More About MathsRepublic's Problem-based Lessons

Students should not just learn about mathematics, they should do mathematics.

This can be defined as engaging in the mathematical practices: making sense of problems, reasoning abstractly and quantitatively, making arguments and critiquing the reasoning of others, modelling with mathematics, making appropriate use of tools, attending to precision in their use of language, looking for and making use of structure, and expressing regularity in repeated reasoning.

The purpose of the Lessons available to Teachers after Registering for a Free Trial (www.MathsRepublic.com.au) is to impact student learning and achievement.

First, we define the attitudes and beliefs about mathematics and mathematics learning we want to cultivate in students, and what mathematics students should know and be able to do.

Attitudes and Beliefs We Want to Cultivate

Many people think that mathematical knowledge and skills exclusively belong to “maths people.”

Research shows however, that students who believe that hard work is more important than innate talent learn more mathematics.

We want students to believe anyone can do mathematics and that persevering will result in understanding and success. We want students to develop a “productive disposition—[the] habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy.”

Conceptual understanding: Students need to understand the why behind the how in mathematics. Concepts build on experience with concrete contexts. Students should access these concepts from a number of perspectives in order to see maths as more than a set of disconnected procedures.

Procedural fluency: We view procedural fluency as solving problems expected by the standards with speed, accuracy, and flexibility.

Application: Application means applying mathematical or statistical concepts and skills to a novel mathematical or real-world context.

These three aspects of mathematical proficiency are interconnected: procedural fluency is supported by understanding, and deep understanding often requires procedural fluency. In order to be successful in applying mathematics, students must both understand and be able to do the mathematics.

What T&L should look like

How teachers should teach depends on what we want students to learn?

To understand what teachers need to know and be able to do, we need to understand how students develop the different (but intertwined) strands of mathematical proficiency, and what kind of instructional moves support that development.

Principles for Mathematics Teaching and Learning

Active learning is best: Students learn best and retain what they learn better by solving problems. Often, mathematics instruction is shaped by the belief that if teachers tell students how to solve problems and then students practice, students will learn how to do mathematics.

Our signature mathematical language routines (MLRs) offer detailed guidance for developing students into mathematical thinkers. Facilitate and assess students’ ability to communicate mathematical thinking verbally, visually, and in writing.

Every lesson plan contains topic-specific professional learning resources. Our materials speak intelligently and professionally to educators, meeting teachers where they are in their practice and advancing them.

Whether it’s for below-benchmark students or accelerated learners, MathsRepublic’s Lessons provide content-specific resources within lessons, from warm-up to cool down.

Discussion-filled classrooms beget deeper learning. Our materials encourage student communication and the development of problem-solving and reasoning skills.

Teachers Love the ‘Anticipated Misconceptions’

Educators can easily prepare to recognize, analyse, and respond to common student struggles thanks to the scaffolding provided with each lesson.

Each course contains nine units. Each of the first eight are anchored by a few big ideas in grade-level mathematics. Units contain between 11 and 23 lesson plans.

Each unit has a diagnostic assessment for the beginning of the unit (Check Your Readiness) and an end-of-unit assessment. Longer units also have a mid-unit assessment. The last unit in each course is structured differently, and contains optional lessons that help students apply and tie together big ideas from the year.

The time estimates in these materials refer to instructional time. Each lesson plan is designed to fit within a class period that is around an hour long. Some lessons contain optional activities that provide additional scaffolding or practice for teachers to use at their discretion.

Students can work solely with printed tasks distributed by their teacher. Later when all students have access to an appropriate device, students will be able to view look at the Task Statements on that device and write their responses in their workbook.

Teachers can access the teacher materials either in print or in a browser.

A classroom with a digital projector is recommended for Teachers to project the Student Task Statements.

Many activities are written in a card sort, matching, or info gap format that requires teachers to provide students with a set of cards or slips of paper that have been photocopied and cut up ahead of time.

Teachers might stock up on two sizes of resealable plastic bags: sandwich size and a larger size. For a given activity, one set of cards can go in each small bag, and then the small bags for one class can be placed in a large bag.

If these are labelled and stored in an organized manner, it can facilitate preparing ahead of time and re-using card sets between classes. Additionally, if possible, it is often helpful to print the slips for different parts of an activity on different color paper.

This helps facilitate quickly sorting the cards between classes.

Invite Your Students Into the Learning of Maths

By Morgan Stipe, a Middle School Maths Teacher. This is an edited version of a Blog published in www.NCTM.org. Morgan refers to format and features of Lessons in this Blog which may be accessed through www.MathsRepublic.com.au

How are you inviting your students into the learning within your classroom?

That invitation is important for all learners to receive from you the Teacher, every day.
My favorite ways to engage my budding mathematicians in the content play directly into the interests of the learners. 
Sometimes I include:

•  a really funny GIF that sparks their curiosity. 
• Other times it’s a pop culture reference or current event that draws them in. 
• Short video clips pique the interest of my students. 
•  Or I’ll throw out an interesting or obscure picture to consider. 
• I might also pull in some related mathematical expressions to allow students to be purposeful about making connections, while practicing mental math and building their language skills.

One classroom favorite is Which One Doesn’t Belong? In this routine, learners are shown four pictures, each with a reason for being different than the others. The images are sometimes shapes, numbers, equations, graphs, etc., but they are always relevant to the tasks and learning from the daily learning goals. 

This is their access point and their invitation. I’ll prompt students by asking: “Which one doesn’t belong."  I ask them to Jot down some ideas, then start thinking about why the others are different, too? After quiet think time, students share in teams, then as a whole class. 

I challenge students to think of as many reasons as they can! All students participate in some level of sharing. Again, all students are invited to engage and actively participate.

Number Talks are also a solid routine to get students doing mental math in class! Students can draw ideas from what they know or from previous work in the activity to evaluate each expression. The work ties into maths which is coming up in the lesson. 

Allowing students to justify their thinking in this activity is crucial ... they’ll learn from any misunderstandings each time. 

Here is a sample of work from a task from a Lesson which can be found in www.MathsRepublic.com.au:







 Using the Notice and Wonder instructional routine gets students picking out important information, asking unanswered questions, and making solid connections. 

I show my students an image, graph, table, photograph, or sometimes the task itself, and ask, “What do you notice? What do you wonder?” After quiet think time, ideas are shared and recorded for the whole class to see. The door is open to all possibilities which the open-endedness of the routine brings. 

And the language development is incredible! Students carefully or wildly put together statements that begin with “I notice…” Then they phrase questions to extend their thinking into the unknown parts of the task, which sometimes make us ponder or smile or laugh. All ideas are valid. All thoughts are recorded. 

All ideas are authored by the students in the room. All students have access and are invited into the learning. Here is one of my favorites from a Lesson.

 

My young mathematicians in my class noticed that the pink socks were balanced and the blue were not, the socks are on hangers, and something is in the blue sock on the left! They wondered: Why are the socks on hangers? What is in the blue sock? Are the socks clean? Would they turn purple if you washed them together? 

We went on to discover solving equations using hanger diagrams, and the socks were a persistent reminder that equations must stay balanced. After this lesson in particular, one of my wide-eyed, curious tweens exclaimed, “This is the most sense a math lesson has made on the first day of learning something!” Mission accomplished.

There are so many resources out there to establish these routines in your classroom. When all students have access, when students are engaged, when students are invited to the maths table, they’re on their way to making sense of mathematics and finding connections to known and new ideas, while reeling peers into the conversation as well. 
How are you inviting your students into the learning?

Change The Way You Teach When Using Technology

The past 10 years have seen a surge in student-centered learning, and the integration of technology into the classroom makes it increasingly easy to create engaging lessons that reach a variety of learners in a variety of ways.

1. Education Now Happens in Classrooms Without Walls

Teachers can continue to communicate and teach outside the classroom. 

This means that teachers have to be increasingly more communicative, more plugged into the community in which they teach or live, and be willing to showcase connections between the classroom and the world around the students. 

2. Textbooks May Be Obsolete

Thanks to technology, many schools are no longer ordering or relying on traditional textbooks.  Instead, it is up to teachers to sift through the content on the internet, or on education websites, to find real world materials that showcase the content being taught in the classroom. Teachers can no longer rely on reading a chapter and then answering the textbook questions.  Instead, technology is encouraging educators to become more proactive in find reading materials that are authentic and relevant, and engage students on a deeper level.

3. Technology Makes it Easy to Flip Classrooms

Instead of teaching the content and then assigning homework, technology enables teachers to provide instructional materials (presentations, recorded lectures, PowerPoints or presentations, YouTube videos, etc) for the students to peruse on their own time and at their own pace. This means that teachers then become guides and resources for the practice work – classwork now that used to be homework – showing the students how to best use the information they took in.  The function of the teacher is no longer to impart information, but to guide students in making the best use of the information they read and learn.

4. Collaboration is Increasing: MathsRepublic is a Leader 

Teachers no longer need to teach in a vacuum! Teachers can collaborate across content areas, grade levels, even across geographical distances. Teachers can communicate with one another to make cross-curricular experiences that will solidify student learning and find experiences that will help their students in real-world situations. It also means that they can give their students opportunities to learn from others in both similar and different life situations, cultures, and locations.  Teachers become facilitators for students’ experiences.

5. Learning Can Be More Personalized

Technology makes it easy for teachers to tweak lessons and materials to each individual student’s’ needs and interests. With technology, a teacher is responsible for differentiating his or her lessons so that every student receives the greatest depth and breadth of understanding.

6. Teachers Can Give More  Meaningful Feedback

Teachers can now measure individual student learning through communication and the near-constant feedback of computers. 

7. Classroom Management Strategies are Shifting

With technology, students always have the opportunity to be engaged, even when a teacher needs to deal with one individual student. In the past, if a teacher needed to stop class to address a student behavior, everyone else had to wait until the teacher had returned to the task at hand to move forward. Now, forward progress continues, regardless of to whom the teacher is speaking or why. But more than that, technology can impact how classrooms are managed.  From planning to engaging to monitoring, teachers can use apps and technology to make sure that students are on task and engaged, thus reducing misbehaviors.

Technology is here to stay. And even though it presents its own unique array of challenges, it pushes teachers to stay creative, to meet students on their home field, and to innovate. From the information taught to the method of delivery to managing the students’ behavior and achievement, technology helps teachers make the most of class-time.

Australian Teachers Resist Change ... PISA Survey

The 2015 PISA test cycle included a survey of principals. They were asked to report on the extent to which five teacher-related behaviours – including teacher absenteeism, teachers not being well prepared for class, and teachers being too strict with students – are hindering learning. Our latest infographic looks at the results from Australia and nine other countries.

Thirty-five percent of Australian Principals said their teachers resisted change. 

Students Don't Need to Know What to Think; But How to Think

For more than a century, educational institutions has been creating an industrial workforce of human automatons, built for the purpose of performing non-routine labour to Scott Santens, a writer with Reddit. 

And we've been operating with the mindset that we should teach students the same way we program actual machines. This is the view also of renowned education guru Sir Kenneth Robinson presently so strongly in his talk relating education to the factory environment: https://www.youtube.com/watch?v=K9C0KNtqiHU

In the 20th century, schooling became a process of information upload in which students are to be filled with all the appropriate data and applications to function as cogs in the machinery of factories and offices, or in the parlance of today, as walking hard drives.

Modern mass production did more than increase efficiency. It chopped up work into simpler components in which one small task could be repeated day in and day out without thought and without knowledge of the whole.

Centuries ago, shoes were made by hand, by one skilled person who made the entire shoe. And then hundreds of people became involved in making one shoe with one relatively unskilled person doing nothing but attaching soles all day, every day. 

Now that machines can perform all the tasks in making an entire shoe, what happens to the humans who were programmed to operate the machines?

 Human-automaton creation must end. 

To succeed in a world of automation will require being as unmachine-like as possible. The entire education system will need to be retooled around no longer teaching kids what to think; but how to think. 

Memorization of facts is pointless in a world where everyone carries around the entire knowledge base of the human species on their person.

The challenge is not information storage but information processing. It's not about information itself but how to use information. 

The teaching of creativity, curiosity, critical thinking, analytical thinking, problem-solving, and a love of learning itself will be critical to transitioning from the industrial age to the automated age. 

Learning how to collaborate and empathize with others will be key. To be human is not to be a lone robot performing a singular task in a vacuum but to be a member of the whole of humanity contributing in countless interdependent ways, including even entirely unpaid ways. This will require nothing less than a redefinition of work itself.

To succeed in the future will require rediscovering what it means to be truly human. Mark Twain once said the two most important days in your life are the day you are born and the day you find out why.

In the decades ahead, our jobs as humans will be finding our ways to our "whys." And education must be re-imagined.