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?


Thursday, April 5, 2018

Fraction Division

I tried some new things with fraction division this year.  I want to document for it myself as well as those who ask me about it. 

I started this unit by making claims.  I wrote about that in this blog post
For the fraction division I used Cuisenaire rods:

Each student has a set of each of the 10 different rods.  Students were instructed not to shout out answers, but to set the “answer” rods aside and keep quiet so everyone has think time.  Allow time for students to figure out each task.  Ask for students to answer and ask several students how they know.

  • Orange:  “This is 1. Find ½.”  
  • Dark Green:  “This is ⅔. Find 1.”  
  • Purple:  “This is ⅖. Find ⅗.”
  • Brown: “This is ⅘. Find ⅗.”  

To explore reciprocal relationships that are needed for division:

Held up and showed students the red rod.  
  • “Now think back to the red. Let’s pretend it’s a granola bar. This is ⅓ of a granola bar. Find 1 whole.”  “How many times does ⅓ of a granola bar fit into 1 granola bar?”  “What equation can you write to describe this situation?”  Students may come up with several that are correct but ask for a division equation if they are not getting it. 
  • Purple: “This is ⅖ of a granola bar.” Find 1 whole.” How many times does ⅖ of a bar fit into the whole bar?”  
When we divided with the Cuisenaire rods it looked something like this:
I have tried to divide using number lines before and it was a hot mess.  Using the rods we were able to understand division as division (as opposed to multiplying by the reciprocal) and used a measurement model for fractions.  

I also pulled some questions out of my Developing Fraction Knowledge Book.  I love this book and highly recommend it.  
Here is an example:
  • A landscape architect bought 5 bags of soil, knowing that each flower bed in the garden she was designing required 3/5 of a bag of soil.  How many flowerbeds can she fill with soil?
Students made different sketches to figure out the answer, but all of them focused on dividing.  Some drew flowerbeds and had 3/5 in each.  Others drew 5 bags and divided them into fifths and colored 3 at a time for each flowerbed.  
After a few questions like this we revisited this question:
  • This time determine the number of flowerbeds the architect can make with 1 bag of soil.  Show your reasoning.
This brought us back to the idea of the reciprocal relationship and viewing it as a unit rate.  Once we knew the unit rate we could just multiply by 5 because we had 5 bag.  

What surprised me about the reciprocal relationship is that I thought it was just what you multiply by to get one whole.  In working through this activity we are able to see it is also a unit rate.  We can see the relationship between multiplication and division.  I do not feel like understand all of this enough to get all students understanding it yet.  It did help in understanding why we multiply by the reciprocal when we connect it to the unit rate. 

We also explored division with a common denominator.  Fawn Nguyen has a great blog post about this.  She used graph paper and rectangles.  Like the rods, it helps students understand fraction division as it relates to whole number division.  These 2 strategies helped deepen my students' fraction sense around division.  They were now beginning to understand why the number would get smaller or larger than the starting number.  

Christopher Danielson also has some great posts about fraction division:
Partitive fraction division - I used some of the visuals from this blog and had my students see if they could explain the "students" thinking.  
Common numerator fraction division  - I challenged my students to see if a common numerator strategy was possible.  Most of them came back to say that they were able to figure out a strategy for common numerators but were not sure why it worked.  It was a great one to explore together.  



Monday, November 20, 2017

Mathematicians...A Follow Up

Last week I had parent teacher conferences on Tuesday and Wednesday.  On Monday I realized that I hadn't had students complete any sort of end of quarter reflection like I have in the past.  This year, after reading Tracy Zager's book Becoming the Math Teacher You Wish You'd Had, I have been helping students focus on what we do as mathematicians.  I decided to take the posters I made (blog found here) and turn them into our reflection sheet. 
Here is what I came up with:

I had students mark which areas they felt were strengths and which ones had room for growth.  There are many bullet points for each characteristic and many students wanted to just put marks down the middle for all.  I made them choose and offered room for them to explain their thinking.  I only gave them 20 minutes so many students were not able to explain all of them.  They were really being thoughtful about their choices and I appreciated that.  To be honest, all of my students have strengths and room to grow in each of these areas.  I was more concerned about their perception.  

I absolutely loved reading the students' responses.  What I loved more was sharing these reflections with parents.  My conferences had a different feel and the conversations were more meaningful.  Here are some examples of responses and conversations that surrounded them:

 Many students interpret "Ask Question" as asking when they don't understand.  I want students to tap into their natural curiosity.  Noticing, Wondering, and asking Why does that happen? are characteristics of lifelong learners.  I want this for my students.  (Insert plug for Table Talk Math Here - flyer for conferences)

 Yea! The vertical non-permanent surfaces are a success!  We often use the whiteboards to show our proofs.  That way everyone can see our work when we are explaining our reasoning and can compare strategies. 

Many students think they are bad at math because they don't get it instantly.  They want the teacher to explain it step by step so they can follow those steps.  That is not how I teach math because I don't believe that is math.  Math is what happens in the confusion.  That is where students are working to make sense of numbers and relationships and looking for patterns.  Of course we need to come out of this fog at some point but some days it is just to move to a different fog of confusion. 

 We do a lot of sense making in class.  Often times the models that parents don't understand (because we didn't learn with them) are not necessarily how we will solve the problem, but instead they show us the number relationships and deepen our understanding.  This helps engage our intuition when working so we can see if our process and answers make sense.

Some students make errors because they don't understand the concept.  The fix? Help them understand the concept.  Then there are students who make errors even though they understand the concept.  What is the fix?  Focus on precision.  A benefit to learning multiple strategies is that we can go back and solve another way to see if we get the same answer, or use a model to check our thinking and engage our intuition (see above).

Another strategy for checking over your work is to go back and tell yourself it is wrong.  It is one thing to look over work to see if there is a mistake.  We are much more analytical if we go back and look at the work as if there is a mistake. 


These two make me so happy.  Both students recognize that they need to grow in these areas, but these responses show me that understand their depth of understanding is important.  They can no longer have just surface level knowledge (memorizing rules).  We are moving to variables and more abstract math.  We need to understand these number relationships so we can apply them to problems where the numbers are not as concrete. 


Most students marked that they needed to improve their ability to prove and reason.  At conferences I often get a peek into the past.  "Last year they had to show their work in a specific way and would lose points if they didn't have it exactly as they should." "He always was marked down in his ability to explain."  I do think that different teachers have different expectations in terms of work and what constitutes as an explanation and that is a vertical discussion that districts should have.  Here is what I tried to focus on at conferences:
  • Showing work is part of what mathematicians do to prove their answer.  They need to show their thought process.  It also help us understand mistakes.  It is hard to reflect on an error if we cannot see where it happened.  
  • Explanations cannot just have the steps taken: first I multiplied then I added.  The explanation is a proof.  It must explain why those are the steps that make sense for this problem.
  • How do I help students to improve their explanations?
    • Models: Some students need models to figure out the answer.  Because models show the number relationships they can show what operations make sense for the problem.  
    • Vocabulary: A good explanation is not determined by its length.  We focus on vocabulary so we can be concise and clear in our explanations.  Many students understand the relationships in their head but struggle to put it into words because they don't have the words.  
    • Skeptic:  Your explanation is persuasive writing.  It has to convince someone who does not agree with you.  You aren't just explaining to your friend.  You said you multiply and you are convincing someone who thinks you should divide.  In class we share our reasoning and help each other find holes in our reasoning.




I did pull out some samples of student work at the conferences.  After going through the reflection sheet, we were now looking at these characteristics of mathematicians in their work.  It was a different conversation.  I was in my nerdy math zone but still able to explain things to parents in ways that they were able to connect with and understand.  

As always please feel free to continue the conversation in the comment section below. 
If you are interested in my other  Becoming the Math Teacher You Wish You'd Had blog posts you can find a complete list here.


Saturday, October 7, 2017

Making Claims - Play to Student Strengths

Any of my former students can tell you, I do not like it when students quote math rules to me.  If we are discussing strategies and a student simply tells me the steps to solve, there will be follow up questions.  I do not accept "because that's how you do this type of problem" or "because a teacher told me to" as an explanation of a strategy.

I still feel this way about math rules, however, reading Tracy Zager's book, Becoming the Math Teacher You Wish You'd Had, gave me a new perspective on how I handle these situations in my classroom.  Mathematicians look for patterns.  Mathematicians make generalizations to extend what they know to other problems.  The problem isn't the math rules that my students know, it is that they overgeneralize (use them when it is not appropriate) or have lost the mathematical reasoning that led to the generalization.

Tracy suggests making a "Claims Wall".  As students start to make these generalizations add them to the claims wall and continually revisit them to be sure they continue to work in other contexts and with other number sets.  A colleague recently told me, we need to play to their strengths.  If rules are what students know, then that is our entry point into the conversation.

With this in mind, I started my review of fraction multiplication and division by having students make claims.  I gave each team a post it to write their claims and made it clear they did not need to explain them at this time.  Then we looked at everyone's claims and made a combined list.  Here is what we had:



Most of these claims came from the post its.  If you look at the multiplying fractions there is a claim in blue.  This one came up after we wrote about fractions less than one.  One boy started making a claim and I knew it was going to be good so I started writing as he was talking.  "You can't multiply with mixed numbers." He paused, thinking he was done, then added the word "easily".  Then he backtracked and added "because I guess you could".  At this point he realized that I was writing what he was saying.  He adjusted his final claim by saying "You should use fractions greater than 1".  I love that the entire thought process is written on our claims poster because now we can analyze not only that we can multiply different ways but we are going to need to have conversations about efficiency.
The other claim that was not on a post it from students is the last one on the division poster.  It is one I added.  After we had the claim that you can use multiplication, I put forward the claim that you can use division.  I asked the students if they thought that was true.  They all thought that seemed like a reasonable claim, but with only a minute or so to think about it none of them felt they knew how to use division to divide fractions.  I can't wait for that conversation!

Starting the unit this way was a big risk for me.  I started class knowing we would do claims and had no idea of where it would go from there.  I did not know what I was going to need to have prepared for our exploration of these claims (because I didn't know what claims we would have).  I had a few things on hand  based on claims I thought would come up.  It was a bit of an unsettling feeling.  I decided to analyze the claims in an order I was comfortable with, not the order they are on the sheet. That allowed me to plan our approach and prepare more for certain claims.
In case you don't read beyond this I want you to know this risk was definitely worth the pay off for me.  Because we were starting with what students knew they all had access to the activities.  Then as we explored, they were able to self assess and realize what they didn't know about the topic.  The learning about fractions became authentic (without made up "real-world" problems to explore).

Here is the breakdown of what we did and what I plan to do:

  • Multiply across: I had the students go to the white boards (I have some new non-permanent vertical surfaces in my room).  They needed to come up with mathematical reasoning why this claim works. Many groups drew unit rectangles and used an area model. One group used a coordinate grid (still an area model).  Even though they knew the visual representation, through discussion we realized they didn't know how that proved their claim.  They just knew that they had used it before in math class.  (This is an issue I have with visual representations.  We cannot just teach them to kids or it is just more math to memorize.  Students need to be coming up with authentic representations and making sense of the math.) 
  • Cross canceling: We used this worksheet to analyze why this strategy works.  I have used it in the past and had it prepared in hopes that there would be a claim about this strategy.  This strategy helps as we work with proportions throughout the year and it's a great discussion about the commutative property.  
  • Multiplying mixed numbers: I sent students back to the whiteboards to show how to multiply without converting to fractions greater than 1.  I want to pull some different problems to focus on efficiency with fractions greater than one compared to keeping them mixed numbers. We didn't get a chance to discuss yet, but here are some pics of what they created.  I am very excited for these conversations (Spoiler Alert: when we talk distributive property we will also discuss their love for the order of operations and how it doesn't need to be followed if you have mathematical reasoning.)


  • Multiplying by numbers less than 1: For this we will play the role of skeptic and see if we can come up with counterexamples.  Then try to prove it will work for every case without having to list every problem that exists.  
Division of Fractions:
I will try to add to this blog next week.  Last year I took a course through the US Math Recovery Council that deepened my understanding of fraction division.  We used this book:
I want to revisit the chapter on fraction division to prepare for analyzing our division claims. It is a fabulous book and I highly recommend it for all ages of fraction work.

Here are my thoughts after this class activity.  
In class we give students tasks to allow them to explore a concept and build their mathematical reasoning behind it.  Then we pull together everything that has been discovered and come to these generalizations.  Some students get to these math rules because they work through the mathematical reasoning and it leads them there.  Others understand the mathematical reasoning but do not connect it with the rule that the class has determined.  (I have many students who feel they struggle with math but actually have great math reasoning.)  Others never understood either piece but continue to work to memorize the rules because the class has moved on and they need to be able to solve the problems.  In the end most students know the rules. They do not care about the mathematical reasoning because they can answer the problems.  How do we make them care about the mathematical reasoning when they can successfully solve problems?  Does it matter?  (I would argue yes.)  Everything we do in my class is about mathematical reasoning.  Explain.  Justify.  Prove.  Always, Sometimes, Never. Notice/Wonder. Question.  The focus is never on the answer.  But if the answer is what the student can do, I do not want to minimize that.  For them to want to know the mathematical reasoning I need to ask them to do things that require more than just calculation. If I can get students to continue to question, then they will not be satisfied with simply knowing an answer or a rule.  


If you are interested in my other  Becoming the Math Teacher You Wish You'd Had blog posts you can find a complete list here.