My MNMathNerds have learned, and continue to learn that:
Everyone Can Learn Math to the Highest Levels. There is no such thing as a “math” person. Everyone can reach the highest levels they want to, with hard work.
Mistakes are Valuable Mistakes grow your brain! It is good to struggle and make mistakes.
Questions are Really Important Always ask questions, always answer questions. Ask yourself: why does that make sense?
Math is about Creativity and Making Sense Math is a very creative subject that is, at its core, about visualizing patterns and creating solution paths that others can see, discuss and critique.
Math is about Connections and Communicating Math is a connected subject, and a form of communication. Represent math in different forms, examples words, a picture, a graph, an equation, and link them. Color code!
Depth is much more Important than Speed Top mathematicians, such as Laurent Schwartz, think slowly and deeply.
Math Class is about Learning not Performing. Math is a growth subject, it takes time to learn and it is all about effort.
We also know that maths is so much more than memorizing facts and procedures. With all of these ideals under consideration, this week we discussed and discovered again, that there are some mathematical ideas and procedures that we just have to be told. When we were young we could figure out that if we had 3 red marbles and 2 blue marbles we could combine the groups and count them to find out how many. We were adding, but we did not know what the addition symbol was until someone showed us + and demonstrated symbolically 3+2=5 as our expression for our marbles. Similarly, when we are doing multi operational maths problems we can find a multitude of solutions if we are not following an agreed upon order of operations when computing. Mathematicians know that some numerical situations might be interpreted in more than one way. Therefore, they agreed on an order for simplifying expressions called the Order of Operations. There are a plethora of mnemonics and tricks for Order of Operations, but after much research and readings I have settled on my favorite, GEMA as endorsed by Tina Cardone in Nix The Trix. Once we reviewed the agreed upon Order of Operations and completed an interactive notebook entry the fun began!
Enter the high cognitive demand math problem! I could have given my students several naked number mathematics problems to practice, but they would have just been following procedures without connections. While sometimes these kinds of practices are good, I wanted to impress upon them the importance and affect the order of operations can play in outcomes.
So instead of giving them a problem like: 2(5×30) + 2(2×70) +10 = I asked them to notice and wonder about the following, and then turned them loose to collaboratively strategize and solve.
- Alicia is making a reading plan for her book club selection. She has 14 days to finish the book. She plans to read 30 pages on weekdays and 70 pages per day on Saturday and Sunday. Following this plan, she still has 10 more pages to read at the end of 14 days. How many pages long is the book?
It was a blast to hear the conversations,thinking, and convincing in the working groups! We have been exploring the distributive property and the different ways you can notate the expressions. I loved seeing the kids explore different ways to to notate and solve and notice how they were distributing in different ways. If I had just given my students naked number math problems, I would have deprived them of a multitude of rich experiences. They would not have had to explain their strategies, justify their thinking, listen to and critique the reasoning of others, and they would not have had the opportunity to process the ideas, strategies and thinking in a variety of ways, thus deepening their understanding. I was beyond thrilled with the myriad of ideas, strategies and paths to solutions explored! I try to anticipate possible strategies and ideas, and I have to admit, my students surprised me with a few I had not considered!
I teach sixth-eighth grade students with disabilities, and I always read aloud the tasks and problems we investigate in class a multitude of times before we notice and wonder. In addition, I always record and leave our ponderings on the whiteboard for reference during work time. For one of my sixth-grade classes, I needed to take this wordy problem one sentence at a time so that they could notice and wonder deeply. If I were doing this problem again, one way I might modify my delivery would be to leave off the last sentence, “How long is the book?” I realize by having this question given, I have taken the opportunity from my students to formulate a question to explore for themselves. All in all, this exploration was a smashing success! The affirmation came when three of my classes stood at the board, taking pictures of their amazing mathematical thinking so they could post it on social media! YES! I have created some awesome math nerds indeed!
Oklahoma Academic Standards:
6.A.2.1 Generate equivalent expressions and evaluate expressions involving positive rational numbers by applying the commutative, associative, and distributive properties and order of operations to solve real-world and mathematical problems.
7.A.4.1 Use properties of operations (limited to associative, commutative, and distributive) to generate equivalent numerical and algebraic expressions containing rational numbers, grouping symbols and whole number exponents.
7.A.4.2 Apply understanding of order of operations and grouping symbols when using calculators and other technologies.
PA.A.3.2 Justify steps in generating equivalent expressions by identifying the properties used, including the properties of operations (associative, commutative, and distributive laws) and the order of operations, including grouping symbols.
Common Core State Standards
Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.
Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.
Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?
Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.–