I still find each day too short for all the thoughts I want to think, all the walks I want to take, all the books I want to read, and all the friends I want to see.
-- John Burrough

Number Sense Worksheets

When I started teaching nearly two decades ago, the big controversy was phonics versus whole language. I believe we have finally reached a compromise, understanding that "authentic language experiences" must also be grounded in a firm foundation of sound-letter correspondence (explicit phonics instruction). We realize that a strong grasp of phonics leads to more solid reading skills. Unfortunately, I do not see that same realization in mathematics instruction regarding number sense. Too many textbooks (and teachers) skip right over number sense development (or give it fleeting lip service) and go straight to skills using numerals, to the detriment of many students.

Jump right to the worksheets!

Understanding numbers, their relationships to one another, magnitude and scale, and how they can be manipulated is essential to being able to grasp and apply higher math concepts. It seems to me that our current math curriculum (in my California school) does not have enough activities to develop number sense in all students. This, coupled with the atmosphere engendered by NCLB, causes many students to be shoved into working with numerals before fully understanding numbers. Because of tightly scheduled tests and curriculum pacing guides, some of these students never fully develop real number sense.

This year, I am tasked with helping students that are struggling in mathematics. These are 4th and 5th graders that have spent their whole academic career so far receiving the state-approved, district-adopted mathematics program, which has been "faithfully implemented" by their teachers (as shown by their peers' increases on state test scores?). In working with them, I find that they have poorly developed number sense. They do not recognize the difference in "good" estimates versus "bad" estimates, they cannot easily visualize or recognize large numbers of items, they think that fractions are made up of two numbers rather than understanding that a fraction is a single number representing a portion of a whole, and they do not easily transfer information from an equation to its inverse (23 + 45 = 68 -> 68 - 23 = ?). These are all symptoms of poorly developed number sense.

A number of years ago, I remember encountering this dilema with another group of students that had also had all their math learning from this particular publisher (in a previous edition of the textbook). They, too, exhibited poor number sense. At the time, I began to develop a series of web pages to help develop their number sense. Projecting these pages onto the whiteboard using a digital projector, we used them as math warmup most days. For many of the students, their understanding of magnitude and fractions increased greatly. For this year's students, I decided to revisit those pages.

I also recalled some second grade classes from when I first started teaching. At the time, I had the pleasure of taking a series of inservice classes offered by one of the district's top math teachers, Gregg Nelson. Because of this training, I restructured my math class. Instead of plowing through the book from front to back, I "spiraled" through the chapters, revisiting concepts using more advanced lessons as the year progressed. I began each math period by doing math warmups: I had a set of index cards on which I had printed basic math concepts (time, fact families, fractions, etc.), and every day we did 3 or 4 problems to develop the concepts from each of 3 or 4 cards. I looked for reproducible homework pages that included weekly review of all the concepts we were learning. I could only find books with whole pages of single concepts. So, I cut up those books and created my own homework that included addition, subtraction, time, fractions, skip counting, money, and word problems on every page. For the next few years, I was getting comments that the students coming from my classes were "bored" in third grade because they had to "wait for the other kids to catch on".

Upon recalling these memories, I immediately implemented the web pages on the projector and the math concept index cards for my 4th graders, the group most in need of remediation. I also borrowed some 4th grade math reproducible books from some other teachers. I was planning to cut-and-paste some homework pages, as well. But when I looked through the books, I wasn't finding the kinds of work that I felt would most contribute to developing number sense. Also, I was really not looking forward to standing at the copy machine for hours making pages to cut up. Instead, I wrote a program to generate the pages just the way I want.

This online program generates pages of problems to help build number sense and practice basic operations (to develop automaticity). I am able to set how frequently a particular kind of problem appears. I am able to include or exclude concepts and set number ranges to tailor the pages per grade level, per group, or even per student. The problems are generated randomly based on start, end, and total days, so I can regenerate a given day and the original problems show up (in case a student lost that particular page). Some concepts are adjusted throughout the year, progressing from simple exploration to more complex implementation as the year progresses. Finally, after "polishing" the program over a couple weekends, I even built in the ability to save and load settings so I don't have to re-set all the options for my different groups. I offer this tool to any and all educators that might find it useful!

My hope is that those of you who would use this tool would also agree that the benefit of these pages comes not so much in the doing but rather in the reviewing, during which the teacher can explain and further develop the concepts being practiced. The teacher should demonstrate why 967 is a little less than halfway between the half-tick and the 1000 tick on the number line. The student should be able to verbalize how he or she arrived at an estimate, or explain why 9,600 is a closer estimate but 9,580 is not "round" enough to be very useful for mental math. During my review of the types of problems on these pages, concepts are explained, understanding is refined, and the students' development of number sense can be more sublimely assessed. Hopefully, these pages can help our students become as proficient as possible by deepening their relationships with numbers.

For more information on number sense, check out:

  • Number Sense — Right Now! - A "call to arms" from the former president of the National Council of Teachers of Mathematics (NCTM)
  • Number Sense Every Day - A description of some aspects of number sense and some ideas for developing it (Univ. Of North Carolina, Chapel Hill)
  • Understanding a Child's Development of Number Sense - NCTM Resources, a description of number sense and videos of children demonstrating various levels and aspects of number sense.
  • Pedagogy That Makes (Number Sense) - A research paper from San Diego State University on the positive effects of number sense development (re: mental math) and the improtance of teachers' awareness of student number sense and development techniques.
  • Number Sense - Short description and activities focusing on the concept of probability. (Math Education Program, University of Virginia)
  • Wikipedia - Concise number sense description.
  • Bit by Bit - Day by Day - Presentation slides from NCTM Annual Meeting, by Amy Mayfield of Math Solutions. These are just what was being shown onscreen without the presenter's explanations, but  some information about number sense can be gleaned.
(Last modified:09/06/2011)

Comments

Rosa Flores - 06/26/2012
Hi Mr. Flores, my name is Rosa Flores and I am preparing myself to become 7th grade math teacher and I found your website that it's great and right to the point that it is true that in most districts do not teach students the proper math. Thank you so much and this website is great.
Thanks Rosa! Congrats to you for becoming a teacher. It$#39;s a hard but rewarding career. We need more teachers like you — you know, teachers that like my website! ;-)
Mr. Flores

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