This week teachers around the globe are asking the question, “How can I use technology to engage my students in meaningful learning opportunities whilst they are at home?”
Creating a community of learners is no easy task, even within the context of a conventional classroom. Whilst we could view our
This week teachers around the globe are asking the question, “How can I use technology to engage my students in meaningful learning opportunities whilst they are at home?”
Creating a community of learners is no easy task, even within the context of a conventional classroom. Whilst we could view our current circumstances as yet another roadblock to building community, we can also choose to see it as a new opportunity for creativity and innovative teaching.
Below are 5 suggestions (from educators) to help you foster classroom community via Mathspace distance learning.
Intentional question prompts are a powerful way to engage students in collaborative discussion, especially via online learning. Whether you’re teaching on an existing learning platform (such as GoogleClassroom), or using the “Chat” feature on a video conferencing platform (such as Zoom), here are some ideas to consider when posing questions for your students:
Before instruction/learning
Ask students a question prior to maths instruction to tap into prior knowledge and natural curiosity. These initial thoughts and wonderings will act as a formative assessment, giving you insight into students’ existing knowledge, and also serve to engage students in the learning to come.
Examples:
During instruction/learning
During online Mathspace instruction, pause your teaching to pose a question. Invite students to respond in the “chat box” or on the “discussion board”  the great news? Most students are so tech savvy that they’ll be very familiar with this! After a little practice it becomes fun, much like a conversation via text. Students can answer questions, share challenges that they’re having, and even celebrate moments of new learning.
Examples:
After instruction/learning (reflection, new learning)
After maths instruction, pose a question that fosters reflection of new learning. Once again, this can provide an opportunity for you to formatively monitor student learning. It also will hold students accountable and increase metacognition.
Examples:
Create a space for students to discuss Mathspace content, topics and questions without any evaluative teacher response. Students can post technical or content related questions and help one another without teacher input. This will promote student agency and build relationships among students. Discussion boards also give shy or quiet students a chance to “speak up” and share ideas, often building their confidence and improving interactions with peers.
There are many options when it comes to discussion boards; go with what you're most comfortable with. If you're already using Google Classroom, continue using posts as discussion boards for students. If you're using Zoom for online learning, invite students to engage in discussion using the "Chat Box," visible to all participants. If you've explored using Slack, there are options to create "Channels" (or chats) for each class. Students will also have the option to send you private messages via Google Classroom, Zoom, or Slack. This is one way to cut out email traffic ask your students to message you via your online learning platform to simplify classroom communication!
Suggestions:
Prerecord videos for your students. This way, you can share information with multiple classes, prepare for instruction ahead of time, and cut down on planning time in the long run. Try using a video recording platform such as Zoom, or the video feature in Google Classroom.
Consider sharing a Mathspace Textbook lesson page with your students, then talk through the content as you would in a classroom setting. You can also use prerecorded videos to address common misconceptions, inform students of upcoming tasks, give positive feedback, or simply say hello! Your students love to hear from you. Giving students the option to respond via video can continue to build community among learners.
Suggestions:
Students love competition  create opportunities where your students can work as a team to accomplish a common goal. Leverage technology and Mathspace Data Reports as a way for students to selfmonitor their progress and engage in friendly competition as they practise maths skills.
Examples:
Suggestions:
Students today love to engage with their peers, including those that they’ve never met before! Use this opportunity to connect with students in another class, another school site, another city, state, or country.
Do you have friends who are teachers in another place? Know any professors or parents of students who use mathematics in their daily careers? Reach out to people and let them know that your students would love to learn from them, you might be surprised who you connect with!
Suggestions:
Example:
How might you connect your class with others to spark new ideas, foster collaboration, and engage students in a new type of community? We would love to hear how you build community whilst your students are learning from home!
]]>Last year we released a new version of the Mathspace iOS app to the Apple app store. This new iOS app
]]>If you have students using Mathspace on a Windows or an iOS touch device, we have made some changes which might impact them.
Last year we released a new version of the Mathspace iOS app to the Apple app store. This new iOS app give students access to all of the latest new features on the student workbook.
The new Mathspace app needs to be installed. Note it is not an update to the existing Mathspace app.
You can find it on the app store (AU link here, and US link here) it's called Mathspace for Students. Read these instructions for more information
We strongly recommend that students delete the old app, and start using the new app on their iPads. All of your students' normal account settings will be available when they login to the new app. Whilst students can still use Mathspace on our old iOS app, we have stopped our engineering support for the app. This means that the old app does not include any of the new product features.
We have made the decision to stop support for our Windows 10 app. This decision was made as we have an extremely small Windows 10 user base, and time spent supporting the app has prevented us from working on other exciting new features for our teachers and students.
Those who have already downloaded the app can continue to use it but we can not guarantee all functionality will work into the future.
The Windows app has been removed from the Windows store. From 31 March 2020, the app will no longer be supported.
We recommend users on Windows devices use Mathspace on one of our supported browsers. You can read more here.
]]>My name is Paulina and I am a Product Manager here at Mathspace.
Today marks the official launch of our brand new suite of teacher reports. After many months of research and development, I’m very pleased to be able to present this range of reporting tools which gives you practical and easytoaction insights to help you in the daytoday running of your classroom.
First some background…
At the start of the year, we set out to begin a major overhaul of all our existing reporting features. In designing the new reports, we worked very closely with teachers across both Australia and the United States to make sure we were focusing on the right needs. You can watch our Webinar on how we leverage user feedback as an integral aspect of our design process here: Shaping Digital Resources: How we turn teachers' ideas into product features at Mathspace.
As a result of interviewing teachers from various different backgrounds and methodologies, we identified a common pain point across all: lack of time. Sound familiar? Teachers were asking for tools to make their day to day more efficient and to help them individualize where needed.
After many interviews and prototypes, we identified four key areas where teachers wanted more information:
Reports on these four areas would help teachers to monitor classrooms efficiently, identify knowledge gaps and support struggling students. These kinds of reports would also help teachers with how they report, giving important information for report cards and parent teacher nights.
The result?
We’ve released a set of student and class reports that will hopefully empower teachers in their learning environment to achieve their goals and help schools move towards a highly personalized approach which focuses on individual student growth.
Read on to see the new features we’ve added to Mathspace:
Class and Student Insights provide a quick snapshot of what the students worked on, and where they need help. In less than 2 minutes, teachers can identify the questions that the whole class struggled with and also students that have not done their work. Insights reports work across both custom and adaptive tasks, and it’s the ideal tool to open up right before a new lesson.
Class and Student Activity display information around the effort that students are putting in. Teachers can select a time frame to view time spent in and out of school hours, questions completed and points gained, to check that students are staying on top of their expected effort. Teachers can also use the Live Mode to verify that students are on task during a lesson and identify if anyone needs assistance. Activity reports are a useful tool to have conversations with the students.
Class and Student Mastery reports show the proficiency that students are achieving with their practice. Mastery Levels were introduced to make the report more approachable. It is now easier to identify which topics have been mastered and find knowledge gaps by looking out for the white areas in the heatmap. Whether teachers use custom or adaptive tasks, it is possible to view proficiency of the whole class across the complete textbook to monitor progress across school years.
Class and Student Tasks display the status and results of assignments. They are designed to provide a quick birdseye view of whether the class and students are completing their assignments and how they are doing. It also includes a really useful view of how each student in a class is progressing with their own tasks, to better support teachers that are individualising and have each of their students doing different work.
Imagine you’ve spent hours designing stations to help your students learn or practice a challenging concept. You’ve planned every last detail, expertly crafted the work at each station, and even carefully divided your students into groups based not only on academics but also their unique personalities.
On the day, you explain the rules to your students: who starts where, how long they spend at each station, which way to rotate and then you start the timer.
Maybe at first everyone is on task and learning, but it’s only a matter of time before a student finishes before the timer goes off and starts distracting her neighbor, and while you’re trying to redirect the offtask students, another student wanders from his assigned station to hang out at his best friend’s station. As you go to direct him back to his group, the timer goes off and half the class doesn’t remember which way to rotate. Eventually, you get through the class and your students have learned something, but you’re exhausted because it didn’t go the way you planned at all.
Most teachers don’t have to imagine, because they’ve experienced a similar situation at some point in their careers. It may have even deterred them from ever trying stations again.
So let’s talk about some of the challenges of implementing stations in the classroom and ways to overcome them. We are going to focus on the math classroom but these same principles can be applied across all subject areas.
It’s completely normal to be overwhelmed when the word differentiation comes up. Everyone loves to tell you that you should be doing it, but no one seems to have a howto guide on the topic.
In the context of stations, you’re already creating multiple stations with different assignments or activities at each. Are you also supposed to be creating different versions of each station to meet the needs of every learner?
The answer is: you could but you don’t have to.
Use each station as a way of differentiating. Remember, not every student needs to visit every station, especially not for the same amount of time. Use the data you collect daily in your classroom to inform what topic each station should address. A student doesn’t have to go to a particular station if the data shows that they are either not ready to learn that content, or have learned the content well enough that they will receive no significant benefit from working at that station.
The Weekly Insights and Student Insights features on Mathspace enable you to quickly pinpoint which content needs to be addressed by which students. Teachers can then quickly assign individual assignments to students to meet their specific needs.
Traditionally, when teachers create stations they’re very specific and intended to be used for one particular class period and never again. This means teachers need to create completely new materials every time they want to use stations in their classroom.
To avoid putting in a lot of effort for limited return, create stations with a general theme that can be reused on a weekly or even daily basis. Stations don’t have to be completed in one class period. They can include ongoing learning experiences that students can go back to class after class.
Some examples of stations that can be used fluidly throughout the year are:
Many of these stations require no more planning than traditional classroom instruction, and they can all become regular fixtures in your classroom yearround with only slight adjustments as the content changes.
One of the main deterrents for teachers when it comes to using stations is classroom management. If you’re not careful, even the bestplanned stations can quickly devolve into chaos.
Fortunately, some of the strategies discussed above will already help by increasing student engagement. By using data to ensure students are at a station that is appropriately challenging, students are less likely to be off task. Keeping stations the same throughout the year will also help students know what to expect.
However, we know engaging work doesn’t eliminate all behavior problems, so some other strategies you can implement are:
Stations can be a great way to differentiate instruction and make learning more meaningful to students. They don’t have to be chaotic or timeconsuming. Start small and teach your students to manage their learning. Before you know it, you’ll have your stations running like clockwork.
The Utah STEM Action Center is in the second year of a five year evaluation program, assessing the impact of digital adaptive math programs on student learning. A report presenting
]]>The Utah STEM Action Center is in the second year of a five year evaluation program, assessing the impact of digital adaptive math programs on student learning. A report presenting the findings and recommendations of the 201718 implementation year has been released, showing positive results for Mathspace.
The Study compared students’ 2018 SAGE mathematics raw scores, attainment of proficiency, and standardized growth percentiles (SGPs). Software vendors provided 201718 student usage data to the Utah Education Policy Center (UEPC) on a monthly basis through a secure platform.
In total, over 94,000 students in grades 312 were involved in the study, of which 6,588 were Mathspace users.
The results showed that students using Mathspace experienced significant growth over the period. In particular, students who were not proficient in the prior year demonstrated exceptionally high growth. This was measured using Student Growth Percentiles (SGPs), which are a measure of student growth calculated by the Utah State Board of Education. This measure assesses student growth by assigning each student to a percentile within an academic peer group. Academic peer groups are created with quantile regression using each students' available SAGE scores in the subject area from previous years.
The report also showed the increase in likelihood of a student testing as proficient in mathematics on the 201718 SAGE if they used one of the math software programs. The percentages were provided for all students as well as for students who were nonproficient in the previous year.
Students using Mathspace showed a 53% increase in likelihood of math proficiency. In particular, students using Mathspace who were nonproficient in the previous year showed a 66% increase in the likelihood of math proficiency.
Founder and CEO, Mohamad Jebara, said that the results were a humbling affirmation of the Company's commitment to moving schools away from a onesizefits all approach, towards a highly personalized approach which focuses on individual student growth.
"We built Mathspace because we believe that every student can excel at math with the right help at the right time. From the outset, we've been focussed on building a platform which can be the right help at the right time for every student.
Our goal is to help teachers to increase student growth. The strong results in the recent report by the Utah STEM Action Center are testament to our commitment to building a worldclass edtech platform for mathematics, which helps us to achieve our overarching goal"
You can read the entire Report here; find Mathspace on slides 108 to 114.
]]>One of the most common reasons people become teachers is to make a difference in the lives of as many students as possible. This means teachers are never happy when one student falls behind.
With Response to Instruction and Intervention (RTI) emerging as a bestpractice model in the United States to help support of struggling learners, students do not have to fail before receiving support.
The Response to Intervention model focuses on 3 main components:
This model requires teachers to think carefully through the course objectives, anticipate where students are more likely to have difficulty, then craft instruction that takes this into account. Teachers are essentially planning support for students that may struggle ahead of time.
The big questions is: How can one teacher do all of this and not fall behind in the course?
I've put together this article to show teachers how Mathspace can help to support teachers in implementing RTI practices.
Mathspace eBook lessons are designed to support all learners with engaging interactive applets that allow students to explore concepts. By reviewing the Worked examples and Practice questions sections, students can test their newly acquired content without penalty – make mistakes fearlessly. Because Mathspace provides teachers with multiple grade levels of content, teachers can scaffold their instruction to encompass the anticipated areas of difficulty. Teachers have all of this at their fingertips to assist them with crafting engaging and exploratory lessons for all students.
To determine whether a student is struggling, the use of data is essential. Tier 2 Instruction is targeted for students who are identified with a potential weakness in a certain area. The phrase “in a certain area,” is important because a student can need Tier 2 Instruction for Geometry and Measurement Topic but not for Equations and Inequalities. Mathspace helps teachers determine who may need additional supports in different areas. By using the Diagnostic Testing series, teachers receive a detailed report on each student’s potential strengths and weaknesses across 3 grade levels. With this information and other data points, determine which areas need intervention for different groups of students. Teachers can monitor students’ progress weekly on Mathspace’s Weekly Insights and students can monitor their own progress by using the Topics page.
Tier 3 Instruction is in addition to Tier 1 and provides students who are significantly below grade level with the intervention that fills in the instructional gaps within the students’ mathematics foundation. During Tier 1, teachers can help fill these instructional gaps by providing connections to prior taught concepts from previous grade levels – making connections explicit. Mathspace allows teachers to create tasks that are ongrade level, above grade level, and below grade level and assign these tasks to the whole class, small groups of students, or an individual student. By using this feature, students can have assignments that are appropriate for them without students knowing that there is a difference. Teachers can use the Diagnostic Strand Tests to monitor student progress on targeted areas of deficits. Teachers who are providing Tier 3 Instruction separately from Tier 1, have access to the problems completed in Tier 1. This allows the teacher to see how this content is connected to belowgrade level subtopics by using the Mastery page and changing the Curriculum focus.
Element  Mathspace Tool 

Screening  Diagnostic Course Tests and Reports can be given as a Pre, Mid, and Post test to identify potential strengthens and weaknesses. 
Instructional Assessments  Tier 1  Custom and Adaptive Task can be given to all students on current grade level content. 
Progress Monitoring  Tier 2 and Tier 3 
Diagnostic Strand Test can be given to assess student progress or performance in those areas in which they were identified by universal screening as being atrisk for failure. 
InDepth Instruction  eBook lesson and applets can be used to augment direct instruction. 
Systematic and Explicit Instruction  eBook lesson, investigations, and applets

Solving Word Problems  Tasks allow students to solve problems with different approaches and provide hints and videos that focus on how to conceptualize a problem situation and identify needs, analyze factors contributing to the problem situation, design strategies to meet those needs, and implement and evaluate the strategies. Investigations allow students to solve situational problems using newly acquired knowledge and skills. 
Visual Representations of Math Concepts  Applets provide students access to abstract mathematical ideas through the use of items that can be physically/virtually manipulated. 
FluencyBuilding Activities  Task Templates teachers can create common assessment fluencybuilders using content from grade 3 through grade 5 and share throughout the school or district. 
Motivation  Tasks are designed to reenforce growth mindset by providing encouraging words when students correctly answer a question and offering hints/videos as an optional resource. 
Mathspace strives to support educators in being the right help to students at the right time. If you want to learn more, join our community on Facebook and schedule a professional learning session with us. We can customize a professional learning session around your district or school initiatives.
The illiterates of the 21st century will not be those who cannot read and write, but, rather those who cannot learn, unlearn and relearn.
 Alvin Toffler
References:
ACT. (2008). The forgotten middle: ensuring that all students are on target for college and career readiness before high school. IA City, IA.
Gersten, R. M., & NewmanGonchar, R. (2011). Understanding Rti in mathematics: proven methods and applications. Baltimore: Paul H. Brookes.
Jackson, R. R. (2011). How to motivate reluctant learners. Alexandria, VA: ASCD.
McCook, J. E. (2009). Leading and managing Rti: five steps for building and maintaining the framework. Horsham, PA: LRP Publications.
SchoolBoard. (2015, May 11). College Success Starts In Math Class. Retrieved from https://www.forbes.com/sites/schoolboard/2015/05/08/collegesuccessstartsin8thgrademathclass/#4f9257b8248e
Stipek, D., Schoenfeld, A., & Gomby, D. (2019, February 21). Math Matters, Even for Little Kids. Retrieved from https://www.edweek.org/ew/articles/2012/03/28/26stipek.h31.html
Consider, far out in the depths of space, two identical objects, alone but for each the other’s twin. Let’s call one of them Alice and the other Bob, as is the tradition in thought experiments.
To complete the setup, we note that Alice and Bob are both spherically symmetric, that they have the same mass, that there is no net charge which would lead to any kind of electrostatic attraction, and that they are initially at rest with respect to each other. This is, more or less, the simplest thought experiment we could possibly imagine: a couple of stones sitting still some distance apart.
Now we ask the obvious question. If we keep watching Alice and Bob, what do we see? It turns out that, regardless of their mass or the distance separating their initial positions, what we would observe is that Alice and Bob begin to move toward each other. Ever so slowly at first, then quicker and quicker. They will be seen to accelerate towards one another until they ultimately collide.
This is no trick question, and you may be thinking that the observed behavior is entirely what we would expect. Maybe so, but suspend your intuitions for a moment and reflect on just how strange this situation actually is. Place Alice and Bob in an empty universe and they will eventually find each other. Without even looking, they come together — not by doing, but simply by being.
What is going on here? Recall Newton’s first law of motion:
Stuff that isn’t moving will keep on not moving, and stuff that moves uniformly will keep on moving uniformly, unless you force it to move differently.
That is, objects at rest can only start moving (and objects moving in a straight line at a constant speed can only change direction or speed) under the action of some force.
The way an object moves in response to a force is described by Newton’s second law of motion, F = ma. Given the force, we determine the acceleration. So, having observed the acceleration of Alice and Bob, can we infer a force?
To help us visualize the motion we are dealing with in this thought experiment, let’s draw a diagram that shows the position of an object at different points in time. A third object, which we’ll call Charlie, will join us for this part of the experiment.
If Charlie is initially at rest, and is acted upon by no forces, we observe the very unexciting timeline shown below.
Pretty tame, but what more would we expect from a single stone out in deep space? Well let’s give Charlie a big shove to boost his speed. Now we see Charlie move off in one direction at a constant rate, since there is nothing else out there in deep space to slow him down. Charlie’s timeline is now more sloped.
So it looks like forces make for more interesting timelines. What if we apply a constant force, by strapping Charlie to a rocket? We would get a constant change in speed — an acceleration.
As a final example, we can see the effect of a dynamic force, where even the acceleration Charlie experiences is constantly changing. Attach Charlie to a spring and we observe him bounce back and forth.
Now that we are experts with this type of visualization technique, we can introduce some formal terms for the things we have been talking about. A diagram that has a space axis and a time axis is called a spacetime diagram. The path of an object through a spacetime diagram is called a worldline.
Spacetime diagrams are really very simple, but they have some interesting features. You may have noticed that Charlie was always moving through spacetime, whether or not there were any forces pushing or pulling him around in space. Even when Charlie was initially at rest (in space) we saw that he was moving consistently through spacetime. Most notably, when no forces are present we see Charlie move along a straight worldline (as in figure 3 and figure 4).
If we do away with thinking of motion in terms of space and time, and we fully absorb the implications of spacetime thinking, then it becomes apparent that being at rest is no longer a meaningful concept. How can something stand still in space and time? Further, it can be shown (through more sophisticated treatments than are worthwhile here) that when forces are present, they only ever act to change the direction of an object’s worldline, never the speed of that object through spacetime.
We can summarize these ideas in what I’ll call Einstein’s first law of motion (which you should compare to Newton’s first law above):
There is no stuff that isn’t moving. Stuff travels at a constant speed, and will move along a straight worldline unless you force it to move along a curvy worldline.
Now you may be wondering, if everything is moving at the same speed through spacetime, what speed is that? Unfortunately this also requires a treatment outside the scope of this article. But suffice it to say that Einstein’s law is a universal law applied to all objects, and the universe itself has a “speed limit”  the speed of light. So if everything is moving at the same speed, what other speed could it be, other than the speed of light?
With all this in mind, let’s now return to our original scenario with Alice and Bob. What we want to do is to be able to describe the situation we observe in space and time (in figure 2) on some kind of spacetime diagram. Starting with Einstein’s law of motion, we have two possible paths to arrive at a satisfying description; we can use forces, or we can make do without them.
The first path is easy. Suppose that there is an attractive force between Alice and Bob, analogous to the electric force between two oppositely charged particles. We see Alice and Bob accelerate, so we know that their worldlines through spacetime will be curved. This suggests that the force acts continuously. The spacetime diagram for this case is shown below.
There is a robust understanding of how electrically charged objects interact under the influence of electric forces, so it is not too far fetched to think that “mass charged” objects like Alice and Bob can be influenced by “massive” forces. Nevertheless, uncovering the exact nature of this massive force will be a considerable challenge. So it is worth our efforts to see if we can find a way to describe the motion of Alice and Bob without calling upon any forces.
We are now on the second path, and though we proceed confidently we are quickly met by significant obstacle. If there are no forces acting on Alice and Bob, then both their worldlines will be perfectly straight. In particular, since they are initially at rest in space, their worldlines will be initially parallel, and will stay parallel as they both move through spacetime.
How can it be that two objects, initially traveling in parallel, can continue each moving in a straight line and yet end up meeting one another?
If you try sketching out a few worldlines on paper, you will soon find that it is very difficult to make Alice and Bob collide. Which is not too surprising, since in fact it is impossible to draw on paper two parallel lines that meet at a point. Paper, tabletop, door, wall or floor – try any flat surface and the result will be the same. In some sense it is the definition of flat space that parallel lines do not intersect.
Luckily for us the world is also full of lots of very unflat things. Why not try drawing a spacetime diagram on a curved surface? An apple seems like an appropriate choice.
Imagine Ash and Bernhard are two ants exploring the surface of an apple. We encounter them both standing at different points along the apple’s equator, both facing the apple’s stem. As they begin to walk stemward we notice that their paths are initially parallel. Being very rational and economic ants, Ash and Bernhard are each following the shortest possible path from their respective starting positions on the equator of the apple, up to their common goal of the stem.
By taking the shortest possible path, both ants walk in what is essentially a straight line along the curved surface of the apple. If calling it a straight line seems a bit odd, just think about what a straight line in flat space represents; it is the shortest possible path between two points. It’s just that Ash and Bernhard are confined to the apple’s surface, they can’t dig down and chew their way through the middle of the apple. Their path is straight, it is only the space they travel through (the surface of the apple) that is curved.
And so we see Ash and Bernhard, initially traveling in parallel, can continue each moving in a straight line and yet end up meeting one another. The big idea here is that their paths were allowed to eventually intersect solely due to the curvature of the surface of the apple, not because of any force pushing or pulling them away onto anything other than the shortest possible path to the stem.
Are we not now ever so close to our solution? Have you made the connection already? Of course Alice and Bob could never meet traveling along straight, parallel worldlines in flat spacetime. But just as Ash and Bernhard “collided” because they were walking along a curved surface, so too can Alice and Bob collide if they are moving through a curved spacetime!
Finally we are able to draw a spacetime diagram that faithfully describes the motion we observe, and with not a force in sight.
Let’s double check that this description is consistent with Einstein’s law of motion. Alice and Bob move at a constant speed through spacetime, and since there are no forces present, they move along straight worldlines. But since the spacetime is curved, they end up accelerating toward each other and eventually colliding.
People all over the world have been thinking about gravity for a long time, and although we have made substantial progress, there are still many things about the nature of gravitation that we do not fully understand. Some of the most profound insights came from the work of Isaac Newton in the 1600s, then later from the work of Albert Einstein in the early 20th century.
Newton conceived of gravity as an attractive force acting between all masses. Celestial bodies and earthly ones are all governed by those same laws of motion. Many of his ideas were so groundbreaking that they couldn’t be described by any of the mathematics that existed at the time. Newton had to invent calculus himself just so he could formalize his theories.
Einstein reimagine gravitation not as the action of any force, but as the coupled interaction of mass and spacetime: masses cause spacetime to curve, and in turn spacetime governs the way masses move. Einstein had very capable mathematical abilities, but his real talent was his highly creative imagination.
He also famously loved thought experiments. It is appropriate now to compare side by side the results of our exploration of the motion of Alice and Bob, first from the perspective of Newton gravity, then from the perspective of Einstein gravity.
While Einstein’s is considered the more accurate theory of gravity, there are many situations for which Newton’s approach is good enough. Which would you use to describe the way an apple hangs on a tree, and the way it falls? Which would you use to describe a long distance flight? What about the orbit of satellites around the Earth, or the Earth around the Sun? Which would you use to describe the image of a distant star as it is eclipsed by a black hole?
After all the successes of these theories, there are still phenomena we observe in our world which are not explained by Newton gravity or Einstein gravity. For these we need a better theory, and this will almost certainly require much more creative and unorthodox thinking to develop.
]]>Math is an extremely polarizing field. I’ve found that most people either hate math or love the subject endlessly and the reasons for both opinions vary immensely. Since the social norm seems to be hating math, I’ve started thinking more about the contrary for those who do love math, why do they love it?
I happen to be one of the ‘loves the subject endlessly’ people. My own love for math is difficult to express. I cannot really say what led to my vast appreciation or when it began, but I do know one thing for certain: I find math to be incredibly beautiful. It is elegant, mysterious, challenging, and is without a doubt, my favorite topic to study. I find that many of my mathematically inclined peers would tend to agree. So my question at hand has deviated: rather than pondering why we love math, I want to know what makes math so beautiful?
By looking to various features of math, I’m hoping that an overarching explanation of the mysterious beauty will emerge.
When you learn math at any level, you’re taught an abundance of definitions. For example, the word gradient or slope, being the inclination of a line, is something we learn at a young age. As we continue to learn more math, we learn more and more of these definitions and can begin speaking in “math words”.
Later on in your math career, you may start to learn theorems and corollaries. These will be comprised of multiple “math words” in a logical and entwined way which will allow you to speak in “math sentences”. After that, you’ll learn how to build proofs, comprised of the definitions, theorems and corollaries. Your proofs will resemble paragraphs and you’re now able to talk in full “math paragraphs”. You might even go on to write a thesis in which you have many consecutive proofs. You’ll then be able to recite an entire essay in “math language”.
To put the theory to the test, just see what happens when you read your thesis at the dinner table. Unless you’ve grown up with a group of mathematicians, it’s most likely that you’ll be met with a sea of confused and glazedoverlooks. Of course, this is completely understandable. Without knowledge of the definitions and theorems that make up your proofs, it would be nearly impossible to make sense of it all. Thus, I would consider learning math to be like learning a new way to speak, or a new language.
I find this linguistic connection to be not only interesting, but a beautiful way of thinking about math. Math is so often thought of as a terrible subject that we’re forced into learning. When we think of learning math as learning a language, it makes it all the more enticing.
When we think about the pictorial representations of math, I think it’s most common to imagine a boring, linear graph. Although this is an accurate depiction of what we can do with math, there are so many other beautiful ways of visualizing math.
For example, pictured above is what we call the Mandelbrot Set. In layman’s terms, the image represents a set of numbers which all follow a specific set of rules. What’s amazing about this image is when we zoom in for as long as we can, the main pattern of the image will always be a constant. The colors of the image also have a purpose. They represent the various amount of steps required to fulfill the associated equations. We’ve taken an extremely complex, hard to digest situation and made it look incredibly cool  how amazing is that?
Another nice visualization in math is anything relating to geometry. Above are examples of some 3D, geometric shapes that represent various sets of rules in math. Although we are talking about some complex, math ideas, we can represent them artistically and derive beautiful images once again.
There are so many instances in math where we can represent a mundane looking math equation with a beautiful, artistic image that appeals to mathematicians and the general public. This is an easytodigest and very clear connection as to why math might be considered beautiful.
When we do math, we are always maintaining balance. To give you a clearer idea of how that is, think about solving an algebraic equation for a variable. In order to deduce what our variable, say x, is equal to, we perform various operations to both sides of our equation. Remember your year 7 teachers saying “You must do unto this side as you do to that side”? Well, what they were telling you in other words is “maintain the balance”.
This idea of maintaining balance is not often how we perceive solving our math problems. In general, these processes are seen as some of the driest and most tedious calculations that we do. So, next time you solve an equation, think about how you feel when you balance on one leg in a yoga class, or when you spin in a quick circle in dance class and don’t want to topple over. The motions we go through to achieve balance are inherently calming. The same can be said for solving math problems and balancing out the two sides.
If we think about some of the processes of math in this way, it makes math appear much more calming and beautiful than it is generally perceived.
I think most of the world can agree upon the fact that math is hard. It is notoriously difficult to understand, tedious to carry out various procedures, and it’s almost always impossible to feel confident while interacting with the field in any capacity.
Although many people might be turned off from math because of the challenge aspect, many mathematicians thrive in the subject for this reason. Working day and night in order to master certain math topics can be very exciting for a mathematician. Even better yet is when you find yourself in a group environment where you can encourage each other throughout the tedious learning process. As much fun as the process can be, once you are able to finally crack understanding, it can be utterly euphoric.
As humans we seek gratification from our accomplishments and math is really no different in that regard. As mathematicians, we may face some of the most time consuming and laborious tasks, but upon solving we receive immense and immediate satisfaction. This sense of accomplishment makes math all the more appealing.
Similar to being a challenging subject, math can also be much more complex than what we initially perceive. As a student learning math at a young age, we learn the many basic operations that will carry us through the rest of our math careers. We learn how to add, subtract, add fractions, and many other procedures. While we learn and do these various steps, we don’t often stop and think about the derivation or reasons for why these particular things work.
When you continue to study math, you are introduced to the world of proofs. Proofs essentially prove the various accepted facts and ideas of math. So as we learn more, we get to build on our previous knowledge and in turn, learn how and why certain concepts came about thousands of years ago. Essentially, some of the easiest and most basic functions of math have incredibly deep and complex derivations.
This fact about math makes it all the more mysterious and interesting. Knowing that there is always more to know and discover, and that it will continue to evolve, gives math a beautiful sense of mystery and excitement.
Someone in your life at one point or another has probably told you that “math is everywhere”. If you were a young student at the time, you probably rolled your eyes and thought ‘whatever, my teacher just wants me to do more of my homework assignments...’. Well, I’m here to tell you that your teacher was right!
Everywhere we look, we can spot some form of mathematics. Whether it be the measurements of buildings around us, the angles of your car park, or the calculation of the total price of your meal out, we are always seeing and doing math.
It’s incredible that such a notorious subject is so prevalent in each one of our daily lives. Whether we like it or not, we will be doing math for the rest of our lives.
I find this last point more fascinating than anything. Math could almost be considered a universal descriptor for everything we know. Math really is everywhere!
In my opinion, math is undeniably beautiful for such an array of reasons: it is its own language, it’s pictorially stunning, it’s balanced, it’s challenging, it’s more complex than we initially think, and it’s a way for us to describe almost everything that we know. Its beauty is not due to one overarching factor; but rather, a conglomerate of so many. Although I might define beauty by these 6 ideas, another person might find beauty in a completely different set. It is entirely up to each individual person’s perception.
If you’re someone who isn’t too keen on math, try to think about some of these reasons next time you do a problem. If you think of learning math as learning a language or remember that there are reasons for the tedious procedures you’re carrying out that haven’t become obvious yet, will that make the process more enjoyable? And if you’re someone who also thinks math is beautiful, what makes the subject stand out to you?
Writing this article has led me to believe that the reasons for math being beautiful are infinite.
]]>My name is Andrew and I’m the Lead Content Developer here at Mathspace. I am very excited to take you through the big changes that are coming your way!
Throughout this year the Content Team has embarked on the biggest overhaul of our 710 content in the history of Mathspace. In direct response to the feedback we received from teachers and students, together with our years of experience in delivering education online, we have completely revised and reimplemented the Mathspace experience for our largest audience of teachers and students.
In this post I'll take us through the various aspects of our process  how we reorganise a course, how we rewrite questions in response to feedback, the way we dealt with geometry, the changes we made to our lessons, and the new visual design system  before sharing our choice of the funniest feedback we received at the end.
Mathspace has launched its Free PD Webinar series, and I will be presenting the 2020 editions during its second session. If you would like to see and hear me talk through the new editions, join me at 3:30pm (AEST) on 31/07/2019  sign up using this link.
Let's get started!
Previous versions of Mathspace have had a very finegrained structure. Sets consisted of between 8 and 12 questions that targeted a specific skill. This organisation led to difficulties in finding content, and made the prospect of piecing together a task quite timeintensive. Our goal in the new edition is to present each subtopic as a predictable amount of work (between 1 and 2 class periods), and group the subtopics in a more thematic way.
Overall the content in year 7 is in fewer Topics (20 → 13) and far fewer subtopics (384 → 93). As a result each set is typically longer than it used to be, and you may also have noticed the introduction of reference numbering. We hope this makes it much easier for you to find what you're looking for.
This reorganisation was drafted at a high level before the next, most critical stage of the process.
The content team receives feedback from both students and teachers, quite a lot of it in fact  a typical school week sees 10,000 reports, climbing towards 20,000 during more busy periods. Obviously a team of six content developers can’t read every piece of feedback as it comes in!
We do take the time to read and personally respond to any feedback sent by a teacher, and student feedback is aggregated to help us find broken content so we can fix it immediately. But most of the time student feedback sits unread for a few months at least.
It is also true that student feedback is probably the most highlyvalued resources available to us on the content team, and every piece of feedback is read by us at some point.
So, how does it work?
Let’s look at an example, where we are creating the new subtopic 1.03 Checking reasonableness. Here is an existing question in this year’s edition of Mathspace:
The answer in our system is “No”, and fewer than 30% of students agreed. Here is some of the feedback we received (as it appears on our end) for this question:
ANSWER_INCORRECT: I put “yes” and it was rong??
CONFUSING_QUESTION: give me back my marks
ANSWER_INCORRECT: this is not reasonable
CONFUSING_QUESTION: WHY
ANSWER_INCORRECT: this game so trash i got it wrong fix youre game people u suck
ANSWER_INCORRECT: REEEEEEEEEEE
UNHELPFUL_HINT: i still dont get it
ANSWER_INCORRECT: CORRECT ANSWER IS YES
HINT_WITH_MISTAKES: Answer is Yes, result is reasonable
This list is heavily curated, gentle reader. It is an occupational hazard for all teachers that their work is continuously reviewed by highschoolers, but throw a feeling of anonymity in the mix and… well… students do love to tell you exactly how they feel. Please don't worry for us  we have thick skins, we know all the memes, and genuinely want to know when students are frustrated.
So, what does this feedback tell us on the Content Team? Considering the question’s context, we draw the following conclusions:
Now there are plenty of questions in Mathspace where students learn to perform subtraction, and so we took steps to distill the notion of reasonableness here in this context.
This is the question as it appears in the 2020 edition:
The answer is now “Yes”, and specifically targets the skills of estimation and rounding (rather than strict arithmetic) and appeals to those strategies in the hints (rather than the equationlevel thinking required in the first iteration).
There are 2,144 questions in the year 7, 2020 edition, and this level of consideration and care was given to the creation of each and every one.
Overall, I can confidently say that the value in receiving feedback is as important to us as it is to our student users, and we hope you encourage your students to always report any issue they come across in Mathspace so we can continue to improve.
Over the years we have learned that schools use our geometry content the least, and that the students and teachers who do use it were more frustrated and learning less effectively than they were in other topics. Each topic received a lot of attention, but we think the difference will be most obvious here.
Let’s take a look at an example in the current edition:
Our feedback helped us understand the following:
And here is the new version:
The most important feature I want to highlight here is that the characteristics of a kite are presented to the student, rather than it being written down in words. We invite the student to notice that it is a kite, and engage their reasoning implicitly to be able to proceed. Any student who can determine this answer quickly can enter the final answer without being forced to show working or state reasons  we can leave that to later years!
This was just one example of the changes to geometry. I encourage anyone who has struggled with Mathspace’s treatment of the topic to check out this new implementation and let us know what you think.
Every textbook has lesson content, and Mathspace is no exception. We aim to provide a predictable and reliable reference resource for students and their teachers. Each lesson in the 2020 edition is written to a basic formula:
Explore  Summarise  Watch a master  Practice
Here are some excerpts:
2.03 Adding and subtracting fractions
6.03 Writing rules for relationships
10.08 Volume of rectangular prisms
Explore for yourself  and let us know which ones you and your students like the best.
We spent considerable time developing a new design language to present our visual assets. We want to ensure that our diagrams are accessible, accurate, and engaging.
I’ll let the pictures speak a thousand words!
The 2020 editions of the year 7 content are complete and ready for use in your class. While I’ve done my best to highlight the amazing efforts of the Content Team, we have only scratched the surface of the work we’ve done to ensure our content is of the highest quality  and more like what you and your students want to see.
We are hard at work on years 8 through 10, and want to hear from you! Please take a moment to complete this feedback form to let us know what you think of the year 7 content. Alternatively, you can get in touch with us through the help center or by reporting an issue, as always.
We seriously do read every piece of feedback sent to us. Here are some of our favourites from year 7 students from the past year.
8x5 = 50, i'm typing 50 but it keeps on saying it is wrong!!!
Not sure we can fix your problem :(
Or I have is of the Rogan?
We tried to guess what you meant, but autocorrect was probably doing the same
it is carrect
No, it is nat :(
i am dis custard
Tasty!
I GOT IT WRONG AND IM ANGY
Sorry to hear that Angy :(
im reporting you to spiderputin
Was Putin bitten by a radioactive spider, or a spider bitten by a radioactive Putin?
IT SAYS THAT I AM WRONG BUT I KNOW THAT I AM RIGHT BECAUSE I AM SMART!!!!!!!
Selfconfidence is half the battle! But only half… ;)
You guys stuffed up big time buddy I'll get my dad to tell Kim Jin on
Friends in high (and remote) places!
math space is the reason I didn't get a pop sickle
We are sorry to hear that :(
Please actually give advice you chicken nuggets
We know all about the chicken nuggets meme. Well played, kids. Well played.
my sister texted it by accident and she texted it sorry
Nice save…
I accidentally typed in 150 and pressed enter cause i went with the SBS News beat while the question was loading and it entered the answer.:(
We’ve all been there
my friend told me the wrong answer twice. he is not my friend anymore
Sounds like you made the right choice!
Thanks for the feedback, everyone  I'll share our favourites from the 810 project sometime next year.
]]>Jamie Walker
Math Teacher and Year 7 Leader
School name: Sacred Heart College, Kyneton
School
Jamie Walker
Math Teacher and Year 7 Leader
School name: Sacred Heart College, Kyneton
School sector: Nongovernment
School grades using Mathspace: Years 7 to 10
Device type: Laptop
Device policy: BYOD
Primary or supplementary resource: Primary resource
Key Mathspace features used:
Approach to education: Giving students the resources and skills to encourage independent learning
Sacred Heart College, Kyneton is a coeducational Catholic Secondary School located in Melbourne, Victoria. One of the College’s core educational values is courage. To achieve this, Sacred Heart focuses on fostering learning environments that build students’ personal efficacy, resilience and confidence.
Sacred Heart has been using Mathspace since 2015. “We wanted to try something that would enable our teachers to spend more time assisting struggling students, whilst also allowing students at the top end to be extended,” says Year 7 Leader and Math Teacher, Jamie Walker.
Jamie Walker explains that he often preprepares custom tasks ahead of a classroom lesson. When creating custom tasks, Jamie has the ability to select the exact questions that he wants students to work through.
Often I will create new custom tasks according to what we’re working on in class. I also sometimes recall tasks that were developed in previous years.
I can access these via the ‘shared tasks’ feature which lets the teachers of Sacred Heart share tasks with one another.
When Jamie starts the lesson, he explains the concept or area of study to students and then asks them to work on the Mathspace task during class. He encourages collaboration and knowledge sharing in this time.
I find that this keeps the students interested and interacting in a collaborative way in a group setting. They discuss the questions with one another, and often the students help one another.
Jamie says he’s seen firsthand the impact of Mathspace on students’ confidence and independence.
Initially students were against Mathspace because of the online nature. But now students really value the support within Mathspace.
Jamie says that Mathspace helps him to instil the school’s policy of “ask 3 before me” in the math classroom. Many of his students use Mathspace’s videos and hints for extra support when solving questions.
Jamie says that one of the big benefits of Mathspace for teachers is the data. He says that the data has helped him to better differentiate his students within a classroom.
After he sets a custom task in class, he’ll review each student’s progress from the Mathspace data dashboard. This helps him to identify knowledge gaps at a class level, as well as for individual students. He then uses these insights to shape the next classroom lesson.
Mathspace has expanded my ‘toolbox’ and allowed me to be more aware of individual student needs.
Once he has the data on how students are performing, Jamie will determine which students are struggling and which students are excelling. He then might assign another custom task for struggling students.
I can look at the types of problems they struggle with (which are generally worded problems). I then will create a new task with similar questions, and assign this as a quiz at the start of the next lesson. This is a great way to discuss and review knowledge.
For students that are excelling, he will often assign an adaptive task to deepen their understanding.
Three hundred and fifty teachers from across the United States have been named as Mathspace's Esteemed Educators for 2019, recognized for their vision and drive to innovate mathematics education.
Building on the success of the inaugural 2018 program, and joining an impressive alumni, this year's winning teachers embrace innovation and are shaping the future of mathematics education for the next generation.
Daniel TuHoa, SVP Mathspace North America, said he was impressed by the way in which this year’s winners are using technology to transform teaching and learning.
These teachers have demonstrated an outstanding commitment to integrating Mathspace’s adaptive technology into the way they personalize each student’s math journey.
We learn so much from our teachers, particularly teachers like our Esteemed Educators, who are really redefining the math learning experience for their students.
The 2019 Esteemed Educators have been invited to work closely with Mathspace as advisors, sharing their expertise and insights to shape future developments of the program. Several of the Esteemed Educators have also shared their stories, and these will be put into best practice documents and case studies to share with fellow Mathspace teachers across the globe.
Here are some interesting takeaways from some of the Esteemed Educator stories:
Mathspace correlates with how I teach. It provides guidance. It allows for mistakes, and encourages perseverance.
Sarah Enz, Wilmington Middle School, Wilmington, California
"I had one student come up to me and tell me that he used to hate math, but Mathspace makes it fun. It lets him see how hard his classmates are working and he can see where he stands compared to it. I have always rewarded effort, and this is a great way for students to see how, even if they didn't do great, that they are working hard, and to keep going."
Steven Geiger, Campbell Middle School, Daytona Beach, Florida
I use student data to monitor concept mastery, assign homework grades, and create custom tasks and coursework for both struggling and excelling students.
Tim Sarver, Harrison School, Wonder Lake, Illinois
"I see students going beyond the curriculum to teach themselves new topics using Mathspace. In study hall or down moments, students log in to do a few problems. It's easy access, which causes them to do more valuable practice. They always have their iPads on them, whereas they may not have a pen and paper to sit down and do purely written work. In this way, I believe they get more done."
Michelle Vieira, Columbia Middle, New Jersey
“I have many students for whom English is their second language, these students like Mathspace because I can customize it to limit the amount of reading and they can watch a video to help them to understand. Three of my top performers are from the highneeds ESOL students.”
Jean Stoney, University High, Orlando, Florida
“I have a really high student who I feel is hard for me to push to his limits. I am able to have him work on the next unit and be ahead of the game as well as the next grade. This program helps me be able to push him to his limits instead of allowing him to stay steady with where he is.”
Jenna Rossi, Knox Middle School, Salisbury, North Carolina
“My district started a Mathspace challenge and one particular student that had stated they hated Math was one of the county’s highest point gainers over a course of 10 weeks. And now the student is fully engaged in class.”
Darrin Turner, Knox Middle School, Salisbury, North Carolina
“Since our students don't come to school daily, they rely on having access to their material in different locations. I find that having an internetbased curriculum works well for many of my students who travel frequently, like our gymnasts and dancers who perform and compete around the world.”
Courtney Selby, Natomas Charter School Virtual Learning Academy, Sacramento, California
To learn about how Mathspace could work at your school, find out more here.
]]>We've launched a brand new feature on Mathspace called Task Templates.
Setting tasks for your students is now easier than ever. You can easily browse our selection of templates created by teachers for teachers. The most exciting thing about Task Templates is that our team of inhouse teachers and content specialists has created them.
We know how busy teachers get during term 2, particularly with half yearly exam preparation, which is why we have created lots of topic revision templates for the launch of this new feature.
We've focused on creating topic revision templates for every topic in Year 710 of the Australian Curriculum. We want to help you prepare for assessments by making it easy to access the right content at the right time, with no searching required.
You’ll find the ‘Templates’ tab at the top of your navigation bar when you first log in to your Mathspace account.
The Task Templates are listed either individually or in groups. Task Template Groups consists of a collection of individual Task Templates that belong to a specific category  e.g. Year 7 Revision. You can also find specific Task Templates by using the search bar located on top of the list.
Once you click on a Task Template, you will be able to see a preview of the questions. You can then assign this Task Template as a Custom task by clicking the assign button in the top righthand corner of the page. This will take you to the existing Create Task screen within Mathspace.
For most Task Templates you will still have the option to add or remove questions from the ‘Create Task’ page, so you can tailor the content for your students. The only templates that are not editable are the diagnostic and standardized testing materials.
What topics are your students revising? Try the Task Templates now!
I have a pair of pants that have a pocket that has a pocket. Not the kind of small change, house key specific pocket you sometimes get sitting up near the belt. This one is deep down within the softer fabric of the lining. It is so well hidden, so well integrated, that I sometimes forget it’s there. It’s only those occasions when I want to warm up my hands a little, by sliding into my empty pockets, that I get to wondering whether they really are all that empty.
The inner pocket has nothing in it. But the outer pocket does, namely, the inner pocket itself! The inner pocket is also exactly the same shape as the outer pocket, just a scaled down version. If I put my index finger into the lowest corner of the inner pocket, the very fabrication of the peculiar nesting means my index finger is at the same time also resting at the lowest corner of the outer pocket.
It is strange the way that putting nothing inside itself can create something. Take an empty pocket, copy it, and shrink it, then put that miniature copy inside the original pocket. With this hierarchy we can distinguish two distinct objects; the inner pocket that contains no things, and the outer pocket that contains one thing. It’s not too difficult now to imagine putting this twolevel hierarchy of pockets into another, bigger pocket. Or even going the other way and sewing a smaller pocket within the inner pocket, to make an inner inner pocket. Then an inner inner inner pocket, and so on...
This notion of recursively nesting identical copies of a thing within itself is at the heart of the formal construction of the whole numbers, from which every other object in modern mathematics can be built. A theory of pockets could get us there too, but mathematicians have already developed a far more sophisticated theory of things they call sets.
We can think of a set as an abstraction of a pocket. It’s a thing that you put stuff in. In the same way that the pockets on your pants come for free, we generally take for granted that we already have access to a set with nothing in it. That is, we make the claim that there exists an empty set. After that we just have to outline the rules of how two sets can be combined, then we’re free to define our way through the rest of the mathematical landscape.
While this origin story has a tidy selfconsistency, it doesn’t really reflect the way many mathematical concepts have naturally developed. This can lead to a confusing experience when searching along the genealogy of a particular mathematical object.
For example, a good understanding of quadratic equations would involve, at least, an understanding of the Cartesian plane. But the Cartesian plane is built from the number line, so we might first want to get a handle on that. Then again, the number line is built from the whole numbers, which we know are built from sets. And sets are built from ... nothing? What then of quadratic equations? We seemed to have stepped back through the family tree to find that there never was a tree to begin with! Having emptied our pockets we find only a pile of empty pockets! What would Descartes say about his eponymous plane?!
Of course, looking to sets for the meaning of quadratic equations is like putting your face right up to a painting and wondering where the art went. We all know that fruit is made of tiny cells, but we don’t expect to see anything that is uniquely characteristic of an apple when we look at it under a microscope. Water can be cool or warm, slippery and sloshy, but a single molecule of water can be none of these things. An ecologist doesn’t need to know about quarks to understand trophic cascade. Even though large, complex objects have smaller building blocks, the properties of those larger things belong to that scale, and cannot be sought at the foundations.
Indeed, Descartes and many others made substantial contributions to coordinate geometry in the centuries before Cantor created the theory of sets. The Cartesian plane was a response to the desire to understand the space we live in, not purely a derivation from axiomatic set theory. We see quite often a mathematical object or concept will be initially inspired by our intuition about how we experience the world. Then an evolution similar to that of handheld tools takes place. The more a concept gets used, the more precise we want to make it, and the more powerful and useful it becomes.
In this way, we can picture a particular mathematical idea as a tool, and the whole body of mathematical knowledge as a toolbox. We fashion a tool to meet a need from our environment. We might then find that this new tool looks like a combination of things we already have, or that we are able to use it to build an even more specific tool. The process of identifying and creating dependencies between formalized mathematical concepts makes them more robust, and helps us to imagine beyond what we might be capable of with our senses alone.
I think that it is curious, but not too surprising, that there is a bottom that we eventually reach when rummaging through the mathematical toolbox. The unpacking must end somewhere, and we have seen that the end is a fully unpacked box. What more would we expect, the metaemptiness of “not even an empty box”?
But to turn around and look back up at the sprawling, unfathomably complex menagerie of mathematical variety, all bootstrapped from thin air, it seems there is no end in this direction.
We are limited only by our imagination.
At Mathspace, we’re committed to creating resources that go beyond simply testing students. So this year, in addition to creating sample papers, we have also built a NAPLAN Report which provides outcome reporting
]]>We’re pleased to announce that our NAPLAN resources for 2019 are now available in Australia.
At Mathspace, we’re committed to creating resources that go beyond simply testing students. So this year, in addition to creating sample papers, we have also built a NAPLAN Report which provides outcome reporting and band predictions for your class and individual students.
We have three fulllength NAPLAN sample papers, created by our team of inhouse curriculum experts. They will be accessible on Mathspace in the Australian and state curricula for Years 7 and 9, under 'Topics'.
(Tip: Make sure you set the sample paper as a custom task. We cannot generate your report if you assign adaptive tasks.)
Check down below for 3 simple steps on how to assign NAPLAN sample papers for your students.
The papers are available under the Australian and state curricula for years 7 and 9.
You can also share your NAPLAN sample papers with other teachers once you have created your task.
Once you start setting sample papers, we will send you your NAPLAN Report based on your students’ results.
A NAPLAN Report will give you:
The report will show you the skills your class and individual students are struggling with, and recommend tasks tailored to reinforce those concepts to assist the growth of your students’ maths skills in the longrun.
Math is an itch that I just can’t seem to scratch!
What excites you about mathematics?
To understand mathematics is to understand the universe. Patterns exist out there on their own and they are just waiting to be discovered!
Who inspired you to pursue mathematics?
My parents. Growing up, they would show me ways that math was both important and beautiful. Like, my mom would always point out the advantages for women who understand mathematics. She was a strong female role model for me. And my dad would always ask me questions to ponder math puzzles, like “Does an odd plus and odd always make an even?” That was a question I remember thinking about in early elementary school and wondering how I could explain the answer.
What needs to be done to encourage more women to participate in STEM?
We need to build a culture where young people  both men and women  are encouraged to explore their passions, whatever those may be. Our youth need safe opportunities to try new things, test the boundaries a bit, and figure out what works and what doesn’t. It should be the collective mission of all educators  parents, teachers, school administrators, government officials, social media stars (yep!)  to provide these kinds of opportunities.
What is the biggest accomplishment in your math career?
I’m hopeful that it’s still on the horizon.
I have a distinct memory of learning that eiπ = 1 and just being in absolute awe of the subject.
Which single word do you most identify your pursuit of mathematics with?
Logic
Who inspired you to pursue mathematics?
My high school math teacher. His passion for the subject was contagious. I was initially going to pursue a career in Music education, but he showed me my true potential and how cool math really is.
What excites you about mathematics?
The interconnectedness of math never ceases to amaze me. I have a distinct memory of learning that eiπ = 1 and just being in absolute awe of the subject.
The fact that this cool number with amazing properties in differentiation, with complex numbers, along with a number I think about being related to circular functions all come together to make something as simple as 1 blew my mind. The question, “did we discover math or create it” continues to keep me guessing.
What needs to be done to encourage more women to participate in STEM?
I first want to acknowledge the good progress on this front in the last 10 years. I think more female role models is an important part of the journey to eliminating the stigma of women in STEM.
We need to encourage women to have the courage and the grit to pursue their passions. Step up, do things that are hard, be BRAVE.
Which single word do you most identify your pursuit of mathematics with?
Bravery
Who inspired you to pursue mathematics?
My Geometry professor at University, Dr. Janet Woodland. When I registered for this class, I knew I was going to struggle and I did, but I loved every minute of it. She had this amazing ability to make things make sense. She took the time to answer all the questions, redraw the diagrams and find ways to help us see the why. As a teacher, I always tried to model her teaching style.
What excites you about mathematics?
I love that you can find mathematics at the root of everything. I especially love finding it in the beauty of nature.
What needs to be done to encourage more women to participate in STEM?
We need to encourage women to have the courage and the grit to pursue their passions. Step up, do things that are hard, be BRAVE.
What is your biggest accomplishment/most satisfying moment in your maths/teaching career?
It is funny, I have never considered myself to be good at math. I always struggled to understand the deeper meaning. Once I became a Math teacher, I realized a lot of my students had the same issue. My biggest accomplishment was being able to empathise and find ways to help build their selfconfidence and to encourage them to take risks in solving problems without the fear of failure.
What does working at Mathspace mean for you?
The Mathspace mission of being the right help for the right student at the right time, was basically the mission I strove for in my classroom as well. Each year of teaching, I was able to try to be that person for 112 students. Working at Mathspace gives me the opportunity to be a part of a team that has the capacity to reach so many more than I ever could alone.
I love how difficult mathematics can be and how elegant it is.
Which single word do you most identify your pursuit of mathematics with?
Challenge.
Who inspired you to pursue mathematics?
My Dad inspired me to pursue Mathematics. I still remember the first time in early primary school when he cut up fruit to explain fractions to me, those types of experiences stick with you.
What excites you about mathematics?
The knowledge that I will have to work hard to get to an answer, and then know when I get that answer right. I love the thrill of the chase. I love how difficult mathematics can be and how elegant it is. To explain such complex ideas in sometimes such a dignified and eloquent way makes mathematics beautiful.
What needs to be done to encourage more women to participate in STEM?
Having strong role models will help encourage more women in STEM. Having those role models present early in life is crucial. Those role models don’t have to be scientists or Maths teachers but people who have a genuine interest and passion for STEM. It’s the passion that makes an impact on people, they can feel your genuine enthusiasm and it resonates and excites.
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