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fall off for the students in 4th and 5th grade when the numbers get bigger and more complex and those strategies are no longer applicable or efficient.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 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)

Screen Shot 2018-10-28 at 11.02.28 AM Screen Shot 2018-10-28 at 10.58.15 AM

  • 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.



Math Identity

I have a great passion for mathematics education. Some might even describe it as an addiction or an obsession. I have ever since high school. Although until recently, I really did not know why. What I did know, was that I enjoy few other things in life as much as I enjoy learning about it and trying to create a better environment and experiences for all educators, students, and parents.

I have the best job in the world. Not many would agree and I am not sure that I could convince others. I can explain why I feel that way. I can explain why I feel like I get paid to do what I love. I can explain why when other things do not consume my time, I can be found on my computer “working.” I do not consider it work though. I consider it do what I was meant to do. I didn’t find my calling, it found me.

Let me explain. My life started as a baby born to a teen mom in 1971. I was placed into foster care. It was the times and that is what we did. Before my first birthday, I was placed back with my mother and we were living with family in another state. Most of the details are not important. I went to elementary school and was that kid that had to go to speech class because I stuttered until like 4th grade. It might have been earlier but I remember it was a really long time and I have always struggled to be able to be able to clearly speak and am not confident speaking.

Middle school and high school brought its own set of challenges. I was an at risk student, had chronic absenteeism because of anxiety of being in both environments home and school. School was torture with my little lunch card colored that “poor” kid color and in “those” clothes or even hand-me downs as the oldest in my family. It wasn’t that I cared about material things, anyone that knows me, knows I don’t and loathe shopping. But I felt judged, I felt less as the other kids, I wanted to be treated special like those “other” kids were.

They would often try to put me in the “smart” classes because of my math scores. Not my reading! I was the reverse of what we hear today. I had some trouble with spelling, structure, and grammar and grew to despise it. All the rules, exceptions, and inconsistencies, never made sense to me and I couldn’t remember them all. Sound familiar! Anyways, because of many stressors in my life, none important enough to mention, I could not succeed in the “smart” classes and was returned to the “regular” classes. The tried in MS and in HS and both times ended in failure.

In HS, I had great mathematics instructors. Teachers were also like magical creatures to me because I found that I always felt safe and cared for by them. They seemed fancy and I wanted to be like them where someone would think I was special.

I made it to graduation, escaped my current reality and joined the active duty military for 6 years. (I know I am supposed to “spell” that out but I like writing number as symbols and not in letters.) I found my way back to math education, earned a B.S in mathematics, taught MS and HS, earned a M.A. in School administration and Leadership, and continued to teach. I returned to earn 30 additional hours in mathematics and education, began as an adjunct professor at a community college in the evenings, and then came to the Iowa Department of Education.

I do “get” it when we refer to “those” kids. I was “that” kid.I was that kid that they put into shorthand and they hoped that someday I might be a secretary and then they would have marked me as a success. But they did something else too. They gave me access to become.

They gave me access to great mathematics teaching and learning that they gave to all the “special” and “smart” kids. They didn’t limit my potential based on what “they” thought I could do or my current reality. The prepared well enough that mathematics was not going to be the reason or factor that I didn’t “make” it and be deemed a “success.”

I often say, the active duty military got me out of the gutter and mathematics education was has kept me out of it. Could they saved me in the moment, honestly, I have pondered this question. The answer is NO.

As a HS teacher, I have struggled at length with this. There is nothing the school could have done to change my life. I had to abandon most of what I knew to be on my current pathway. That is Grit. For those of you old enough to remember “An Officer and a Gentleman,” there is a scene where a trainee responds, “I got nowhere else to go!” (30 sec video clip)

Yet, I made it. I made it, because others believed and gave me mathematics instruction like we ALL could. Not what ALL could in the moment, but eventually, for what I was to become. Did they still put me in shorthand, yes, did I still have to take remedial math in college, yes, but only one! I was always pushed to be more, not hindered or told to settle for what I was. I was prepared for what I was to Become!

So, there it is. That it is. That is the story. I do not share to say garner any other reaction than understanding for my motives and drive for what I do. I am not a Brainiac. I had to be taught and I had to learn mathematics. Still had to learn Number Sense two years ago. My point is, they gave this little kid from the NY gutter a future, they saved me.

Mathematics education saved my life. It is why I took this position. Nothing more, nothing less. It wasn’t for the public speaking or being in front. #notafan It was because if I can help save another person from poverty, then it was all worth it. If I can help to make sure mathematics is not the reason they do not succeed, then I honored and repaid what was done for me.

The world is not getting less technical, everyday it gets more. I would not need or use short-hand in today’s world. Just the other day, Gmail started anticipating my email text and all I must do is hit tab. WOW! We need to over teach every student every day. That is how we will prepare them for the future.

I have the privilege and honor to work at the Iowa Department of Education as the mathematics consultant. Some know me as the State Supervisor of Mathematics. The State Supervisors of Mathematics is the professional organization for those at state the DE level. It is how we recognize each other. It doesn’t mean I supervise anyone or anything. At best, I supervise mathematics, which is not possible since it is not a person.

But, as the designated State Supervisor of Mathematics, what that title does give me is access. It gives me access to people and information that not everyone has access to. Which means I can connect! I can connect other educator to the information. The information we can use to make better decisions to save kids.

So, this blog is going to be different than other ones. This is a Q&A to generate access to information, conversations, and other educators that I have access to. It seems like a crying shame not to share it when we have the means.

So, give me the question and I will tell you the best “we” know at this time. My responses will not be fancy, there might be typos, or something. Okay. I also will not give you answers or “the” solution. What we do is complex. There is only information to use to make better decisions.

I hope you all will give me some questions so I can answer them. The information will determine the length and my schedule will determine how often.

I want to dedicate this post to three educators:

First: Ms. Stagner from Camden, NY for being my magical creature. It was her first year of teaching and she prepared so hard for each lesson to be so understandable and precise. She gave me the access to great mathematics instruction and also showed me the hard work and care that goes into it.

Second: Is Mike Flynn. Last week I attended a virtual book study on Becoming the Mathematics Teacher you wish you had. I think it was the impetus to this revelation in my understanding of my own journey. (to be honest, I didn’t get my reading done before class so it was really class.)

Thirdly: To Tracy Zager for writing Becoming the Mathematics Teacher you wish you had.

Feel free to submit your questions here, hit me up on Twitter @apriliowamath, or email me your question at