Tuesday, September 3, 2019

Creating an Intervention Course - Planning Activities

This is the third post in a series about creating an intervention course.  The first post can be found here.

The pacing guide discussed in the last blog post, gave me an outline for the first unit.  I have a better understanding of what concepts we want to teach.  This in and of itself feels like a huge improvement from what we have done in the past because we now have some direction.  There are so many great resources and activities.  I could spend a whole month on some of these topics: area and perimeter, fractions, percents.  In looking for activities I have some criteria in mind.  

  1. It must allow the students to play and explore ideas.  
  2. It must elicit conversation so that teachers get a better understanding of student thinking and misconceptions.
  3. It must help create understanding and make connections (not practice skills and procedures that are memorized).
  4. It must have some support for teachers who are facilitating the activities.
The teachers I work with are great.  I am truly blessed to be surrounded by dedicated, passionate math teachers.  When we adapted CPM curriculum 6 years ago we were all in.  It was a very tough year, but we embraced the teamwork, collaboration, and hands on activities.  We have truly transformed our math classrooms.  I do not need to worry about doing any PD around what math class should look like.  Now that we know what it should look like, we also know that it can take a lot of time to plan quality lessons and tasks.  And as much as the teachers understand math at their grade level, we do not always understand what conceptual teaching looks like for grade levels below ours.  I have dedicated a lot of time over the past 4 years learning elementary math so that I can be a better middle school teacher.  The downside of that is that I lost my work / life balance and wore myself too thin trying to do it all.  Because of this I want to find resources that provide that professional development for teachers.  

The 2 main resources I plan to use are Jo Boaler's Mindset Mathematics books and Illustrative Math Open Up Resources.  These two resources have lesson plans for teachers and resources at grade levels below 7th grade.  These are both great resources that allow students to explore the concepts with quality tasks. I don't want to overwhelm teachers with too many resources at this point, so while there are many other quality resources that fit my description, these are the two I intend to use the most.  I will supplement with other resources if I can't find what I am looking for in these to places.  

As I am filling in the pacing chart with activities and learning targets I am also linking to notes below the table.  One of the 5 Practices for Orchestrating Productive Mathematics Discussions is anticipate. I think it was Jon Orr, in one of the first Make Math Moments that Matter Podcast, who said  


he tries to think about the least amount of math a student could use to solve a task.  

So this is what I have been doing.  Not only anticipating the least amount of math, but thinking about what skill sets they need in order to choose a more sophisticated strategy.  This is going to be important for teachers to make note and ask questions to elicit student thinking.  Often times in middle school interventions students know other strategies, they just don't feel confident using them or they have a go to method that they use whether it is efficient or not.  Some students have actually become so efficient at strategies that should not be efficient (like count by ones) that it can be hard to get them to move forward in their thinking.  These are all things that we need to uncover through conversations.    

With the start of the school year, I am doing the anticipating and writing my notes.  The goal is to do this with the teacher team once we get rolling and have time to meet.  Here is an example of some of the notes I've written:


My first meeting with the teachers will be tomorrow.  We need to discuss success criteria and the first few activities.  

I will let you know how it goes.  

Thursday, August 29, 2019

Creating an Intervention Course - Standards and Pacing

This is the second post in a series about creating an intervention course.  The first post can be found here.

Last school year the 7th grade team worked to determine essential standards for the grade.  When it comes to intervening with students we need to have a focus, so we worked to determine which standards were essential in 7th grade.  For this process we started by unpacking the standards and determining what we called "essential understandings".  Basically we were answer the questions, "What do we want students to remember after we teach this standard?"  Turns out it is actually a great question and shifts the focus from what students need to know how to do (procedures) to big ideas and understandings they need to have.  For example:

For example, for the standard 7.NS.A.1.B
Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts.

Some things students need to remember are

  • the definition of “additive inverse” and how it can be used to add integers.
  • The difference between adding +17 and adding -17
  • The significance of a negative quantity and how to represent it.


After unpacking all of the standards we looked at 3 criteria to help us determine which standards are essential:

Now that we have our essential standards determined, our goal for this year is to create what we are calling an Essential Standard Tracker for each chapter.  This is part of our PLC goals for this year.  We are using Tim Kanold's Mathematics PLC at Work resources to help us.  


There was no way to get a good screen shot so you could read the words, but the headings are:
  • Description of Standard (in student friendly terms)
  • Description of Proficiency (this one is an important discussion)
  • Vocab and Pre-requisites
  • When Taught (which is a link to our pacing guide which I will write about next)
  • And then some common formative and summative information


Again, the focus of this document is the essential standards.  We still teach all of the standards but for interventions identifying the vocab and prerequisites for the standards with endurance, leverage, and readiness for the next level is where we want to focus our time.  If we can front load these skills and help strengthen students background knowledge before we reach that standard in class, they are much more likely to engage in work we do in class.  

As part of our PLC process this year we are creating pacing guides.  Honestly, we already had this information in different formats and different places and I questioned the need to copy and paste things into a different format.  What I found however, was that when we put in together in one place we started to notice areas that needed improvement.  For example, some targets were too general and spanned several days.  We relooked at our learning targets and made them more specific to that lesson.  We also added success criteria.  I love that part and am excited to see how it goes in the classrooms.  

Here is what the pacing guide looks like at a glance:

And here is a close up of one of the learning targets:

I really think the success criteria is going to help teachers with the close of the lesson and help students be more reflective on their progress with the concepts.  

After we had the pacing guide for the chapter created I made a pacing guide for the intervention course.  I thought it would be overwhelming, but our interventions are every other day.  So with an 18 day unit I only had 9 days to plan for the intervention work.  Suddenly an overwhelming task seemed much more reasonable.  

The goal for the year is for the 7th grade team to continue creating pacing guides for each chapter.  Then I will use that to create the pacing guides for the intervention course.  In Unit 1 of Math Investigations (the name we have given the course) we will work on chapter 1 concepts and chapter 2 prerequisites.  Then in Unit 2, when the class is on chapter 2, we will work on chapter 2 concepts and chapter 3 prerequisites.  

The tricky part will be to stay ahead on our essential standards trackers and pacing guides so that when I create each intervention unit, we have the next chapter for math class already mapped out in terms of the essential standards.  

In my next post I will discuss my process for selecting activities and planning for work with students.  




Creating An Intervention Course - The Origin Story

I don't blog as often as I'd like, but when I do I love how it helps me process and reflect on the work I am doing.  This is my second year as a math coach / interventionist and I have decided to create an intervention course for our 7th grade.  It will be an ongoing project all school year and I decided I should blog about the process for a few reasons.
1.  I want to remember what I am doing in case I want to try to do something like this again.
2.  I want to share it with others in case it helps you with your work.
3.  A blog is a great place to share ideas and concerns to get feedback from others, and I know I am going to need some help along the way.

This first post is just going to give you some background on why I have decided to create this course and what I hope to accomplish.

Our teachers run intervention groups during study hall time.  They work with small groups every other day.  We have spent years floundering through these interventions.  Not sure who to work with, what to work on with them, or if what we were doing was making any difference.  Because the bulk of our day was spent teaching math classes, the intervention work was always somewhat of an afterthought in the planning process.

One of the struggles we have experienced in my middle school is that the intervention work is often many grade years below what we do in class.  Students might be making some progress in interventions but they would continue to struggle in math class and receive failing grades.  It is difficult to keep a student motivated to do extra math when they are not feeling successful in that class.

Listening to the Making Math Moments that Matter Podcast last year I started to think that maybe we were taking the wrong approach to interventions.  We were selecting activities below grade level to help improve those skills.  But what if we chose grade level tasks.  We could engage students in open tasks that explore grade level concepts, and help build and strengthen the prerequisites along the way.  In the podcast, Kyle and Jon, talk about the importance of anticipating how students will solve a task.  If we could anticipate possible solutions, we could anticipate discussions about those background skills along the way.

Last year was also a big year for our 8th grade math program.  Two of our teachers were able to pilot the CPM Support course with some our our students.  Here is the description given by CPM:

The course is unique in that it focuses on problem solving, building relationships, building student confidence, while also focusing on some key 8th grade standards like ratio and proportion, solving equations, and numeracy. 

I was able to observe this course being taught and it was great.  Our teachers saw great results.  Many students who had been disengaged and overwhelmed in math class were participating in discussions and more willing to try new things.  The teachers were frontloading some concepts and the students were more confident in class when those concepts were discussed.

Because of the success of the support course, I decided I wanted to try to create something similar in 7th grade.  Keeping in mind that CPM has done a lot of research and training to create their course that I will not be able to do, I want to use the same philosophy as CPM's support course and the framework*  laid out by the Making Math Moments that Matter team to create an organized curriculum for 7th grade interventions.

*Framework is not really the right word.  You can follow the link and decide what the right word choice should be. 

And so this year my I am going to attempt to create what we are calling Math Investigations.
Wish me luck!


**I should also note that 2 teachers will be using this intervention course.  There is still another layer of interventions that allow for small group targeted interventions.

Next Post in the Series: Standards and Pacing

Tuesday, February 19, 2019

A Reflection about Universal Instruction

After reading Mark Chubb's recent blog post about interventions, I started reflecting on the learning I have been doing this year around universal instruction.
I am in a newly created middle school math coach / interventionist role this year.  My job description is pretty open ended as my district recognizes the need for support and change in math and wants the role to evolve as we tackle the many issues surrounding math education.
One of my main focuses this year has been universal instruction.  As Mark's blog points out, universal instruction should meet the needs of 80-90% of the students.  Before determining what interventions should be done, it is important to reflect on our effectiveness in universal instruction.

Having been a math teacher this was not a new topic for me.  I have had professional development and spent lots of time with my PLN on Twitter.  Rich tasks with a low-floor and high-ceiling help us make the necessary shifts in math instruction.  (Love this shifts in math self-assessment.)  I have used 3-Act tasks, Robert Kaplinsky's problem based tasks, notice/wonder, and many of the resources identified in this document from Margie Pearse.  I quickly realized, however, that it is one thing to fumble through these in my own classroom, taking risks (thanks to Classroom Chef for the encouragement), and improving my craft through experimentation.  It is completely different to be in a coach role and try to help teachers navigate this process and encourage them to take risks.

Enter Jon Orr and Kyle Pearce and their amazing project, Making Math Moments that Matter.  They have created videos, an online workshop, online communities on social media, and my favorite, a podcast.  It was early on in the podcast series, when they were discussing the idea of low-floor/high-ceiling tasks that I had an aha moment.  My understanding of these tasks was that they have an entry point for all students.  All students can work through the task at their level.  I don't remember which episode and I don't remember who said it, but Jon or Kyle made a comment about the purpose of the low-floor in the task is to hook the students.  Now they are engaged and more willing to go further into the math because it isn't about the math for them anymore.  They are curious and want to explore.  This comment has been lingering in my thoughts.  I realized that my understanding of low-floor/high ceiling-tasks had students working at different levels.  They all get to the answer but some students might use lower level math to get there.  The statement in the podcast made me realize that this was not the case.  Going back to Mark's blog, he mentions access and equity.  While students are not tracked into different classes, my interpretation of the purpose of the low-floor was a type of in-classroom tracking.  I had a conversation earlier this year about the purpose of using different strategies.  Is the purpose of discussing different strategies a means to hopefully get each student to have at least one strategy that works for them so they can find the answer?  Or is the purpose of discussing different strategies to help students think flexibly and to deepen their understanding of a concept?

Not that I ever thought it was ok to have students stuck in some type of lower level math land, but the comment got me thinking about how we can help students with what are often considered intervention skills during these tasks.  Pam Harris has a blog post about the development of mathematical reasoning.
Non-classroom interventions at my school tend to focus on the first 3 ovals.  Often times in class we give students "access" to higher level math by giving them algorithms and calculators. The interventionist part of my role has me thinking about how we can use the low-floor of these tasks to help with student growth in these areas of mathematical reasoning.  

In a Twitter discussion about strategies versus algorithms and the visual above, Pam provided this question. 
"Where is the opportunity to build students' relationships?"And it ties back to the low-floor/high-ceiling task.  Students enter the task at a place that is comfortable for them.  They are hooked.  Now we can have discussions that move students thinking forward.  We can show them the connection between what they know and the higher level math we are working on.  One of the big struggles that teachers voice to me about interventions with students is that they don't see the impact of it in class.  Interventions work on lower level skills and the students are still struggling in grade level math even with extra math time build into their day.  I believe that there is so much potential here to bridge the disconnect between intervention work and grade level math.  But how?  What does that look like?  

It definitely seems like something that is easier said than done. I would say there have been moments of this in my teaching, but how can I make this the norm instead of a happy accident?  I've been mulling this over since my aha moment about low-floor/high-ceiling tasks.  As if they were reading my mind, episode 12 of the Making Math Moments that Matter podcast answered my question.
The short answer: Anticipate
The long answer: Listen to the Podcast 😉

The highlights and my big take-aways: 
The task is selected based on a specific learning goal, but we can be intentional about making connections to other areas of the curriculum. 
Prior to task, anticipate strategies and student thinking.  *I would anticipate strategies to highlight, but I don't think I ever took the time to anticipate as thoroughly as Kyle and Jon discuss. * 
Consider background knowledge.  Think about the different entry points and the levels of mathematical reasoning.  Identify if the task needs to be modified so that there is an entry point for all students.
Allow students to see their strategy through.  Value their work, stretch their thinking, and elevate the new learning during consolidation.  
Plan an extension and challenge students to use the new learning in a related task/problem. 
Visuals and tools students are familiar with, such as a double number line, can help students connect to new concepts and ideas.
If we give students time to explore and experiment with tasks they will help us to see the connections between big ideas that we may not have planned or even seen ourselves.  

The big ideas are a web.  I know this.  As I teach, my brain seems to shift into linear thinking and I again need the reminder.  The big ideas are a web.  Perhaps its the name low-floor/high-ceiling that gives the illusion of a linear progression.  Perhaps it's because the students go through standards by grade level that we see student thinking as "below grade level".  The big ideas are a web.  

Questions to continue asking myself:
How well do I know that web?
What connections do I see?
What visuals and tools have students used that could be used to connect to the ideas in the task?
How might this be solved with the least amount of formal math?
Where is the opportunity to build relationships so strategies become natural outcomes? 

Other notes:
It seems like it would be beneficial to have teachers discuss the web together.  We use CPM and it does a beautiful job of connecting concepts.  Each year I taught I noticed more.  What connections did other teachers notice that I did not? 
When it comes to tools, visuals, strategies, etc.  I feel I need more vertical conversations.  I do not know what students used in previous years.  I also am not always sure when/how to introduce them and how to get students to use them flexibly.  If it is a another students strategy, how do I help other students connect with it without it becoming a procedure (ie. multiplication on a number line, area model for multiplying fractions).  

I think it is worth noting that if the idea of creating and implementing these tasks every day in you classroom feels overwhelming, Episode 8 of the Making Math Moments podcast is worth a listen.  It is a mentoring moment with Katrien Vance and she shares her realization that these tasks don't need to be elaborate, time-consuming tasks.  This can be accomplished with a simple problem as well.  Be sure to check it out. 


Friday, February 15, 2019

Critical Math for the ACT


I recently went to a workshop on Critical Math for the ACT at the Wisconsin Math Institute.
I wasn’t really sure what to expect.  I had been told that there is a lot of middle school
math on the ACT, so it seemed like I should know more about it.  I was really quite
surprised by the information and it got me thinking, not only about the foundation for
the ACT, but about acceleration practices.  This blog is really a place for me to try to
process all that I learned.


I did not know that the ACT has created its own College and Career ready standards.  

ACT College and Career Ready Standards to view online for most up to date version

PDF version Feb2019 - I like the layout of this one better but could become out of date



For a benchmark score of 22, most of the standards align with 6th, 7th, and 8th grade math.  

There are still many up to a score of 27 and some for scores higher.  

So it turns out the critical math for the ACT is middle school math. Middle school math is foundational.  Not just learning it, but retaining it.  That is an issue of focus, coherence, and rigor, but that is a different blog.



The ACT® Test User Handbook for Educators Online Link, PDF lays out the
Content Covered by the Mathematics Test (p. 46)
Preparing for Higher Mathematics (57–60%)
This category covers the more recent mathematics that students are learning, starting when they began using algebra as a general way of expressing and solving equations.
Integrating Essential Skills (40–43%)
This category focuses on measuring how well you can synthesize and apply your understandings and skills to solve more complex problems. The questions ask you to address concepts such as rates and percentages; proportional relationships; area, surface area, and volume; average and median; and expressing numbers in different ways. Solve non-routine
problems that involve combining skills in chains of steps; applying skills in varied contexts; understanding connections; and demonstrating fluency

This means that approximately 40% of the test requires the application of concepts learned prior to 8th grade.  


Achieve the Core did independent research found here. They found:

In mathematics, fewer than half of items on the assessment were judged to be aligned to the claimed Common Core mathematical content standards for high school.

Clearly there is far more value in middle level math than most people realize. We are not just laying the foundation for higher level math, we are the core. (If someone has a better word choice to offer I would appreciate it.  I struggled to come up with the right words.) The ACT can be a gatekeeper to higher education. In middle school, with preadolescent hormones running rampant, many students don’t see the point of school.  They figure they can pull it together when it is important, you know, high school. I think all the above information speaks to why middle school math is important and why teaching it in a way the promotes conceptual understanding and sense making is key.  


But what about the high students?  The students who skip a year of math or work at
an accelerated pace?  When students are accelerated they have less time exploring
these core concepts and therefore do not have as deep an understanding as they could.
 And what is the trade off? Supposedly they get to high math in high school. Depending
on the options in high school this may or may not be the case.  It is quite possible for a
student to take Algebra freshman year and with the right plan, get to AP Calculus senior
year. It seems to be the mindset of parents and students that the higher you get in
math the better.  I’m not sure if that is true.
MAA/NCTM Position

Although calculus can play an important role in secondary school, the ultimate goal of the K–12 mathematics curriculum should not be to get students into and through a course in calculus by twelfth grade but to have established the mathematical foundation that will enable students to pursue whatever course of study interests them when they get to college. The college curriculum should offer students an experience that is new and engaging, broadening their understanding of the world of mathematics while strengthening their mastery of tools that they will need if they choose to pursue a mathematically intensive discipline.

Digging deeper in the background information on the statement it states:

...the pump that is pushing more students into more advanced mathematics ever earlier is not just ineffective: It is counter-productive. Too many students are moving too fast through preliminary courses so that they can get calculus onto their high school transcripts. The result is that even if they are able to pass high school calculus, they have established an inadequate foundation on which to build the mathematical knowledge required for a STEM career. Nothing demonstrates this more eloquently than the fact that from the high school class of 1992, one-third of those who took calculus in high school then enrolled in precalculus when they got to college, 8 and from the high school class of 2004, one in six of those who passed calculus in high school then took remedial mathematics in college.



Within our schools, there is tremendous pressure to fill these classes, accelerating every student who might conceivably be ready for calculus by the senior year regardless of whether such a student might benefit from a slower and more thorough introduction to the traditional topics of high school mathematics.



It makes me wonder if we are we taking time to consider the individual? Do we provide opportunities for parents to discuss this so they can help make the decision for their child?  I’m not talking about the parents who are obsessively asking that their child be accelerated. I feel confident that those conversations are happening, but what about the parents who simply put their child in an accelerated course because the student met the school’s requirements.  As someone who dropped out of a middle school accelerated program my mother never wanted me in, I feel strongly that there are so many factors the school does not know. And the research around retaking and remediation shows that an experience like mine can alter career plans. We need to reach out to parents to help equip them to better make the decision with their child.  



Should we support acceleration? This question, like many questions in mathematics education, does not have a binary answer. The answer is “it depends.” Sometimes acceleration is appropriate and sometimes it isn’t. What does the answer depend on? Here the answer is clearer: it depends on the student’s demonstrated significant depth of understanding of all the content that would be skipped. If a student demonstrates significant depth of understanding of some but not all the content that would be skipped, then this is more appropriately an opportunity for enrichment rather than acceleration.

There is evidence that students who speed through content without developing depth of understanding are the very ones who tend to drop out of mathematics when they have the chance (Boaler, 2016).  Acceleration potentially decreases student access to STEM careers if it results in students dropping mathematics as quickly as possible, rather than cultivating and developing the joy of doing and understanding mathematics. This is important to point out to parents, as dropping out of mathematics is clearly not an outcome parents want to encourage.



I honestly could have copied and pasted the whole article.  It is fabulous and you should click the link and read the whole thing.  

My goals coming out of this workshop:

  • Create a parent communication night for students who selected for our compacted Math ⅞ course to help them better understand their options.  
  • Analyze our districts data to see which content students are not retaining and evaluate our instructional strategies to see if there are shifts we need to make to better build conceptual understanding so students retain and can apply these skills.  
  • Have vertical conversations with the high school.  The middle school teachers cannot be solely responsible for these standards that students are expected to apply on a high stakes test their junior year of high school.  We need to look at when skills are introduced and when we expect proficiency. ACT provides this helpful tool that I am hoping will focus these discussions.  


I’d love to hear about work you are doing in your district around this topic or if you have articles or links to share.  Please use the comment section below.


Thursday, September 27, 2018

Units Coordination and Proportional Reasoning

This morning I was in a 7th grade math classroom.  The class was reviewing for a test and working their way through problems at different stations.  I sat down to talk to a group that was working on this problem:

When shopping at Target, Sam found pizzas priced at 4 for $14.00.  He was having a party with 16 friends to watch the MLB playoffs. 

  • If he wants to make sure to have half a pizza for each person, how many pizzas should he buy?
  • How much money does each person need to contribute to cover the cost of the pizza?


As we discussed the strategies students were using to solve this problem, I started to notice levels of units coordination embedded in the problem and how the strategies students chose were related to how many levels they were able to coordinate.  (If you have not yet read my blog posts explaining units coordination you can find them here.)

Here is a visual I created to show the levels of units involved in this problem.
There actually are 4 levels of units as well if you take each pizza in the lower left visual and slice them in half.

My discussion with the first group was so enlightening that I continued to discuss this problem with other groups as they rotated through the stations.  I found that for many students they could easily determine that we need 8 pizzas.  One pizza feeds 2 people, so 8 pizzas feed 16 people.  Most students relied on proportional reasoning, thinking about it as doubling.  They focused on whole numbers and multiplying rather than fractions and dividing.   

The second question is what caught my attention.  Most of the strategies students used helped them find the price per pizza.  Once they had that, they divided by 2.  There was even one student who worked from 4 pizzas for $14, divided by 2 to know 2 pizzas is $7, and divided by 2 again to find 1 pizza is $3.50.  I realized that many of these students were working through one or two levels of units at a time instead of being able to jump from $28 for 8 directly to $1.75 for 1/2.  

I think this is an important distinction to make when discussing units coordination.  All of the students were able to get the correct answer.  All of their strategies relied on their mathematical reasoning and number sense.  From the teacher perspective, however, it is important to notice the sophistication and efficiency of their strategy.  We need to strive to understand where students are in their ability to coordinate units so that we can continue to move them forward with this type of thinking.  Otherwise as the math continues to get more complex and students begin to struggle, we might not understand the root cause of the confusion.

The other interesting thing I discovered in my discussions is that while most students did not divide by 16 to find the price per 1/2 piece, they checked their answer by multiplying by 16.  This was a great reminder to me that units coordination is reversible.  Perhaps if students had been given the price per piece they could have more easily worked their way to the cost for a group.  (I'm a little tempted create some questions and draw a visual for the levels of units I could work into a problem like that, but I am short on time.  Perhaps another day.)

Here are some questions I want to reflect on.  I hope you do too.
  • Does units coordination play a role in our next unit? 
  • What questions can we plan that will allow us to see how students are working with and coordinating units?
  • Do the questions I have planned ask my students to coordinate units both forwards and backwards?
  • What supports can we put in place within the lesson for students who struggle to coordinate units in order to move them forward in their thinking?




Tuesday, May 1, 2018

An Abstract Understanding of Concrete Things

Recently Kyle Pearce blogged about concreteness fading.  It got me thinking about a session at the Math Recovery Council annual conference with Dr. Robert Wright on the topic of settings and distancing the setting.  I am constantly reflecting on the best ways to move my students forward in their math thinking and it seems to me that this is an area in which I need to be more purposeful.  I decided to read a little more, ask a few more questions, and this blog is my reflection on all that I am thinking and learning about these topics, focusing more specifically on the role of manipulatives in my classroom.

The concept of concreteness fading as described both in Kyle Pearce's blog and in this article by the Learning Scientists is about the progression of moving students from concrete to representational to abstract.  The idea of this makes sense to me.  I feel that too often I introduce a manipulative, like integer tiles, for example, only to take it away too soon.  I allow the students to use them, but if it isn't specifically in the directions many students struggle through without them thinking they should be able to work abstractly.  This is where many kids will latch on to rules and memorization in order to work abstractly with something they don't fully understand.

What stood out to me in Kyle's blog was this quote, "We must understand that concrete manipulatives are still more abstract than using the actual items in the quantity being measured."  Just the other day I was working with a student and using cubes to represent something.  It made perfect sense to me and I realized as we were working that this student did not connect the cube with the item from the problem.  I need to keep in mind that it is not just a 3 part continuum from concrete to abstract, but there is a continuum within the concrete category.  There is so much to be purposeful and thoughtful about when selecting manipulatives and concrete examples.  I don't think I do enough to modify and adapt the manipulative choice if a student is not able to connect with it.  Plus Kyle's blog really helped me think about choosing manipulatives with the end result in mind.  Where do I want the students and what manipulative/tools get them there?  Perhaps it will be different if I am focused on adding mixed numbers with a distributive strategy versus a fraction great than one strategy.  I love the vertical number line for proportional relationships that Kyle shows to then connect to the table that shows the numbers vertically.  It's a simple change that makes a big difference! (It was both an aha moment and a face slap when I saw it. How did I not think of this?)

I also started to think about this process in reverse.  By 7th grade, many of my students have memorized rules or tricks and can solve problems abstractly.  The issue is that they lack the depth of understanding.  They can work abstractly but don't understand the number relationships enough to show what is happening concretely.  They are unable to mathematically justify what they are doing and it affects their ability to transfer their understanding to new problems.  This video of Ruth Beatty (that Mark Chubb shared with me) speaks to this.


In the constructivism definition that is given in the video, the word concrete is being applied to the students understanding.  It is not just about the concrete example or having something concrete to manipulate.  This video changed my thinking about the goal I am trying to reach with students.   I need to reverse the concrete to abstract model and work with what my students already know to help them understand more concretely.  Sometimes as a middle school teacher it can feel like I have to start from the beginning with these concepts, but I need to keep in mind that the starting point for my students is their abstract understanding.  

So how does this model of concreteness fading compare with what I know about distancing settings?  This paragraph is taken from the book Teaching Number in the Classroom.  Distancing the setting is part of progressive mathematization (which is the progression of learning and thinking in terms of mathematical sophistication).  


I am not sure if there really is a difference between concreteness fading and distancing the setting, however, the definition laid out here seems to include more.  When planning a setting, it isn't just thinking about the concrete representation, but it's selecting what verbal language and number range are appropriate, deciding how to move students through the concrete section of the continuum while interweaving the representational and abstract.  I know I have talked about concreteness fading as a continuum, but it is also cyclical so students are not just moving from concrete to abstract and done.  After giving it much thought I feel that distancing the setting is something that is done within concreteness fading.  There is one student who comes to mind with all of this.  We have worked to build strategies and his multiplicative reasoning and yet unless directed his default is to count on his fingers.  It is by purposefully distancing the setting that I can help him progress to naturally select more sophisticated strategies and thinking. 

In thinking about the examples Kyle laid out, I feel that those are the first steps in lesson planning.  I need to map out the models and progression through them.  Once that is done I need to be thinking about how to distance the setting.  Screening and flashing are great strategies.  Showing students the concrete and then covering it/part of it or showing it only for a brief moment.  This will challenge students to start creating that mental picture.  They don't have to create a mental picture themselves yet.  It is given to them, then covered, and they continue the problem with that picture in their head.  

Since I feel that distancing the setting is something that fits inside of concreteness fading, I wanted to go to Kyle's middle school example in his blog and consider ways to distance the setting as I move students from concrete to representational.  

The original picture prompt is really a screened image.  I can see part of the problem but 4 of the boxes are not pictured.  
My thoughts about distancing the setting for this example are all over the place and I have typed and retyped many times.  So many options and thoughts running through my head.  I am going to try bullet points:
  • For some students, this number range will be too high.  They will need enough cubes for all 7 boxes.   
  • Some students may need boxes to put the cubes in.  Connecting them together may not be concrete enough.
    • If a student has boxes and cubes inside of them (or cheerios to look like donuts), one way to distance the setting is to have cubes in the box and then close the box.  This could also be recreated by using cardstock that covers the cubes, acting as a box. (which I would totally print the donut logo onto)
    • Closed boxes could be added, one at a time for students who need to think additively and the through discussion and activity we could get to some multiplicative reasoning. 
    • Using boxes that show the outer donut (one row and one column) would be a way to distance the setting to help students use additive or multiplicative reasoning over counting strategies.
  • As we move to the number line in Kyle's visual example, I am loving this number line picture from Sara VanDerWerf's NCTM presentation.  I am now thinking about how to add donut visuals to the number line and table to help students connect the concrete ideas to the representations.

There is a lot to think about with just this one example.  It is a little overwhelming to think about doing this every day in my classroom.  I think, however, that once I get into the habit of thinking about these things (and dedicating time to think and reflect on it)  the ideas will flow and I will see how ideas for one concept transfers to other concepts.  It will be a progression and continual goal in my teaching.  The main questions floating through my head right now are this:

  • Do I really offer my students enough strategies to understand concretely?  
  • Is my teaching limited by my limited knowledge of "tools"?  
  • How do I build my concrete understanding so I can help my students?