Education and a Global Society

I recently watched Gordon Brown’s Ted Talk,

“The power of our moral sense allied to the power of modern communications and our ability to organize internationally. That in my view gives us the first opportunity as a community to fundamentally change the world.” -Gordon Brown

As I reflect on that quote and the competencies required of individuals to change the world in this way, I am reminded of how educating young people can play an important role in developing a truly global society. I see 3 competencies that are vital: ethics, communication and action; with this comes the need to understand multiple perspectives and cultural differences/similarities. What does this look like in elementary school?  How do we develop empathy and as Jason Silva mentioned in The Big Picture, extend our gaze to develop a massive transformation of consciousness?

I hear it everywhere… the purpose of education is to foster the ability to think critically and develop skills to be a productive member of society. But, I have also heard (especially in this age of high stakes testing) that the purpose of school for your children is to develop  basic skills.  Students absolutely need basic skills, but can’t we help students acquire those skills in ways that build a moral compass, empathy, communication and the desire to act? 

What I very much appreciated in the article Setbacks Aside, Climate Change Is Finding Its Way Into the World’s Classrooms by Beth Gardiner was the idea of focusing on, “the “handprint” of individuals’ beneficial actions, rather than the harm suggested by an environmental footprint.”  I sometimes worry that through news and current events we are depicting the world as a very scary and unjust. Yes, we absolutely have so many inequities and many global problems, but I think if we start using the lens of what “handprints” are being left and what can we learn from those actions we might be able to turn some of these injustices into better learning opportunities for ways our young people can act.

On Wednesday, my class participated in a Google Hangout with Paul Salopek, a Journalist with National Geographic who is walking around the world for 7 years, following the path of early migration. If you have not heard of the Out of Eden Walk, check it out. And, as an educator check out Out of Eden Learn through Project Zero at Harvard.  It has been a great way for my students to connect their world with students in Dubai, Australia, and Eastern United States (the countries represented in our walking party).

Screen Shot 2016-04-30 at 12.02.13 PM

But, back to our Hangout with Paul.  One of his messages was human kindness is everywhere.  He stated, ”I’m being passed from kindness to kindness”. We have to remember this as much as discuss global problems.

Another way that we have focused on the “handprints” we can leave is when we studied teen activists.  I got this idea from Lucy Calkins’ Units of Study in Writing .  While students were reading about such issues as slavery, girls who weren’t allowed to go to school due to radical leaders and religious beliefs, or child labor, the focus that there were people out there acting and making a difference was powerful. Yet, while I was hoping they would be inspired and choose to act (I felt like I shouldn’t force them to and I’m not sure if this was the right choice), they have not done so yet.  My question is how do we move from knowledge about global problems to acting to change and rid the world of those problems? As I think about that, while they showed a desire to act, they have no idea how to start.  

We are starting small.  This next week, students will be choosing one thing that they can do to make the world a better place.  It can be small or large, but they need to commit to something. We will see how that goes, but I really want it to come from them, so I’m still thinking through this aspect.

Unfortunately, inequities are not just a global problem, they are a problem in our schools as well. When Gordon Brown spoke about how Olof Palme desired to abolish the poor and to let everyone have the chance to realize their potential to the full, it made me think about all the inequities in our schools: tracking, access to technology, access to powerful teaching and learning, etc. How are we doing at providing equitable learning experiences to all our students and are we providing all our students the chance to realize their potential to the full?  This reminds me of a blog post I just read by Kristin Gray, a math coach on the east coast, regarding RTI. I agree that we have to develop strong interventions for students. It is vital, but are we being equitable and just in how we are doing it? This is an area we need to focus on. The article Money, Race and Success: How Your School District Compares was yet another reminder of this.  We need to do a better job.

One way to improve equitable access to learning is through global education. When we provide all students with opportunities to interact with others and understand multiple perspectives, in addition to learning how to communicate in person, in writing, and online, while also developing empathy for others and providing all students with rich, relevant, meaningful tasks that foster critical thinking skills, we will be one step closer to developing a more just and equitable educational system and improved global society. 

Ahhh…always more work to be done.

Ratios, Rates, and Proportional Reasoning

Every day I think…how do I help my students connect mathematical ideas and make sense of the math we are studying.

My class has been developing an understanding of how to use ratio tables through various tasks I have brought in to supplement CPM Core Connections 1. They have demonstrated a good understanding of this, though some students are having difficulty moving from additive thinking to thinking multiplicatively. (On a side note…this is now their favorite word…they love saying it for some reason… cracks me up.)

———–

We all have the moments, or maybe it is just me, when our planning isn’t as complete as it needs to be. I had glanced at lesson 7.1.3, but had not looked at problem 7-26 as closely as I should have.  We had only been using tables that were proportional, starting at (0,0) and then students came across this situation.

Screen Shot 2016-03-25 at 10.18.10 AM

They kept telling me that there was a mistake with Tamika’s table.  “Ms. S the relationship  between 1 and 7 is not the same as 2 and 9. This doesn’t make sense.  ”

So, we reread the problem and thought about the context. What do you notice?

“They are knitting. They have a constant rate.”

Ok, what does it mean to have a constant rate?

“It is the same. But, Ms. S, it is not the same. ”

Can we look at the table in any other way? They nodded and I left them to work. Then I heard them…

“Well, we can compare 7 to 9 and 1 to 2. Oh maybe it is increasing by 2 and by 1….”

“Every hour she knits 2 inches.”

They were starting to make sense.  Then other groups were having similar struggles, so we discussed as a class. We continued on discussing, giving them time to think with their groups and process, and realized that Tamika had started knitting with 5 inches already completed. But, I could see many of them were still a bit confused about the differences in the two situations.

We had been doing a lot of linear visual patterns and making tables and graphs from them, in addition to graphing ratio tables and noticing that ratios which are proportional create a line through the origin (even though this really is formalized in 7th grade, the discussion has grown organically).

Last weekend, I engaged in a conversation on twitter about visual patterns with  .  Their thoughts and comments made me think a lot.

I realized that my students had not been able to connect all of these situations and see the relationships between them. My goal had been to let them develop reasoning skills, looking at relationships, and connecting visual patterns to expressions in order to be able to formalize their explorations in the coming grades, but seeing them struggle with the knitting problem, I thought, how can I help them connect the visual patterns to a situation like the knitting problem?

So, I came up with two patterns that I knew they would easily be able to generalize and then we could compare and connect. Which one shows a proportional relationship?

After students had time to process and look at the patterns, think about how they were growing and the relationships, we then started our discussion.

I stated, one of the things I would like you to think about has we look at these patterns is how do these patterns connect or relate to the knitting problem we had done a couple of days ago. Remember, we are always looking at math as connected and looking for relationships to help us better understand.

Screen Shot 2016-03-25 at 10.00.16 AM

 

We first looked at how the patterns were growing and then we engaged in a “Compare and Connect Talk” (see Intentional Talk, by Elham Kazemi and Allison Hintz).

They made some great connections and there were a lot of “ahhhhhs” “oohhh, I get it.”.

Then we looked at the differences.

“Look at the bottom left and the right. They both have arms that are the figure number, but the left one has a square of 4 and the right one doesn’t have anything.”

“Oh, so if we subtract 4 from each on the left, then we get the right pattern. “

Can you make a connection to that in the table? They turned and talked and analyzed.

Screen Shot 2016-03-25 at 10.58.32 AM

One student said, “Look, if you subtract 4 from the 7 and the 10 and the 13, you have the same numbers in the other table.”

I then asked them, what if these patterns represented Jelly Beans and Cupcakes?  Turn to your partners and try to come up with a situation that each pattern would connect to.  Remember the knitting problem. Do you see anything related?

They talked and some struggled, but through their discussions and a class discussion, we came up with the following situation. They were seeing the patterns connected to a context and were really starting to make sense.

Screen Shot 2016-03-25 at 10.09.47 AM

Where do we go next? Do we keep playing around and looking for connections? Do I formalize some of the language?  I’m not sure… Next year, I think I would do the comparison of these two patterns before we do the CPM Lesson, as I think the visual connection would help them understand that situation more. But, as for this year…I think we keep playing and exploring and connecting. They will be ready for formalizing these ideas in 7th grade.

We need to formalize division of fractions! Ahhh! But, all this work with ratios has really started to help students understand division of fractions more as we make that connection. Wow! There really is so much to think about when designing the best way to facilitate student thinking and sense making. It is amazing. I thought I really understood ratios, rates, and proportions.  My depth of understanding is growing everyday with my students.  There is so much depth to math. It is just brilliant.

Every day I think…how do I help my students connect mathematical ideas and make sense of the math we are studying.

 

Solving Current Events: A Follow Up

Last week I wrote about a lesson I had developed from a video I saw about a volunteer in Lesbos using the materials of life jackets for bags. This is a reflection on how the lesson went with my class.

First of all, it was motivating, inspiring, open, and students had to persevere.

Screen Shot 2016-03-15 at 6.40.46 PMThe day after I wrote the lesson we were not going to have math since we were going on a field trip, but I had 15 minutes before we left on the trip and I was able to engage students in some wondering. We went through the first slides and noticed and wondered, not thinking about math at all. Though, my students are now mathematizing the world, so a few mathematical wonderings came through.

 

Then that evening they read this article and were able to answer some of their wonderings.

The math extravaganza began. We took on a math lens and thought like mathematicians. Noticing, wondering and determining what information we needed.IMG_7390 (1)

I am always so amazed with the mathematical questions students are developing now. Determining what they needed was a lot harder than in other lessons we have done. Why? It was a big scale problem with multiple questions to solve. But, through conversation they came up with some good ideas.  I then gave the students the information they asked for and they were off!

IMG_7382IMG_7375

It was fascinating to watch how students approached the problem. Some just started tracing complex shapes, others grabbed rulers and drew precise lines, others drew free hand lines, but then as they saw their classmates using rulers, changed their minds. IMG_7387IMG_7380

They helped each other and after awhile all students were looking for shapes that they knew how to find the area of. I had a few students that really struggled with this and I wanted them to get to finding the area, so I provided them with my shapes.  Then they were off. It was a great way for some students to get additional reasoning with finding the area of shapes and it was through this problem that some really saw the relationships of different shapes to one another.

“I know how to find the area of triangles, so I decided to turn all these into triangles. There are triangles in every shape. ”

And then they solved….IMG_7386IMG_7388

Thinking, connecting, and then we needed a discussion. Can we find the actual area if we use the ratio 1 mm: 2 4/5m?IMG_7400We discussed how the ratio we were given was a linear ratio and we could use that for finding length, but to find area, we needed to create a new ratio of square millimeters to square meters.  And then we had some solutions…IMG_7402

We shared different strategies: converting all the lengths in millimeters to meters and then finding the area compared to finding the area in square millimeters and then converting to different units. We then found the median and mean of our results.

Then came the question: How many lifejackets are there in 40,000 square meters?

It was great to be able to review the use of ratio tables and to see the different ways students approached the problem, connecting the three different strategies. We engaged in the “Compare and Connect”  math discussion that is introduced in Intentional Talk by Elham Kazemi and Allison Hintz

And then there were the students who extended the problem…

I clearly am a terrible photographer… But did you know that 40,000 square meters is equivalent to 7 1/2 football fields, 91 1/2 NBA regulation basketball courts and about 1/2 of Buckingham palace?

IMG_7398IMG_7397

And if you were to lay pencils down in 40,000 square meters, you can fit 3,007,518 pencils. hahaha

Finally came…Screen Shot 2016-03-09 at 12.22.19 PM

Hahaha, this is when I was reminded I work with 11 and 12 year olds. What would they do? “I would make a pile and build a huge slide on it.”

“I would jump in the lifejackets.”

“I would take them back to Turkey and resell them.”

“I would make a boat out of them and sell the boat.”

Clearly, we have fun seekers and entrepreneurs.  Next time, I would change the question to… What would you MAKE with 450,000 life jackets?

I then shared with them the video about how they are making bags out of the lifejackets and left students with the question. What extraneous materials do we have lying around and what useful object could we make out of it?

It was a two day lesson, but a lot of learning: using a ruler, calculating area, solving proportions, developing a plan, comparing and connecting, and persevering.

 

 

Solving Current Events: Math, Design, Global Connection

Who knew Facebook actually could be inspiring?

A friend recently posted this video

http://www.rferl.org/media/video/upcycling-refugee-style/27587634.html

And then the wheels started to turn when I saw this picture…

30c8feb800000578-3426430-image-a-75_1454341622703

How many lifejackets are there? What area is taken up by life jackets? Ooohhhh…I see rectangles, trapezoids, and triangles.  I can connect what we have been studying in geometry to what we are moving on to next: ratios and unit rates. Not only is there math, but  we can write, do some design thinking, and develop a greater understanding of the greater world. Here is the lesson. Please feel free to comment. All ideas are welcome.

Before doing the math lesson, I might have students read one of the two articles and notice and wonder about some images.

time.com/4023601/migrant-crisis-greece-lesvos/

https://newsela.com/articles/migrants-techsavvy/id/12077/

Screen Shot 2016-03-09 at 12.20.46 PM

I would then show students this image, asking what they notice and wonder. We might record their noticings and wonderings using Poll Everywhere or Pear Deck, or I might chart their ideas on chart paper. I like using technology because then everyone is able to share at once, but I also like when students share ideas orally because it tends to generate even more and creates excitement in the classroom.Screen Shot 2016-03-09 at 12.19.34 PM

We would then zoom in and add new noticings and wonderings.

Screen Shot 2016-03-09 at 12.19.41 PM

and zoom in even more to finally see what is actually there.

Screen Shot 2016-03-09 at 12.19.48 PM

Now enters math… I would start our math class telling students it is time to think like a mathematician and have them ask some mathematical questions.

From the list we generated I would hope that these two would arise. Screen Shot 2016-03-09 at 12.21.13 PM

Then I would ask what information do you need to answer those questions. We would create a list of information that is needed and then give them the following information.

Screen Shot 2016-03-09 at 12.21.21 PM

Screen Shot 2016-03-11 at 10.32.47 AMNow comes the fun…planning, persevering, and solving. After students have been working for a bit, if I notice a lot of struggling, I might stop them and ask them…Screen Shot 2016-03-09 at 12.21.50 PMScreen Shot 2016-03-09 at 12.38.27 PM

If students are still struggling, or for those students that need additional supports, I might share with them the shapes I saw and let them work form that.Screen Shot 2016-03-09 at 12.23.57 PM

After students were given time to solve for the two questions, we would share out strategies and engage in Mathematical Practice #3: Critiquing and Justifying.

I would then share with them the data that was in the article:

Screen Shot 2016-03-09 at 12.22.12 PM

At another time, I would then ask students…Screen Shot 2016-03-09 at 12.22.19 PM

We would engage in some creative design thinking and come up with some uses for 450,000 life jackets. I would then share the video that inspired this lesson and we would read. this article.

Lastly, here are a few extensions…

Screen Shot 2016-03-09 at 12.23.16 PM

Here is the Ratio and area lesson Refugee if you would like to use it.

As I write this all down, it feels long.  I wonder what I might change. What do you think?

Time and Patience is Vital to Learning

There has always been too much for students to learn each year, but now there is too much for teachers to learn too.  Too much learning? Is that possible?  I revise that statement. There is much for students and teachers to learn and the only way that we can learn it all is to look for connections, study, try, fail, study, and try again. My new favorite word is iteration.  Learning takes iterations. We have to be patient, but persistent and unrelenting in our pursuit of getting to where we want to be.

As for teachers… we have a lot of “new” on our plates: NGSS, ELD -integrated and designated, and continuing or learning with CCSS. Not only is there all this new content to teach, but we have new pedagogies too.  The only way we are going to do it all is to integrate and understand how content relates. I love the idea, but to do it well, it takes time. Time to understand subject matter. Time to process and connect ideas. Time to develop lessons that are cohesive. Time to implement those lessons. It takes time. It takes iterations.

practicesvenndiagram_page_1_000

When you do make the time to see the connections in your curriculum and integrate subject areas, it makes a difference for student learning.  This past month, we read A Long Walk to Water by Linda Sue Park.  While the reading level was on the easier side,  the subject matter was not. The easier reading level really allowed us to focus in on some of the more complex literary standards, such as analyzing the craft of the author and the structure of the text, and developing an understanding of theme.

alongwalktowaterThe content of the book connected both to our study of ancient Egypt/Nubia and to science and the cycling of Earth’s water. We started out the book by  learning about the geography of that part of the world in ancient times and comparing it to the time period of the book and how it has changed over time.

As we finished the book and the students learned about the wells that Salva Dut was building, they began to inquire about water and how it ends up in the ground. I am counting that as a natural phenomenon and from their questioning we then started exploring the water cycle. Science teaching from phenomenon and student exploration…I’m exploring it, and not even close to doing it well, but after a few fails with ideas and activities trying to facilitate my students’ learning and my own in what they are suppose to do according to NGSS, I might actually be on the right track. More to come about this later.   

But, what I want to share about is this notion of patience. Patience for ourselves as teachers  and acceptance that as long as we are trying and moving forward, we will have a few fails here and there. But, also patience for our students. We all have a lot to learn, but we will learn with time.

time

I had taught my students when we were writing essays about teen activists that there are different types of leads and that an introductory paragraph needs a lead and a thesis statement.  We worked on it and 1/2 the class created some decent introductions and the other 1/2… we won’t talk about it.  Then last week I had them write introductions to an already written essay. As I read these introductions, I was so frustrated. Literally, there were only 3 that were decent. We then looked at the 3 examples that were well done and analyzed and critiqued them, noticing what made them strong.  Then it was time to write essays about the themes of A Long Walk to Water.  It took a full week for students to process and develop outlines, but their discussions were brilliant.

“We are working on the theme of persistence, but it seems to overlap with overcoming obstacles and survival. Is that ok?”  They processed. “It is ok. It is with persistence that you overcome obstacles and it is will persistence that they were able to survive. Persistence helps to achieve goals and obstacles can get in the way.”

Time…if I had said, “hurry up write your outline. It is due today.” They wouldn’t have come to the understandings that they did. Ok…fine…I might have said hurry up a few times. But, I extended the time. I am thankful I did.  We all need processing time. Time to think through ideas, talk about ideas, and make sense of complex ideas.  We have to give our students time to think.

I reminded students of types of leads and we discussed which leads might be best for this type of essay. Appealing to universality was shared and possibly questioning and or an anecdote.  They then wrote.  This weekend I read through their essays to see where they were and was no longer frustrated  The introductions were brilliant and especially for 1st drafts.

“Imagine being just a small scared Sudanese child in your War torn country. You’re starving and feeling like you are about to faint. You just keep walking and walking in the middle of the desert with a big group of people not knowing where you’re going. There are dangers big and small all around you. You don’t stop even though you’re on the verge of death. The only thing you know is that the group is walking to a safe place. You remember what you’re uncle always kept telling you, one step at a time. Well you’re life isn’t even close to this! This is what the main characters, Salva and Nya, have to face in the story, A Long Walk To Water. They have to overcome difficult and almost impossible obstacles. They use smart thinking, group support, and sacrifice to survive the difficult conditions.”

“Imagine walking through an African desert with thousands of people you are unfamiliar with. Hot and tired. Walking constantly. No stopping. Searching for food and water. Waiting to find your family who might have been killed. Running from the war. In the book, A Long Walk to Water, by Linda Sue Park, walking like this happened extremely frequently. This and many other examples add up to one trait, persistence.  Persistence develops strength, it overcomes obstacles, and it completes goals. There is a notion of these ideas throughout the story.”

“Have you ever wondered how important kindness is, or what it can do? Thousands of people rely on other people’s kindness. In the book ,A Long Walk to Water, there are many examples of kindness. Kindness helps people to survive and improve living conditions. “

“Everyone has been there, when you or someone in your life are alone, confused, and helpless. That all changes the second you bring a leader into the picture. They could be a coach, a teacher, friend, or a parent. Maybe it’s even you. Whoever they are, they’re a leader. There are many demonstrations of leadership throughout the book, A Long Walk to Water. More importantly, it shows how leadership has the power to change. It can change people, the environment around people, and it can do this in positive or negative ways.”

 

Time. Patience. Persistance. Leadership. Lesson learned….pushing and expecting high level work is important and it will come in time. This goes for students and teachers.  We need to expect high level work from our students and ourselves, but also patience and forgiveness when it is not there yet. With time, effort, a goal, and drive we can do anything. I will be able to be an amazing science teacher too…in time.

fb19657e04c767112cc7df27dc760440

 

 

Global Math

I love to travel.

I love teaching math.

I love when I can turn travel into a math lesson.

On December 26, I had an urge…an urge to travel, so I jumped onto Kayak to see where in the world I could go that would be a decent price. Ecuador? Nope…too much. Japan? Whoa…this might be it.  Leave tomorrow? Sure, why not? So I went for it. I booked my flight and was ready to go.

But, that is not all. I was celebrating my most random last minute trip with a friend and her husband said, “Babe, why don’t you go too?”

“Yes! Yes! Yes!” was all I could say. Short story…we booked her a flight too!

celebrate

On December 27, we were off on our adventure to Japan. For the global wanderer in me, I have 3 words…I love Japan!  I never knew I would, but I am definitely going to go back there and spend some time observing schools and learning about their educational system, especially math instruction. I have always been fascinated, but now I HAVE to do that!

Well, our trip was amazing, but when I can combine the mathematics that my students are learning with a trip, it makes it even better. First idea…travel distance to Japan using rates. We had just started working on ratios before I left on the trip, what a perfect introduction to unit rates using miles per hour.

My plan…make it a game. Where in the world did Ms. K and Ms. S go? Giving students the average speed, the time to travel, they can determine the mileage.  We will do a little noticing and wondering. Have them determine what information they might need. Then I will give them the data and off they will explore. After they figure out the mileage, I will give them maps and they can try to guess where we went using their knowledge of ratios. My goal will be for them to understand what a rate is and how unit rates can be useful. I definitely made it complicated, but complexity is what makes it fun and my students would not expect anything less. See the video below.

A Japanese Holiday Room 15  on Vimeo. Password: japanmath

 

The second idea came when we were eating lunch and looking at all the Japanese coins. We came across1971japan5yenobv400

We had no idea how much it was. All the rest of the coins had the value shown, this one did not.

jpy_coins_2

We finally finished and paid. Ahhhh,  what did we get as a change? Yes, the unmarked coin. We paid with 2,000 yens. The bill was 1,682 yens. Our change….

147794b147794b147794b220px-juendamaevergreentreewp

japan-1-yen-1991japan-1-yen-1991japan-1-yen-1991g896

Ahhhh, a little algebra… 3 (100) + 1 (10) + 3 (1) + x = 318 yen

The unmarked coin is worth….

This will be a perfect warm up to keep working on writing equations with variables. Yay!

Lesson learned…staying home is relaxing, but traveling is fun and inspirational! Where will I go next?

 

 

 

Thinking and Engaging…A Week of Math!

What would school be like if every so often (or always) we didn’t race in the Daily Academic Decathlon: teaching  reading, writing, math, social studies science, art, engineering, music (oops….), PE, and technology?

dan-obrien-and-dave-johnson-before-the-1992-olympic-gamesI grew up watching Dan and Dave, but I bet they didn’t train for every event all day every day.  I bet they focused and went deeper in their training.

Yes, yes, yes…I don’t believe those are all isolated subjects and most aren’t subjects, but we do this in school all the time. I have completely gotten pulled into this and because of trying to do it all, at times we are accomplishing nothing (ok, that’s not true, but sometimes it just doesn’t seem feasible to do it all and then we are always a dollar too short.)

What would happen if we took a week and focused on becoming a scientist, an author, an engineer, a mathematician? What would happen if we just dove into a content area, topic, current event and engaged our entire being in it? What if we became mathematicians for the week? What if we, “play(ed) the whole game”  as stated by David Perkins  in Making Learning Whole? What if students came to expect that, “a large part of their learning in the subject area involved acquiring the thinking abilities and processes of the discipline, not just learning about it for the test.” as stated by Ron Ritchart in Creating Cultures of Thinking? What if parents would focus on the learning and not just the test grade or the completion of the HW assignment?

ritchhart-cultures-239x300

Well, we went for it. We spent the last week as Mathematicians investigating math. Where did this idea stem from?

  1. I always feel crunched for time. I have 1 hour (ok…1 hour 15 mins…I never finish in the hour) that we must get our math in.  What would it be like if we could work on a problem until we were done instead of waiting to finish it the next day? What would school be like with no time limits? What if we took a break when we needed a break instead of when the bell rang? Who lives their life on a preplanned schedule?It is a little bizarre to be honest.
  2. I had just had a lot of conferences and was reflecting on the need for parents to experience the math that we do. I kept hearing, “I’m not good at math. My husband is the math person.” Ughhhhh….Makes me want to throw up.  Everything was about the answer.  I wanted them to stop and see that math is anything but boring and their children are completely engaged. So I thought, let’s open up the class for the week. Come in and do math with us any time you can.

So what did we do? We emerged ourselves in math. We played. We modeled mathematics. We looked for patterns. We read about Mathematicians. We explored the mathematics of ancient civilizations.

As Jo Boaler wrote in Chapter 4 of Mathematical Mindsets: Unleashing Students’ Potential Through Creative Math, Inspiring Messages and Innovative teaching,

When students see math as a broad landscape of unexplored puzzles in which they can wander around, asking questions and thinking about relationships, they understand that their role is thinking, sense making, and growing. When students see mathematics as a set of ideas and relationships and their role as one of thinking about the ideas, and making sense of them, they have a mathematical mindset.

51nfg6tuwgl-_sx399_bo1204203200_

And I think that this “mathematical mindset” is really a thinking mindset.  No matter what we are doing, the fun comes when we think, question, and reflect.

As always, there were things that I loved and things I wish I had done better.

What I loved….

Each day we had a Big Idea and Details to help us focus

Examples…

Screen Shot 2015-12-22 at 10.07.41 AMBig Idea:  Mathematicians create viable arguments and critique the reasoning of others.

DetailsDetails:

  1. Use examples to prove a claim is true and non examples to prove a claim is false.
  2. Explain with words and evidence why a claim is true or untrue.
  3. Question others to find holes in their arguments or to better understand their argument.
  4. Use drawings, tools, and representations to support a claim.

Another day we focused on…

Screen Shot 2015-12-22 at 10.07.41 AMBig Idea: Mathematicians look for patterns and relationships to make generalizations they can use.

DetailsDetails:

  1. Look for things that are repeated, things that are the same, things that are related.
  2. Question…How is what I am working on similar to….?
  3. Does what I am working on remind me of something else I have done? What is the relationship?
  4. Look for smaller parts that compose the larger problem. Are there patterns or relationships in the smaller parts?

We truly engaged in the Standards for Mathematical Practices and made time to think. As I look at this list from Ron Ritchart, I do think we have a classroom culture of thinking.  The biggest gift that this week brought us was the gift of time.Screen Shot 2015-12-22 at 9.46.22 AM.png

from…http://www.ronritchhart.com/ronritchhart.com/Welcome.html

My other love was having parents join us.

father-son By the way, why is it that I could not find a single positive clip art of a parent and child doing math together. Reading? It’s all over the place. Math? It is always an image of frustration. Ughhh…

“Wow, I am having fun. I wish math was like this when I was a child.” Pretty much, what this mom was saying, was that she wished she was allowed to think and make sense of math.

After doing Andrew Stadel’s Post-It Three Act Task http://www.101qs.com/518-file-cabinet–act-1, a parent responded,”That was great. I went first to thinking I need to find the number of post-its and what is the easiest way for me to do this. Then I looked over at my son who was finding the entire surface area and then dividing by the post-its surface area. At first I thought, that is so inefficient. We had a great conversation at our table. Then I realized. Actually, he would have a better model. If the size of the post-it changed, he would easily be able to adapt for that. I on the other hand would have to recalculate everything.”

How awesome for students to learn with their parents?

After doing a visual pattern (from Mathematical Mindset), a Physicist (parent) said, “Wow, this is what I do all day long. We look for patterns and make generalizations.” Ahhh…thinking. Then he continued, “I need more time to wrap my head around this one though.”

Screen Shot 2015-12-22 at 10.26.04 AM

Another love…allowing our next learning be driven by student inquiry. 

During the first week of school, we worked on the Four Fours: Use four 4s to make the numbers 1-20.  They loved this, but were having trouble finding expressions equivalent to some numbers so I introduced to them factorials. Minds blown!

A couple of weeks ago, we were working on a visual pattern Screen Shot 2015-12-21 at 7.06.26 PM from http://www.visualpatterns.org (amazing amazing blog of patterns that Fawn Nguyen has compiled) and one student said,

“I wonder if we could use a factorial. I see 3, 2, 1 and then 1, 2, 3 with the 1 used twice.”

Another student responded, “No, it is 3+2+1 not 3 x 2 x 1 so that won’t work.”

“Oh, right”

“Ms. S, is there something like a factorial but for adding numbers in a row?”

Oh boy, I thought. “Yes, there is. We will definitely explore that. But, we need to do that another time.”

Well, our other time came during the week of math.  And, NRich came to the rescue with two great tasks to explore (Mystic Circles and Picturing Triangle Numbers ) and then an article the read.

We noticed and wondered for Mystic Circles, noticing the adding of sequential numbers. Screen Shot 2015-12-22 at 10.28.12 AM.pngand then we looked for relationships in Picturing Triangle Numbers

Screen Shot 2015-12-22 at 10.34.25 AM

After, we went back to look at the two strategies for solving the Mystic Rose Screen Shot 2015-12-22 at 10.47.53 AM

and students developed arguments for each strategy.

What I wish I had done better at….

Preparing students for having their parents in the class.

While we as a class have developed a classroom culture where mistakes are important and our goal is to learn and not just perform, that does not mean they have that culture with their parents. I had one child breakdown due to wanting to “perform” for his parent. He has never broken down in class, but I see this desire to “perform” and the pressure (self imposed?) really got to him.  This was a good wake up for me, as while we might be on the way to creating this culture, it is going to take a lot more work to bring it into every home.

I also wish more parents could have joined us. I know there was some fear, as one parent shared, “6th grade math is intimidating”. I just wish more could have come to see how open it really is. I need to figure out how to share this with more parents.

Connections, Connections, Connections…

We didn’t do a bad job on this. The connections we made throughout the week were incredible. We were connecting ratios to graphical representations and then questioning/wondering and analyzing if the visual patterns we have been working on are proportional. Do all graphs of equal ratios create a line through the origin?  We were connecting our work with visual patterns, to the work we did with triangle numbers.  We were connecting and comparing the many different strategies students used when solving Sugar Packets (which I adapted and probably made a bit too hard. Though, the students persevered and definitely engaged in some productive struggle.) and the Leaky Faucet. We were connecting the representations to equations to words. We connected and compared our base ten system and place value system to the Babylonian’s base 60 system and the Egyptians base ten, but lack of place value.We were extending ideas and connecting to the math practices.  We were deepening our learning by creating arguments for a claim being Sometimes, Always or Never.

Screen Shot 2015-12-22 at 11.01.58 AM

But, it was mostly oral.  I wish I had recorded the connections so that they were more explicit, and had the children record the connections they were making. I wonder if when we come back to class, we can look back on the various problems and explorations we did and make those connections explicit. They were explicit in my mind when I chose each problem, but I wonder how explicit they are in every students’ mind. I think this will make the week even more powerful than it was.

Mytakeaway

 How can I create this type of week in my class every week?  Can I extend math on some days to allow for more processing/thinking time? Can I extend our time to write, so we aren’t constantly starting over? Can I really help focus our learning more on one topic, so we can go deeper and allow connections to be made? What do I need to add to make all the connections and thinking even more visible so that students can transfer to other parts of their lives?

It sure is a good thing I have 2 weeks to process this and develop a plan. Time to reflect is good. I will  make it happen.

 

 

My Love for Students’ Understanding of Math Grows Deeper and Deeper

I am so grateful that I get to begin each day with math! Every day starts with playing! It is incredible.

We explore and talk every morning while making connections, looking for patterns and engaging in mathematical arguments in the form of a Number Talk; a Mathematical Conversation from Intentional Talk; a Sometimes, Always, Never discussion; a True or False Reasoning activity; choral counting; or estimation activity. My biggest problem is keeping track of time.

This past week, it was time to explore ratios. I looked through our CPM Book, the 6th grade Standards, Jason Zimba’s coherence map, and didn’t feel like the Text Book was going to meet our needs exactly the way it was set up.  So, I started to think about what my goals were and what my students need to understand.

Throughout parent conferences, I kept hearing myself speak to parents about the importance of connecting mathematical concepts. Too many students were seeing math concepts as isolated and I really want them to see mathematical relationships and see all the connections.   I realized that because some hadn’t been making these connections for years, my most important job is to create opportunities and engage them in conversation so they can make these connections.

I really wanted students to understand the language of ratios, understand the different types of ratios: part to part and part to whole, and see ratios as a relationship while connecting what they knew about fractions.

I remembered a Math Assessment Project Lesson that I thought might be perfect.

  1. It allowed for multiple entry points, allowing all students to have access to it.
  2. It would allow us to make connections to their understanding of fractions, but also allow students that need to develop more understanding with fractions that opportunity as well.
  3. It would allow us to investigate what a ratio is and come up with various notations and the language of ratios without being directly told them.
  4. From this task, I would learn what my students know and where we would go next.

As I mentioned earlier, we start each day my playing with numbers. That morning we had started with an Always, Sometimes, Never for these three statements: 1/4 = 25%, 7/12>8/14, 3<4x. We had recently been comparing fractions to percents, I wanted them to make note of when comparing fractions to use a benchmark number such as 1/2, and since we have also been working with variables, I wanted them to think about what would happen if you used a fraction versus a whole number in place of the variable.  I loved that we got on the topic of the importance of the whole. “Ms. S, 1/4 = 25% only if they are both out of the same whole. 1/4 of 8 is not equal to 25% of 4.” Love!

I wanted to give them something that they might be able to connect to  and the comparison of the two fractions was perfect for that.

For the task of Fizzy Orange Drink, Screen Shot 2015-12-06 at 2.08.00 PM.pngwe first started out with some Noticing and Wondering. This I have come to believe is one of the best habits my students can develop. It creates engagement and as Jo Boaler states in Mathematical Mindsets, “the desire to understand it and to think about it.”

I then gave them the information that they had requested and we were off.Screen Shot 2015-12-06 at 2.08.27 PM

We came up with these three possibilities of most orangey to least orangey. Then students justified and explained, caught mistakes, and connected to the second problem in the warm up.”Remember in the warm up we said 7/12 was 1/12 away from 1/2 and 8/14 was 1/14 away from 1/2 and because there were the same number of parts we can compare them easily. We can do the same thing here.” Yes! The student’s noticing helped other students who naturally weren’t making the connections. We agreed on the Middle row, comparing the fractions or part to whole ratios using a common numerator or unit rate.Screen Shot 2015-12-06 at 2.20.38 PM

It was then time to stop, so for HW they were asked to respond on our classroom forum and reflect about the day’s work.Screen Shot 2015-12-06 at 2.24.18 PM.png

The next day they got into the big task of sorting and matching. But first they made some Noticings and Wonderings. They noticed that some cards were related, asked about the blank cards and decided themselves what they thought the task should be. It was awesome!Screen Shot 2015-12-06 at 2.08.50 PM

I told them, as they sort and order, to be thinking about what they think a ratio might be.IMG_1778

IMG_1781

Then it was time to Troubleshoot and Revise. So for HW, the students had to analyze two strategies that had mistakes. Then the next day in class  we held a Trouble Shoot and Revise discussion.  This was so important because we were able to surface some misconceptions and then students had an opportunity to revise their sorts.

Finally, it was time to share and compare strategies. One or two people from each group was to stay with their poster and explain their process and reasoning. The other members walked around and were to ask questions and critique their peers’ reasoning.  We then discussed mistakes and corrections, analyzed the dictionary definition of a ratio and related it to what we had just done and created a list of notations and language of ratios.

IMG_1800They then had a follow up task of mixing drinks but with 3 ingredients. My job now is to analyze their strategies and depending on how they did, we will look at my 3 favorite mistakes and Trouble Shoot and Revise or Compare and Connect Strategies.  So what comes after that? Good question.

That is what I am trying to figure out now.  In our CPM book, they are enlarging and reducing pictures and shapes. That is really a 7th grade standard, but focusing on it as scaling up or down it could be a great connection to what they were doing in 5th grade. So, I think I will have them engage in that, but with the focus of connecting scaling up or down to the work we just did. I like that it is a completely different context and think that could lead into some great conversations and big understandings.

The other thing I want to do is introduce some representations: double number lines, ratio tables, and graphs to help them see the multiplicative reasoning. I’m thinking of possibly showing them a problem solved with the three representations to see if they can connect each representation and determine how they work instead of me directly teaching. This will help place more of the cognitive load on them and require them to really engage actively.

Some are also still struggling with the notion that a fraction is division. So, pulling in some equal sharing problems with proportional reasoning is also going to be helpful. I just need to determine which I do first. Ideas?

Their minds were firing all week and they are starting to make some good sense of the math. I think we have a good start. There is a curiosity and a desire to want to make sense of it all now.

845e77e0e22e91eff5b90cf9dfe5416c

 

Goal Setting, Thinking, and Understanding

It has been some time since I last posted, but that does not mean there has not been a shortage of things to think about. While preparing for conferences, my inner conflict about grades versus feedback was in full force. The tension was great.

rope-tensionI had to assign a numeric score to report cards, but it often felt conflicting to our classroom focus on learning. Then came conferences and instead of always just focusing on students’ thinking, learning, how they were improving, and focusing on the new goals they created, I could feel our conversation more focused on the evaluation of skills.  It is really fascinating how difficult it is to change our culture of school from performance to true learning.

One of the things that helped us stay on track though, was that each child reflected on their learning over the trimester and we then created learning contracts. We focused our goals on reading, writing, math, and Listening/Speaking and our Norms. Each student filled out a Learning Contract Template.

Screen Shot 2015-12-06 at 10.20.13 AM.png

This was definitely not easy for all students to fill out, but the process of doing it was extremely valuable.

Some examples…Screen Shot 2015-12-06 at 10.31.20 AM.png

Student BScreen Shot 2015-12-06 at 10.28.12 AM.png

Student CScreen Shot 2015-12-06 at 10.27.15 AM.png

Student DScreen Shot 2015-12-06 at 10.29.03 AM.png It was clear who were independent learners and had internalized how to learn and which students still need support in thinking routines that can help them to learn and understand.

I am trying to figure out how to help them understand what actions they can take to meet their goals. That was really hard for a lot of students, which is really the most important component. What can I do to understand and meet my goal? It will be interesting when we revisit them, to see if they can develop any more specific actions to help them. I think I need to really make their thinking explicit and visible as they are doing it. I need to take the time to allow them to reflect and process.

Then came Thanksgiving break, I read Jo Boaler’s new book, Mathematical Mindsets: Unleashing Students’ Potential through Creative Math, Inspiring Messages and Innovative Teaching, and it was such a great time for me to read it.

51nfg6tuwgl-_sx399_bo1204203200_

It was very similar to her previous books, her articles and website, but no matter what, the message is so important and great to keep repeating.  This quote really hit me.

“Successful math users have an approach to math, as well as mathematical understanding, that sets them apart from less successful users. They approach math with the desire to understand it and to think about it, and with the confidence that they can make sense of it. Successful math users search for patterns and relationships and think about connections”

Excerpt From: Boaler, Jo. “Mathematical Mindsets.” iBooks.

As I mentioned, during my conferences I was getting pulled into talking with parents about math as discrete skills students needed, yet my goal is really the quote above. My goal is for all students to approach math and all subjects with the “desire to understand it”.  This is not an easy task. For some, the change is happening, yet for others they have spent 6 years in school passively doing school and to change that mindset has proven to be much more difficult than I was hoping. But I am not giving up.  See the next post.

This past week, I finally made time for the students to reflect on their learning  3 days.. I ran out of time not the others. I am so bad with time!!!! My goal is to get to 5, but 3 was a good start.

Each day, they were to reflect on our Norms by posting to our class LMS: Schoology. Screen Shot 2015-12-06 at 10.48.53 AM.png(adapted from Jo Boaler’s Math Norms…I added in what each might look like in different subjects as well since the students are in a multi-subject classroom.)

17316244565_c220ecebaf_z

What is great, is that students are able to read each other’s reflections and their thinking is visible for all to learn from. Below are some of their reflections.

“I liked how ______ explained himself in math and that made me think a lot harder and able to figure out the problem.” -Student A

“Today I questioned my teammates work in math and I overlooked it, got into a conversation about what was right, and then figured out the right answer. I appreciate _____ for really trying hard in math and helping me and _____ find the right answer.” -Student B

“Today I thought about if it was always, sometimes, or never. On one question I said it was sometimes and it was always, so I learned from that and got the next one right. ______ made it make more sense when he proved his answer and that helped my.” -Student C

“Today during read aloud, I asked questions about words that I wasn’t familiar with in our story and learned lots of new things about Egypt.” -Student D

“I made a connection from the example paragraph to my Mesopotamia paragraph that helped me understand how to transition to different details.” -Student E

“The trait I engaged in was being perseverant. I was being perseverant because I kept going on finding a way to put a key word in my questions.”- Student F

“…But now I learned from my mistake and my brain just grew. : )” -Student G

I really think being consistent (consistency is so hard for me) with this reflection time might help some of my more dependent learners become more independent.

And then I started to read Ron Ritchart’s book, Creating Cultures of Thinking: The 8 Forces We Must Master to Truly Transform Our Schools.

ritchhart-cultures-239x300Wow!, I have so much to process and discuss with others about this one. Where do I start?

Ahhh, thinking…the culture of thinking. It really is the only way that students are going to succeed in life and I feel like it is my most important job to facilitate learning opportunities for them to engage in thinking.

More to come… reflecting on engaging students in the whole subject, thinking like a disciplinarian, and making connections.

 

Choral Counting: So Much More than Counting

When you hear the word “counting” your mind quickly goes to primary age students. You see the kindergarten students counting, the 1st grade students counting, but what about 6th graders?

Our school is trying to target students’ needs by having tw,o 30 minute target times (RTI), to help move students forward.  My Tier 3 group needs a lot of work with number sense and they had been asking me to work with fractions, so I decided to do some Choral Counting with them. (The next day I did choral counting with my whole class. I love things that all students learn something from.) I thought it would be a great way for them to work with fractions, while also engaging students in SMP 3.

To be fair, we have done a lot of work this year with patterns: number patterns with the 100s chart, Pascal’s triangle, the multiplication chart, visual patterns and growth, and structural patterns in equations. Actually, we look for patterns everywhere…in the books we are reading, social studies, science, and across disciplines…we are becoming pattern seekers. This was completely evident during a choral counting lesson that I did with fractions.  Patterns and relationships are so important to learning.

We started off…” We are going to count by 2/3 starting at 0.” I had planned how I would record the pattern and thought about what relationships the students might notice: patterns in a column, the denominator remains the same, the numerator is changing by 2 (I wanted to start easy, to build success and ease their working memory so that they could really focus on the relationships.) We started counting. (I was nervous that this was going to be too easy for them.) Hmmm…they were getting stuck in a few places and then I was reminded how important counting is. We are working on fractions, but at the same time we are reviewing multiplication facts, skip counting, and number sense in general. We then had 3 rows completed and I stopped. “What do you notice?”

“We are skip counting by 2.”

“Turn to your math partner. Do you agree or disagree with that statement?”

We then discussed this as a class. Many students agreed and then we got into the discussion of the numerator and denominator. Whoa! Misconceptions have surfaced!

Then a student stated, “We are not counting by 2s, we are counting by 2/3.”

“No, we are adding by 2s. See,” and a student points to the numbers in the numerator.

“If we are counting by 2s, then we should be able to add 2 to the previous number to get the next number. That doesn’t work.”

Ahhh…we have an argument going on. Great! So, I ask the students to sit with their partner and prove or disprove that we are adding by 2.

Screen Shot 2015-10-24 at 3.20.15 PM

After some drawings and equations the students come to the realization that we were not counting by 2, but we were counting by 2/3. Though, they all agreed that the numerator is changing by 2.

Then a student said, “I noticed that if you look diagonally down toward the right, the numbers are increasing by 14/3 .” I asked “Is that true for every diagonal?” We tested it out? Yes. “But, why does that work I asked?” Again the students turned to their partners and discussed.

“Well, we are counting by 2/3 and so from 2/3 to 16/3 we keep adding 2/3. That adds up to 14/3.

I write what she said… 2/3+ 2/3 + 2/3 +2/3 + 2/3 + 2/3 + 2/3= 14/3

“Actually if you just multiplied the fraction by the number of times that is much faster,” a student shouts out. Ahhh, yes now we were connecting their previous knowledge about addition and multiplication with whole numbers to fractions.

IMG_1726

We continue to look for patterns and notice the relationship down the column, moving diagonally from right to left. We talked about adding and multiplying fractions and then also subtracting fractions if we moved backwards.

Then I placed some boxes in some areas below the 3 rows and asked students to determine which numbers would go into those boxes? They used all the patterns we discussed earlier to determine which number fit and critiqued each other’s reasoning and pointed out more efficient ways.

What I love about choral counting is that it surfaces some misconceptions, allows us to discuss operations, vocabulary, and relationships between numbers. It is not threatening. Every child feels successful and we can really work on justifying their thinking and critiquing the reasoning of others.

Lastly, it is so important how I record students thinking. I have been thinking about this a lot and color has become my best friend. The best statement at the end of our counting session as we looked at the finished board.

“We all know what we were doing, but if someone walked in right now, they would have no idea what all those circles and lines were. But, it looks so cool.”

It looks cool to me too because it reminds me of the rich discussion we just had.

Making connections and looking for patterns. It is so much more than fractions and counting. It is learning how to make sense of the concepts we are exploring.