Some students in the math methods class were pissed. I read the comments to find out why. One comment summed it up, ‘She wanted everything to be so exact even though we were just doing elementary.’ They were talking about teachers being precise when writing and speaking about math with elementary students.
I’ve been thinking about this comment for a couple days now. I did a little digging. I talked to students and faculty. I looked over the handouts and did some self-eval. I understand why the students were pissed. Students were doing this for the first time. Precision was never stressed in the teacher education program before.
Most students in the course felt that they were not math majors (2 of them were) and that it was ‘unfair’ to ask them to talk and write about math in this way. Further, students live in a culture where close enough is the same as close. Anyone can be awesome (breath-taking, fearsome, awe-inspiring) in exchange for doing very little. To insist on precision when speaking or writing is annoying to most students. A huge ‘what’s the big deal?’ deal.
That said. In the future, I plan to let students know, from the beginning, that precision when writing and speaking about math is an important part of elementary math teacher preparation. But I don’t plan to teach the course differently. I believe that precise use of mathematical terms, concepts and notation forms a strong math foundation. It prepares students for more advanced math understanding. It prepares students for learning in STEM subjects like chemistry and physics. It encourages the use of mathematical language in daily life.
When working with pre-service and in-service elementary math teachers, I tend to focus on 4 main areas of precision: definitions, correct use of terminology, notation and units.
1. Precise definitions: In math, figures, numbers, algorithms and other entities are defined by a specific set of properties. A property is a characteristic or quality that is ‘always’ true, except in certain specified situations. Definitions of math concepts are often made up of a list of properties. Definitions of concepts should be grade appropriate. Or appropriate to students’ levels of understanding.
“Knowing the properties of figures, numbers and algorithms is a powerful tool for advanced math and engineering.”
As student understanding increases, teachers move from basic properties to more advanced properties. Basic definitions may include a few properties. For example, when geometry is introduced, a square is a four-sided with four equal sides. More advanced definitions include things like angles and parallel lines. As such, when students learn about angles, teachers add the notion of four equal angles of 900. Then the concept of parallel sides…you get the idea.
Knowing the properties of figures, numbers and algorithms is a powerful tool for advanced math and engineering. Properties and property notation form a kind of language that eventually becomes second nature. And it begins in the primary grades.
2. Precise use of terms: Students should learn to use terms correctly. In writing and speaking. The teachers can facilitate this in 2 ways:
Way1: Modeling- Teachers should try to model correct ways of speaking and writing about mathematical terms when interacting with students. Ensure that the notation on the blackboard or smartboard is correct (more on this in the next section). Ensure that when you talk about math, you use terms in the correct way. Students tend to do what you do and say as you say. Even when you’re not teaching, you’re teaching.
Way 2: Communication- Encourage students to express mathematical ideas in speech and in writing. Ask them to explain to classmates how they solved a problem. Ask them to describe, in writing, how they would solve a problem. As students talk, listen for the way they use terms. Make corrections if they falter. Do the same for writing. For this to work, the classroom has to be a safe space where students and teachers feel comfortable making mistakes or being correct.
“Mathematical expressions and equations can be read like sentences. And these sentences must make mathematical sense.”
3. Notation: This is the writing system for recording mathematical concepts. It’s made up of symbols with precise meanings. A single symbol, say ∞, infinity or undefined quantity, is easy to deal with. The issue of precision comes up when groups of symbols and numbers are used to form mathematical expressions and equations. Mathematical expressions and equations can be read like sentences. And these sentences must make mathematical sense. These sentences must be ‘true.’ Often, while trying to solve an equation or find an answer, a student writes mathematical expressions and equations that don’t make sense. The final answer, solution or proof may be correct, but along the way, the student used expressions and equations that were, what my favorite math teacher from Trinidad called, ‘ratch.’ Ratch means a ‘scheme or trick to do a patch-up job.’
For example, a common ratch for solving an equation with a single variable:
Please solve for x: 2x + 3 = 25
Ratch solution:
2x + 3 = 25
– 3 -3
2x = 22 = 11
x = 11
The solution is correct. Statements like: 2x = 22 = 11, don’t make mathematical sense. The notation for subtracting 3 from both sides doesn’t make mathematical sense. It’s essentially an image that uses numbers to help the solver visualize what’s happening. And this does have a place in math. But it’s important for students to move beyond the ‘image’ stage and use good math grammar when solving problems.
Further, when I asked the student to explain, she didn’t understand why she performed some steps. She didn’t seem to understand that =, is the same as, was the main reason that operations had to be performed on both sides of the equation to keep ‘sentence’ true. The expressions made sense in the student’s head, but not on the page.
The mathematical solution is:
2x + 3 = 25
2x + 3 – 3 = 25 -3
2x = 22
2/2x = 11 22/2
x = 11
This solution has great ‘math grammar.’ In time, students can skip steps. But solving the equation in this way, fosters excellent math habits that can serve students well in their math and education futures.
“A great way to help students understand how units work is to maintain units in mathematical equations and expressions.”
4. Units: Students should use them. Period. You’re calculating the area, volume and circumference with unit quantities; your answers should contain appropriate units. You’re adding 6 apples to 5 apples, the answer is 11apples. Units are a kind of mathematical notation too. Units tell you if a quantity is a vector (magnitude and direction) or scalar (magnitude) quantity.
A great way to help students understand how units work is to maintain units in mathematical equations and expressions. Especially in the early stages. This will help students understand how units are derived during operations.
For example:
If the area of a rectangle is 14cm2, and the breadth is 2cm, what is the length?
A = l . b
14 cm2 = 2cm . b
14 cm2 = b
2cm
714 cm2 cm= b
2cm
7cm = b
You get the idea. That’s all folks. Please share this post. #heidiholder #redloheducation
