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.

Students Need Freedom of Learning not Gonski

I always had trouble with Gonski. The latest arguments and disagreements about Gonski 2.0 brought my feelings back again. The trouble I have is not just about the discussions about the funding per student and there was no information about how to utilize this funding.

Everything in the past turbulent weeks seemed to be concentrated on the Gonski plan to provide a school with specific amounts of dollars per student.

That’s it!

But that’s not it … where were the discussions about teacher quality; far more and better professional development of teachers; ongoing preparation for the digital classroom with technology training; less students per classroom; integration of subject areas; and what’s dear to my heart collaborative learning.

My real problem with Gonski is that fundamental philosophy of needs-based funding. This has the aim of providing equal opportunity for every student. Underlying this must be a belief that given the same opportunity all students will end up at the same place. 

This is clearly nonsense. One bad teacher in one year will wreck all that.

Who got brain-washed through all this government bickering and one-upmanship? Who addressed those fundamentals concerned to the fact that students are not, nor will ever be, the same even with similar monetary commitments.

It’s not about that. It’s about freedom of learning. Streaming. We want the brightest students totally independent of age to learn at their capacity. Not with the learning restrictions that education systems place on them.

We need learning to be aligned to the student ... not imposed on them in age-groups.

Have we not learned anything as teachers and education authorities and thought-leaders over the past 100 years. Well some of us probably did but we have no power to change anything unless we're a community.

Australia will now continue to slip down the international comparison tables … guaranteed!

Fortunately there’s the digital education classroom available now and the world is becoming one mega-classroom. No need for Gonski there.

Personalised Learning May Imply Student-to-Student Learning ie Collaboration

I found this article interesting. If you are trialling personalised learning and collaborative learning like we are at MathsRepublic then please read and pass to colleagues:


http://www.thetechedvocate.org/personalized-learning/

Inquiry or Discovery? Are You Confused?

A colleague of mine has been struggling recently to understand how to develop inquiry in her mathematics classroom. Despite having secure content knowledge and a vast experience of teaching she finds it challenging to devise teaching activities that truly manifests as inquiry. After observing her a few times, it became clear to me that she confused Inquiry with discovery.

David Perkins, Professor at Harvard Graduate School of Education, proposes the idea of “playing the whole game” in his book Making Learning Whole. The central thrust being that learning should be provided in a whole form, rather than in topics as is more normal. This is particularly intriguing for mathematics instruction. His metaphor of a baseball game being a different experience compared to learning different skills in a training setting is curiously poignant.

So what is the difference? I can almost hear teachers going “yeah, what is the difference?” Or even “what’s wrong with discovery?”

For many teachers, this “inquiry method” is just another name for discovery method. But is it? I beg to differ and I can only support my view with experience and the tsunami of research out there.
Discovery teaching relies heavily on two things in my opinion:

• The learner is unaware of the learning intention
• A logical activity that is well planned leading to a predetermined outcome

The ignorance of the learner is critical to the success of discovery methods since if the learner is familiar with the content, the long process of proving sometimes disengages.  How many times is the annoying learner encountered who chooses to blurt out the intended destination of the lesson?

Another pitfall of discovery teaching in Mathematics is the lack of transfer in weaker learners. They logically follow through the process, but struggle to connect it to the Mathematics on the board. We have all been there.

A popular topic is the sum of interior angles in a triangle. The logical process of drawing a triangle, cutting out the ends, joining them and voila a straight line relies almost totally on the teacher guiding – telling perhaps – the learners. Despite the process being well supported by logic and the outcome undisputed many learners still do not remember the simple fact. Transfer is lacking. Or at least independent transfer.

Inquiry teaching takes another route to learning, as it is largely inductive and demands questions from the learners for it to thrive. Questions are promoted with further questions, one answer by a learner is followed by another question probing understanding, questions are focused on concept formation and the learners are leading the learning by virtue of their own curiosity.

As a teacher in an International School, I have found inquiry to be the safest and most effective method to introduce learning in Mathematics. My main reasons were:

1. International students are normally second language English learners
2. International students have covered different aspects of the syllabus before
3. Variety of teaching experiences and methodologies seen by learners
4. Differentiation required since most are largely non-selective

Questions are the key and the versatility of the teacher to guide the learners, deeper into thinking, thus probing understanding and challenging certainties. I for one have been known to not provide answers, or as my students say, answer questions with questions. This is a great atmosphere to have in your classroom, it makes everyone equal and the pursuit of knowledge joins everyone together.

It means that the learning in your classroom becomes more deliberately focused on conceptual formation rather recognition of content validity. Proving that the angles of a triangle add up to 180, only adds another bit of content but it doesn’t explain why. Why does this happen? Why is there a limit to all triangles?

Understanding the triangle explains all polygons including the circle. Is the circle a polygon? Ask your class to think about this, but with colleagues probe this deep thinking? Is the reason because it fits on a straight line? Surely there is something more conceptually profound that commands it to be so.

Inquiry teaching will challenge your thinking and understanding but it fosters a positive learning atmosphere in any classroom. I must admit that many teachers find the adjustment challenging, but in the modern world where knowledge is free, analysis and synthesis are invaluable skills. Deliberately pushing every interaction with your learners into a questioning atmosphere will develop your own inquiring skills.

Inquiry implies involvement that leads to understanding. Furthermore, involvement in learning implies possessing skills and attitudes that permit you to seek resolutions to questions and issues while you construct new knowledge.

Here are some pointers, to promote real inquiry in your MYP classrooms

1. Write down the conceptual questions you want to cover in your lesson. This will help to keep you on track if you tend to lose it a bit in the questioning phase. Having a written record means that you would have thought about the activity and expects to discuss this as priority.
2. Avoid answering learner questions. Develop a habit of not answering any question. Reward the questioner by saying good question and either follow up with another or pass it to someone else. This will promote good questioning and make the curiosity element in your class active. At first your learners will find this weird, but eventually will feel the benefits in their conceptual understanding.
3. Ask at least two questions to the same responder. Begin with a factual question and follow up with a conceptual one. As a rule I try to ask at least two questions to probe understanding. Why did you say that? How do you know?
4. Keep some question stems that promote deeper thinking. I like using “Can you think of another example?” “Is your answer always correct?” “Is there any value in knowing this?” You can see my point.
5. Promote lateral thinking and transfer. Allow your learners’ questions to veer off as long as it naturally follows the topic being explored. Having a prepared list of questions will help to guide the conversation back to the learning. Thinking is deeply rewarding if freedom reigns in your learning space.
6. Accept only written reflections. I always let my learners write their reflections down before sharing. As soon as the concept is covered, I ask them to write their reflections and then read them out loud. This forces deeper thinking and allows them to really reflect since writing is a more involved process. This works well if you ask them to “write three things they learnt today”. Do not ask for one! Three pushes them deeper.
7. Focus the learning on concept rather than content. Keep the concept formation as the guide to the questions, be alert for opportunities to delve deeper in ideas proffered by learners. Concept acquisition is far more valuable than content.
8. Video yourself and count the number of questions in your classroom. Check to see who ask the most questions. Try and improve the quality of your questions, watch for opportunities missed and learn to capitalize on them. An audio recording is equally useful if a video intimidates.

These are just some of the pointers I have garnered over the years. The list of suggestions, are by no means new and are not intended to solve every problem. However, the aim is to share some ideas as to how to convert your classroom into an inquiry one. As with most changes, it begins with you.

___