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Just me explaining what I did with a few samples building toward our goal

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Kids and Toddlers Education Cartoon Videos Learn Colors

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Rubber Duck Wooden Toys for Kids Children Toddlers

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Best Learning Toys for Kids Baby Learn Colors

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Surprise Eggs Wooden Ball Hammer Educational Toys

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Link to Explain Everything Files to access virtual manipulatives

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Virtual Manipulatives - Instructional Support Services - Lake Elsinore Unified School District

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Lake Elsinore Unified School District serves grades K-12 and is located in Lake Elsinore, CA.

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K-5 Games

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planning and progressions slides

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Inside Mathematics

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5 Practices for Orchestrating Productive Mathematics Discussions

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The post title is a book by Margaret S. Smith and Mary Kay Stein. It's a wonderful little book that can make a big impact in a math classroom.

I'm lucky to be part of a team of presenters - there are four of us: Chris is a math professor at UCSB, Maria is a high school math teacher, Jeff is a district math coach, and then there's me.

This past summer we started an Institute called "UCSB Mathematics Project - California Common Core State Standards (CaCCSS)." We have about 30 participants, most are schoolteachers, who attended a one-week workshop in late July and will continue to gather again for three days during the school year.

While the Institute's focus is on the new Common Core Standards in Mathematics, we also present and discuss leadership and equity issues, we engage teachers in doing math activities that model the new standards, and together we examine the 5 Practices for Orchestrating Productive Mathematics Discussions chapter by chapter. I was assigned to cover the Introduction, Chapter 1, Chapter 5, and Chapter 7. (Hey, there are only 8 chapters, how did I get so many?)

The book persuades and guides teachers to lead more thoughtful and productive discussions during a math activity. It outlines the necessary steps:

1. Find a math problem that is high-level with multiple strategies.

2. Launch the task to students by telling them what tools are available and what the nature of the task products are to be expected from students.

3. Discuss and summarize by using these "5 Practices":

1. Anticipate - the teacher must do the problem ahead of time and anticipate the different strategies and solutions that students may come up with.

2. Monitor - the teacher needs to pay close attention when students are working in groups, listen to their mathematical thinking and observe their strategies.

3. Select - the teacher needs to select which groups or which member(s) of a group will share with the whole class.

4. Sequence - the teacher must arrange the order in which the selected people will share. (No fun if the group with the "best" strategy shared first!)

5. Connecting - the teacher is responsible for asking students to connect the solutions presented by the different groups and what the key mathematical ideas are in the problem.

I like to practice what I preach, so I've been using the 5 Practices whenever appropriate. My algebra kids were working on a problem involving systems of inequalities. I monitored closely to learn the following:

Luis' group was talking about so-and-so-dude-on-YouTube,

Bella's group was on task but they were too focused on one strategy that may not take them to the solution they needed,

Miranda's group was drawing graphs, but everyone focused on the same graph instead of branching out to get more done,

Martin's group smiled at me and covered up their empty papers,

Eliana's group had the first part completed correctly but had a hard time going further,

and Dean's group was... "Dean, what are you doing over there?... Did you ask me if you could get out of your seat?... Nobody cares that your back is hurting right now, Dean!"

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5 Practices for Orchestrating Productive Mathematics Discussion

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The post title is a book by Margaret S. Smith and Mary Kay Stein. It's a wonderful little book that can make a big impact in a math classroom. I'm lucky to be part of a team of presenters - there are four of us: Chris is a math professor at UCSB, Maria is a high school math teacher, Jeff is a district math coach, and then there's me.

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Illustrative Mathematics

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Providing instructional and assessment tasks, lesson plans, and other resources for teachers, assessment writers, and curriculum developers since 2011.

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blog about differences in pedagogy

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My school district has been piloting math materials for the past two years. Now that the second year is coming to a close, tensions are rising between teachers. Differences in pedagogy are bubbling to the surface. Amongst other things, it is very apparent that inquiry-based instruction is on one side and direct instruction is on the other. It's a battle of good versus evil, the Dark Side versus The Rebel Alliance, or is it?

Which side is evil and which side is good? Is evil really evil, or is it misunderstood? After all, even Darth Vader had the best interest of those he loved in mind when he went dark. Is one side entirely correct? Is there only room for one style of teaching? Every day? For every lesson? For every In the instructional strategies portion of the California Math Framework it states that direct instruction, "... has proved to be effective for teaching part of every the lesson? Direct instruction is described as, "...the use of straightforward, explicit teaching techniques, usually to teach a specific skill. It is a teacher-directed method, meaning that the teacher stands in front of a classroom and presents the information." information and basic skills during whole-class instruction." This has me wondering, what does it mean by information? How does it define basic skills? (It's important to note that the framework was not suggesting which strategies are best to use exactly, it is just a list explaining 7 instructional models.) Is direct instruction necessarily better to learn those things? Reminds me of Van de Walle's advice on, "What to tell, and what not to tell." Telling things like conventions or what is needed to gain access to the math, but not over scaffolding and removing the problem solving for students.

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Good vs. Evil: Pedagogy in the Classroom

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My school district has been piloting math materials for the past two years. Now that the second year is coming to a close, tensions are rising between teachers. Differences in pedagogy are bubbling to the surface. Amongst other things, it is very apparent that inquiry-based instruction is on one side and direct instruction is on the other.

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introducing fractions, double number lines, proportional reasoning & algebra

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Okay, so I knew this "Math Fight" was coming. (Thank you, John Stevens) The standard I was going after was 1.G.3

Although, this is the performance standard (a very simple standard at first glance) that students need to be able to do, my mathematical goal was different. Further down (like about 3 pages later) in the CA math framework, I find this beauty of a mathematical goal: (Mathematical goal meaning - What is it about math you want students to understand?)

"Students need experiences with different-sized circles and rectangles to recognize that when they cut something into two equal pieces, each piece will equal one half of its original whole (1.G.3). Children should recognize that the halves of two different wholes are not necessarily the same size. They should also reason that decomposing equal shares into more equal shares results in smaller equal shares." Van de Walle also chimes in about early fraction work and said, "Given children's experiences with fairly sharing items among family and friends, sharing tasks are a good place to begin the development of fractions. The sharing tasks allow children to develop concepts of fractions from an activity that makes sense to them, rather than having the structure imposed on them." (Teaching Student-Centered Mathematics K-2)

So I wondered, "What can I do to create a little controversy?" Then I had an idea. Yes, it concerns cookies... Older grade teachers don't check out yet, this lesson is not limited to the idea of parts of a whole. I began double planning teacher questions and some anticipated responses.

Fast forward to today's lesson... I am wearing a leopard apron.

Kids kept asking me why I was wearing an apron. I told them I had been up all night baking cookies, which was true. So, I called two students up to the front of the room to be my helpers. I told the kids I had two halves of cookies. I asked them if it would be fair if I gave them each half of a cookie. The cookies are hidden in a paper bag. The class all shouts out, "Yes!" So I give half of a small cookie to Amore and half of a huge cookie to Shawn.
The uproar was on the border of out of control, but it's in the name of math, so... we kept going. This argument begged a student created definition of what half is and what half is not.

Don't worry, this is only the launch of the lesson. I know from listening to upper grade teachers and reading Van de Walle that students have a difficult time understanding fractions as numbers and not only parts of a whole. So I stole an idea that Andrew Stadel shared and used it in a different way. (BTW - this is an example of how elementary, even primary, can steal ideas from middle and high school teachers!) I put zero and one on a string number line and asked the kids where half would go. They partner talked their disagreements, but in the end it came out that half should be equidistant between zero and one.
I chose to record the language of half in words. See the last sentence of the pic below.

Still not done.
Time for a little estimation...
I held up one small cookie and one extra large cookie and asked the kids to choose a just right brave estimate for the question, "How many small cookies equal a large cookie?" Students wrote their estimations on their desk with a marker.

Partner talk...
Then we laid out small cookies on the large cookie to see how many it would take. It turned out to be 10 so we added 2 - 10 to the existing number line.

Then I strung up another number line directly above it and hung zero up above the other zero. This number line represents the large cookies.

"Boys and Girls, if the yellow number line represents the small cookies and the pink represents the large cookies, where will the one go on the large cookie number line?"

Kids whispered to each other. Brave ones called out, "Above the 10!"

Kids, "Because it takes ten cookies to equal one big cookie."

Hmm "Well, if that's true then where would one half of a large cookie go on the number line?"

Kids, "Above the 5! Half way between the zero and the ten!"

Kids, "Because 5 is half of ten. 5+5 is 10."

Push: "If half goes here, where would half of half go?"
Kids disagreed on this one. Some thought between half and one, but after some kids convincing each other, and a demonstration of breaking half into half they came to consensus that half of half is half way between zero and one half. Kids created some examples and non examples of half and divided half into half again.
Tomorrow students will reflect on what they know so far about halves and wholes in their math journals.
The next goal will be: "They should also reason that decomposing equal shares into more equal shares results in smaller equal shares."
Thinking - What is the pattern and/or relationship between the number of parts and the size of the parts. We will continue to work on equal sharing tasks as Van de Walle suggested.

I'm wondering how shallow my student's understanding would be if I didn't read the whole framework, and just glanced at the standard.
This lesson planted seeds for fractional reasoning, proportional reasoning and algebraic understanding. I'm sure I'll continue to chip away at this lesson, but I think we are off to a great start.

P.S. Of course, the kids had to have cookies!

Thanks Matt Vaudry and John Stevens for the fun inspiration!

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First Grade Math Fight... Fractions, Proportional Reasoning, and Algebra, Oh my!

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Okay, so I knew this "Math Fight" was coming. (Thank you, John Stevens) The standard I was going after was 1.G.3 Although, this is the performance standard (a very simple standard at first glance) that students need to be able to do, my mathematical goal was different.

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A First Grade Teacher Enters The Math World

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A First Grade Teacher Enters The Math World

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quick blog about a transition into teaching math meaningfully

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My name is Jamie Duncan and I am a first grade teacher in southern California.

I have been teaching for about 14 years, all of which have been spent in kindergarten through second grade. The last 8 years have been in first grade. I've been reading some tweets about us elementary folk being left out of the math loop, and it is true for many, so I thought I would weigh in and share my journey.

As most primary teachers, I have spent most of my efforts on language arts. I think many would agree that the importance of literacy ＞ math in the early years of school. It makes sense, right? Kids need to be able to read to be able to do any other subject area? It made sense to me at the time so that is why I put all my effort into being the best reading and writing teacher that I could be. This worked out well for me as I didn't have a "key to the country club." That is how Dan Meyer described students or people who "get math." I was locked out of the country club. Nobody gave me a point of access. Don't get me wrong, I did my best to teach math as well. I did what they taught me in school: I do, we do, you do and used manipulatives (all the good teachers used those) that I chose for my students because I thought they made sense or that's what the curriculum company told me I should use. Most kids got the answer this way, right? They must understand it, right? Then common core came along, and along with it came performance tasks. My kids could get the multiple choice items correct and some of the constructed response, but they bombed the performance task portion of the assessments. What the heck was I going to do? Oh don't worry... I can reteach it! While I'm teaching new standards as well, no problem... I don't have to spell that out for you do I? Please say no.

Sometime after Christmas break during a staff meeting our principal introduces Ryan Dent to the staff. He is saying all kinds of crazy stuff like I do, we do, you do is backwards! Seriously? How are my little first graders supposed to come in and teach each other math they haven't even learned yet? He wants teachers to talk as little as possible and the kids to do nothing but talk?! Classroom management nightmare, right? This guy is nuts... but he is really passionate... Then something weird happens, I go to a training at the county office and someone else (who I know is not crazy) says the same thing. I guess I better learn a little bit more about it.

Later in the spring I attend my first Standards of Mathematical Practice training offered by Ryan and Kristian Quiocho at my district office. I don't know how else to explain the training other than it was like a giant epiphany. A huge Aha! Now it all makes sense. I see it now. It will take getting used to, but I get it.

After the training I start trying out some of what I learned. Success after success. I guess these guys are on to something. Later in the year they announce that our district is going to write our own math modules, but they want teachers from each grade level to come together and write tasks. I figure, this is new for me, but I know I will learn a lot so I do it with another first grade teacher and we work ALL SUMMER, and by all summer I mean from the day school gets out until we went back. We both felt like we had gone to college for the summer.

When school began I decided there was no better way than to dive in. I gave students tasks, I watched them struggle (NOT RESCUING THEM WAS EXCRUCIATING), I used samples of their own work, students explained and critiqued the work of others, they made connections, they talked to each other about math, they wrote about math in their math journals. I decided to use the same assessment I had used the year before to compare data. This is what happened:

They must have gotten lucky, but then this happened...

Seriously? I went to college. I got my bachelor's and master's degrees, and took more classes after that and my kids are better teachers than I am??? My students learn better when I hardly say anything at all??? How are my kids able to figure out how to do this math without me telling them how to do it? Then it hit me. It was that thing Ryan and Kristian shared at our SMP training only maybe I wasn't quite ready to fully understand it yet... It was problem solving. My kids can problem solve. What is problem solving? Simply put, problem solving is what you do when you don't know what to do.

What was I doing while the kids are running the show? I give them a task, I listen for common misconceptions, I choose work samples to fuel discussions,

and I do my best to ask probing questions that focus on conceptual understanding. (It's easier to funnel them into the answer.) As Phil Daro would say, "What is the math they need to learn here?" I worked with my co-module writer to train other first grade teachers in our district on using the modules. It was exciting to see other teachers open up and try teaching math this way. It was even more exciting to read about their success in emails.

After all of this I begged my district to go to NCTM's national conference in Boston. For some reason, they said yes. My district sent 3 of us to represent our district. I was representing K-2, Ryan was 3-8, and Chris was 9-12. (Both Ryan and Chris's mathematical intelligence intimidates me, but I have made peace with that and I choose to look at it this way. I am the lucky one, because I get to grow the most.) I couldn't believe it! I was so shocked... and thankful... and humbled... and a little scared... Teachers don't often (next to never) get to go to conferences out of state! I truly wanted to learn as much as I could and then use what I learned to help better any teacher willing to listen. While I was there I was given the opportunity to learn from the very best. I went to sessions by Phil Daro , Jason Zimba, William McCallum, Dan Meyer (I took a pic with him! There was a line! I'm not kidding. Math nerds unite at NCTM),

Marilyn Burns, Sherry Parrish, Ruth Parker, Karen Karpe, and Maria Blanton. Not only were the presenters amazing to learn from, but I got to meet other professionals like Steve Wyborney who is just amazing. All in all, it was phenomenal. I wish I could share every last detail of the training with people, but it's exactly how Dan Meyer describes math class in the beginning of Math Class Needs a Makeover. (see video link above) Our math team (Ryan, Chris and myself) took back the message that we had received at NCTM and reported to the district and used that learning to plan future professional development for our teachers. I am soooooo looking forward to these!

I couldn't be more pleased with the growth my students made last year. They are my math superheros. They taught me so much! This year my group of students is very different. My room is now the inclusive classroom in first grade and it's more homogeneous, at least at the moment. My math goal as of right now is to focus on meeting these kids where they are right now. At NCTM Jason Zimba said, "Don't let the standards be handcuffs for the children placed in front of you." So I'm not. I'm also setting a goal to really work on my questioning technique with a strong emphasis on conceptual understanding.

As far as Literacy ＞ mathematics, I can't say I agree anymore. Yes, students need to be able to read to do just about everything else, but they also have to use reasoning skills. They need to be able to choose tools to solve problems in and outside of a mathematical context. They need to be able to analyze and critique the reasoning of others to help make decisions for themselves. In the real world, people are not going to spoon feed them and give them everything they need. They are going to have to problem solve. Why not start cultivating this when kids are little?

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A First Grade Teacher Enters the Math World

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My name is Jamie Duncan and I am a first grade teacher in southern California. I have been teaching for about 14 years, all of which have been spent in kindergarten through second grade. The last 8 years have been in first grade.

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Common core standards have brought with it many new expectations for teachers and students. Most of the new expectations aren't due to the content standards themselves, but the standards of mathematical practice.

This is where most, if not nearly all, of the publisher's curriculum gets it wrong. Most of the tasks they give do not ask students to make sense of problems and persevere in solving them nor attend to precision. There is far too much suggestion about how a student should solve a problem and there is no room for students to work things out.

Take a look at the first grade problem above. It is giving the students all of the information and even telling them which tool would be best to use. Of course, it makes sense to me to to use the 100 chart, but what if it doesn't makes sense to the student? What if they still require concrete objects to physically manipulate or move around. What happens when we tell students to use certain tools or strategies that makes sense to us? Is that student centered teaching? Does that promote conceptual understanding? Will it give a solid foundation for procedural fluency to be built upon?

Take a look at Bella's work below. How might her thinking been different had I just given her the curriculum pages and gone through it the way the curriculum company says to?

When we did this problem in class I only showed the first three lines of the problem using my document camera. Students were asked to use tools of their choice to solve. Take a look at some of these representations from students.

So, what is a teacher to do? Considering the long hours many teachers put in, it can be all too tempting to just pull out the next math lesson and let the curriculum do all of the thinking for you. I imagine this is a bigger problem in elementary where we have to plan all subject areas. The curriculum of course, does not allow for flexibility of your students' needs and rates of learning, it is likely over scaffolded removing the opportunity for everything you really value in a math lesson.

Of course, you don't have to use the curriculum. Teachers are the key to students' learning, not the curriculum. Teachers can search online and find some great resources or create their own.

Instead of using the publisher's curriculum to plan out a unit, we need to ask ourselves how can we use resources like the California Math Framework and supplementary materials like Van de Walle's Student Centered Mathematics and Fosnot's Landscapes for learning to build a sequence of rich tasks that promote reasoning and problem-solving, while maintaining the focus and coherence of the mathematics and standards themselves, then looking to the curriculum for any holes that exist in that sequence. For the moment, my district is using Envision, and it's weak. Most of the lesson is procedural. It follows the old faithful, "I do, we do, you do" format. It usually does have a question or two that can be made into good tasks. The "Do you understand?" and "Problem Solving" problems are usually a good place to look.

Identify a standard:
1.NBT.1-2
Take a look at this 1st grade lesson.

What jumps out at you? Is there anything worthwhile here?
How about the, "Do you understand?"

I could see that being a good "launch" or starting point of this lesson.

How about #10?

What else can you do to it to strengthen it?
What might happen if you just showed this first?

Notice and Wonder
I had the pleasure of attending Annie Fetter's session at CMC-South this past weekend and I can't say enough about taking the time to notice and wonder. I had seen a video of Annie before, but it was awesome to see her in person and really listen to her ideas about notice and wonder. She said, "Noticing changes the dynamic of racing to finish. You have to do some sense making before you make your first steps in problem solving."
What would happen if you showed that to your students and asked, "What do you notice? What do you wonder?" then, you charted everything students shared during this part of the lesson and put their name next to their contributions?

What you're hoping for is that someone in your class would wonder how many people can ride in the buses. If they don't give you that question, can you fake it and say you overheard someone say it, or someone from you class last year wondered that question? How would using students questions for investigations promote a positive disposition toward math? Of course, you already know the question before you begin, although your students may surprise you and come up with more interesting and engaging questions.

(If you don't have a visual to "notice and wonder" about you can use a word problem. @bstockus shared the idea of " numberless word problems" with me and I think it is genius! Truly! It promotes a great deal of reasoning so that by the time the students get the numbers, figuring out what to do with them is so much easier. I am so excited to try this when we get to compare problems!)

Estimate
This is a good time to ask students to make an estimate. Van de Walle said, "An estimate component adds interest, makes the activity more problem based, and contributes to number sense. Listening to children's estimates is a useful assessment opportunity that tells you a lot about children's concepts of numbers."

Now that you have a question to solve, you can ask your students to solve it (still without the information provided on #10). Students will quickly become unsettled. You may see them look a little anxious or others may simply yell out things like, "I can't do it. I don't know how many people fit on each bus."

This is a really important moment in the lesson. Students have attempted to solve a problem without the necessary information which has now created a need for that information.

In my room I say, "What's wrong? Why can't you figure this out?"
Kids will respond telling me they need more information, and they're usually pretty great at telling me exactly what they need.

At this point I would give them the needed information.

I scribble out or cover any suggestive ideas from the curriculum. At this point I still do not give students the consumable page. I'm just projecting it. I find that the curriculum doesn't give students adequate space to work out their thinking. Even students who would normally express their thinking in two or three different and creative ways will limit themselves to the worksheet if it's given to them.

Anticipate what might students make of this problem? What misconceptions may arise?

Have students think/pair/share what tools or strategies might help you solve a problem like this.
Then, let them go. Encourage them to use a tool or strategy that makes sense to them to solve the problem. While they're working, you are monitoring. Listening for common misconceptions to bring those to the light to help progress the thinking of the class.

Select a few samples to share with the class. Project one and ask a different student (not the author) to explain it to the class, further encouraging students to critique the reasoning of others.

Sequence the work in order from the lowest level thinking to the grade level way of thinking or beyond.

Connect - encourage students to make connections between the different representations and the context of the problem. Constantly going back and forth, showing as many at the same time as you can. What is the same? What is different? How is this one like that one? How does that relate to the problem? Why did they choose these numbers? Did you go back into the context of the problem? Does it make sense? How do you know? Will it work every time?
What is the role of the teacher here? How about the students? Who is doing the teaching? Who is doing all of the talking?

Here are some pictures of student work in my classroom where students are encouraged to use tools that make sense to them to solve tasks.

After the CONNECT, which is the most important piece of the lesson, I like to ask students, "Which tool(s) make the most sense to you? Which one is the most efficient? Which strategy will work every time? This is a great time for math journals. There are a lot of different directions you can go in math journals, but one idea is to ask students to solve the same problem with a different tool/strategy and show it in their math journal. I would ask, "Who saw a different tool/strategy that they would like to try?"
In doing this, students are deepening their own connections between their thinking and another way of thinking. As the teacher, I am looking at their math journals to see if they chose a tool that is showing a progression in their thinking. Are they moving from concrete to representational, representational to abstract? Did they correct a misconception they previously demonstrated?

Here is some of my data comparing when I let the curriculum do the thinking for me (blue) and when I took over (red).
3=meets standard, 2 approaches, 1 far below
Please note, not all topics are there because in the 2nd trimester of the year I used modules that were written by teachers in my district (myself included) instead of the district given curriculum.

The problem of over scaffolded math tasks that remove opportunities for problem solving isn't limited to Envision. Here is another example from another publisher.

I wonder how differently the lesson might have gone if the teacher had only shown this to start?

I truly don't understand how so many publishing companies can do such a poor job. Creating tasks to help students become mathematically proficient is their job, that's it.

They don't have duty, IEP meetings, SST meetings, staff meetings, professional development, conferences, they aren't training student teachers, or new teachers working through Induction, they don't plan all subject areas, they aren't tired from teaching all day and then having to recreate their math lessons. They write tasks, and they aren't good enough... yet. (That yet, is me being hopeful.)

I hope this post helps someone somewhere who is feeling like they can do better, but aren't sure where to start. I was that teacher about 2 years ago and I'm very grateful to all the people who have helped guide me to where I am now.

The skeleton for this lesson and most of my lessons is based on the Five Practices for Orchestrating Productive Mathematical Discussions.

The "Notice and Wonder" component was learned from Annie Fetter. You can find her on Twitter at: @MFAnnie

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Common core standards have brought with it many new expectations for teachers and students. Most of the new expectations aren't due to the content standards themselves, but the standards of mathematical practice. This is where most, if not nearly all, of the publisher's curriculum gets it wrong.

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Scroll down and choose a problem type.

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This is the very best to understand 3 acts, but it's about an hour long.

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A talk I gave in Durham, ON, in 2012, asking why students hate word problems and what we can do about it.

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Math Class Needs A Makeover

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By Graham Fletcher

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The number one resource

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Interventions

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How to plan interventions & instruction

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Fluency Game

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Fluency Game

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Working to help teachers think outside the traditional way of doing, and teaching, mathematics.

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Fluency Game

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Fluency Game

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Number Talks, Estimation Tasks, WODB

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Great tasks you can do in 10-15 minutes every day...click the image and they're yours.