Thursday, November 12, 2015 / 13 comments

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:
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!)

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.

The picture on the top left is my favorite! haha

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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  1. Awesome! This is exactly the message we've conveying to teachers in my district since we adopted our new digital textbook last year. Our district has never advocated following a particular textbook for the very reasons you mentioned, but because we also had new standards last year, many teachers felt drawn to using the textbook. Unfortunately they felt pushed away from it just as quickly because they didn't feel like it did a good enough job. I love how you show that that's pretty much to be expected, but that doesn't mean we need to abandon these materials just yet! There is gold to be found there if you dig, if you pay attention to what your students know and need to know, and if you use what you already know about good teaching. Thanks so much for putting this together and sharing it!

    1. You're welcome! I think it's pretty awesome that your teachers take a close look at the curriculum though! Many just tale it and run with it.

  2. Hi, I teach secondary math and was also at the "Notice and Wonder" session at CMC South 15. It was a great session with a lot of take aways. Love this post.

    1. Thank you! I really enjoyed Annie's session too! It made the conference for me!

  3. So many good points. Thanks for sharing. I am pushing Noticing and Wondering with my teachers as well.

  4. Thank you! It makes a world of difference!

  5. Jamie, I shared this post with my daughter's teacher today and she is SO EXCITED! It's great for her to have examples from the grade and curriculum she teaches. Thank you so much for compiling it!

    1. I'm so glad I could help someone! Makes my heart happy! I know it's nice to have samples from your exact grade. We don't get a lot of that in math. We usually have to break everything down to our level.

  6. Hello! This post was recommended for MTBoS 2015: a collection of people's favorite blog posts of the year. We would like to publish an edited volume of the posts and use the money raised toward a scholarship for TMC. Please let us know by responding via email to tina.cardone1@gmail.com whether or not you grant us permission to include your post. Thank you, Tina and Lani.

  7. I emailed you to give permission. :)

  8. I happened upon your blog and I am in love with what you have to say. I am a Math Instructional Coach and am wondering how to get your ideas out to our staff. You have a wonderful way of writing what you do that is easy to read and understand. We have transitioning to a new series that is to be utilized as a "resource" and your post is an excellent example of how we can begin this process. Thank you for sharing your experiences!

    1. Ahh Thank you so much! I love learning about math and doing it with my kids, but writing about it has been the hardest part! I need to get writing again...


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