3. Numeric Symbols
This moving back and forth between the use of visual models and verbal language “has to happen for quite a while,” she said, so children develop a context for those number words and quantities. Only when they have a solid understanding of these words and quantities should they be introduced to the numeric symbols associated with them.
“When we’re teaching children about numerals, we have to be mindful of what experiences they are bringing into the classroom,” Bobo said. For instance, they might have learned or been exposed to non-Arabic representations of numerals, such as if they come from a Chinese household. Other common numeral misconceptions include…
- Confusing 1 with a lowercase “l,” uppercase “I,” or a vertical line.
- Confusing 6 and 9, 6 and b, or 9 and g or q.
- Confusing 8 with B, or a snowman or racetrack.
- Confusing 7 with an upside-down L, “greater than” sign (>), or an arrow tip.
- Confusing 11 with two lines, a road or train track, or a pause symbol.
- Confusing 5 with S.
Just as educators need to make connections between visual models and verbal language, they also have to form connections between both of these methods of representation and symbols, Bobo said. For example, they might use baby jars with numerals drawn on the lids, and ask children to place the correct quantity of items inside each jar.
4. Operations Confusion
When teaching basic math operations, the same logic applies. Consider how many misconceptions can arise if symbols are introduced too early: The expression “5 – 1” easily can be mistaken for a date, or the number “511” with the middle numeral lying on its side.
Teaching by Early Math Language Stages
There is a proper trajectory of language that educators should use to make sure students have a solid understanding of basic math operations, Bobo said. One reason ORIGO’s core math program for grades preK-6, Stepping Stones, is so successful is because it’s built on this research-based progression.
The first stage, student language, is the use of the child’s natural language. For instance: “There are two bunnies, and one of them hops away. How many bunnies are left?” Instead of using mathematical words to suggest subtraction, you’re speaking the way young children do and leveraging the knowledge they bring with them to school, using descriptive phrases such as “broke,” “lost,” “fly away,” “hop away,” and so on.
Once children have mastered the concept of subtraction in this first stage, the next stage is materials language. “These are the new words that are used to act out a story with resources,” Bobo said—such as “take out,” “remove,” or “cover up.”
The third stage, mathematical language, is where educators would introduce math terms such as “subtract” and “equals.” At each successive stage, the language narrows until students learn to associate the correct mathematical terms with the operation in question. Students would learn that “remove” is another way of saying “hops away,” and that subtract” is another way of saying “remove.”
“We are building them up to be successful when we get to the final stage, the symbolic language stage, and they see this,” Bobo said, showing the expression: “10 – 2 = 8.”
“For a child to see this without going through the proper trajectory of learning can be scary,” she noted. Without this progression, she explained, children might confuse the minus sign for a dash or hyphen, a horizontal line, a sideways 1, part of an email address—or even part of a plus sign.
“But if they’ve gone through the proper progression,” she said, “you can explain to them that this means ‘ten subtract two equals eight.’”
She concluded: “Just think about how many misconceptions students can have if you don’t take them through the proper trajectory of language. If we rush too fast to the symbolic language, how many conceptual gaps are we leaving the kids to swim through?”