This has been a really popular question lately as we are well into the school year. What to use for a math intervention? Sometimes educators are looking for a strategy, sometimes a process and sometimes a thing.  I always assure them that if there was a “magic” mathematics anything, I would know about it and be sure to tell them but there isn’t. There are some better rabbit holes for us to go down as we try to find what helps our students learn mathematics and what educators can use to help students “get it,” or get “caught up.” Let us just agree, that those terms in quotes are ambiguous at best.  In  this blog I am going to help you find  a few rabbit holes you may want to explore.

In previous blogs, I have shared about Number Sense and Learning Trajectories.  I first learned of Learning Trajectories directly from Doug Clements at math SCASS (Math State Collaborative on Assessment and Student Standards), which is the mechanism nationally or how people who have my job get “smart” together about the good stuff to bring back to our individual states. It is put on by CCSSO (Council of Chief State Officers). The same group that brought us the mathematics standards.

Anyways, I didn’t “get it.”  It wasn’t that Doug did a bad job of teaching me, it was  because I had a secondary mathematics background and I didn’t know what I didn’t know.  Which the experience made me realize, I do not know what I need to know about K-5 mathematics. I needed to get “smart” about K-5 mathematics and quick. So I asked some of the AEA consultants, how do I get smart about what I need to know about K-5, and they said take Number Sense from Christina Tondevold.

After taking the Number Sense training, then and only then did I “get it.” Eureka, I understood that I had not had Number Sense myself!  No wonder I didn’t understand the Learning Trajectories !  I didn’t even understand the 4 early Number Concepts and the 4 Number Relationships (subitizing, verbal counting, object counting, cardinality, spatial relations, 1/2 more/less, benchmarks of 5 & 10, part-part-whole.)  This was the reason that I didn’t understand the Learning Trajectories and how to use them to help students.  But by connecting the  Number Sense learning with the research based Learning Trajectories activities, I realized how powerful they could be to use with students.

This brings us to the topic at hand. Can they be used as a math intervention? An intervention is a combination of program elements or strategies designed to produce behavior changes or improve health status among individuals or an entire population. Educators learning more about Number Sense  and then using Learning Trajectories, to give students experiences to increase their Number Sense  along research based Learning Trajectories could help teachers take students from where they are at to where we want them to be more quickly than perhaps what we are currently doing.

Example: A student still counting on their fingers may need more subitizing. (I know this because I understand the information in Number Sense .)  I know I need to give the student experiences so I go to the Learning Trajectories and select subitizing and I can get close to where I think the student is at and then work them forward along the subitizing trajectory.  See the screencast here.

A few last tidbits. Number Sense is closing for purchasing individual accounts on October 28, 2019 and will not re-open for a year. Schools purchasing multiple accounts can do so at any time.  The Learning Trajectories are open sourced and you just have to create an account.  I encourage you to watch the video  snippets.  Lastly, for those who are looking for Learning Trajectories more bells and whistle or sophistication in computer program, diagnostics, intervention, be sure to check out Building Blocks or Number Worlds which packages the work of the Learning Trajectories.

I share this information as a potential rabbit hole that you may want to explore. Will we do a better teaching and learning with increase Number Sense ? Yes.  Will we have researched based activities to use with students if we use Learning Trajectories ? Yes. Could this help teachers love to teach math, help students love learning math and help them learn more math? You be the judge. Let me know your thoughts in the comments.

I receive several questions about how to count high school mathematics.  Years, courses, standards, etc.  Students need three years of mathematics and all of the standards. In Iowa, we have local control which means high schools can decide how to “bundle” standards into courses.  Which can be confusing to count.  We know counting matters.  So, let’s think about counting in regards to standards.

Again, how schools bundle the mathematics standards is up to them in high school.  The mathematics standards do not designate specific courses.  If you count the high school mathematics standards, you will find there are over 150 of them.  For simplicity, we will just say 150 standards.  Next, I am going to say something that will surprise many – the number of the high school mathematics standards does not matter.  That’s right – it does not matter how many there are.

This is because, they are not the same things or the same size. Just like the ducks and pizzas. They cannot be counted together because they are not the same.  What I mean by this, is the fact that there are 150 standards means just that, there are 150 standards.  Some of them are big and some of them are smaller.  If most of them are small, then they do not amount to very much.  But we all know that even if they are all small, there is still a lot of high school mathematics content so we must have Focus which is usually missed in implementing the standards.  (I explain this further below.)

The bigger question I get asked is about counting courses of different length that contains the same number of standards.  So, let’s do some math.  If all students have to take three years of mathematics and all students need all of the standards, then most would say, it would be a good idea to divide them into three equal parts and handle it that way.  But when if we take the standards and divide them into 1/3’s and stretch that over two years, that still leaves 2/3’s of the standards to fit into the last year. Which does not seem feasible because if it takes me two years to get through 1/3 of the standards, then getting through (hopefully learning) 2/3’s of the standards in half the time seems like it would not work.

How is this all supposed to work?  It comes down to Focus.

All standards for all students bundled in a way that that honors the “importance,” the “weighting,” or the Focus of the standards.  Since high school was structured differently than K-8, then many miss the Focus Documents for high school.  To say it another way, most of the time in high school mathematics should be spent on the standards that matter the most to prepare students for the futures they want.  Take a look at the Focus Documents, notice that it doesn’t say we are skipping any standards.  It says, spend most of the time on algebra and functions because if student can solve some things and graph some things, it will serve them well.  Perhaps an algebra heavy geometry would be the right path for some students.

When students do not get an opportunity to learn all of the standards, are “we” becoming the deciding factor for their future? Is this an access and equity issue? Are students belonging to a certain sub-group more likely to end up not getting all the standards? Is this contributing to the achievement gap? Are we able to predict which students will not get access to all of the standards?  What are the solutions besides not giving students access?

Lastly, the (+) are the additional standards and are not intended for all students.

Let me know your thoughts and questions.

How much time is “required” for mathematics instruction?  This is probably the question that I get the most.  It is also the topic or question that I have the least amount of resources or research to support.  However, I think there is a common ground that most would agree makes sense. The first one being that without quantity, it is hard to have the quality mathematics instruction that we know is required for all of our students.  The second is that quantity does not ensure quality.  The third, we will get to in a moment.

First, let’s jump in. In Iowa, there is not a “required” amount of time. So, we provide what we have as a suggestion.  This is from the publication Adding It Up from the National Research Council 2001. This is a free download from the link above and this is the suggested citation: Adding It Up: Helping Children Learn Mathematics. Washington, DC: The National Academies Press. https://doi.org/10.17226/9822,

In Chapter 11: Conclusions and Recommendations, Section: Giving Time to Instruction it says,

“Giving Time to Instruction

Research indicates that a key requirement for developing proficiency is the opportunity to learn. In many U.S. elementary and middle school classrooms, students are not engaged in sustained study of mathematics. On some days in some classes they are spending little or no time at all on the subject. Mathematical proficiency as we have defined it cannot be developed unless regular time (say, one hour each school day) is allocated to and used for mathematics instruction in every grade of elementary and middle school. Further, we believe the strands of proficiency will not develop in a coordinated fashion unless continual attention is given to every strand. The following recommendation expresses our concern that mathematics be given its rightful place in the curriculum:

Mathematical proficiency as we have defined it cannot be developed unless regular time is allocated to and used for mathematics instruction in every grade of elementary and middle school.

• Substantial time should be devoted to mathematics instruction each school day, with enough time devoted to each unit and topic to enable students to develop understanding of the concepts and procedures involved. Time should be apportioned so that all strands of mathematical proficiency together receive adequate attention.”

Some will argue with “regular time (say, one hour each school day),” and claim there is no evidence.  I think we can all agree that “substantial” time is needed.

What does this mean? Substantial means “of considerable importance, size, or worth.” I tend to think of the amount needed to “reasonably get the job done.” I know some will say, well then it can be as little as 30 minutes.  I don’t know if that would pass the considerable importance, size, or worth test.  I have others ask, “can it be less than 60 minutes?” I think the answer is it depends.  It depends on so many factors which is why there isn’t research that makes this topic clear.

So, the third common ground point is we have to make a decision that make sense based on what students need and not for the convenience of schedules and systems.

Option one: follow what the recommendation has been for almost three decades, which is 60 minutes.

Option two: go with local data and evidence. What does your data say? What do the teachers say? What does the evidence say?

Please let me know what you think in the comments below and what is working, is it 30, 45, 60 minutes or more?

It recently occurred to me I have been chasing my tail and did not realize it? 

While talking to some educators the other day, I found myself intrigued how differently we were describing and framing a topic.

I thought to myself, “if only we had a unifying network to help understand better. Like a mathematics and mathematics educator network.” Then I had a “seriously” moment. A moment when I say “self, how did you not “get” that before, seriously?”

We do have a network of mathematics and mathematics educator. I realized that it has been right in front of my face the whole time.

This network of professional mathematics organizations is known as the Council Board of Mathematical Sciences or (CBMS). It is the “umbrella” organization that connects all of the other formal mathematics groups.  I like to think of it as our “math solar system,” with “math planets.” These 18 organizations or planets if you will, offer a network and access to the best of the best.  Each planet has a national president and some even have regional or state affiliates like National Council of Teachers of Mathematics (NCTM) and National Council of Supervisors of Mathematics (NCSM).

I realize now that this is our collective efficacy, which has an effect size of 1.57 according to John Hattie.  I also realized that I (we) collectively are underutilizing this vast resource. I can tell from only scratching the surface of only a few of these planets, most of what I have needed and others have needed (if not all) is from one these planets. This is what caused me to say, “self, seriously?”  The hours of time I have wasted going down “rabbit holes,” “google holes,” and “e-mail holes” when what I needed was in front of me the whole time.

Now, I have made the commitment, that before I look outside of the mathematics solar system, I am going to make sure, the resource, information, knowledge, research, position statement, white paper, practice, strategies, tool, etc. is not on one of our math planets.  Here are the 18 professional organization or planets.

         

        

         

      

Select a link below to see details and explore. The links to their websites are on the pages below.

1. AMATYC American Mathematical Association of Two-Year Colleges
2. AMS American Mathematical Society
3. ASA American Statistical Association
4. ASL Association for Symbolic Logic
5. AWM Association for Women in Mathematics
6. AMTE Association of Mathematics Teacher Educators
7. ASSM Association of State Supervisors of Mathematics
8. BBA Benjamin Banneker Association
9. INFORMS Institute for Operations Research and the Management Sciences
10. IMS Institute of Mathematical Statistics
11. MAA Mathematical Association of America
12. NAM National Association of Mathematicians
13. NCSM Leadership in Mathematics Education
14. NCTM National Council of Teachers of Mathematics
15. SIAM Society for Industrial and Applied Mathematics
16. SOA Society of Actuaries
17. TODOS Todos: Mathematics for All
18. WME Women and Mathematics Education

As I previously stated, I have just begun to scratch the surface of the vast resources that this mathematics “universe,” network, community has to offer. I wish when I would have realized this three years ago and even longer ago when I was just a teacher. I hope that through this sharing of this realization, it will encourage educators who are seeking for mathematics “best” practices to look at the 18 planets to find the tools you need.  My call to action is for you to pick a planet and explore. You will be surprised by what you find.