The 5 Big Ideas In Teaching For Mastery

 

When we teach, we are teaching students with the idea that they will master the material.  Sometimes, we need a reminder of what it involves and ways we can manage it while other times, we stroll along.  For many of us, school has started and we are doing the first of our assessments. So today, I'm looking at the overall picture of teaching for mastery.

First is coherence which is the way topics are taught so they create a coherent learning progression throughout the grades.  They hopefully will end up with a deep understanding of the concepts and topics so they apply math across a large range of contexts.

Second, is carefully selecting representation of the concept or idea to show the structure.  This results in students "seeing" math rather than using it as a way of doing the math. The representations become mental images for students that they can think about as they work on the math and which helps them develop a deeper understanding.

Third, encourage and help students develop mathematical thinking which allows them to make connections, develop conjectures, perform reasoning, and generalize.  When planning any lesson, students should actively be involved in thinking mathematically and communicating using mathematical language. 

Next, all students should have developed fluency of number facts and use of operations so they can spend more time learning the concepts. In addition, fluency also means that students have the flexibility to move between various contexts and representations, be able to make connections as they recognize relationships and choose the best method to solve a problem.

Finally, students need to develop both conceptual and procedural variation.  Conceptual variation is when students are able to vary the way a concept is represented to show a critical feature.  Procedural variation looks at how students "proceed" through a problem. 

These are the 5 big ideas in teaching for mastery and are an overall look.  Over the next few weeks, we'll look at each idea in more detail and see how they can be applied in a post Covid world.  Let me know what you think, I'd love to hear.  Have a great day.

Mathematical Modeling On Body Clock Distruptions

 

It happens to everyone.  Their sleep patterns are disrupted due to time changes through daylight savings time, jet lag, working night shift, or playing on your phone into the early morning.

Researchers from two universities created a mathematical model to help explain the resilience of the body's master clock found in the brain.  There is a cluster of neurons in the brain that work together to manage the internal rhythms of the body.  They also hope the results lead to suggestions on how to improve the internal clock in people who do not have a good internal rhythm.

The reason for this study is because continued disruptions to a person's circadian rhythm can cause diabetes, memory loss, and other issues.  Since society has created a situation where many people end up working hours outside of the normal day light hours, it can be be hard on being exposed to light, eating, and sleeping.  

The researchers created a mathematical model that looked at the suprachiasmatic nucleus (the internal clock) as a macroscopic or big-picture system made up of an infinite number of neutrons.  They looked at the connection between the neutrons in the system and how they manage to have a shared rhythm.  It was found that frequent and sustained disruptions to the circadian rhythms can weaken the connection which weakens the shared signals. 

The mathematical model indicated that a small number of disruptions can make the connection closer but they are still working on how this relates to the fact that a lot of disruptions can cause a weakening of communications between the neurons.  Mathematical modeling allows scientist to study situations that cannot be explored through normal methods. It creates a situation where they can change variables without hurting anyone. 

This is interesting because I know from personal experience that jet-lag can really mess up your internal clock especially if you live in a location that does not have nights that get dark. I also find that my sleep patterns take a while to get back to normal.  Let me know what you think, I'd love to hear.  Have a great day.

Heading Home

 Heading home after month of travel. Back on Monday.

Roman Numerals

 

Today’s thoughts will be on the short side as I am getting ready to fly home from Sydney, Australia. Today, I explored a part of Sydney that was new to me. I found the courthouse in Darlinghurst, on Oxford street. 

As I looked at two of the buildings, I realized that they has]d the year of construction marked in Roman numerals. It took myself and my travel companion several minutes to determine the values. We are both pretty good up to 100 but a bit shakier with 500 and 1000.

Although we introduce students to Roman Numerals, we really don’t do much with then to place then in some sort of context. For instance, do we work with the history teacher when they are exposing students to Ancient Rome. This would provide them with a bit more context and students could see how they were used.

We could show a clip of a movie that shows the year of publication. I have not checked recently but several years ago you’d see the dates like MDCCCCLII. I don’t know if students have ever caught that. I noticed it and tried to figure out the year. In addition, some clock faces Wes done in Roman numerals but I am not sure how many now are done in this way.

The other place, I regularly see Roman numerals used is on buildings that have been around a while. As mentioned earlier, I saw them on a court building. 

As you can see 1884 and 1888 are the dates on these buildings. I have no idea why they used roman numerals but it may be either tradition, or they thought it made buildings look more official. Let me know what you think, I’d love to hear.

Growth Mindset In Math.

Last time, we looked at misconceptions regarding growth mindset versus fixed mindset. Today, we will look at the topic specifically in regard to math since it is a subject where many parents perpetuate a fixed mind set. We know that students with a fixed mindset believe that their abilities, intelligence, or talents are fixed and cannot be changed.  Students with a growth mindset believe this talents and abilities can be developed with good teaching, effort and perseverance.

It has been found that students who have a growth mindset are more willing to put in effort even when they struggle or fail. In addition, they are able to stay focused on what they are learning. Students with a growth mindset tend to do better in math and are more engaged in the lessons. 

There are ways to help students develop a growth mindset. First of all, Jo Boaler’s website Youcubed.org has some really nice material on working towards a growth mindset there. There are videos, printed materials and more that can be used in the classroom.  In addition, there are some great rich open ended tasks  on the site to be used in class.

In addition, there are things one can do to help encourage a growth mindset in our students. The first thing is to introduce students to the idea that the brain has the ability to grow. This can be done by showing a clip from the YouTube video “The Neuron Song” for younger students or a clip from the BBC documentary “The Human Body” for older students. This introduces students to brain neuroplasticity.

Second, show how mistakes can be used to promote brain growth. If you as the teacher make mistakes, take time to show students how you can learn from the mistake. Spend time during class to analyze typical mistakes and show them how to analyze their own mistakes. Create a culture that celebrates mistakes and learning from them.

Third, instead or relying on closed traditional problems, expose students to open ended, rich tasks. Ask students if they can solve a problem in two different ways. Learn how to take traditional problems and change them into open ended problems for students  to practice. Give them challenges such as take four 4’s and any operation to find 1 to 20. An example for that would be 4/4 times 4/4. That is not the only solution, just one.

Remove the emphasis on speed. Teach students strategies to use and emphasis that learning to use these strategies are more important than quickly finding an answer. Assign few problems and insist students justify their answers or find multiple answers. Throw in a couple of reflective questions that make them think about what they’ve learned, etc.

Finally, be mindful of your attitude and watch the wording you use in class. In addition, teachers need to make sure they do not show a fixed mindset even vaguely because students will pick up on that. If we want students to change their mindset, we have to model the appropriate ways so students see the growth mindset in action.

With the new school year starting, it is the perfect opportunity to begin the year with activities designed to help students develop a growth mindset. This is especially important because many students have gotten behind, are aware of it, and feel as if they cannot do the math. Let me know what you think, I’d love to hear. Have a great day.

Warmup.

If you get 250 peanuts per square foot, how many peanuts will you get from 43560 square feet?

Warmup

 

If you raise 6 peanut plants per square foot, how many plants will you need for one acre. (One square acre = 43560 square feet).