The world needs thinkers, doers, creators… Your memories of school when you were a kid are just that, memories. And we all know how notoriously unreliable memories are!!!
Why is it that despite evidence that clearly indicates we can ENGAGE STUDENTS in rich experiences that make them THINKERS AND DOERS we continue to focus on dumbing mathematics down to a series of mindless rules, meaningless tricks and “just remember this long enough to pass the test” tips?
IN THE 21st century, we need citizens who can think…. and more important who want to think... want to understand why, want to evaluate information and to tackle the complexity and ambiguity of the “real life “problems they will confront on a daily basis.
YOU DO NOT LEARN TO ENGAGE IN COMPLEXITY by drilling sets of isolated facts..
Learning is not being told what to do, in how many steps, then repeating those steps mindlessly until you can do them without thinking….
Learning is not sitting alone at a desk for 45 minutes filling in blanks and counting on your fingers…
This is additive thinking… it will not help this student learn to multiply and divide if there is not a guided discussion that leads him to compare his solution to one that is premised on multiplication. This student does display perseverance, focus, and the ability to perform multiple steps in solving a problem (He had to remember to come up with 19 as his answer, but again likely just counted. No one told him to do this, this is how he solved the problem he was given.
Learning is active, engaging, pushes you to talk, build, think, reflect, debate, re-consider, compare.
Learning to think, discussing, critiquing and analyzing thinking, teaches you to take responsibility for, and be highly critical of your own thoughts, your responses to the thoughts of others, the way you respond to the challenges from others, the work you do and the way you react to the world around you. LEARNING teaches you responsibility and stewardship.
The current Alberta Math Curriculum is focused on teaching students to communicate, make connections and learn to take charge of their thinking. It does not ignore skills like addition and subtraction, multiplication and division BUT it states that we not teach them as sets of unrelated “facts.” The brain is ASSOCIATIVE. It stores information in richly connected webs. The more associations it makes, the more likely it will recall information with ease. When you learn the facts in an unrelated manner, there are no rich connections to associate. When you slow down and learn facts in a connected manner, one fact triggers another.
This Grade 2 student can perform these calculations in her head, and very quickly. But I asked her to show me what her thinking would look like if she wrote it out. She is using connections she knows from decomposing numbers. I REPEAT… she answered this problem orally in less time than it took other children to record the numbers and start trying to remember what to do with the 7 + 8.
DO NOT CONFUSE CURRICULUM with a textbook, a set of resources, PINTEREST, OR a SILLY ABSURD MADE UP PICTURE ON THE INTERNET WHICH HAS NOTHING TO DO WITH WHAT TEACHERS ACTUALLY DO IN CLASSROOMS. CURRICULUM IS NOT THE PROBLEM.…. no time and energy invested in keeping teachers up to date and current, no time and energy invested in giving teachers the resources and support they need to stay abreast of the ever increasing volume of information, to unlearn ineffective strategies, to practice, become proficient with better practices… that’s the problem… Would you use a doctor who said “I have not had any updating or professional learning for several years so while I hear there is some kind of laser that reduces the risk and time for recovery in this surgery, but I have not had time to learn how so I’ll just cut you open the “old tried and true” way.
Did you know you could literally see a pi “r” square?
Problem solving is the teaching approach this curriculum focuses on but unfortunately most of us have had little or no experience with teaching or learning through problem solving so we have no idea what it feels like to engage in rich investigations of number facts.
Teachers are being challenged to unlearn and re-learn ways of doing math. As a teacher, have you accepted the challenge? As a parent, do you understand that teachers are constantly engaged in learning?
I teach with the intent that every student will build personal confidence, experience personal success and want to pursue thinking and problem solving. Problem solving is not reading contrived story problems trying to pick out numbers and guessing what operation to apply. Problem solving means actively and deliberately working at understanding a situation, coming up with an approach to use, talking, listening, comparing, checking results with others…. confidently reaching a solution you can explain.…
These students were considering the following: I had 136 Sweettarts and I had to share them into 6 goody bags for a party. How many in each bag? Below is some student thinking, my job is to direct the discussion to move them to understanding how multiplication and division are related and what will we do with the remainder…
This boy solved the problem in his head and raised his hand with 22 as the answer and 4 leftovers. This is what he wrote as an explanation when I asked him to share with the class.
After he explained the girl behind him said,
“Why not start with 6 x 20 = 120?”
How are these two solutions similar? Did they get the answer that the first boy did.. Look at all the practice and thinking these students engaged in… Where did they make mistakes? What in each solution really helps you understand the problem? These are the questions that engage the entire class in sense making. Seeing that you can find your way through a problem then learn to tighten up your thinking for next time. These discussions allow every student to stay engaged in the lessons… every student!
IN THE 21st CENTURY IT TAKES INCREDIBLE COURAGE, PERSEVERANCE AND ENERGY TO TEACH ….. It is not for the faint of heart and it is certainly not 9 to 5 with major holidays and summer off. Teaching is constant and continual learning and growing… I teach 3rd and 4th year education courses and I am concerned that many candidates are not prepared for this fact and therefore should not consider teaching as a profession. I have invested my life in coming to better understand how children think, learn, play and grow… I don’t think I will ever feel I know enough but I certainly know that the more I know, the more likely I am to instill a love of learning in others .
I teach for understanding. I teach for thinking because I want to stop the cycle of failure and despair that drives so many of our students away from learning.
There is no “fuzzy” curriculum or “fuzzy math” in our curriculum but there sure are some “fuzzy” ideas about how rapidly shouting answers to some very low level knowledge questions somehow constitutes “knowing” math. The teachers I work with seek rigour, depth and connectedness. They want to understand how to teach so that EVERY student is empowered. They exhaust themselves trying to study, plan, implement and then adapt their lessons and inquiries to more effectively engage all students. They work tirelessly and often thanklessly to find ways to engage their students, to provide opportunities for their students to explain their thinking, to choose the right question so as to prompt and guide student thinking, to encourage even the most reluctant that they have the power to learn. They do not rely on computers or “textbooks” to “teach” because they know that inanimate objects cannot teach. Teaching and learning involve relationships and collaboration.
THEY DO NOT BLAME STUDENTS FOR NOT LEARNING. Instead they look to themselves and their practise… they listen to, collaborate with, and study their students, they study themselves… they constantly and continually learn….Remember, elementary teachers have 7 to 9 subjects to teach….. They do not get to specialize…
This is the result of just learning your facts. This Grade 6 student has no idea what multiplication is. Multiplicative reasoning forms the basis for work with rational numbers in Junior High. She has none after 3 years of “just learn the facts” approach.
I teach for understanding. I have done so for more than 30 years. I teach parents, children, teenagers, administrators, teachers, University professors, businessmen and women, professionals from all walks of life… I ALSO TEACH FUTURE TEACHERS… the stories I am told can be heart breaking…. Adults who have hated math all their lives,
I always felt stupid in math because I could not spit out the answer to 8 x 7 as fast as the kid across the table. The teacher called it a game, but I sure didn’t find it any fun.
I always wanted to understand why things work. I was curious and liked to think. That is the key to an engineering or science career but instead I was told it doesn’t have to make sense it is “just the way you do it”.
I never knew you could actually see and think about ideas in math. I thought you were just supposed to memorize and remember. I feel cheated now that I see I could actually understand it… why didn’t I get a chance to learn this way? Maybe I would be a different person today.
I am tired of the buck passing: blaming the kids, blaming the curriculum, blaming the textbook, ….. Teaching is HARD WORK. It is complicated and complex, it involves the heart and the mind… Many who come to the profession are surprised and disappointed to find there is no manual or quick fix or easy way to teach.
Learning as a teacher or a student is complicated, challenging and requires us to search for meaning in what often begins as confusion.
This grade 1 was asked to explain how many ways a sweater could have a combination of blue and yellow buttons when altogether there are 7 buttons. He has all the ways. My job is to help him learn how to organize in a more logical way. As we do so, the entire class learns about organizing data, seeing the pattern and using it to build understanding of the facts.
People who can “do” math often have absolutely no idea how it is taught or learned… they just know how to do procedures or remember how to follow an algorithm. In the last 40 years, since I began teaching, a solid body of research on how mathematics is learned has been developed…And I have spent hours and hours and hours studying it… Mathematics is not learned by mindless memorizing or by endless counting.. it is learned by deliberate and active thinking and doing and explaining and comparing and testing out and thinking some more…
I have to laugh out loud every time I hear the phrase “traditional algorithms” . The variations on how to do multiplication and division and “column arithmetic” are as varied as are the lay persons’ opinions on how teachers should teach. There are multiple ways to perform any operation with numbers… no GOD of MATH created a set of tablets with THE ONE TRUE WAY chiseled into it. I want my students to understand what they are doing when they solve any kind of problem with any set of numbers…. because the problems they will face in the years to come will not magically disappear if someone can just shout out the answer to 8 x 7 in under 2 seconds.
This curriculum is set up to focus on building understandings of KEY RELATIONSHIPS between numbers as students learn to work with numbers, not after they have memorized a set of facts.
NOWHERE IN THIS CURRICULUM will you see a statement that suggests we do not want our students to end up knowing BASICS.. in fact this curriculum challenges teachers to KNOW and TEACH more BASICS than ever before.
- Equality is introduced in Grade one and contrasted with inequality in Grade 2
- The commutative property, the associate property appear as early as Grade 2
- Expressions to describe quantities are called for as early as Grade 2
- Understanding and applying inverse operations begins in Grade one and carries through from addition and subtraction to multiplication and division.
- The Distributive Property is introduced in Grade 4 and repeated in Grade 5.
These are BASIC RELATIONSHIPS THAT make the facts make sense and they are all relationships that teachers need to study and learn themselves. Basic relationships many teachers did not learn when they were in elementary school, that were not recognized as important until we moved into the “Information Age.”
The BASIC SKILLS for mathematical thinking and reasoning begin with sorting, matching, classifying, comparing, organizing data, searching and describing pattern, generalizing then testing and justifying those generalizations.
THIS IS THE SAME WAY YOU PREPARE TO LEARN A LANGUAGE, PREPARE TO LEARN TO READ, DO SCIENCE..… IT IS learning how to think, organize your thinking then apply your thinking skills to learning more effectively. It comes before you memorize words or facts.
We know too much to say that what was good enough for us is good enough for our kids in learning math… I am arguing that unless our students know how to learn and how to think, they become trapped in the same cycle of memorize, regurgitate, quickly forget, that has left the majority of the population “hating math” or “not being much good at math.” I vote for creating a generation who can think their way through a problem and that means teaching them to think.