If a student responds on an end-of-third grade, world-class assessment that 704 — 356 = 348, then that’s an indication of being on track to graduate from high school on par with students from high-achieving nations. This question involving subtraction within 1000 also indicates that the student meets Power-Start Goal #1 (sense number size & order); Goal #2 (know addition and subtraction facts); and Goal #3 (understand place value).

**Arithmetic of the 20th and 21st Centuries**

In 20^{th}-century school mathematics, most students would probably have been taught only one method for solving this problem, a right-to-left process using place value to “regroup” or “borrow” from other columns to the left. This method is known as a “traditional algorithm,” like the one pictured for subtraction here. An algorithm is a pencil-and-paper process for performing an operation. With an algorithm, one uses the same procedure regardless of the values of the numbers. Furthermore, throughout the 20^{th}-century a majority of teaching time in elementary school years focused on practicing traditional algorithms with pencil and paper. Now that we have abundant 21st century technology, operations with three-digit numbers are probably performed more efficiently by using a calculator. Even so, a deep conceptual understanding of subtraction – knowing when to use it, as well as knowing when to capitalize on an efficient mental strategy – is an ability of mathematically proficient students. Below is a visual representation of 3 strategies for solving this subtraction problem other than the traditional algorithm: a count-backward, number-line strategy; a repeated-subtraction algorithm; and a base-ten-block model emphasizing place-value concepts.

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

**Where might we start in preschool to prepare ****children for this test question?**

** I. Number Sense: Size and Order**** (Goal 1)**

Verbal counting forward and backward helps children start early in developing a sense of size and order for numbers. You can do this riding in the car or almost anywhere. Count objects or just count aloud yourself to help the child get started and make counting sound exciting and fun.

- Count forward and backward from 1 up to 10 and also from 10 down to 1.
- Introduce the idea of zero and include it in your counts forward and backward.
- Skip count by 5s (5, 10, 15, 20, …) and by 10s (10, 20, 30, 40, 50,….)
- Practice start-and-stop counting. For example, count forward from 4 to 8 and then backward from 8 to 4.

When children begin to recognize numerals, make “counting tapes” from a cash register ribbon. Counting tapes differ from “measuring tapes” because a measuring tape includes lots of numbers between any two consecutive whole numbers. Use arrows at each end to suggest that more numbers exist on either end of the tape you have created. Counting tapes can start and end with any number. Make tapes that leave blanks and have children fill them in. In my drawing I have left two blanks below zero. Consider doing this at some point if children seem interested in numbers less than zero — negative one, negative two, etc. Also, as children learn how to write numerals, encourage them to make their own tapes.

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

**II. One-Digit Addition and Subtraction (Goal #2)**

So that children can master the one-digit addition and subtraction facts before leaving third grade, begin early showing them concrete examples of addition and subtraction situations, such as a plate of cookies. Solving this problem produces algebraic understandings as well: If 4 + x = 7, then x = 3. Give preschoolers numerous experiences with a variety of one-digit problem types for addition and subtraction. Use everyday examples or make up story problems. Try making up addition and subtraction problems relative to the storyline of a favorite picture book. Here are some examples of four different types of addition and subtraction scenarios.[1] Encourage children to draw pictures or use concrete materials to explain their thinking processes.

____________________________

[1] The story problems presented for you here are 4 different types on an addition and subtraction learning trajectory from the work of early math researchers Dr. Douglas Clements and Dr. Julie Sarama. I learned about their methods for classification of addition and subtraction problems from their book, *Learning and Teaching Early Math: The Learning Trajectories Approach* (2009; Routledge Publishers)

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

**III. The Big Idea of Ten-as-a-Unit (Goal #3)**

Once children are learning to count to 20 and above, show them how easy and fun it is to use base-ten patterns in extending their counting skills to 100. Display a 100-grid poster in a place where children can have sustained interaction with the important foundational idea of ten-as-a-unit presented on the grid. An understanding of unitizing with ten leads naturally to later extend thinking to 100-as-a-unit (356 is 3 hundreds, 5 tens, and 6 ones). Avoid the tedium of saying every number by reverting the count to a skip-counting-by-tens process:

* 10, 20 , 30, 40,….100,…*

It would be really great if children entered kindergarten counting being familiar with counting numbers up to 100. Deep understandings result when a child can connect a number’s name in words with the number’s name in words; the number’s name in symbols; and a concrete representation of the number. The following illustration shows multiple representations for 43.

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

After revamping my website, this is the first blog post (June 18, 2018). I plan to publish my second post next Monday, June 25. It’s named “Trading Cards” and will be about memorizing the multiplication tables by the end of third grade. For each post, I would greatly appreciate your questions and comments.