UPDATE 9/20/2014: It appears the new Common Core State Standards (CCSS) may actually (implicitly) support my call for a revision of the traditional hundreds chart. The CCSS calls for emphasis on building number sense. Students are expected to understand numbers and number relationship, not merely be proficient with manipulating them through externally taught (versus internally developed) algorithms. In fact, students are not expected to even be proficient with (addition and subtraction) algorithms until the end of 4th grade! In other words, number sense should be a solid foundation upon which we build our mathematics house; don't start building until the cement has cured! Hopefully, my idea for an "upside-down" number chart will help in building that foundation! (Maybe it can be the rebar lattice. See, what I did there, with the lattice, and the chart being like a vertical lattice?)

(Originally published 4/30/2009) – Recently, during a planning meeting with my first grade team, we were discussing useful skills and techniques to teach our first-graders here at the end of the year. Coming from an upper grade experience (I taught 4th, 5th, and 6th for the previous 5 years), I felt that developing number sense* would be most beneficial. I explained how I had worked with so many students who had been "raised" on the current mathematics program but who could do no problem solving and were unable to transfer skills from one situation to another. This was, I believe, due to poor number sense.

I suggested to my colleagues that we teach our students to develop proficiency in "skip-counting" by tens. I explained that "counting by tens" (10, 20, 30, ...) was all well and good but that skip-counting (17, 27, 37, ...) would serve to develop number sense. Then I pointed out that I hate our traditional "hundreds charts" (1-100).

For years, I have been bothered by our hundreds charts. The only hundreds charts I had ever seen had started on the top left with 1, like this:

Occasionally, a student might come across (what I consider to be) a superior version starting with 0. At least the "0-99" chart has all the "20s" on the same line.

I explained to my team members that we should have hundreds charts that started at the bottom and went up. Initially, some of them thought I meant a "backwards" chart, but after I clarified the format of the chart, they thought it was interesting and could see the utility of such a chart, but they were still unconvinced that such a chart should be used. A major argument against using such a chart was "nobody else does". I see some logic in this. If we use the "upside down hundreds chart" when 99.9% of classrooms use the "normal" chart, the students will be at a disadvantage when they move to those other classrooms. It was late in the day when this point arrived so I did not attempt to offer a retort. I left the meeting a bit frustrated so I sought out other teachers upon whom I could foist my "wisdom". When I encountered a couple of willing participants, I gave them the shpiel. The more I talked about it, the more excited I became about it. These teachers agreed that it would be a useful and possibly more understandable chart, but they too had their qualms (thanks, Alex) about implementing it. They suggested I blog about it and see what other teachers thought.

I believe the "upside-down" chart would do wonders for our students. It would build number sense*, teach graphing skills, remove pet hair, and clear up acne! Okay, maybe not the last two so much, but it would do a better job at getting students to understand quantitative increases more easily than the common hundreds chart. When we have more stuff, the pile goes higher. When we measure height, we start at the floor and go up. Yet, using the traditional chart, when our number increases, our position on the chart goes down, instinctively opposite and lexically opposite the direction we should expect it to go! You may argue, "It is instinctual (or logical) to go from left to right, and from top to bottom because that is the way we read." However, you must remember that directionality in writing is arbitrary. Hebrew is a right-to-left written language and many Asian writings are vertical! The more I think (and talk and write) about the "upside-down" chart, the more I convince myself that this is the way to go. *EDIT: I did make a poster and have been using it in my classroom for a number of years now. Here is a link to my version.*

(For a little more on this type of chart, see Hundreds chart - Thinkmath and Hundreds chart - Guruparents.com.)

As mentioned in the note below, I was not looking forward to "wording" about why the upside down chart might be better. I think it was because, though I had come up with the idea many years ago, I had never pursued it or even thought deeply about it, for exactly the reasons my colleagues had presented: "nobody else was doing it". Today, I am ashamed that my passion for teaching has not led me to take more initiative in instigating "maverick" practices. I regret that I have succumbed to the pressure to increase test scores, teaching breadth at the sacrifice of depth. I have been afraid to "rock the boat" (though some of my friends may not be convinced of this statement) and so even when I have stepped away from the scripted curriculum, I have always kept a toe on the safety of the pier.

After writing about this, I believe I will introduce the "upside-down" chart to my students, if not this year (with only 6 weeks to go) then surely at the beginning of the next school year. I think I will also try harder to impress on my colleagues how useful it is in the hopes of convincing them to implement it as well.

As always, comments are appreciated and encouraged.

NOTE: I was SO not looking forward to writing this entry because I didn't feel I would be able to explain my position eloquently enough. I only considered writing it after I discussed the subject with teacher friends who thought I was making some kind of coherent statement about this idea. Then I found the web pages mentioned above and a great weight was lifted! Seeing it in print (even if it was only digital print) kind of proved (at least to me) that I wasn't a complete nut-job.

* - Number sense, to me, is the understanding that numbers (not to be confused with numerals) work together in ways that make sense and follow logical rules. This applies to amounts. A child who looks at group A:[*****] and B:[* * * * *] and thinks B is more because it is "bigger" does not have number sense. It also applies to operations: a child who understands that "7 + 8" is the "same" as "8 + 7" has number sense. There are many more facets, but you get the idea. Visit some of my online exercises to develop number sense.

(Last modified:09/20/2014)

## Comments

teaster- 04/21/2011I commend you. I am teaching fourth grade this year but originally I have taught middle school math. My fellow teachers think that basic fact memorization will cure all of their math woes with kids. I have been stressing that a keen sense of numbers is far more helpful and a universal strategy to use in solving problems. I was excited to find your proposal for an upside down hundreds chart and your number sense activities reflect what I am now trying to incorporate into my classroom. Thank you for taking a stand for kids.

Barry Chlypavka- 09/20/2014This is a great idea,did you ever put one together or find one? If not I may do it myself.

Mr. Flores- 09/21/2014Actually, I did make one immediately after writing about it. I used it in my math intervention classes (4th and 5th graders), and later when I moved back down to 2nd grade, I used it with them. Presumably, they had used a traditional chart in 1st grade, but they had no issue using the "upside down" one. I was able to get my fellow second grade teacher to also adopt it. With the new Common Core State Standards emphasizing number sense and foundational concept building, maybe we can get more to join the movement!