Showing posts with label math. Show all posts
Showing posts with label math. Show all posts
Wednesday, May 17, 2017
Presenting Math as an Art
Presenting Math as an Art
I was disconnected from the internet for most of the August and got a luxury of time to read a few deep-hidden non-fiction books from our library. One of these books was a shocker - a light, eloquent, captivating and constructive critique of the ways mathematics is percieved and presented at schools, universities and most of our households. It is a very refreshing and convincing analysis from a former scientist and experienced math teacher and I am going to recommend it to every teacher, parent and high school student. The book is: "A Mathematical Lament" by Paul Lockhart. It is subtitled: "How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form."
Dr Lockhart suggests that math is an art and should be treated exactly like music and painting. We should not be learning math because it is applicable to our daily lives or essential in our future occupations, but rather because the process and the result bring joy to our soul like listening to or producing music does. Forget all the bad math associations you have nurtured and think of a pleasure that discovery of any pattern brings to us. And the sweet mental game of attempting to explain this pattern.
Dr Lockhart suggests that math is an art and should be treated exactly like music and painting. We should not be learning math because it is applicable to our daily lives or essential in our future occupations, but rather because the process and the result bring joy to our soul like listening to or producing music does. Forget all the bad math associations you have nurtured and think of a pleasure that discovery of any pattern brings to us. And the sweet mental game of attempting to explain this pattern.
Take for example this curiosity:
1 = 1
1+3 = 4
1+3+5=9
1+3+5+7=16
1+3+5+7+9=25
Fascinating! Sum of the subsequent odd numbers produce results that are all square numbers.
But why?
Can any square number be represented as a sum of subsequent odd numbers?
If we let our mind wonder about this for a while we may be able to come with an explanation, perhaps even as thrilling as this one:
Any square number can be drawn as a square, such as this big and colorful 5x5. Any square can be split into such colorful parts.
Count the number of little pieces in these parts. They are 1 (purple), 3(green), 5(yellow), 7(blue) etc. These are our subsequent odd numbers. Looks like any square can be split into such pattern.
Lockhart writes that in school "the rich and fascinating adventure of the imagination has been reduced to a sterile set of facts to be memorized and procedures to be followed." Being math expert, math lover and experienced math teacher he suggests a number of directions to improve such math education:
- "Mathematics is an art of explanation." This means that students should be allowed to pose their own problems, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble togather their own explanations and proofs."
- Math history is fascinating and provides a great base for captivating storytelling. Instead of handing kids formulas to memorize retell them stories of ancient and contemporary mathematicians and their search for solutions to the fascinating problems that are wrapped in these formulas today.
- Like other arts math should be subject to debate and critisize in schools. "Is this argument sound? Does it make sense? Is it simple and elegant?"
- Forget proper notations that are frequently a source of frustration. "Math is not a language, it is an adventure." DaVinci, Pollock,and Warhol each created art in their own way.
The author also cleverly warns that although math is and should be presented as an "art" subject, such shelving may be dangerous as "useless" art disciplines are frequently the first to be cut off from the curriculum in time of budget cuts.
This book will definitely change the way you think about math.
It may boil your blood from anger with the author or on opposite, the current math curriculum against which he convincingly stands.
It may explain you why you always hated math classes but liked riddles and puzzles.
It surely will inspire you and teach you a few tricks that you will rush to share with your kids as I did.
Available link for download
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Tuesday, March 21, 2017
Problem Based Learning for Common Core Math FREEBIE
Problem Based Learning for Common Core Math FREEBIE

How many of you are still confused about Common Core? I just learned this week that my state will not test on Common Core for a couple more years. My students will still be tested over the old state standards which Im no longer teaching. CRAZY! I think the thing that concerns me more than anything is the differences Im seeing in implementation. The literacy seems to be consistent across the board BUT the MATH is has me very concerned. Ive been trained exclusively in the Problem Based Learning (PBL) method. This is a stark difference from the old way of teaching. Before you had a set of standards that you taught and checked off the list. (By the way, Ive seen numerous checklist for Common Core on TpT.) The PBL method is the complete opposite of the checklist method. The skill and drill is definitely OUT with this method. Students are asked to solve real-world problems. Students will use various methods for solving the problem and no one way is correct. Teachers will no longer teach students a set of rules or algorithms. Students will "discover" these on their on by being immersed in student discourse. The teachers role will be that of a facilitator. The problem Ive been faced with is there is absolutely NOTHING on the market to help with implementing this type of approach, so I was forced to create my own. While spending numerous hours looking at student work samples, I noticed the lack of consistency. In many cases it was hard to even know what answer the child came up with. Even among trained professionals there was disagreement over what the child was thinking. In the PBL method student thinking is the key. Again I came up with a method to help reduce the confusion. I needed to create a standard for solving problems, so we could put our focus on the more important things. Thats how the A.L.I.E.N. method came about. The students choose their own method for solving the problem but are then asked to make sure it is in A.L.I.E.N. format, so we get a clear understanding of their thinking. A.L.I.E.N. stands for Answer, Label, Isolate, Explain, and Number Sentence. I love it, and it has helped tremendously when it comes to math discourse.
Ive had the opportunity to look at several examples of student assessments for the Common Core Assessment that is coming sooner or later. If these examples are anything like what the real assessment will be like, those teachers not teaching the PBL method are doomed. Seriously these assessments are ridiculously hard. A child will need a lot more than the ability to add, subtract, multiply, and divide to score proficient. Im very curious to know if any others states are being trained in the PBL method.

In my freebie for you, Ive given you everything you need to implement this method in your classroom. simply click HERE or on the picture above for your freebie. You can create these problems all on your own. If you are interested, Ive created a TpT product with 110 problem sets (3 problems per set) for a total of 330 problems. This is enough problems for an entire year of teaching.Ive also included an overview of the method and given you question stems for conferencing with students. Im also having a 24 hour Flash Sale for my readers. If you want to pick up a copy from my store, click on the picture below.


Available link for download
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