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John Plencner wrote, "In the haste to embrace the "practical" uses of mathematics, we (the math community) seem to have forgotten the abstract nature of the material, and the need to really learn the methods and rules." I take a slightly different perspective. In teaching elementary school mathematics teachers need to think of the word "abstract" as a verb, not as an adjective. Most, if not all, of elementary (K-5) mathematics can be abstracted from everyday/physical phenomena. Mathematics, I believe most people agree, is built up on previously learned ideas. As students grow older and study more "advanced" mathematics, they will have to start abstracting ideas out of what they had previously abstracted. Thus, the nature of the subject becomes more abstract. However, in elementary schools, mathematics teaching should be built on concrete phenomena.

I also think the CCSS' treatment of fractions and decimal numbers are rather ineffective. I would rather introduce decimal numbers first, develop arithmetic with decimal numbers based on whole number arithmetic. Because decimal numbers are simply an extension of whole number (decimal) notation, the same place-by-place approach works for addition and subtraction. With multiplication and division, students can make use of properties of multiplication and division to use their prior learning of whole number arithmetic. Fractions can come after students' learning of decimal numbers. Students then can build their understanding of fractions on their knowledge of decimal numbers.

In theory, the order can be reversed - fractions first, then decimals. However, in the CCSS, fractions are introduced first, but then students are expected to learn decimal arithmetic before they complete their study of fraction arithmetic. I think we are making learning of these ideas more difficult by putting these topics in a rather incoherent sequence.

Thanks for joining the conversation, John. What jumps out at me are your comments about math attitudes...

Negative perceptions of math are often driven not by the subject itself, but by an “I’m not a math person” kind of mindset. Research from Jo Boaler and Carol Dweck tell us that this affects students as young as 5 years old. Also, we know that “I’m not a math person” is fiction, without any basis in science. Luckily, educators across the country are changing this stigma, and hopefully soon, “I’m not good at math” will be as strange a statement as “I’m not good at reading.”

As for your thoughts about the abstract nature of math and need for outstanding resources and tools, check out my response to David B. :-)

*In the spirit of full disclosure, I’m CEO and co-founder of Happy Numbers, which is mentioned below for illustrative purposes*

David, I see your point -- I have similar conversations all the time: teachers and educators want to see how hands-on exploration and math modeling will help with tests that are procedurally-oriented.

The way I respond is by showing that hands-on learning and math modeling empower procedural practice, not substitute it.

Just because the assessment is composed of abstract addition facts doesn’t mean that drilling students on abstract addition facts is the best approach. Here’s an example of how students can practice an addition fact with solid context and meaning behind it:

https://www.youtube.com/watch?v=I69PTocXwgw

In this video, building the model comes first to establish context for the addition fact that is composed afterward.

The challenge of this approach is the appropriate use of manipulatives -- using manipulatives in a way that will build meaning behind math and connect to abstract thinking.

And I believe that technology is what will help teachers meet this challenge, especially when we are not only expecting hands-on learning, but also that it is tailored to each individual student’s needs (differentiating instruction is important).

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In this vein, here’s a recent case study made together with educators from NJ:

https://blog.happynumbers.com/hitting-math-benchmarks-in-k-2-case-study/

The case study questions whether mind-growing lessons that focus on conceptual understanding lead to success on abstract tests that only measure procedural fluency.

I also agree, and have seen evidence of this in every school I have taught in over my 26 year career. I have taught grades K-8, so I have seen this from the perspective of the primary teacher, and then the results of this in my current position as a 6/7th grade math teacher.  We know the research points to teaching a balance of concepts with procedures, but this is not a reality at the primary level. Most primary teachers don't enjoy teaching math as much as ELA, and most of us were not taught this way.  In the end, teachers are relying on two things- the way they were taught, and the curriculum their school or district has adopted. If we have no model or curriculum (I have yet to see a curriculum that does it efficiently) to support what we know works, then this results in students having poor understanding of math concepts, and then always being behind. It's hard to know where to start when you meet multiple 6th graders who don't understand subtraction with regrouping, for example. I venture to say most teachers are teaching procedures because it's what they know, and our texts are filled with them. It also takes time to develop conceptual understanding, time many teachers don't feel they have. Primary teachers also have flexibility in their day, and most of them spend more time on ELA than math.

I agree that the problem seems to be occurring before high school.  I have taught Freshmen for most of the last fifteen years.  Students that do poorly, for the most part, have well established negative attitudes towards mathematics.  In visiting Kindergarten and 1st Grade classrooms, I find that almost all students are excited about numbers and mathematics, and are eager to learn the material.  There is definitely something happening in between.  We urgently need to figure out what is the problem.
Some thoughts on the problems. 
1.  Teacher lack of confidence and/or dislike of the subject communicates to the student, no matter how zealous the teacher is about not communicating the lack of confidence or dislike.  Similarly, parental attitudes on the subject are communicated to the students ("Oh, I was never very good at math." has just given the student permission to not be good at math.)
2.  The vast majority of the school systems have not developed a means of addressing deficits in mathematics.  The systems have been in place for reading for years.  We do not do the same for mathematics.
3.  We are demanding too much of our elementary teachers as far as lesson planning.  Good, complete, detailed lesson plans are available for K-8 mathematics (for example the jumpmath.org teacher binders).  Objectives, detailed rationals, lists of materials, instructions for manipulatives, leading questions, hand gestures, expansion activities, etc.  For every single lesson!!!!  Stop having every teacher have to reinvent the wheel.  Give them an outstanding resource that they can modify.
4.  In the haste to embrace the "practical" uses of mathematics, we (the math community) seem to have forgotten the abstract nature of the material, and the need to really learn the methods and rules.  A prime example of this is long division.  In Algebra II, we address diving rational expressions using algebraic long division.  This is suppose to scaffold onto long division from fourth grade.  The numbers in my classes have been getting steadily worse, to the point that close to 80% of my juniors no longer remember how to do a long division problem.  There are many other topics where the student retention of the material is also bad.

Thank you, Evgeny and Megan, for responding to my post.  The consistent message from NCTM and the writers of the CCSS is that math teachers need to place a much stronger emphasis on developing conceptual knowledge and not just focusing on procedural knowledge.  Math teachers and their administrators may know and believe this--in theory.  But in practice, teaching for deep understanding through hands-on experiences and free exploration requires patience and time--and this necessarily conflicts with the all-important testing regimen.  NCTM and CCSS emphases notwithstanding, manipulatives are routinely scorned because, the reasoning goes, students won't be able to use them when they take the computer-based test!

Teacher accountability models and rubrics for grading schools are tied to standardized test scores, and this results in higher stakes for the teachers and administrators than for the students.  Perverse incentives are now in place that reward short-term performance at the expense of developing a strong foundation for future math learning.  The year-on-year obsession with "learning gains" is big business for computer-based test preparation, computer-based practice, and even computer-based lesson instruction.  Much of this preparation, practice and instruction is  focused on performing computations and not developing conceptual understanding.

I have enjoyed following this thread, and I appreciate the engagement by Evgeny around clarifying possible questions and / or commenting and sharing insight. While the focus of the post has specific research around what is happening, and perhaps when, along learning trajectories - I can't help but wonder about bigger ideas shared in the most recent post: 
"There is a solution.  It starts with teaching fraction arithmetic operations within a meaningful context and without resorting to memorized rules.  Rules taught and practiced through hands-on experiences in the same way that whole number operations were taught, practiced, and, most importantly, mastered."
These ideas are heavily supported by NCTM Principles to Actions and the teaching and learning beliefs we hold (specific to mathematics). Which then lead to the 8 Math Teaching Practices (not to be confused with the CCSS Mathematical Practices). Good stuff all around, and I think - if I 'm reading and following correctly - we are all in agreement here! Keep the conversation going!

Megan 

David, thanks for your comment and for sharing the research confirming that difficulty with algebra has roots deep into elementary school.

Conceptual understanding of fractions and operations using them is important and thus, no surprise, it affects success with algebra. And indeed, the “rules” / shortcuts for operations with fractions (remember those butterflies drawings to help remember those rules? ufff...) are harming the learners instead of leading them to explore and understand the meaning behind the math.

Instead of drawing butterflies, students could model fractions (and later operations with them) using a number line, tape diagrams, and area model. Thus, it’s important that before it’s time to learn fractions, students are able to use a number line, tape diagram, and area model as a tool -- and this comes in PK-2, when students are working with whole numbers. That’s why it’s important not to limit modeling just to such (great) manipulatives as base-ten blocks and fingers, but also extend it toward modeling with drawings and geometric shapes.

The math story starts in early grades; you can’t miss that part and be successful later.

Everything I read in each of the prior comments is consistent with my experience as a public school teacher, private tutor, and parent over the past 25 years. The problems documented at the college and high school level are clearly rooted in the lower grades, particularly with respect to skills and concepts taught, retaught, and retaught again--but without ever leading to mastery.  Between teaching, tutoring, helping my kids with homework, and collaborating with my preschool-teaching wife, I have been immersed in the full spectrum of math instruction up through college.  In my personal opinion, the problem is all in the instructional methods employed--particularly in the upper elementary grades and middle school. 

Research by Seigler, et. al, (http://journals.sagepub.com/doi/abs/10.1177/0956797612440101), and Brown & Quinn (https://eric.ed.gov/?id=EJ776592), among others, cite difficulty with fraction arithmetic operations as a primary cause of later difficulty with algebra.  Difficulty with algebra leads to difficulty with all of the math that follows.  The poor test scores recorded at the college and high school levels are indicative of the fact that these students haven't mastered the fundamental concepts that they saw repeatedly throughout middle and high school.  Practically every topic in my daughter's "college algebra" course is just a rehashing of what is taught in middle school and  beginning of high school:  Mostly pre-algebra and algebra with a dash of algebra 2.

In my reading of the research, the problem lies in the transition from whole number operations to operations with fractions--which is soon followed by integers (negatives), and, eventually, irrational, and complex numbers.  Prior to the introduction of fraction operations, students have learned and mastered whole number operations over the course of many years including in pre-school and at home.  Much of this early learning was through hands-on experiences and took place in authentic contexts that gave meaning to the numbers and purpose to the operations.  Whether manipulatives were employed or not, all of these early calculations involving whole numbers could have been easily modeled and confirmed with physical objects and representative sketches to improve understanding, correct misconceptions and instill confidence both in the choice of the operation to perform and in the result.

Contrast the learning of whole number operations with the manner in which we generally teach fraction operations.  Or, operations with integers.  These are taught primarily through abstract rules for students to memorize.  Rules that are very different from whole number operations.  Rules that are easily confused and misapplied.  Rules that are presented in quick succession without the abundance of time or hands-on experiences that were allotted to practicing and mastering the rules governing whole numbers.  This is the crux of the problem. 

For many students, mathematics beyond 4th grade is nothing more than a collection of confusing and contradictory rules that are used to manipulate numbers for no apparent reason.  Bogus "real-world" and "application-type" problems that are completely foreign to students' lives add confusion instead of aiding understanding.  It's no wonder that one of the most repeated question in math classrooms is:  "When I am ever going to use this?"

There is a solution.  It starts with teaching fraction arithmetic operations within a meaningful context and without resorting to memorized rules.  Rules taught and practiced through hands-on experiences in the same way that whole number operations were taught, practiced, and, most importantly, mastered.  It continues with placing the focus of fraction operations on the critical role of factors.  Prime factors in particular--despite their near absence from the CCSS.  Everything learned about prime factors with proper instruction in fractions transfers directly to operations with powers, roots and polynomials in algebra. 

Interested in a brief preview of how this approach can work?  Check out this link to a blog post I wrote for Didax in support of the revolutionary new manipulative I have created:  http://www.didax.com/blog/prime-factors-for-fraction-success/

Tad, thanks for following up with the questions!

For your question related to measuring college readiness in math -- I referred to NAEP because:

1) According to the National Assessment Governing Board, "NAEP should report 12th grade students’ readiness for college-credit coursework."

2) Also according to the National Assessment Governing Board, "NAEP/SAT linking study for mathematics, the average NAEP score for 12th grade students scoring at the SAT college readiness benchmark for mathematics was 163, which is lower than NAEP's Proficient cut score of 176. Percentage of students scoring at or above a score of 163 on the Grade 12 NAEP scale in mathematics is a plausible estimate of the percentage of students who possess the knowledge, skills, and abilities in mathematics that would make them academically prepared for college. A score of 163 in mathematics is between the cut scores for the Basic and Proficient achievement levels in 12th grade mathematics. In 2013, 39% of 12th graders nationally scored at or above 163 in mathematics."

So while the % depends on the cut, and finding the "right" cut is the question, the NAEP assessment result is a plausible way to measure the dynamics over the years and see it decline or improve.

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For your question on individualized instruction: It's not about some students getting more/less instruction than others, it's about them getting different instruction. As I mentioned in my previous comment, this subject needs another post(s), but getting a bit upfront -- there are studies showing that an individualized model of instruction benefits all students compared to a one-lesson-fits-all model.

I appreciate and enjoy your engagement in the conversation -- thanks for taking the time!

So, are you saying that "college readiness" means at or above proficient level" on NAEP?

When you "individualize" instruction, why wouldn't more advanced students make as much (or more) progress? Would "lower" students be getting more attention from the instructors?

Thanks for your thoughts, Tad!

To answer your first question about the criteria for (math) college readiness, this data comes from the most recent assessment by the National Assessment of Educational Progress: https://www.nationsreportcard.gov/

If you check it out, you can see that only ¼ of the students performed at or above proficient level by the end of high school.

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Regarding the gap remaining through K-8, there is some evidence that this gap could be improved by individualizing instruction for each student. I’ll write a separate post on this later, but here’s something to think about:

Research shows that there is no such thing as a math / non-math person. All kids have abilities to perform in math. It’s about fostering it correctly. For example: the average tutored student scored higher than 98% of students taught in a conventional classroom. This was reported by Professor Bloom in 1984 showing the power of personalized learning.

I agree, students will perform differently, but with such huge differences? No educator wants to see a 3-year lag between the lower quartile in the class and the middle two quartiles (to say nothing of the upper quartile achieving this level 5 years prior!)

An interesting post, thank you. A couple of questions came to my mind.
First, do we have any agreed upon criteria for (math) college readiness? I'm not quite sure.
Second, in some ways, it seems reasonable that the gap remains through K-8. Even if we have a set of wonderful curriculum and a cadre of great teachers, if "low" students learn so much, you would think "high" students would learn just as much - unless we put an arbitrary ceiling. I guess what we are aiming for is that even "low" students are performing at a higher level than they currently are.
Finally, in part responding to Carlotte's comment, a part of an issue in (K-8) teacher education may be that colleges/universities are not requiring appropriate mathematics content courses for prospective K-8 teachers. Taking College Algebra/Precalculus/Calculus does not necessarily prepare them to be better K-8 mathematics teachers. A lot of schools requires prospective teachers to take "math for elementary teachers," but common textbooks for those courses may not be hitting the target.

Thanks for sharing your thoughts and personal experience with us on this topic, Charlotte!

Indeed, professional development for teachers is very important -- and it’s great that schools and districts are investing in it, ensuring the elementary classroom experience is a deeper dive into math understanding, and thus preparation for algebra.

In addition to school/district efforts, thanks to the Internet, there are many opportunities right now for PD -- such as this community, Twitter, blogs, webinars, etc. -- and most of it is free!

A few years ago, I sat in a testing room for the CSET, which is California's Multiple Subjects Exam that k-8 teachers must pass. It is the type of exam that if a test-taker fails one part, he or she can come back and take the one section. The room was filled with people who were there only to take one section because they failed it on an earlier attempt. Nearly everyone who was there, for one section, was retaking the math section!

When I opened up the math section, I was shocked that so many people were retaking it. I had taught 4th-6th grade in NJ, concentrating on teaching math. My top 6th grade students could easily pass the math section of the CSET. Here is where one of the biggest problems in k-8 math is found: Teacher Preparation.

Years before, I worked as a Sixth Grade General Studies Teacher in  middle school. It must have been 1992 or 1993 when the district adopted a rigorous textbook that had a number of teachers worried. They did not have the background to teach a more rigorous form of grade six math. I was always strong in math--didn't always like it--but, I was usually a top math student. I also was a business major for 2.5 years. So, I was selected to be one of the two sixth grade math/science teachers, not because of my science background but because of my math background. 

Something strange happened when I started teaching math. I actually decided I like it. So, I took a math placement exam at the local community college and did well enough to place into the engineering level Calculus track, something I could have done years before. 

I was moved into my former NJ district's Blue Ribbon School, which had/has an amazing math program (pre-high school). I became an  Elementary Specialist and I worked with talented Math Specialists. The best part is that there was block scheduling so, there were at least three days where we could deeply focus on math content.

Right now, as I read your blog, I want to scream. My son attends an SFUSD high school. This California District does not want students to take Algebra before grade 9. It is ridiculous. My son is allowed to sign up for an online geometry course this summer or he can spend 5 hours a day over the summer taking Geometry so he can be on target for Calculus before he graduates high school--the district does NOT care how high school course is taken. But, they care about 9 Algebra. It is bizarre. And, it is upsetting to me because I think this is far more a k-7 issue than a high school issue.