For the past few days I have been going over and re-working my Minnesota Essential Standards for 8th Grade Mathematics. One thing that I struggle with when looking at the standards and allowable notation is the phrase “items must not have context”. To me, instructing students in the ways to manipulate mathematical expressions and equations is similar to explaining to my 5 year old how to drive his Power Wheels Truck. He will understand what will make the truck move, and what has to happen to navigate around the yard with it, but will he be a safe, competent driver? I am pretty confident in answering no to that question. Information without context is meaningless to too many of our students. The days of handing out computational worksheets are over yet this approach to mathematics is still embedded into our state assessments.
Recently the debate over how we present mathematical concepts to our students to promote engagement and mathematical thinking appeared in the following blogs:
[Fake World] Real-World Math Proves Tough To Pin Down by Dan Meyer and Changing the Unknown by Graham Fletcher.
In order for mathematics to spark interest in the bulk of our student population, it needs to be linked to something in their lives. Drawing from the student’s experiences they can then understand what we are presenting and make mental connections that will not fade after a couple of days. One of my essential standards is for students to “generate and evaluate expressions using algebraic properties.” Consider the following problem: 100 + 4x = 25 = 7x. To solve algebraically (while showing your work) you would do this:
100 + 4x = 25 + 7x
100 + 4x – 4x = 25 + 7x – 4x
100 = 25 – 3x
100 – 25 = 25 + 3x – 25
75 = 3x
75/3 = 3x/3
25 = x
While I enjoy this practice of properties of equality and combining like terms, does this make any connection for students? More importantly, is this type of abstract work cognitively appropriate for a student twelve years of age? I would approach it this way:
You and your brother both want to buy the same game that is on sale. You have $1 and your brother only has a quarter. You go and ask your Dad for money since he always has change in his pocket. If your Dad gave you 4 coins and your brother 7, what type of coin did he give you? How much was the game on sale for?
Even with a problem of this type, I still get bombarded with a bunch of questions that cut this example apart. Why ask me how much the game is when it tells me on the tag? I could just look at what my Dad gives me and tell you what type of coin it is! These questions are valid, and what students do not realize is that each question is just a variation of the problem I gave them. If the tag tells you how much the game is, how many quarters do you and your brother need to purchase the game? If your Dad gives you nickels, how many did he give you and your brother so that you have the same amount? Normally I write down their questions on the board and state that we will look at each of their questions after we find our answer, it’s a great way to get students to analyze the problem fully. The best thing about this example is that it automatically instills a context the students know and have had experience with. They will work on this problem because they believe they can find an answer, and that it can be solved fairly easily.
But what about all those important mathematical properties of equality that you are leaving out? Students will not need or use them with an example of money!
The answer to this question is that I am a professional, as well as a teacher, and I will use student thinking and examples to illustrate the properties of equality. My students automatically tell me that they have $.75 more than their brother. How did they know that? They compared their money and found the difference.
Student talk: I have a dollar and my brother has a quarter, but Dad gave us money so we had the same amount.
Example on board: $1.00 + 4 coins = $.25 + 7 coins.
Student talk: I have $.75 more because I subtracted his money from mine.
Example on board: $1.00 – $.25 + 4 coins = $.25 – $.25 + 7 coins.
What are you doing when you compare? Do you and your brother have the same amount of money still? This is called the subtraction property of equality (if you don’t have any students who can recall that fact). Using student’s thoughts and examples to illustrate a mathematical property is the type of instruction that our students deserve. My biggest hurdle as an 8th grade Algebra teacher is that all of my students do not think they their ideas are Algebraic Thinking, when in fact almost all of it is.
So as we get closer to the dreaded State Assessment date, I find myself moving from problems that make sense to problems that do not. Students move from problems that have meaning to them to problems that are just symbols on a page that represent a foreign language. It is important that all students can write a mathematical equation and solve it symbolically? I do not believe that to be so. I do think that all students can and should be able to think mathematically about a situation, present a solution and justify their thinking. To me, math is a whole lot more than if x + 3 = 5 then x = 2.