Why do you keep saying Number Sense?

Whenever someone asks me how to impact mathematics instruction, I say Number Sense. Yet, some may not be truly sure what I am referring to.  Is she talking about a program, standards, pedagogy, professional development?  What I am talking about is plugging the hole with a finger.

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In the “The Boy Who Saved the Netherlands,” the lockkeeper’s son saved the town by placing his finger in a leak in the dike when the hole was small.

That is what I mean when I say Number Sense. I am talking about plugging the hole when then hole is small.  In K – 2nd grade, the hole is smaller than it will ever be for a student.  That is when we need to “plug it.”  This is the moment to make a difference.  Can we do it later, sure… but understand the hole will be bigger. It will take more fingers.  If we want to plug the hole when it is small, then Number Sense is how we do it.

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So, what is this Number Sense I am talking about?  First, it is helping teachers to develop it for themselves.  Number Sense is a thing that if I as an educator do not possess it, then I cannot help students to have it.  I know this because I did NOT have Number Sense as a secondary math teacher. I had no Number Sense.  I, as a math educator (ALL levels), have to have it so I can make good instructional decisions for students so they can develop it.  It is about giving them experiences that allows their brains to grow and to develop it.  It is not a worksheet or a computer program.

It is how do I make sense of quantity in the context of the world around me.  It is can I look at 3 fingers and know that is 3 without counting?  You just tried it. Now, try it again. Yep, still 3.  Now, what about 5?  Did you know 5? How?  What about 3 fingers and 4 fingers?  What is the maximum number of things that you know the quantity of without counting?

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So that is step one of Number Sense and some of our students cannot do that.  That is when the hole is small.  They just didn’t get as many of the experiences they needed to have their brain grow and develop that.  But the good news is that it is not too late.  This can be developed at any time and any age.

The truth is, if students cannot know how many of something is without counting, then it is really, really hard for them to know 5 and 10.  If they do not know 5 and 10, then they will not understand our number system which is base 10, based on 10.  Which means it is almost impossible for them to know part-part whole which is foundational for fractions and algebra.  There it is!

Why can’t our students do algebra, because they have no Number Sense.  Algebra is not the gatekeeper for college and career readiness, Number Sense is.  If I cannot see quantity, then how can I see “structure,” which is what algebra is, representing quantity abstractly.  Then if I cannot see structure, then I cannot see space – Geometry.   Number Sense is what is limiting access to algebra.

This is why I keep saying Number Sense. I want us to plug the whole when the hole is the smallest.

IM Resource: What’s the difference?

Illustrative Mathematics (IM) recently announced a few more partnerships which has left some educators wondering, who, what, when, why, huh?  Users are now faced with choices and it can be confusing. This post is to help so educators can go back to what is important, focusing on all students having access to aligned instruction in mathematics.

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If you are using IM resources then it is probably because of the review in EdReports.  The IM resources has the highest rating on Ed Reports for the Shifts: Focus, Coherence and Rigor. It is also rated high for usability.  In addition, it contains instructional practices that are held near and dear to mathematics educators.

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But, one shoe does not fit all!   We want options and lots of them. Consider something as simple as buying acetaminophen or Tylenol.  We have rapid release, PM, enteric coated, extra strength, 24, 50, 100 count, tablets, capsules, etc., you name it.  Some people are okay to take the generic and some want the brand name.

The same way with a highly-rated aligned resource.  There are options. Plain and simple, options.  Users can decide what is important to them and what version they will use.  Each will have strengths and challenges.  It is up to educators to pick which option is best for them.  Some will use exclusively from one partner while some will use different parts from several.

In Iowa, we know alignment to the Shifts matters. We know Focus, Coherence, and Rigor matters. This means EdReports can provide useful information when considering which resource to use.  Since the IM resources are highly aligned, then it is important we use a “certified” version. Yes, it matters.

If you have ever talked to me about making changes to the IM resources, then you know I make no bones about owning the fact that, “I do not have the skill-set to know how changes to the resource will impact the alignment and I cannot tell you how it will/will not impact the scope and sequence. ”

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I am not embarrassed to admit it and do not think I should know. What I do know is students deserve the best we can give them and aligned materials matter.  Hard Stop! Students deserve resources which are aligned to the Shifts: Focus, Coherence, and Rigor, and instructional practices. Teachers deserve to teach with highly aligned instructional materials.

So, whether you use the free version or want the additional bells and whistles that come along with the paid version, students and teachers deserve the guarantee that the version they are using is the genuine, authentic, original, “certified” resource. They deserve to know the review from EdReport is intact. They deserve the assurance that any changes made will not alter the scope and sequence or hinder the Shifts. They deserve the changes to be “certified” by IM.

Having access to aligned mathematics instruction should be a right and guarantee each and every student is given. IM is committed to this promise and is providing a guarantee that with the “certified” versions, any changes will not hinder this right and guarantee. This is a very exciting time as the narrative has changed to which “certified” version of a highly aligned instructional resource should we use. It is a good problem to have.


Back-mapping and Learning Trajectories

Basically, we want to help kids think mathematically! It is why we, the adults in mathematics education, do what we do. But, there are so many variables, factors, causes, element, and it is complex.

To equip us the best, to help students, I have listed two nationally recognized tools below.  They are available for use at no charge due to the funding from the Hiesing-Simons Foundation and Bill and  Melinda Gates Foundation. They are aligned to the standards, from credible sources, based on current research, and have been presented and shared with state supervisors of mathematics nationwide.

Before we jump to the tools, consider another question that I am often asked: Do we focus on core instruction or intervention? The answer is both. Prevention and early detection are the best intervention and we will not intervene our way out of good core instruction. We want to continue to improve instruction each year with the goal that we are decreasing the number of students who will need intervention. At the same time, we need to intervene with those students who need it at the earliest possible time to maximize impact on the student. The ultimate goal is to have as little as a need for intervention as possible.

The use of both tools below would be greatly enhanced by user knowledge and understanding of Number Sense. Otherwise, educators may make decision to ignore or skip necessary experiences for students.  For example, students who are struggling to count (including verbal, object, cardinality) may need experiences to develop subitizing.  Not sure what this is. It has been around since the 1940’s in the research and is foundational and critical if we want students to add and multiply.   A common error educators make is to ignore or not spend enough time on developing student subitizing.  Another common error is to focus on a strategy itself and not the cognitive development.  Focusing on strategies for the sake of strategies will not develop the cognitive subitizing our students need. Teaching should be focused on cognitive development that leads to fluency, NOT teaching strategies to get right answers.  This will fall off for the students in 4th and 5th grade when the numbers get bigger and more complex and those strategies are no longer applicable or efficient.

My favorite feature of these tools is that most of the resources can be used with minimal prep time, are turn-key, and can be used with students the next day.

Step 1 – Use to back map: The Coherence Map – Developed by Student Achievement Partners (Jason Zimba lead writer of the standards) and available on the achievethecore.org websiteScreen Shot 2018-10-28 at 10.56.50 AM.png

  • a digital map of the standards that can be used to see the coherence of the standards
  • it can be used to back map and identify at what point in the standards the students struggle or have gaps
  • contains aligned tasks and assessment items to use with students
  • from the authors of the standards
  • understanding of the progression documents will strengthen use of the Coherence Map (be sure to read the Front Matter to understand increase understanding)
  • do not use the Coherence Map to forward map as there are multiple forward paths

Step 2 – Use to forward map: Learning and Teaching with Learning Trajectories– Developed by Dr. Clements and Dr. Sarama.  Doug Clements worked on the progression documents, with the standards writing team, and is nationally recognized as a lead researcher in early childhood mathematics.   (About Clements and Sarama)

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  • a digital resource of the learning progressions that can be used to move students forward in their cognitive development of mathematical thinking conceptually, conceptual understanding and skills for fluency
  • a Learning Trajectory is a content-specific learning path or a developmental progression that can be used to build conceptual understanding.
  • contains
    • 17 learning trajectories (birth to 3rd grade) for students to access the standards
    • short developmental video examples
    • resources for 1 on 1, small group, whole group task and computer
      • only 4 computer games because there is no computer substitution for student experiences with teachers
  • has a document that shows the Gold Standards alignment
  • can set up a classes of students
  • be used to observe and record individual child’s progress on the in each trajectory
  • parents can Access
  • User Guide – screen shots of setting up account






Universal Screening in Math

I have received several questions about Screening in Mathematics. Now what?

When I think of a Screener, I am reminded of a thermometer.  A thermometer can be used as a measurement tool to measure the temperature for an entire class of students.  It will most likely identify several students who have a core body temperature above 98.6 degrees Fahrenheit. Some of those students may have a slightly elevated temperature for a whole host of reasons.  Some might have a very elevated temperature also for various reasons.  What we can do is decide based on the data about further steps.

First, we would want to consider the environment. Is there something in the environment that is causing a higher than expected number of students to present with an elevated temperature. If there is, then we would need to address that environmental factor.  Otherwise, trying to address all the elevated temperatures without addressing the cause will not be able to be sustained long term.  After we have taken care of our environmental factor, then we will have a more reasonable number of elevated temperatures to address.

Next, we would make decisions about which students will be looked at further.  Some students with a mild elevated temperature might just be watched for a few days to see if the temperature returns to normal.  Some students with a little higher elevation might receive some medicine to help control the temperature to see if it resolves itself.  Some students, may get further testing to attempt to diagnose what is causing the students to run the temperature.   Then, after a diagnosis, the students may receive treatments of various types.

What we would not want to do, is to try to find one treatment to lower the temperatures of all students without considering environmental, student developmental changes, further testing, and an expert diagnosis.  None of us would want that for our children with an elevated temperature.   Most of us would want to sue the doctor for malpractice if every time our child had an elevated temperature, the doctor prescribed the same medication, the same dosage, and the same strength as he does for every other child.

We would also be very frustrated with the doctor who only addressed the temperature and did not look further.  If the doctor just kept treating the temperature for the sake of getting the temperature to decrease, and not addressing the cause, I think most of us would look for a new doctor.

This analogy demonstrates a screener in mathematics.  It will identify where classroom instruction can be strengthened so more students can have access and equity to best practice in mathematics instruction.  It will also over identify students who may not need any further testing, diagnosis, treatment, etc.  It will also not tell us what is causing the problem, what the problem is, or how to fix it. Most importantly, we should not look for a “cure all” for all students or give all the same students the same treatment.  And we should definitely not try to manipulate or control the screening data for the sake of the data.  These attempts will only prove to further perpetuate the situation for struggling students’ long term in their mathematics education and achievement.

So, what do we do?  We do some of the same things we did before we had screening data but we do it more accurately.  We reflect. We reflect on our classroom instruction and is it meeting the needs of most learners?  If not, what can we do to strengthen that?  Can we work with an instructional coach, AEA math consultant, etc.  Can we have more Focus throughout the year on the Major Work of the Grade and Clusters? Can we allow students more time and experiences around the mathematics that demand conceptual understanding before procedural fluency?  Can find tasks that promote problem solving and reasoning and use the 5 Practices to increase student discourse and increase engagement and understanding.

What about the students who are really struggling? What did we do before? Are there students who need further assessing to find out what the mathematics is that they need more experiences with? Can we give them more experiences to help them build the knowledge? Can we go ahead and send some information home so parents can work with them earlier? Can we call the parents in sooner than conferences to help get the extra time before or after school with the student?  Can we respond at the first sign of struggle when it is still manageable? There are several things that we can do to help students.

I would love to know what are some of your favorite responses when students do not learn.  Keep the great questions coming.