PUFM 1.1 Counting

Photo by Iain Watson via flickr. In this Homeschooling Math with Profound Understanding (PUFM) Series, we are studying Elementary Mathematics for Teachers and applying its lessons to home education.

Many things in mathematics need to be understood relationally — that is, in relationship to other concepts. But some things just need to be memorized. How do you know which is which? A homeschooling friend pointed out that one thing children definitely need to memorize is the counting sequence from 1-100 and beyond. While there are some patterns that make counting easier, one does just have to memorize which “nonsense sounds” we have attached to each number.

Another sort-of counting that young students should master is subitizing — recognizing at a glance how many items are in a small group. Children do this instinctively, but we can help them develop the skill by playing subitizing games.

[Aside: In writing this blog post, I ran into some nostalgia. Back when we first did these PUFM lessons, my daughter Kitten was only a toddler. I wrote, “I’ve tried to do lots of counting with my youngest, who hasn’t quite gotten beyond, ‘…eleven, twelve, firteen, firteen, nineteen, seven,…’ The numbers tend to start appearing randomly after she gets past 10.” Ah, memories.]

Lesson 1.1 Counting

Mathematics is a language and a way of thinking. Early exposure is enormously beneficial.

— Parker & Baldridge
Elementary Mathematics for Teachers

I thoroughly agree with this. Don’t wait until your children reach school age to do mathematics with them. Begin early with oral story problems and hands-on enrichment activities.

Notice the two models for numbers: set model and measurement model. I would call these discrete and continuous quantities, but the idea is the same. Some things come as separate, countable objects, like fish or apples or children. Other things are continuous, such as applesauce or pudding or a child’s height. Be sure, when exposing your children to numbers or making up oral story problems, to include both types.

Our math textbook uses the term whole numbers, but this is not a well-defined math term. Some people mean the whole numbers to include zero, others mean to exclude it, and a few people use the term to mean all of the integers. If it matters (as in a middle-school math contest problem), you would be well-advised to state your definition or to use an unambiguous label, such as non-negative integers.

The Number Line

The number line is the only model that continues to work as elementary school mathematics progresses through whole numbers, fractions, and on to negative, rational, and irrational numbers.

— Parker & Baldridge
Elementary Mathematics for Teachers

A child’s mental number line starts out with an almost logarithmic scale. The small numbers are spread out, but the bigger the numbers get, the closer they crowd together. For very young children, “100” is almost the same as “infinity”. As our students become more comfortable working with numbers, their mental number line becomes more linear.

A number line is a valuable tool for many mental math calculations. And don’t neglect negative numbers, even with young children.

Numerals in Other Cultures

I don’t think the foreign number systems in the chapter are intended for teaching to young children. Singapore Math doesn’t even teach Roman Numerals, let alone Egyptian ones. These are fun enrichment activities for older children, however, and they’re great for helping teachers understand the importance of place value.

The simplest schoolboy is now familiar with facts for which Archimedes would have sacrificed his life.

— Ernest Renan

The main “fact” he means is how place value makes calculation easier, especially with large numbers. But as our textbook authors point out:

There is a price to pay for this convenience: decimal numerals are more abstract than Egyptian numbers. Place value is tricky to learn and causes many problems in the early grades.

— Parker & Baldridge
Elementary Mathematics for Teachers

Note on writing Egyptian numbers: An Egyptian scribe would never write more than four of any shape in a row. They stacked the rows to make bigger numbers. Five was a row of three and a row of two, and I think I remember that the longer row went on top. Six was three and three, seven was four and three, eight was four and four, and nine was three rows of three. These were all fit on the same line with the rest of the number, so the shapes got smaller as they were stacked this way, but it was MUCH easier to read a number at a glance than just putting all the shapes in one long row. So in Homework Set 1, problem 1b: 648 would be written as 3-and-3 squiggles, 4 humps, and 4-and-4 tally marks.

To learn more about Egyptian numerals and calculation, you might enjoy:

• Egyptian Math in Hieroglyphs
• Alex’s Puzzling Papyrus
• Egyptian Math Puzzles
• Another Egyptian Math Puzzle
• The Secret of Egyptian Fractions

Homework Set 1

As you do homework, bear in mind that the goal is not merely for you to do the problems, many of which are not hard. Instead, the goal is to think about problems from the perspective of a teacher. Teachers must be able to identify the key steps in solving a problem, so they can guide and prompt students. They must also be able to give clear, grade-appropriate presentations of solutions.

— Parker & Baldridge
Elementary Mathematics for Teachers

Normally, I will not be posting homework answers. But I enjoy story problems so much, I will make an exception for them. One of the things that most amazed me in Liping Ma’s book was how the Chinese teachers were able to come up with so many different stories to illustrate one calculation. I can see how keeping these models in mind would help that sort of creativity.

For number 4 of the homework, here are my stories:

• [set model] 7 ducks are swimming on a pond. 5 ducks fly in to join them. How many ducks are there now?
• [measurement model] Mom made punch. She mixed 7 cups of ginger ale and 5 cups of pineapple juice. How many cups of punch did Mom make?

For number 6 of the homework: I always let my children do problems like these orally, just reading the number to me rather than writing it out. I don’t require the written work until 4th or 5th grade. My kids take that long (some longer!) to get their fine-motor skills, which are required for handwriting, caught up with their math ability. To require extensive writing before their muscles are ready for it only leads to frustration.

That’s all I have for this section. I look forward to hearing your thoughts!