deals with pattern detecting and creating rules. Xiang and Passmore write, “There has been increased recognition in the past decades that model-based inquiry (MBI) is a promising approach for cultivating deep understandings by helping students unite phenomena and underlying mechanisms” (Xiang & Passmore 2015), which I think applies to this tool.

Here’s the link to the simulation.

Generate:

Students explore and play around with the simulation. Then, they look for as many different ways as they can make equation or number sentence for two numbers that have a sum of 10. Then 15. Then 20. Then 50.
Students collect information about numbers and the patterns then can detect individually.

Evaluate:

Students are grouped into threes and discuss their findings with their classmates. Collectively, they turn those patterns into relationships and rules for mental maths strategies.
Students collate their rules and move around the class to have a look at how many of their rules match up with others in the classroom. Students take note of rules that looked different from their own.

Modify:

Students review and test out each others’ rules, making room for comments and being ready to explain their thinking.
Students go through the cycle again, assessing their strategies with subtraction, and looking for commonalities

As Finkelstein et al. write, “We do not suggest that simulations necessarily promote conceptual learning nor do they ensure facility with real equipment, but rather computer simulations that are properly designed are useful tools for a variety of contexts that can promote student learning” (Finkelstein et al., 2005). This tool is one such simulation that enhances student learning.

References:

Xiang, L., & Passmore, C. (2015). A framework for model-based inquiry through agent-based programming. Journal of Science Education and Technology, 24(2-3), 311-329.

Finkelstein, N.D., Perkins, K.K., Adams, W., Kohl, P., & Podolefsky, N. (2005). When learning about the real world is better done virtually: A study of substituting computer simulations for laboratory equipment. Physics Education Research,1(1), 1-8.

Constructivist Knowledge Diffusion

How is knowledge relevant to math or science constructed? How is it possibly generated in these networked communities? Provide examples to illustrate your points.

The first question could be answered in two parts based on some of the readings for this week:

1) Teachers rely on constructed sets of knowledge as common understandings or starting points from which to teach students. Teachers put stock in common understandings that can explain the world around them, they rely on findings of the experts in their field, and have a starting place from which to instil curiousity and explain natural phenomena. Yoon et al. write, “… most formal educational experiences are designed for students to participate in belief mode where ideas are investigated and proved or disproved with evidence for or against” (Yoon et al., 2012).

However,

2) These ‘common understandings’ are created through dialogue, experimenting, exploration, and challenging different and opposing views. Driver et al. write that, “… it is important in science education to appreciate that scientific knowledge is both symbolic in nature and also socially negotiated. The object of science are not the phenomena of nature but constructs that are advanced by the scientific community to interpret nature […] Rather, they are constructs that have been invented and imposed on phenomena in attempts to interpret and explain them often results of considerable intellectual struggles” (Driver et al., 1994). So knowledge relevant to science and math is discussed, challenged, and proven with evidence that supports the common claim, however this is not necessarily how scientific knowledge is taught/designed in “formal educational experiences”.

In networked communities however, there are underlying factors that point to a need for collaborative learning. Falk and Storksdieck write about “free-choice learning experiences”, where “adult visitors have considerable choice and control over what they actually attend to and visitors enter with a wide diversity of prior interests, knowledge, and experiences” (Falk & Storksdieck, 2010), and Yoon et al. provide research that “suggest[s] that ability to theorize from the museum experience can be improved through the use of knowledge-building scaffolds such as response forms and the ability to work in groups” (Yoon et al., 2012). If learners are engaged (by free choice and control), sharing and growing confidence in curiousity, conversation, and discovery, this can lead to deeper learning through play and collaboration. The Exploratorium in San Francisco is one such example of a space created to host networked communities and collaborative instructional play. The Science Centre in Toronto is another example. Also, in looking through the Exploratorium’s online repertoire, I’m impressed by the wide range of available “Science Snacks” and “Explore Activities” for teachers who can’t get to the Exploratorium. I’m looking forward to sinking my teeth into this resource.

References:

Driver, R., Asoko, H., Leach, J., Scott, P., & Mortimer, E. (1994). Constructing scientific knowledge in the classroom. Educational researcher, 23(7), 5-12.

Falk, J. & Storksdieck, M. (2010). Science learning in a leisure setting. Journal of Research in Science Teaching, 47(2), 194-212.

Yoon, S. A., Elinich, K., Wang, J., Steinmeier, C., & Tucker, S. (2012). Using augmented reality and knowledge-building scaffolds to improve learning in a science museum. International Journal of Computer-Supported Collaborative Learning, 7(4), 519-541.

Embodied Learning and Math

Though not necessarily tied to the idea of technology, one excerpt from this week’s readings reminded me of this graphic that’s been floating around my social media feeds:

(MindShift.com, 2018)

Winn writes, “Some recent thinking suggests that it is better to consider students to be tightly coupled to the environment rather than embedded in it. Being embedded suggests the student is passive, carried along as the environment changes. Successful students are anything but passive.” (Winn, 2003).

To be brief, Winn argues that “Artificial environments can use computer technology to create metaphorical representations in order to bring to students’ concepts and principles that normally lie outside the reach of direct experience” (2003). Essentially, technology helps the learning and provides a form of adaptation, in that the learner interacts with their environment significantly more than was possible or realized previously.

In another article, I read about the application of a program on handheld devices called TechPALS to mathematical problem solving. It was a great reminder of how the software of the technology does not have to be entirely about immersive experiences within the specific curriculum area for it to be effective. This article used control classes and classes integrating TechPALS to have students work on “repeated practice, feedback, and cooperative learning”, which creates embedded experiences within the content, and affects the environment in which the students interact with the subject matter. Roschelle et al. write that TechPALS is important because, “technology can socialize learning, encouraging positive behaviors such as asking questions, giving explanations, and discussing disagreements. These social behaviors, in turn, may engage students in connecting conceptual and procedural aspects of mathematics content” (Roschelle et al. 2010). The embodiment of their learning is intrinsically tied to what they refer to as “positive interdependence” and “individual accountability”. As far as setting up a similar scenario in my own practice, a mobile app like Kahoot or something comparable but perhaps less gamified? The students should be able to fit pieces of learning together like a jigsaw this could serve similar aims for embodied learning. From my perspective, reading for its usefulness and engagement, the instructional design of the lessons had everything to do with the embodiment of the content, and little to do with the actual technology. As the environment changes, the students interact with it in various ways, and the ability to engage in conversation about those observations, question each other respectfully, and have their views challenged goes back to the “adaptive” learning environment Winn was referring to. The point of technology helping to facilitate those goals is outlined in his idea of learning as adaptation, and the possibility of “us[ing] technology to reduce the limits imposed by our sensory [or cognitive] bandwidth” (Winn, 2003), facilitating more spaces for students to interact with the environment as it happens.

Finally, the last article I read was about mental mathematical strategies by Jérôme Proulx. It was a very interesting take on embodied and embedded learning, as it’s a current article linked to the theoretical ideas of John Threlfall (2002), and not necessarily what I would instinctively teach. I think I have some researching to do! Proulx argues that teaching strategies for mental maths is almost unnecessary, and could be readdressed in education. He writes that, “This is at the grass roots of Threlfall’s argument for the futility of classification and choice of strategies, for no mapping of classifications of strategies produced by students appears satisfactory. This said, even if some authors, as Threlfall highlights, recognize the variety in strategies as too great to contain them in categories and that these would need to be broadened enough to encompass them all, he insists that not even broad categories would successfully account for the diversity in strategies from one author to another. Categories or classifications somehow become useful fictions, that can even be seen to serve a question- able purpose, especially when it comes to teaching these strategies” (Proulx, 2013). His article cites perspectives “grounded in enactivism” where students interact with the problem as it happens and use what is comfortable for them to solve it. So my first question to you is based on his writings:

Q1: Is there value in naming strategies (specifically for mental maths) if Proulx has determined “it does not give much justice or credit to the nature of students’ mathematical activity when they engage in these strategies in a mental mathematics context” (2013)?

Q2: How does an educator monitor differentiation in embodied learning?

References:

Roschelle, J., Rafanan, K., Bhanot, R., Estrella, G., Penuel, B., Nussbaum, M. & Claro, S. (2010). Scaffolding group explanation and feedback with handheld technology: impact on students’ mathematics learning. Educational Technology Research and Development, 58 (4) pp. 399-419.

Proulx, J. (2013). Mental mathematics, emergence of strategies, and the enactivist theory of cognition. Educ Stud Math. (84) pp. 309–328.

Winn, W. (2003). Learning in artificial environments: Embodiment, embeddedness, and dynamic adaptation. Technology, Instruction, Cognition and Learning, 1(1), 87-114.

TELE Synthesis

Apologies for the late post, and thanks in advance for understanding. Here is my contribution to the TELE Synthesis forum:

In reflecting on the four different TELEs for this module, it’s clear that the constructivist approach and inquiry based nature of all four is central to their pedagogy. One of the many benefits of this is the fact that teachers have significantly more opportunities to catch the misconceptions discussed at the beginning of this course, and develop ways to have students recreate/redefine rules for the information in front of them. The more students talk about their thinking and experiences, the more they elaborate on why and how things work the way they do. Should any problematic understandings come to light, teachers can either scaffold to help them adjust their misconceptions, or present them with data that proves their thinking is problematic, and have them modify their understanding.

For my own teaching practice, these different approaches help significantly in terms of resources and instructional design. The variety in the different models can help dictate how best to teach a topic; use a Jasper-like or Anchored instructional model for real-life, problem solving contexts, use an iterative T-GEM approach for inquiry and critical reflection of their thinking, etc. These different TELEs also make me reflect on the use of technology in my classroom, and the fact that while I sometimes wish I used more technology to enrich learning, I need to continue to focus on meaningful integration. Like any other well-design materials, best practice with technology needs to be thoughtful and meaningful to be effective- all of which I feel were represented in one way or another (or in many ways!) with these TELEs.

References:

Biswas, G. Schwartz, D. Bransford, J. & The Teachable Agent Group at Vanderbilt (TAG-V) (2001). Technology support for complex problem solving: From SAD environments to AI. In K.D. Forbus and P.J. Feltovich (Eds.)Smart Machines in Education: The Coming Revolution in Education Technology. AAAI/MIT Press, Menlo, Park, CA.

Khan, S. (2007). Model-based inquiries in chemistry. Science Education, 91(6), 877-905.

Linn, M., Clark, D., & Slotta, J. (2009). Wise design for knowledge integration. Science Education, 87(4), 517-538.

Oh no, Fractions!

As a couple classmates have already identified, I also have noticed that students struggle with fractions as they move beyond identification of basic fractions such as half, quarter, and eighths. Particularly, comparison of fractions is a tricky topic to tackle with students in the lower elementary, and one in which teachers often rely heavily on visual aids. I’m going to add another one to the list, as I think that the app “Oh no, fractions!” has been extremely helpful as a tool to help visualize and compare fractions. Though it has other functions for adding, subtracting, and multiplying fractions, for the age group I teach, it has been most useful for comparing. The function of the app to estimate before they check their work allows them the space to explore and try to find commonalities and relationships within comparison.

Generate:

Students use Oh no, Fractions! to explore initially, before being answering guiding comparison questions:

Which is larger, 1/3 or 1/2?

By how much is 2/3 larger than 1/5?

Evaluate:

Give students fractions to name and sort. Have them order the fractions from largest to smallest. Create fractions on a linear fraction mat, pie chart, and using a variety of other tools. Talk about reasons or strategies for mapping the comparison. How did you get to your answer? How do you know that it is true?

Whitacre and Nickerson talk about a “reference point” perspective, involving “reasoning about fraction magnitude on the basis of proximity to reference numbers and a “components” perspective involving “comparisons within or between two fractions, as in coordinating multiplicative comparisons of numerators and denominators” (Whitacre & Nickerson, 2016). It is these perspectives we aim to get at and explain, in terms of logic used to compare the fractions. Ex. “How do you know one fraction is bigger? Step by step, explain your thinking. Is this the same or different to other people in your group?”

Modify:

Generate a list of rules as a class- we know that: x, y, z about fractions. Get the students to elaborate as much as they can on their reasoning. Once the list has been compiled, ask the student to pair up and investigate:

Can they be applied to sharing anything?

Can these rules be applied to all fractions?

Reference:

Whitacre, I. & Nickerson, S.D. J Math Teacher Education (2016) 19: 57. https://doi-org.ezproxy.library.ubc.ca/10.1007/s10857-014-9295-2

LfU and Adaptations

I am currently teaching a unit on animal adaptations, and how environments can dictate their physical or behavioral adaptations. Looking through the MyWorld resources and arcGIS was timely, in that it’s given me lots to think about for enriching my lessons. One such idea came from browsing the maps, and finding this “Upper Elementary biomes” map, where I could incorporate the idea of mapping into adaptations. The students I’m working with are in second grade, so not quite in the ‘upper elementary’ level, however the map is both simple enough to use with the 8 year olds, and interactive enough that they can glean information for research from it.

Later in the unit, when we talk about migration, hibernation, and other behavioural adaptations, we will come back to the map and track the change in environment. Using scenarios based on the tools this map affords, they can use in a given environment

Motivate: Introduce the activity as a biome research hunt. As a class, using either mind maps written on the board or the Padlet app, we brainstorm what the students think the different biomes are, and what their features include, simply projecting the map as is for the class, with the colours and legend. Students use their prior knowledge and assumptions to fill the brainstorms.

Construct: Students choose an animal to research, and record where they think the animal might live, doing a 321 Bridge activity, describing which biome they believe this animal lives in, and why. They then research the animal’s biome, and use the interactive map to determine features of the biome. Then, they complete the 321, using the map and research sites as the prompt for the bridge.

Refine: By seeing how their thinking changed before and after they used the research and maps to connect the biomes to the features of the animal, they will be able to link their understanding of why the animals have those features, and what potential adaptations might come from learning that the animal lives in a biome they didn’t anticipate. In this way, they verify and research for understanding.

As Edelson describes it, “Refinement of knowledge can also take the form of reinforcement, which increases the strength of connections to other knowledge structures through the traversal of those structures and increases the likelihood that those connections between knowledge structures will be found in the future” (Edelson, 2001). In this activity, by looking at their thinking before and after on the map, they can begin to not only connect features of the landscape to animal’s homes, but also start to look for patterns in the features different environments offer to living things (or patterns in the adaptations animals have developed). Their misconceptions are visible and relevant in being able to map out (pardon the pun) in the 321 Bridge where their thinking changed. This conceptual understanding informs their future analysis of other animal’s adaptations and helps them look for relevant patterns instead of memorizing facts about animals.

Reference:

Edelson, D.C. (2001). Learning-for-use: A framework for the design of technology-supported inquiry activities. Journal of Research in Science Teaching,38(3), 355-385.

WISE Genetics

I worked on the Genetic Inheritance (ID: 23367) project, and had a look at the ‘Unused Steps’ section more than trying to create my own slides for the project. I found some interesting connections and discussion points that must have been taken out for the sake of relevance to the content knowledge of the project, but I thought they were useful inquiries to create connections within understanding genetics. One such example was the ‘observing probability’ slide, where the students make connections to what the likelihood of certain trait acquisition would be. Though the slide takes a simplistic view on probability (observing only 50% chance of getting a trait), this could be extended by the teacher in looking at probability of certain traits over time, and turning this into a mathematics inquiry, which would be more grade level appropriate, as the lesson is targeted to Grade 6-8.

Also, though there are many opportunities for writing their learning and reflection, I found the slide where students were asked to “Turn to your partner and talk about the choice you made. Who do you think is right?” particularly relevant. To me, it follows Gobert et al.’s model that “make[s] science accessible for all students where accessibility has two meanings: to engage students in problems that they find personally relevant, and to engage students at an appropriate level of analysis and explanation, rather than load them down with abstract scientific models of phenomena which do not readily connect with students’ ideas” (Gobert et al., 2002). In getting them to justify their thinking to their peers, they are simultaneously checking their own understanding and building meaning together (using constructivist strategies).

I really enjoy the collaborative format of WISE, using a framework where the lessons are taught (by WISE teachers) and go through various iterations before being inducted into the library. I also find that the effective use of technology makes it a model for good technology integration. Rather than focusing on the hardware, WISE focuses on patterns and pattern detection through technology, where “WISE continuously adds new, proven features in response to user needs and as a result of user experience. WISE curriculum design patterns capture proven inquiry strategies. These patterns can inform design of other learning environments and of diverse forms of instruction” (Linn et al., 2003).

References:

Gobert, J., Snyder, J., & Houghton, C. (2002, April). The influence of students’ understanding of models on model-based reasoning. Paper presented at the Annual Meeting of the American Educational Research Association (AERA), New Orleans, Louisiana.

Linn, M., Clark, D., & Slotta, J. (2003). Wise design for knowledge integration. Science Education, 87(4), 517-538.

NRICHing learning

Anchored instruction by nature is meaningful problem solving, in a relevant environment (Cognition and Technology Group at Vanderbilt 1992). The evidence that exists suggests that it improves students’ cognitive ability to solve multi-layered problems. In general, “Jasper students showed less anxiety toward mathematics, were more likely to see mathematics as relevant to everyday life, more likely to see it as useful, and more likely to appreciate complex challenges” (Cognition and Technology Group at Vanderbilt 1992). So anchored instruction gave positive results with regards to its approach (both in integrating technology, and with its pedagogical practice), with exception: assessments.

Assessments aside, because those concerns were addressed to a certain extent in the articles, my primary concern is balance (especially with the younger students in creating a strong foundation of number sense), and the amount of time dedicated towards explicit instruction of skills in mathematics in addition to skills for exploratory problem solving. I do believe strongly that the anchored instruction the Jasper program facilitates is valuable. The article by Biswas, Schwartz, and Bransford mentions that students “…learn to work smart by inventing tools like graphs, charts, and spreadsheets that help them solve these problems at a glance” (Biswas, Schwartz, & Bransford, 2001). No doubt this is the kind of math students should be learning, and creating time for, but the organization of this knowledge amongst other new skills needs to be explicitly taught, which requires time, scaffolding, and opportunities to build on each others’ learning. Time being the resource most teachers are concerned with.

Because of its global context, Jasper is a closer connection to STEM than many other approaches to teaching mathematics, and its context for real world problems is engaging. Additionally, the ability to switch between different variables (what if we were measuring the speed and distance of a boat instead of an Ultralight, what if the tank was larger etc.) makes it easier to differentiate, and makes the students more fluent in seeing connections between themes, instead of focusing on a particular operation because that’s the unit they’re working on and those are the numbers and variables given, or what the Cognition Group at Vanderbilt call “computational selection” (1992).

In thinking about other resources that are available online for the age group that I teach, I can’t help but think of Mathletics and think that there is a lot within that program that helps for practicing calculation and computation, but not a ton on problem solving. It doesn’t have much to do with an anchored approach to learning, but it does provide lots in terms of differentiation, novelty, and friendly competition to motivate students to feel more comfortable with math. In the same vein as the Jasper model, I tend to gravitate more to resources like NRICH maths, which is designed as group work and explorative math/logical thinking activities. The learning doesn’t have as much of a narrative built in as the Jasper model, but it does have multi-step exercises (based on the age group you’re focusing on). With less video prompts than the Jasper episodes, NRICH starts with minimal technology in their activities (citing classic examples such as the ‘Tower of Hanoi’ mathematical problem), and builds their integration and aides around good practice, much like the Jasper study that focused on “… start[ing] with stone age designs (SAD) environments and to add sophistication and complexity only as necessary to achieve our instructional goals” (Biswas, Schwartz, & Bransford, 2001). It is inquiry-based, focused on using group work, exploring, and noticing patterns, but not anchored instruction- it uses anecdotal tasks but not involved contexts to solve problems like Jasper. For the sake of extended questions within the learning, I would consider looking at NRICH from the perspective of anchored learning as exemplified in the Jasper model, and use problems that allow me to extend variables across many lessons, in addition to identifying and teaching through themes as opposed to specific situations (i.e. idea of calculating speed of that car, boat, train vs. the speed of one specific vehicle) to inform my future practice.

References:

Cognition and Technology Group at Vanderbilt (1992). The Jasper series as an example of anchored instruction: Theory, program, description, and assessment data. Educational Psychologist, 27(3), 291-315.

Addition and Number sense in G2

In second grade, fluency in single and double-digit addition become early stages of multiplication. In my current class, we are now looking at multiplication, though these second graders are a strong group and have made big jumps, thanks to their foundation teacher last year, who is a whiz at explaining and reinforcing number sense.

We begin with looking at the students’ strengths with number bonds (to 10, then 20), and we use manipulatives produced by the Numicon program to help our maths understanding.

Having a strong understanding of number bonds (7 + 3, 6 + 4, 8 + 2, etc.) helps them ‘make a ten’ and recognize and name other mental math strategies. It has been incredible for me, to use and explain the concept of number bonds with Numicon shapes, as simple and as lego-like as they are. The program was originally designed as a remedial resource, but the school I work at has adopted it as a foundation for all students, and I strongly feel that it has improved students’ number sense and confidence with manipulating numbers by a large margin. Here’s a short video on YouTube if you’re unfamiliar with what these manipulatives look like. When dealing with larger numbers, we introduce Cuisinaire rods as manipulatives, and begin to move towards developing and relying on Mental math strategies like these:

We explicitly do not look at the traditional algorithm in maths, until students demonstrate strong understanding of partitioning numbers, and feeling comfortable manipulating large numbers the same way they do with their smaller number bonds. Then, we teach the ‘lining it up’ method, where students can deal with smaller sums as a shortcut. This only comes after they can explain the place value of all of the numbers in the equation, and can explain step by step what they are actually adding. With a strong emphasis on “you solve it however it is easiest for you to see the numbers”, the only thing we ask students to do is to show their thinking. Whether that’s with a number line, compensation, partitioning, or otherwise. Funnily enough, even after using the traditional algorithm, many students in my class in particular still opt to add numbers in a lengthy way, because they feel it’s more visual and understandable. For example:

359 + 492 = 300 + 50 + 9 + 400 + 90 + 2, and

300 + 400 = 700, 50 + 90 = 140, 9 + 2 = 11, and 700 + 140 + 11 = 851

or more recently with respect to multiplication:

202 x 4 = 202 + 202 + 202 + 202

200 + 200 + 200 + 200 = 800, and 2 + 2 + 2 + 2 = 8, and 800 + 8 = 808

Word problems are a part of the teaching all the way through, and identifying the language that commands a certain function (“find the difference, how many altogether”) is recorded and referred back to regularly.

Admittedly, I was afraid of teaching math for a long time because it was something I lacked confidence and fluency in. Thinking about PCK for math scared me as a fresh graduate of teacher’s college, because my foundations for content knowledge were shaky at best. And as Mishra and Koelher point out, “…having knowledge of subject matter and general pedagogical strategies, though necessary, was not sufficient for capturing the knowledge of good teachers” (Mishra & Koelher 2006), which made it even more intimidating. Tools like Numicon and and the GloSS assessment for mental maths have been incredible for laying out processes in problem solving and mental math strategies. I strongly believe that these resources, in addition to seeking out help and advice from those much stronger than me, has allowed me to get a little closer to what Mishra and Koelher describe as “find[ing] different ways to represent it and make it accessible to learners” (Mishra & Koelher 2006). Though I clearly still have a long way to go, I’m confident in explaining different ways of thinking about numbers (at the elementary level, mind you!), and this allows me the perspective of being critical of technology integrated in my pedagogical practices, unless it is truly beneficial for the learning and allows students to further explain their thinking or explore concepts that they could not see otherwise.

Reference:

Mishra, P., & Koehler, M. (2006). Technological pedagogical content knowledge: A framework for teacher knowledge. The Teachers College Record, 108(6), 1017-1054.

Transdisciplinary approaches

Jonassen’s (2000) definition appealed to me the most, because it reiterates the role of the teacher as the facilitator, and the students as the enablers of their own learning experiences. Student agency and collaborative constructivist experiences are at the core of his ideal classroom setting, and I tend to agree with that approach. He writes that, “[S]tudents learn from thinking in meaningful ways. Thinking is engaged by activities, which can be fostered by computers or teachers”. The conceptual knowledge facilitated by technology that would otherwise be unattainable or difficult to visualize is the catalyst for project-based inquiries to explore; thinking and learning from the modelling, processing, refining, and iteration involved in TELEs are what make the experience enriching.

Designers should create spaces with the idea of flexible learning spaces in mind. Though maths and science are typically thought of as specific subjects with specific content knowledge, flexible environments and transdisciplinary approaches (like STEM/STEAM) help students see connections across the curriculum and more importantly, as problems to solve in the real world. Change should always be anticipated, and opportunities to work across traditional ‘subjects’ should be welcomed as a transformation that allows students to see math and science as more than concepts in neat boxes, where they are applications to real world issues.