Math definitions matter! There are many words we use in mathematics that have one meaning in that discipline and another in ordinary life. Take for instance the word “difference”. In regular conversation, if I ask you to find the difference between two things you are looking for some way in which the items are not the same. However, in mathematics, finding the “difference” specifically refers to finding the answer to a subtraction problem.
But we as teachers might be sending some confusing messages to students, sometimes even when we think we are right on track with our definitions. One example of this is the seemingly easy-to-define term “even”. How would you explain to a young child what an even number is?
There are two popular ways this property of even is explained to primary students: First, many teachers suggest that we can do an “even check by examining whether a particular number of items can be split into two equal groups. Armed with this definition, children should see that six is even since there can be two groups of three, but five is not even because there are two groups of two and one left over.
Alternately, teachers often suggest that students look at the value of the one’s place digit of the number in question. If there is a 2, 4, 6, 8, or 0 in one’s place, then a number is even. Using this method children should conclude that 74 is even since there is a 4 in one’s place, but 73 is not since there is a 3 in one’s place.
The problem is that both of these simple definitions are not fully correct. There are exceptions to them that, in fact, that are incorrect.
Concerning the “two equal groups” definition, young children figure out quickly that when sharing 5 cookies between two friends, each can have 2 ½ cookies. There are two equal groups, but the number 5 is still an odd number. What important detail have we failed to communicate here?
Concerning the “one’s place digit of 2, 4, 6, 8, or 0” definition, a student can declare that 74.3 is even since it fulfills the definition stated. Again, what important detail have we failed to communicate here?
Some might argue that these exceptions above the student; that we need not muddy the waters, so to speak, by giving extraneous information that we think is above our students’ heads. I disagree. I feel that if we just mention the restrictions as we talk about the definition, that it becomes part of the language the students are used to. We often underestimate how much students can understand, and we “dumb down” the language as a result. I would love to challenge your thinking along that line. In the case of primary children in particular, they love to learn what I refer to as “27-syllable” dinosaur names, yet we are afraid of using good math language with them!
I hope you will stop and think about the simple definitions you are using with your students and reflect on whether or not there are some hidden exceptions that need to be teased out and exposed.
Mathematically yours,
Carollee
Math definitions matter! There are many words we use in mathematics that have one meaning in that discipline and another in ordinary life. Take for instance the word “difference”. In regular conversation, if I ask you to find the difference between two things you are looking for some way in which the items are not the same. However, in mathematics, finding the “difference” specifically refers to finding the answer to a subtraction problem.
But we as teachers might be sending some confusing messages to students, sometimes even when we think we are right on track with our definitions. One example of this is the seemingly easy-to-define term “even”. How would you explain to a young child what an even number is?
There are two popular ways this property of even is explained to primary students: First, many teachers suggest that we can do an “even check by examining whether a particular number of items can be split into two equal groups. Armed with this definition, children should see that six is even since there can be two groups of three, but five is not even because there are two groups of two and one left over.
Alternately, teachers often suggest that students look at the value of the one’s place digit of the number in question. If there is a 2, 4, 6, 8, or 0 in one’s place, then a number is even. Using this method children should conclude that 74 is even since there is a 4 in one’s place, but 73 is not since there is a 3 in one’s place.
The problem is that both of these simple definitions are not fully correct. There are exceptions to them that, in fact, that are incorrect.
Concerning the “two equal groups” definition, young children figure out quickly that when sharing 5 cookies between two friends, each can have 2 ½ cookies. There are two equal groups, but the number 5 is still an odd number. What important detail have we failed to communicate here?
Concerning the “one’s place digit of 2, 4, 6, 8, or 0” definition, a student can declare that 74.3 is even since it fulfills the definition stated. Again, what important detail have we failed to communicate here?
Some might argue that these exceptions above the student; that we need not muddy the waters, so to speak, by giving extraneous information that we think is above our students’ heads. I disagree. I feel that if we just mention the restrictions as we talk about the definition, that it becomes part of the language the students are used to. We often underestimate how much students can understand, and we “dumb down” the language as a result. I would love to challenge your thinking along that line. In the case of primary children in particular, they love to learn what I refer to as “27-syllable” dinosaur names, yet we are afraid of using good math language with them!
I hope you will stop and think about the simple definitions you are using with your students and reflect on whether or not there are some hidden exceptions that need to be teased out and exposed.
Mathematically yours,
Carollee 
The only restriction that matters here is that “even” is an attribute of counting numbers, whole numbers, integers, and does not apply to fractions, decimals, rational numbers, real numbers or complex numbers. I am in full agreement with you on this one.
Yes!! I left that for folks to think through, but that is the point. We teachers need to think through to a deeper level and not just rely on “simple”. Thanks for responding.