Developing Multidigit Number Sense

Cn you find a reasonable solution for the question mark?
                         Find a reasonable solution for the question mark

Lately I’ve been reading David Sousa’s book and have come across some interesting (at least to me) observations.  Humans have an innate ability to subitize.  That subitizing can lead to estimation and through appropriate practices this leads to bettering number sense skills.

After speaking with a few middle school teachers this summer I’m finding that one area that tends to need strengthening is found in the overarching umbrella of math skills: number sense.  Without adequate number sense skills, students flounder when asked to complete higher level math concepts.  Year after year I find some students have a decent understanding of procedures, but fall apart when it comes to explaining their reasoning for completing particular math problems.  Students are able to identify and repeat processes (usually copying the teacher), but not understand why they’re using that strategy.  I find this particularly a concern when students don’t question the reasonableness of an answer.  In my mind, I think finding a reasonable answer or estimation shows a form of number sense.  This isn’t always the case, but I tend to find it when students blindly follow only a set of procedures. I believe that this can lead to more problems down the line.

This is especially a problem with larger numbers. Researchers Carmel Diezmann and Lyn English found that the activities below seem to help students develop multidigit number sense.  All of these activities can be used at the elementary or middle school levels.  The headings in bold are found in David’s book and highlighted in Diezmann and English’s research.  My narrative is below each heading.

Reading Large Numbers

Placing an emphasis on place value when reading large numbers is important.  Being able to identify and see the value of each digit can help students read large numbers more accurately.  I find that students in kindergarten and even first grade start to combine place values when speaking of large numbers (e.g. one hundred million thousands).  Giving students opportunities to take apart these large number by digit value can help reduce this issue.  I find comparing standard and written forms of numbers can also help students start to recognize the place value of large numbers.

Develop Physical Examples of Large Numbers

Visually observing 100 and 1,000 dots can show students the physical difference between large numbers. There’s so much math literature that can help with this. 100 Angry Ants, Really Big Numbers and How Much is a Million can be used with manipulatives to show examples of large numbers. If you teach in an elementary school, I recommend adding these books to your math library. Using unifix cubes, base-ten blocks, or replicas can also be used to show a physical example of large numbers.  Having the physical example can benefits students as they start to question if their solution to abstract problems are reasonable.  A number line with 1:1 correspondent is also one way to showcase large numbers.

Appreciating Large Numbers in Money

Kids tend to like to talk about money.  Showing how $1 compares to 100 $1 bills can show students a visual scale between the amounts.  Visual representations of money in dollar and coin forms can lend itself to having students become more aware of how place value impacts the value.  A problem that tends to always get students curious relates to how much money will fit in a briefcase.  Will $10,000 in $5 bills fit in a 20″ x 18″ briefcase?  These types of questions can have students start visualizing money and the reasonable of their answers.

Appreciating Large Number in Distance

Maps can be useful here.  I remember having students use Google Maps to calculate the distance from one particular destination to another.  Also looking at the distance from one continent to another, or even from Earth to another planet.  I find that a good amount of scaffolding is needed to help students experience large numbers in distance/measurement.  Comparing skyscrapers to distances can also play a role with this as well. If you stacked one Willis Tower on top of another, how many would it take to reach the moon?

They’re many ways to have students observe and interact with large numbers.  I’d like to add appreciating distance in relation to time to the the list.  Time can also be used highlight and compare large numbers. I’m thinking of the dates in history and the Science involved in evolution. Many of these activities can be interdisciplinary as connections between curriculum content exist. Digital and physical forms play a role in having students conceptualize an understanding of large numbers.  Students should be given opportunities to recognize large numbers in a  variety of contexts.  By doing so, I believe students should be able to better question whether their answers are reasonable or not.

By the way, the answer to the top image is 1,000 dots.

The Real Number Line

Image by Winnond

Approximately two weeks have passed since the new school year has started and I’m finding that the traditional number line (that many teachers have become accustomed to) needs an upgrade.  My math students are benefiting from the number line, but true understanding of numbers doesn’t come from a number line alone.  For the past seven years I’ve used a “typical” number line from -10 to 100 in my classroom.

Don’t get me wrong … the number line is helpful in teaching many number sense concepts.  In my opinion, the number line offers students a visual/spatial representation of the number system.  I  believe many numeracy concepts are built from understanding the system of numbers.  What is often missed, or not necessarily taught, while utilizing the number line are numbers that don’t fit the category of being whole.  For example, I generally don’t see pi or irrational numbers being part of a number line.

Recently I found a “Real Number Line” poster.  I was fortunate enough to find this poster and have utilized it to teach elementary students about the number system. I think it’s important to communicate that square roots, fractions, percentages, mixed numbers, etc.  should be included on a number line.

I actually created a practical follow up activity in response to this post here.

Instead of purchasing a poster, you could have the students create their own.  A few examples are found below:


I believe that Wolfram Alpha does an excellent job of emphasize the importance of a number line in the answer it provides.  The answer can be represented on a number line.  See the example below.


I’ve been reading How the Brain Learns Mathematics by David Sousa.

David emphasis the importance of the mental number line.  All humans have number sense.  For example:  studies indicate that the brain can decide that 60 is larger than 12, but it takes the brain a longer time to distinguish that 76 is less than 79.  It seems that when the digits are closer in value the response time of the human increased.  Visualizing many different forms of number lines would be beneficial and assist in developing better number sense skills at a young age.

 I thought this quote was beneficial:

“The increasing compression of numbers on our mental number line makes it more difficult to distinguish the larger of a pair of numbers as their value gets greater.  As a result the speed and accuracy with which we carry out calculations decrease as the numbers get larger”

 – David Sousa