Kindergarten, Counting and Cardinality 3/3

Opinion – Comparisons in general

The idea of comparing runs deep. We can compare objects that we can measure. Of these two stones, which is heavier, and which is lighter? Two considerations here: we’re only talking about two stones, heavier is the opposite of lighter, where the heavier stone is superior with respect to weight and the lighter stone is inferior. Who cares? We all do because we compare objects all the time. You are taller than I am; you are younger than I am; etc. Help your child to understand what it means to compare two people with respect to a measure such as height or age.

What if we want to discuss more than two stones? Which of these five stones is the heaviest? In this case, heaviest is a superlative, heavier than all others.

One caution: some attributes are difficult to measure; we say they are subjective. Which of these designs is the prettiest? Which of these patterns is most suitable? Subjective attributes, while they may lend themselves to comparison, are not part of math per se. However, it may be possible to measure which pattern is more (or most) popular.

A lot to think about here and we’re still in kindergarten.

CCM Lesson of the Day: Counting and Cardinality (K.CC), 1/3/3

Compare numbers.

Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group, e.g., by using matching and counting strategies.

Comment: Ah yes, definitions. Greater than (or “more”), Less than (or “fewer”), and equal to (the same number). Take some time with this because we are mixing words in with the math. I believe that this is where many people learn to fear and hate math. The problem may be that they never learn the definitions. If English is not your first language, you may want to devote a little extra time to definitions, because in the USA, English will almost certainly be the language in which your child learns math.

Compare two numbers between 1 and 10 presented as written numerals.

Comment: I assume that this means to use the symbols, 1, 2, 3, … Looking ahead, you might want to introduce three more symbols here: < (less than), > (greater than), and =, as in 1 < 3, 3 > 2, and 2 = 2.

Definitions

Equal: of the same quantity or amount. For example, I have an equal number of fingers on each hand. As mentioned, equal can be written “=”.

Greater (than): having a larger number or amount (than). For example, 4 is greater than 2, which can be written 4 > 2.

Less (than): having a smaller number or amount (than). For example, 2 is less than 4, which can be written 2 < 4.

Kindergarten, Counting and Cardinality 2/3

Opinion – How to fight back against the invasion.

COVID-19 has invaded our planet. It’s making some people very sick and killing some of the most vulnerable. It’s almost time for children to return to school in the USA, but parents and teachers are afraid. Their fears are real because we are fighting the virus with too few tests, too little ability to trace contacts, and no vaccine.

So, what do we do? It may be true that the children themselves are at low risk – the data are not yet conclusive on this point – but as with other flu bugs, children are quite capable of bringing the virus home. At a minimum, to send our children off to schools and indoor classrooms, we need reassurance that health guidelines – masks, daily testing, distancing, ventilation, time limits, hand washing, etc. – will be followed. Moreover, we need some confidence that schools are actually equipped to implement those guidelines. However, some level of risk will remain and any breakout at your school could lead to a shutdown. The probability is rising that your child will receive online instruction during the next year or two. Are her teachers ready to deliver that instruction effectively? Do you and the school have the technology you need to support that plan?

Any way we look at it, real learning is likely to suffer. So how do we prepare for the drought? If there is no water on our hiking trail, we must plan ahead to be safe; we must carry our own water. If schools are unlikely to reopen soon or to connect effectively with our children online, home-schooling is one option, but it’s not right for everyone for various reasons. Some charter schools, because they operate under different constraints, have shown themselves to be effective distance-teachers – I leave that research to you. It appears to me that at a minimum, you should consider stepping up, at least in the area of math, as your child’s coach.

Make sense? Let’s continue our journey; we’ve only just begun.

CCM Lesson of the Day: Counting and Cardinality (K.CC), 1/3/2

Count to tell the number of objects.

Understand the relationship between numbers and quantities; connect counting to cardinality.

When counting objects, say the number names in the standard order, pairing each object with one and only one number name and each number name with one and only one object.

Opinion: That’s a lot of words to describe counting. My suggestion: just count stuff. Count fingers, toes, apples, Cheerios, … Then ask, “how many?” Count numbers. That’s right, count 1, 2, 3, 4, 5. If I do that, no matter where I stop, I have counted that many numbers, 5 in this case. This will help to relate the symbols to the words and to illustrate cardinality (see the definition above).

Understand that the last number name said tells the number of objects counted. The number of objects is the same regardless of their arrangement or the order in which they were counted.

Comment: Your child will get this. Count the fingers of one hand from small finger to thumb. Then count them from thumb to small finger. Any bets on the results?

Understand that each successive number name refers to a quantity that is one larger.

Comment: Fingers can be useful here too. As we count, we can interject “plus one.” “One plus one is two; two plus one is three; three plus one is four.” This sets the stage for addition.

Count to answer “how many?” questions about as many as 20 things arranged in a line, a rectangular array, or a circle, or as many as 10 things in a scattered configuration; given a number from 1–20, count out that many objects.

Comment: Sure, ask your child how many apples are in the bowl. Then, ask your child to give you three apples. This will provide a head start on comparing.

Definitions

Ordinality: A number indicating position in a series. For example, consider the series of characters {a, b, c, d, e, f, g, h, i}. The cardinality (feel free to revisit the definition from the previous post) of the series is 9. The ordinality of the letter ‘f’ is 6. Why, you may ask, is this important for kindergartners; actually, I don’t think it is. However, as a parent, you need to understand CCM language. But wait, you didn’t even see the word ordinality above; very true, but if we’re going to discuss cardinality, we really should mention the complementary term, ordinality.

Plus: Plus indicates addition, as in 2 plus 1 is 3. Admittedly, I introduced this term for convenience, but we will need it later any way.

How many? What is the number of objects in a set, as in the number of apples in a bowl? This is just one more example of your child learning English at the same time he is learning math.

Kindergarten, Counting and Cardinality 1/3

Opinion – time to launch

There is a certain satisfaction in following a path to math competence, even though we may not recognize the path. We probably came this way, but we may not have “seen the forest for the trees,” or vice versa. In any event, pointing out a few trees and flowers to students will be satisfying. And as we teach, we will notice that we are also learning. This phenomenon has already been noted, but there is no substitute for the experience. A couple of points to remember: showing and doing count for more than just telling; getting students to show us and do for themselves will be as important. Practice, practice, practice, but the practice should be fun, and when attention begins to stray, that’s usually an early warning that practice is over.

If your student is following a formal course with a professional teacher, try to coordinate these posts with the pace of the class. The teacher should be able to help. Don’t worry if that doesn’t come naturally. There is so much depth and breadth in math that your child is going to be learning no matter what.

If you are home-schooling your child, find his rhythm – neither too slow (he is bored) nor too fast (he is confused). If you don’t wind up exactly where and when the formal classes do, remember that it will be the quality of learning – an appreciation and an affection for math as well as a reasonable amount of retention, especially for problem-solving processes – that will ultimately carry the day.

CCM Lesson of the Day: Counting and Cardinality (K.CC), 1/3/1

Notice that there are four parts to this lesson. Each part will take some time, but you are the coach. You can let your imagination, experience, and observations (of your child) be your guide for how fast to go and whether to combine any of these sections. Make no mistake though, there is a lot to learn.

Know number names and the count sequence.

  Count to 100 by ones and by tens.

Borrowing from an earlier post, consider these complications in our number system:

  - Where do the names of numbers such as ten, eleven, and twelve come from?

  - Why does the spelling change from three and five to thirteen and fifteen?

  - And from two, three, four, five to twenty, thirty, forty, fifty?

  - And then there is zero – is that even a number?

The point is that our number system, from the perspective a child who doesn’t yet know it, is quite random. Therefore, learning the number system is an exercise in memorization. It is very interesting to me that memorization, which some educators devalue, is thrust upon us at the very start of the trail to math competence. That leads me to simply accept that memorization plays a prominent role in learning math just as it does in learning any academic subject.

Yes, there are patterns that will become visible over time, but there also many exceptions to the way we say the numbers in English. I suggest that you and your child practice saying the numbers in order, out loud every day until they are embedded in that formative brain. There is no reason that fingers and toes (or blocks, beans, marbles, etc.) should not be recruited into the process but don’t worry, future lessons will ascribe symbols and meaning to the numbers.

   Count forward beginning from a given number within the known sequence (instead of having to begin at 1).

This exercise helps to reinforce the memorization and to reveal patterns (and pitfalls) as indicated by some of the random aspects above. Don’t forget to count by tens because it is not obvious that after learning “one, two, three,” we would say “ten, twenty, thirty.” That’s just the way we count around here, but I’m sure that took you and me some time and practice to learn when we were in Kindergarten.

   Write numbers from 0 to 20. Represent a number of objects with a written numeral 0–20 (with 0 representing a count of no objects).

This is where we start to link written characters, so-called Arabic Numerals, to numbers. Note that we start at 0 and that the difference between writing 1 and 10 is significant, even though the written numbers only differ by a 0. Most of us have 10 fingers and 10 toes. Any ideas on how to use that fact as part of the teaching the number of objects represented by each and every number from 0 through 20?

One key here is that the ten decimal digits (0, 1, 2, … , 9) must be memorized; no exceptions. A second key is that 10 through 99 are represented by two decimal digits. As we memorize these numbers, there should be many opportunities to point out examples in documents, newspapers, signs, etc. Let your children know that you like them to do this.

Definitions

Count: Count (the verb) means to enumerate the cardinality of a group (or set) of items.

Cardinality: 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.

Example

When I count the number of fingers on one hand, the result is 5 (fortunately for me).