### Mathematics content workshop day five: Technology and workshop assessment

Friday the sessions moved into the PICS dining hall. There I hooked up a television to a ChromeCast unit using ED District One's free WiFi. A shout out of thanks to Senator Ferny Perman for his role in supporting the deployment of Internet to schools in his district. The ChromeCast and Internet allowed me to demonstrate Schoology, open educational resources, and Desmos to the mathematics instructors.

The session led off with a demonstration of capabilities of Schoology. In the brief time I could only scratch the surface of the vast depth and breadth of capabilities in Schoology. I showed the teachers the assignment submission system, online testing, attendance, gradebook, grade setup, the use of custom marking rubrics. I mentioned messaging and the phone app for Schoology.

I then showed the teachers the open educational resource from OpenStax that I used in my algebra and trigonometry course this past summer, OpenStax Algebra and Trigonometry. While OpenStax focuses on collegiate level texts, there is an increasing amount of material that can be used at the high school level.

The Center for Open Education at the University of Minnesota offers College of the Sequoias professor John Redden's Elementary Algebra text that could substitute for an algebra I text. Redden also offers an open source, free on line Intermediate Algebra text that could substitute for many algebra II texts. The CK-12 Foundation offers a High School Geometry text. Math Open Reference is perhaps slightly less textbook-like, but has offerings across the high school math spectrum. More open source material is appearing online each day.

My own students responded with an enthusiastic positive response to their use of their on line text. As I noted to the teachers, this might not be a "right now" technology, but as 2023 looms in the front windshield of the FSM, options such as free online texts could provide a critical cost saving path for the education sector. How one ensures every student has the technology necessary to access online resource is another part of the challenge. Thus the morning session was intended to provide exposure to new options, awareness, not to specifically facilitate immediate adoption of these technologies.

The rest of the morning was spent demonstrating Desmos graphing mathematical engine. I no longer think of Desmos as a calculator. Desmos is an engine for doing mathematics in mathematics notation. No command line, no special commands. Just math notation. Pure math notation. The teachers asked whether the standard form linear had to be solved for y prior to graphing. As can be seen above, Desmos does implicit graphing. Serious horsepower under the hood. I also used the ChromeCast to cast the Desmos Android app to the screen. Desmos app does not need the Internet to work. The teachers appeared quite interested in this capability. A graphing calculator that does more than graphing, runs on equipment some already have access to, and does not need the Internet. One teacher had not understood that Desmos is also a web site that runs in a desktop browser until today. I had mentioned this, but seeing is understanding in this case. I also showed how the above graph can be shared via a URL that I can email. I emailed the Desmos graph link to the class, and a teacher with a laptop received the email and opened the graph above - which they were now free to edit. Those edits would be in their copy only.

In the afternoon I used site swap notation to convey two points to the teachers:

1. Starting with abstract notation on the board without some sort of concrete example makes mathematics particularly hard to grasp for students.

2. There are many more fields and branches of mathematics than one can imagine, and there are some that are quite comprehensible with only a basic knowledge of mathematics. Site swap notation is one of these fields of mathematics.

I started with the notation on the board and then asked if the teachers understood the system. Teachers are braver than my students and all of the teachers made it clear they had no idea what I doing at the board. My college students are far more shy and often just nod "yes" and then hope that I do not ask them a question about the notation. The teachers were correct when they responded that they did not know what that notation meant.

Then I juggled and showed how the 3333 and 515151 notation revealed the landing site pattern for the balls. The letters suddenly had meaning, as did the numbers and arcs. Light bulbs turned on and burst into brilliant light. A 342 swap explanation suddenly made sense.

I told the teachers, if you always start at the board, some of your students will always be lost. Sure, the sharp and fast ones, the math geniuses, will engage with the abstract material and succeed. Other less gifted students will quietly struggle or simply zone out.

At first former physical scientist Lorry-Ann did not recall the symbols on the board, and then she remembered. She also demonstrated the staying power of learned motor skills when she quickly regained her juggling skills.

Melina not only demonstrated an ability to juggle, she also multiplexed four balls, something I still cannot do. If three ball juggling is elementary algebra, multiplexing is advanced calculus.

Her hands are a blur as she multiplexes.

Merlihse starts with a two ball, one hand variation of juggling. Pattern is 22222...

After this session, the workshop moved on into assessment activities. Assessment involved answering three open ended questions:

What did you like about the workshop?

What did you not like about the workshop?

What could/should future workshops do better/cover?

Everything you presented was just perfect and productive.

Learning new strategies and methods needed for teaching mathematics. History and mathematicians who contributed to the fields. Relating math skills with other subject areas.

"I like everything that my professor taught us in class.

a. Using manipulative to teach the lesson.

b. Alignment of the standard and the benchmark to the textbook.

c. Explaining for each lesson loud, clear, and understandable.

d. The professor was kind, friendly, and a good role model.

e. The site (web pages for math) using Internet."

I like the way the instructor presenter explains the topics in math. He is so energetic and fun. Explains most things needed.

I like the timing and demonstrations to help us understand concepts taught. Activities involve moving around which is less tiring. I also like all the links that were being sent to me to use. I appreciate the standards cross--walk. It makes it easier for me to follow along and find chapter in both textbooks. In general, I've learned a long in this short period of time.

I like it when you try to use real life situations and turn it into a math problem. I tend to think that it takes extra work to do that for my classes and only give problems from the textbook.

A lot of things I learned from the workshop. The workshop showed a lot of things that we could apply in our classrooms. The workshop is better than what I attended before.

I learned how to make my own teaching materials instead of buying them from the stores. Also, I like that the trainor was able to open up to fully answered all of the given questions from us HS teachers. Lastly I like that I learned a lot of things that will benefit myself as a teacher and my students.

"That the workshop covered some of the ways we can teach algebra.

a. Using different math properties such as commutation, associative property, and distribution property.

b. Solving quadratic equation by using geometric figures to show relationship between geometry and algebra

c. Use pieces of string and chalk to construct geometric figures and shapes.

d. Use Desmos to work with mathematical problems"

Elementary math teachers should have participated

I was not able to answer some of the questions and forget most the lessons presented. There was not enough time to go over everything

The venue: too hot.

The food."

Everything about the workshop is good except Thursday and Friday's lunch. Also most importantly specialist (math) need to attend this workshop.

Basically I think the only thing I don't like is that the math specialist wasn't present for us to ask question about the textbooks/standard.

All is good except for our food, that could definitely be improved in the future.

I don't like the food that they serve cause one might cost less eight dollar to the workshop given it is better [sic].

The time is too long and lunch is not that really good.

Do not have materials (right) textbook to work with our standards as it should.

I recommend that we spend at least two week so we could cover more topics.

The workshop must be conducted for two to three weeks so more lessons will be covered. Lessons just relate to other subject area and real-life applications.

"The length of the workshop. If the workshop could be a month, we can learn more about it.

Algebra I teachers, Geometry, Algebra 2 and PreCalculus or 9th, 10th, 11th, and 12th grade teachers will do their workshops separately."

Prepare worksheets I believe will be fun for the future workshops. I also believe math specialist need to attend future workshops. Snacks/drinks should be prepared. Workshop assessment sheets (typed) need to be prepared.

I like all the concepts of algebra, trigonometry, and a little of calculus, but next time I'd like it to be simple in terms of high school student level.

If next time we are to be connecting with standards (course), I suggest our math specialist to be present so what are found can be known to the specialist to do adjustments or improvements on.

We should visit or work more on what might benefit the students and also the teacher. The workshop should have technologies related so we can also use it to applied to our students.

Extend the workshop duration and prepare a better place where there is AC and a big space to do activities that teachers can share with students.

It will be useful if our department (DOE) specialist provides tools for the teacher use especially with laptop, calculator, and needed tools in the workshop. Standards and textbook in use not aligned.

I would note that the location of the workshop was my choice. I want to teach the teachers in the exact same environment in which they teach.

First, because I will be using that environment as part of my teaching. The bricks at PICS rock! I used them more than once and I know that if I taught there more regularly, those bricks would appear repeatedly in my lesson plans. I thrive on whatever environment I find myself in.

Second, because we should all feel what the students feel. If the classroom is too hot for me to teach in, then the classroom is too hot for any student to learn in. And I say this as a runner who can run for two hours in the sun. I have a high tolerance for being hot and continuing to perform at high energy levels. So when I say "if a classroom is too hot for me" then that classroom is definitely too hot for those who do not have a tropical runners tolerance for the heat - both students and teachers.

The heat problems at PICS are two-fold, and neither of them is a call for air conditioning. The first is that the roof gets hot which in turns heats the room. Painting the roof with paint that reflects light and heat radiation would be beneficial to learning in the classroom. Although this would cost money, this is a fix that could - in some theoretic universe - be accomplished.

The second problem is that the layout of the buildings blocks cross-ventilation through the windows. This cannot be well fixed. In the future, buildings ought to be laid out to facilitate natural ventilation. The buildings at the college are staggered relative to each other, permitting cross-ventilating breezes.

The above said, the location was my choice, not that of the department of education nor the coordinator.

My thanks to the department of education, the training coordinator, the specialists at the department, PICS, and the many others whose work contributed to the success of this workshop. I would also like to thank Ms. Senoleen S. Booth for the use of her classroom this week.

My most sincere thanks go to the teachers who put up with my rambling on well past any reasonable break time. The mathematics teachers I worked with this week are top notch, an elite crew who produce students with the highest math capabilities in the nation. Teachers who can lift vocational students to math skills levels well above that of college bound academic students at some of the private schools. These teachers inspire me.

The session led off with a demonstration of capabilities of Schoology. In the brief time I could only scratch the surface of the vast depth and breadth of capabilities in Schoology. I showed the teachers the assignment submission system, online testing, attendance, gradebook, grade setup, the use of custom marking rubrics. I mentioned messaging and the phone app for Schoology.

I then showed the teachers the open educational resource from OpenStax that I used in my algebra and trigonometry course this past summer, OpenStax Algebra and Trigonometry. While OpenStax focuses on collegiate level texts, there is an increasing amount of material that can be used at the high school level.

The Center for Open Education at the University of Minnesota offers College of the Sequoias professor John Redden's Elementary Algebra text that could substitute for an algebra I text. Redden also offers an open source, free on line Intermediate Algebra text that could substitute for many algebra II texts. The CK-12 Foundation offers a High School Geometry text. Math Open Reference is perhaps slightly less textbook-like, but has offerings across the high school math spectrum. More open source material is appearing online each day.

My own students responded with an enthusiastic positive response to their use of their on line text. As I noted to the teachers, this might not be a "right now" technology, but as 2023 looms in the front windshield of the FSM, options such as free online texts could provide a critical cost saving path for the education sector. How one ensures every student has the technology necessary to access online resource is another part of the challenge. Thus the morning session was intended to provide exposure to new options, awareness, not to specifically facilitate immediate adoption of these technologies.

The rest of the morning was spent demonstrating Desmos graphing mathematical engine. I no longer think of Desmos as a calculator. Desmos is an engine for doing mathematics in mathematics notation. No command line, no special commands. Just math notation. Pure math notation. The teachers asked whether the standard form linear had to be solved for y prior to graphing. As can be seen above, Desmos does implicit graphing. Serious horsepower under the hood. I also used the ChromeCast to cast the Desmos Android app to the screen. Desmos app does not need the Internet to work. The teachers appeared quite interested in this capability. A graphing calculator that does more than graphing, runs on equipment some already have access to, and does not need the Internet. One teacher had not understood that Desmos is also a web site that runs in a desktop browser until today. I had mentioned this, but seeing is understanding in this case. I also showed how the above graph can be shared via a URL that I can email. I emailed the Desmos graph link to the class, and a teacher with a laptop received the email and opened the graph above - which they were now free to edit. Those edits would be in their copy only.

In the afternoon I used site swap notation to convey two points to the teachers:

1. Starting with abstract notation on the board without some sort of concrete example makes mathematics particularly hard to grasp for students.

2. There are many more fields and branches of mathematics than one can imagine, and there are some that are quite comprehensible with only a basic knowledge of mathematics. Site swap notation is one of these fields of mathematics.

I started with the notation on the board and then asked if the teachers understood the system. Teachers are braver than my students and all of the teachers made it clear they had no idea what I doing at the board. My college students are far more shy and often just nod "yes" and then hope that I do not ask them a question about the notation. The teachers were correct when they responded that they did not know what that notation meant.

Then I juggled and showed how the 3333 and 515151 notation revealed the landing site pattern for the balls. The letters suddenly had meaning, as did the numbers and arcs. Light bulbs turned on and burst into brilliant light. A 342 swap explanation suddenly made sense.

I told the teachers, if you always start at the board, some of your students will always be lost. Sure, the sharp and fast ones, the math geniuses, will engage with the abstract material and succeed. Other less gifted students will quietly struggle or simply zone out.

At first former physical scientist Lorry-Ann did not recall the symbols on the board, and then she remembered. She also demonstrated the staying power of learned motor skills when she quickly regained her juggling skills.

Melina not only demonstrated an ability to juggle, she also multiplexed four balls, something I still cannot do. If three ball juggling is elementary algebra, multiplexing is advanced calculus.

Her hands are a blur as she multiplexes.

After this session, the workshop moved on into assessment activities. Assessment involved answering three open ended questions:

What did you like about the workshop?

What did you not like about the workshop?

What could/should future workshops do better/cover?

**What did you like about the workshop?**Everything you presented was just perfect and productive.

Learning new strategies and methods needed for teaching mathematics. History and mathematicians who contributed to the fields. Relating math skills with other subject areas.

"I like everything that my professor taught us in class.

a. Using manipulative to teach the lesson.

b. Alignment of the standard and the benchmark to the textbook.

c. Explaining for each lesson loud, clear, and understandable.

d. The professor was kind, friendly, and a good role model.

e. The site (web pages for math) using Internet."

I like the way the instructor presenter explains the topics in math. He is so energetic and fun. Explains most things needed.

I like the timing and demonstrations to help us understand concepts taught. Activities involve moving around which is less tiring. I also like all the links that were being sent to me to use. I appreciate the standards cross--walk. It makes it easier for me to follow along and find chapter in both textbooks. In general, I've learned a long in this short period of time.

I like it when you try to use real life situations and turn it into a math problem. I tend to think that it takes extra work to do that for my classes and only give problems from the textbook.

A lot of things I learned from the workshop. The workshop showed a lot of things that we could apply in our classrooms. The workshop is better than what I attended before.

I learned how to make my own teaching materials instead of buying them from the stores. Also, I like that the trainor was able to open up to fully answered all of the given questions from us HS teachers. Lastly I like that I learned a lot of things that will benefit myself as a teacher and my students.

"That the workshop covered some of the ways we can teach algebra.

a. Using different math properties such as commutation, associative property, and distribution property.

b. Solving quadratic equation by using geometric figures to show relationship between geometry and algebra

c. Use pieces of string and chalk to construct geometric figures and shapes.

d. Use Desmos to work with mathematical problems"

**What did you not like about the workshop?**Elementary math teachers should have participated

I was not able to answer some of the questions and forget most the lessons presented. There was not enough time to go over everything

The venue: too hot.

The food."

Everything about the workshop is good except Thursday and Friday's lunch. Also most importantly specialist (math) need to attend this workshop.

Basically I think the only thing I don't like is that the math specialist wasn't present for us to ask question about the textbooks/standard.

All is good except for our food, that could definitely be improved in the future.

I don't like the food that they serve cause one might cost less eight dollar to the workshop given it is better [sic].

The time is too long and lunch is not that really good.

Do not have materials (right) textbook to work with our standards as it should.

**What could/should future workshops do better/cover?**I recommend that we spend at least two week so we could cover more topics.

The workshop must be conducted for two to three weeks so more lessons will be covered. Lessons just relate to other subject area and real-life applications.

"The length of the workshop. If the workshop could be a month, we can learn more about it.

Algebra I teachers, Geometry, Algebra 2 and PreCalculus or 9th, 10th, 11th, and 12th grade teachers will do their workshops separately."

Prepare worksheets I believe will be fun for the future workshops. I also believe math specialist need to attend future workshops. Snacks/drinks should be prepared. Workshop assessment sheets (typed) need to be prepared.

I like all the concepts of algebra, trigonometry, and a little of calculus, but next time I'd like it to be simple in terms of high school student level.

If next time we are to be connecting with standards (course), I suggest our math specialist to be present so what are found can be known to the specialist to do adjustments or improvements on.

We should visit or work more on what might benefit the students and also the teacher. The workshop should have technologies related so we can also use it to applied to our students.

Extend the workshop duration and prepare a better place where there is AC and a big space to do activities that teachers can share with students.

It will be useful if our department (DOE) specialist provides tools for the teacher use especially with laptop, calculator, and needed tools in the workshop. Standards and textbook in use not aligned.

I would note that the location of the workshop was my choice. I want to teach the teachers in the exact same environment in which they teach.

First, because I will be using that environment as part of my teaching. The bricks at PICS rock! I used them more than once and I know that if I taught there more regularly, those bricks would appear repeatedly in my lesson plans. I thrive on whatever environment I find myself in.

Second, because we should all feel what the students feel. If the classroom is too hot for me to teach in, then the classroom is too hot for any student to learn in. And I say this as a runner who can run for two hours in the sun. I have a high tolerance for being hot and continuing to perform at high energy levels. So when I say "if a classroom is too hot for me" then that classroom is definitely too hot for those who do not have a tropical runners tolerance for the heat - both students and teachers.

The heat problems at PICS are two-fold, and neither of them is a call for air conditioning. The first is that the roof gets hot which in turns heats the room. Painting the roof with paint that reflects light and heat radiation would be beneficial to learning in the classroom. Although this would cost money, this is a fix that could - in some theoretic universe - be accomplished.

The second problem is that the layout of the buildings blocks cross-ventilation through the windows. This cannot be well fixed. In the future, buildings ought to be laid out to facilitate natural ventilation. The buildings at the college are staggered relative to each other, permitting cross-ventilating breezes.

The above said, the location was my choice, not that of the department of education nor the coordinator.

My thanks to the department of education, the training coordinator, the specialists at the department, PICS, and the many others whose work contributed to the success of this workshop. I would also like to thank Ms. Senoleen S. Booth for the use of her classroom this week.

My most sincere thanks go to the teachers who put up with my rambling on well past any reasonable break time. The mathematics teachers I worked with this week are top notch, an elite crew who produce students with the highest math capabilities in the nation. Teachers who can lift vocational students to math skills levels well above that of college bound academic students at some of the private schools. These teachers inspire me.