I've always known that my students struggle with number sense and if I could improve their number sense that would increase their understanding of math in all other areas. But how to do that when my students are already in 8th grade? I can tell you the students in my classes that have strong number sense and which students don't. I don't need to look at a test to give me that information. My students with good number sense can pull numbers apart and put them back together in a number of different forms. For example, they know that 25 is the same 5 X 5, 10 + 10 + 5, 10/2 and √ 625. They can recognize number patterns and can spot errors easily. They understand the order of numbers, can compare them and place them on a number line. They just "get" numbers where my struggling students just don't "get it" when it comes to understanding numbers. I know the math, I' m a good teacher so why can't I help those struggling students? What more can I do? I need a template with a step by step process to use with my kids to develop these skills. Some type of check list or something like that to refer to so I am making sure I am covering all of areas the kids needed.
So when I was doing my Action Research Project for my masters, I knew that I wanted to research how to improve number sense in my students. Most of the reading that I found on teaching number sense stressed the fact that you need to start young with developing these skills, giving these young children the opportunity to use manipulatives to understand the different forms of a number. I totally agree with this. However, my students are in 8th grade not pre-school so that doesn't help them at all.
- Well, I finally found what I was looking for when I stumbled upon this article by Valerie Faulkner, The Components of Number Sense, an instructional model for teachers. (Read it, I found it fascinating.) In this article, number sense is divided into seven categories,
- quantity and magnitude,
- numeration,
- equality,
- base ten,
- form of a number,
- proportional reasoning and
- algebraic and geometric thinking. I love the chart that she gave. I am going to print this up and hang it in my classroom as a reference for not only the students but also for me. It will remind me to discuss all of these different forms as many times as possible each period. These are discussions that can be made in every math lesson and are applicable to the middle school classroom. I am going to work to address as many of these categories as possible in each and every lesson.
Here is a brief explanation of the different categories to address with students on a daily basis.
Quantity and Magnitude
Students need to understand that math is about quantity not just
numbers. It is the amount of something whether it is a weight, measurement,
sets of elements or symbols. These are all ways to show quantity. It is
important for students to understand the magnitude of numbers and what they mean in terms of a
number line. I am always amazed when in 8th grade students can't place numbers correctly on a number line. This is very obvious when we get review negative numbers and the number line. Is this something I need to spend more time on with them? They should be able to show where quantities are positioned on the
number line with placement to the right being a higher magnitude and to the
left on the number line a smaller magnitude.
Base Ten
Students need to know that the numbers that we use are based on ten. To understand 16 students need to know that the one stands for one-ten and the six stands for six ones. This is a pre-requisite to then be able to compare numbers according to their magnitude. I need to ask for expanded forms of numbers and again watch the language that I use when teaching sections with tens. When I teach scientific notation I should say the number is in expanded form and have a conversation about this form. I need to stop giving shortcuts, they will figure those out, and have deep discussions about the forms of numbers. Understanding base ten leads to better estimating skills. When the student is a good estimator (Estimation 180 site, love it.),they will be able to use many different forms of numbers to show their understanding of the problem.(Faulkner, 2009, p. 28)
Numeration
Numeration is the act of numbering, counting, or computing. Once
again base ten is crucial to the understanding of the numeration system. I need to take time when discussing numbers and say 23 is two-tens and
three-ones and when adding and subtracting to include this type of discussion
as well. Reflecting on my discussions in class, I am not sure that I spend enough time on this, assuming that my students understand this already. Again, I can't tell you how many times my students just stare at me as if to say I am speaking another language when we discuss the value of a number. 56 is not 5 and 6, it is 5 tens and 6 ones. In a class I took last month, the teacher said when she would ask students to show her the value of a number using base ten blocks, they would take 2 tens and 5 ones for 25 and say they value was 7!
Equality
Equality does not necessarily mean the same as. Students need to
understand that numbers have different forms with the same value. For example:
2/3 = 4/6 are equivalent but represent two different forms with the same value.
Two out of three items and four out of six total items are different
representations of the same value. (Faulkner,
2009) Teachers need to be aware of the language that they use with students
instead of just showing the steps involved in solving a problem. “Two trucks
may be equal in weight to an elephant, but they certainly aren’t the same as!"(Faulkner, 2009, p. 26). (I love this analogy.)
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Form of a number
The different forms of a number should be shown to be equal in
value but shown in different forms. An example of 45% can be shown as 0.45 they
are equivalent but different forms of the same value. It can also be shown as
45/100 and can be written as a ratio as 45:100.
Proportional
Reasoning
When teaching, I should give a variety of activities that allow them
to develop the proportional analogy themselves. (Geometer's Sketchpad will be great for this.) An example would be to have students measure the circumference of various circles and divide the circumference by
the diameter and discover pi. This is a proportional reasoning activity that
will be more meaningful then just plugging pi into a formula and solving
without meaning attached to problem. (Faulkner, 2009, p. 27)
Algebraic and Geometric Thinking
I need to take time to “unpack” the concepts of algebra and
geometry and place them under the previous components listed above. Pi becomes
a proportional reasoning problem, X = Y is an equality problem showing
different forms of the value. (Faulkner, 2009, p. 25). Similar figures and dilations and the list goes on ...are other examples.
I love Andrew Stadel's site, Estimation 180, and I plan on using it next year with my classes. There is an estimation activity for the students every day of the school year. Having the students pick a reasonable guess that is too high and too low is a great way to have students understand the ordering and magnitude of numbers. I can't wait to see the improvement in their estimating after a full year of doing this program.
My goal for the upcoming school year is to use these seven components throughout the year the so students will have the building blocks of number sense. I am going to be mindful of the language that I use and give my kids more opportunities to use manipulatives to get a deeper understanding of the meaning behind numbers. The great thing about blogging is I know all of you will hold me accountable for this. Thanks. I'll try to regularly update this post.
Til next time,
Jan
Click HERE to see activity sheet and lessons from NCTM.
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