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?