Replace the four question marks with the correct numbers in the table below.
The proficient third-grader uses arithmetic, and of course, knowledge that pigs have 4 legs and chickens have 2 legs to complete the table with 6 pigs; 9 chickens; 15 animals; and 42 legs. Another key skill for solving this problem is the mathematical practice of using information in a table (Power-Start Goal #7). Using tabular information to solve a problem can be about working with data, like my July 9, 2018 post about teachers’ bookshelves. Also, another important way of using tabular information is for algebraic thinking, that is, looking for patterns with numbers and their operations. The question marks in the table above announce entry into the algebra domain — the realm in which we work with numbers that are unknown. In later schooling, question marks are frequently replaced by letters such as x and y.
Algebraic Thinking for Preschoolers.
You now may immediately say to yourself, “Whoa! Algebra? — reading tables?– that’s way beyond the capabilities of a preschooler!” But, think about it a little more and you’ll recall how today’s children begin looking at screens from early ages. Accordingly, they are accustomed to getting information from visual images and not just from the voices of adults. Even so, screen time and looking at TV are passive activities and shorten attention spans. So, here, I’m showing you another table containing information about legs for pigs and chickens. The use of a table like this one might follow previous conversations about how many legs (or feet) different creatures have — humans, pets, farm animals, insects, dinosaurs, etc.
Teaching Preschoolers about Money
I also want to show you how you can use algebraic thinking to build foundations for understanding money (Power-Start Goal #6 ). Today, most monetary transactions involve numbers printed on a paper cash register receipt or even just on a computer screen. Even so, children’s understandings will be deepened by starting concretely with coins.
Algebraic Thinking in Middle School
I selected the pigs/chickens and the coins examples because they are typical of problems classified as “mixture problems” in a traditional Algebra 1 high school textbook. However, if U.S. high schoolers are going to attain levels of math proficiency commensurate with high-achieving nations, then algebraic thinking needs to be “kicked up another notch” in middle school. Nations with higher scores on international tests are teaching middle school students how to formulate equations from written information and solve the equation to get the correct answer. Now consider the pigs and chickens problem posed as a typical mixture problem from a traditional high school Algebra 1 textbook. This problem would now be classified as middle-school level on a world-class test:
A farmer has a total of 15 pigs and chickens and these animals have a total of 42 legs. How many are pigs and how many are chickens?
Here, are two tables one might use to think through this problem. Table 1, like the third-grade problem’s table, still has 4 question marks but they are in different places. A solution strategy for Table 1 might be a guess-and-check procedure: a guess of 8 pigs and 7 chickens means 46 total legs, which is a little large but close to the correct answer of 42. The second strategy begins with making a 3-column table for organizing symbolic expressions for the unknown quantities and ends with solving an equation.
Then, using the table, one can write an equation and solve for x.
4x + 2(15 – x) = 42
4x + 30 – 2x = 42
2x = 12
x = 6
So, 6 pigs and 9 chickens will have a total of 42 legs. This algebraic solution for the pigs and chickens problem is an important first step for understanding a mathematical modeling process. A middle schooler who can formulate algebraic equations to solve mixture-type problems is well on her way to math proficiency at the high school level and to a career in a STEM field (Science, Technology, Engineering, and Mathematics). Furthermore, mathematical modeling — describing phenomena symbolically with an equation, graph, etc. — is used extensively by statisticians and researchers.
Additive and Multiplicative Thinking
So, all of these examples suggest that when we think algebraically we must shift back and forth from additive thinking (the number of pigs, chickens, or coins) to multiplicative thinking (the number of legs for 4 pigs and the number of cents in two nickels). Children usually learn to think additively before they think multiplicatively. Thinking multiplicatively can begin with skip counting; repeated addition; experiences with row and column arrays; or considering area of rectangles. Here are a few suggestions for helping children move to higher levels of additive algebraic thinking.
- Adding or Subtracting One or More Say to the child, “I’ll give you a number and you tell me the next number and the number just before.” Then, ask about 2 more and 2 less. When looking at a 100-grid, children can think about algebraic patterns for adding and subtracting 5 and 10.
- Guess My Rule In this conversation let the child select the number, the adult produces the second number and the child determines the pattern.
- Hiding Games Hide a certain number of poker chips or coins under a piece of paper or in a file folder and ask, “How many am I hiding?” Then, add or remove a certain number and ask again.
- Make Sixes or Tens Produce 3 items using concrete materials and ask the child, “How many more do I need to make 6?” Then, reduce the number to 2 items and then ask again how many more are needed to make 6. Work up to patterns for making 10s. Knowing the possible algebraic patterns for making tens supports deep understandings of multi-digit addition and subtraction.