Kindergarten, Measurement and Data

 

Opinion – Back to school, catch me if you can

The challenges within K-12 education are immense in the best of times (i.e., in the absence of pandemics), but in the coming school year, beginning in the Fall semester of 2020, they will be greater than ever. In addition to the uncertainty surrounding whether and how instruction will be delivered – in traditional or modified classrooms, in “normal” or staggered sessions, on-line (synchronously or asynchronously), or some combination of these various options – course content will also be less predictable. It is common knowledge that not all students were able to receive, let alone master, all the intended lessons of the Spring 2020 semester. However, the prevailing wisdom of the education community at present is that course content should reflect the current grade level, rather than remedial lessons, and that any needed “catch-up” material will be provided “just in time.” How well will this work? Time will tell.

What do these observations imply for parents and children? If at all possible, keep up with the grade level. Assume that we want our children to master to the full range of K-12 CCM. No matter how comprehensive (or sparse) the in-class instruction might be, as coaches, we need to assure that our children learn the standards of the current grade level. If they are able to “keep up,” there will be no need to “catch up.”

CCM Lesson of the Day: Measurement and Data (K.MD)

Describe and compare measurable attributes.

Describe measurable attributes of objects, such as length or weight. Describe several measurable attributes of a single object.

Comment: Again, I am struck by the extent of the vocabulary used to describe these K-level lessons. Be sure to take a look at the definitions below, but I leave it to your judgment when it will be most helpful for your children to learn a word such as “measurable.” Having said that, measures and measurability are distinguishing characteristics of mathematics. As mentioned in an earlier post, measurable attributes allow us to compare objects “objectively;” for example, we should be able to agree on which of two children is taller. On the other hand, attributes not so easily measured, such as beauty, might find us in disagreement over whether one fabric is more beautiful than another.

Directly compare two objects with a measurable attribute in common, to see which object has “more of”/“less of” the attribute and describe the difference. For example, directly compare the heights of two children and describe one child as taller/shorter.

Comment: I suggest that you ask your K-level student to do this exercise. If there is only one child around, compare the heights of adults, or the lengths of pieces of paper, etc. By performing the exercise, your child will acquire the concept of measurement and an ability to measure.

Classify objects and count the number of objects in categories.

Classify objects into given categories; count the numbers of objects in each category and sort the categories by count.

Comment: apples and oranges?

Definitions

Object: for our purposes, an object is a thing, such as a table or a dog, that has attributes.

Attribute: an attribute is a characteristic of an object.

Measure/Measurable: an attribute is measurable if we can ascribe a quantity to it. For example, a table has the attribute of height, which can be measured, normally using units of length such as feet and inches or meters.

Compare: We can compare to objects with respect to a common attribute, for example the number of apples in a bowl.

Less/more: In comparing the amount of oatmeal in bowls 1 and 2, when there is more in bowl 1, there is less in bowl 2.

Category: a category is a set or a logical grouping. For example, when counting trees, we may want to put them in two categories, such as deciduous and evergreen.

Kindergarten, Number and Operations in Base Ten

 

Opinion – CCM Language Matters

CC Math asks children in Grade K to learn a lot, and language, not normally associated with our youngest students, is used to group specific skills. While Grade-K students need to learn the skills, they may neither be ready for nor need to be aware of the more general categories.  However, since they are being set up for further learning in these areas, it makes sense to me that we, their parents, became familiar with those terms because first, our kids just might ask us about them and second, we will begin to see the big picture ourselves, that math topics such as counting, arithmetic, algebra, geometry, etc., are closely related. If we show no fear, our children are less likely to be afraid.

Repetition has an important place in education, so let’s remind ourselves of a couple of definitions that we have already encountered. Keep in mind that we adults want to know what these terms mean; our K-level students are unlikely to know or care.

The cardinality of a set is the number of elements in that set.  The set {dog, cat, chair} has a cardinality of 3.  The set {3, 6, 9, 12} has a cardinality of 4.

An algebra consists of symbols and the operations indicated by those symbols.  For example, a simple algebra might consist of {+, -, *, /, superscript, =}

Where  A+B = the sum of A and B

A-B = the difference between A and B,

A*B = the product of A and B

A/B = the quotient of A divided by B

A^B = A raised to the power B (e.g., 3² = 3 squared = 9) and

A and B are whole numbers (e.g., 1, 0, -7, 22, etc.).

Here are two new terms that will help us think about numbers but will probably elude all but the most precocious K-Level students, base and place. The base of a number system refers to the number of symbols in that system.  A decimal system uses ten symbols {0, 1, …, 9}; an octal number system uses eight symbols {0, 1, …, 7}; and a binary number system uses two symbols {0, 1}.  If we write those symbols in a string, place refers to the position of each symbol.  A number written in any base assigns a weight of one to rightmost place, the base itself to the place second from the right, the base squared to the place third from the right, etc.  Confused? See the examples in today’s lesson below.

CCM Lesson of the Day: Number and Operations in Base Ten (K.NBT)

Work with numbers 11–19 to gain foundations for place value.

Compose and decompose numbers from 11 to 19 into ten ones and some further ones, e.g., by using objects or drawings, and record each composition or decomposition by a drawing or equation (e.g., 18 = 10 + 8); understand that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones.

Comment. I am fairly sure that this makes sense to you, but it is a sophisticated concept. Instead of having to mark off eighteen 1s, I can simply write 18 (in base ten) to represent eighteen items. Therefore, I would not expect every K-level student to grasp it immediately. Moreover, while the number “eighteen” does capture, in a single word, the combination of eight and ten, numbers like 11 and 12, which we refer to as “eleven” and “twelve” are more obscure. Once again, K-level students are being asked to learn counting and the concepts of base and place at the same time. Repetition and memorization must be part of this learning process. I believe that it is worth allowing children the time to learn these initial lessons because they will be important throughout their math education. I suggest that you not be impatient to move ahead before they appear comfortable with these ideas.

Definitions

Base: The base of a system refers to the number of symbols in that system.

Place: The place of symbol refers to its position (in the string of symbols which comprise a number)

Decimal: A decimal system has base 10 and uses the symbols 0,1,2,3,4,5,6,7,8,9.

Examples

In the examples which follow, a subscript is used to indicate the base.

Example 1: 234₁₀ = 2 * 10² + 3 * 10 + 4 = 200 + 30 + 4

Example 2: 234₈  = 2 * 8² + 3 * 8 + 4 = 128₁₀ + 24₁₀ + 4₁₀ =  156₁₀

Example 3: 10110₂ = 1 * 2⁴ + 0 * 2³ + 1 * 2² + 1 * 2¹ + 0 * 2⁰

                                = 16₁₀ + 4₁₀ + 2₁₀ = 22₁₀

                                    = 2 * 8 + 4 + 2 = 20₈ + 6₈ = 26₈

Kindergarten, Operations and Algebraic Thinking

Opinion – CCM grouping, helpful or intimidating

At first glance, the language surrounding this domain is far beyond anything that you and I were exposed to when we were in Kindergarten. Whom is such language for?

“Algebra is the study of mathematical symbols and the rules for manipulating these symbols.” (Wikipedia)

Most of us parents were introduced to the subject called “algebra” in middle school (or its predecessor, junior high school). As the above definition shows, an algebra is a set of symbols and the rules for using them. However, our K-level children are going to be learning the most elementary arithmetic here, not the algebra (per se) that we all love (or not). What gives?

The heading of this CCM standard makes it explicit that, starting at the Kindergarten level, various areas of math will be put into general domains. Rather than being a generality that K-level students need to master, calling it “algebraic thinking” allows CCM to group arithmetic operations under that heading. We will see other examples of this type of grouping in upcoming posts.

Will this particular grouping offer an advantage when the algebra that we all know (or have long forgotten) is introduced later in the program? The originators of CCM thought so; for the rest of us, especially our children, time will tell.

CCM Lesson of the Day: Operations and Algebraic Thinking (K.OA)

Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.

Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.

Comment. To me, this is not clear. Represent means to set up the addition (or subtraction) problem. For example, start with two bowls of apples; count the number of apples in each bowl. I imagine that there are 5 in one bowl and 3 in the other. Then, add them together by combining the contents of the two bowls into one bowl.  Solve the addition problem by counting the number of apples in the one bowl; in our example, that would be 8.

Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.

Comment. We explained above how to represent and solve an example addition problem. Let’s decribe how to represent and solve a subtraction problem. From the same bowl (of 8), remove 2 apples. Then count the number of apples remaining in the bowl – I expect that there will be 6.

The larger lesson here is that addition and subtraction are extensions of counting. I suggest that you make sure that your child has mastered counting before taking on addition and subtraction.

Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).

Comment. The bowls and the apples can work for you here. Start with the number you want to decompose, such as 8, in one bowl. Then, move some number of those into the second bowl. The contents of the two bowls represent your decomposition. Notice that “+” and “=” signs were slipped in above.

For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation.

Comment. Decomposition?

Fluently add and subtract within 5.

Definitions

Add: Combine two numbers, represented by “+”. Example, 3 plus 4 equals 7 which is generally written as 3 + 4 = 7.

Subtract: Remove a number of objects from a quantity, as represented by “-“. Example, 7 minus 4 equals 3 which is generally written as 7 – 4 = 3.

Decompose: Divide a quantity into two or parts. Example, 7 can be decomposed into 6+1, 5+2, and 4+3.

Fluently: from memory?