We are a group of instructional coaches and educators who work collaboratively together in classrooms, as well as in this digital space, to co-learn and deprivatize all of the work that we are doing to build our content knowledge and ignite student thinking and reasoning. We also believe, based on research and how our own students feel about math, that we need to change how we are teaching math. Teaching math the way that we were taught isn't really working and there is a growing body of evidence, that didn't exist when we were kids, to show that there is actually a developmental landscape or continuum to how we learn concepts, and that conceptual and procedural learning need to go hand in hand. When we were growing up the emphasis was on learning procedures and tirelessly practicing them until we perfected them. Traditionally, the emphasis was on teaching by telling (Dr. Catherine Fosnot, Young Mathematicians at Work: Constructing Fractions, Decimals and Percents, p. 25, 2002), teachers began lessons with a modelled example, explaining operations and rules first, before letting students try them out on a word problem. Many of us grew up being taught how to use procedures and rules, like when you divide fractions invert the fraction and multiply, without any real understanding of why we were doing that or why it worked. This led to a whole lot of people, in many generations, to not understanding math, and even hating it, proudly declaring, "I'm not a math person." You don't see many people wearing illiteracy like a badge of honour, it rarely evokes emotions of anxiety, unless being asked to read Shakespeare, but it has created something called "math anxiety". Have you seen this video of Patricia Heaton, an actress who stars in a TV show called "The Middle", on "Who Wants to be a Millionaire"?
This is math anxiety at its finest, it's actually fairly common, and there are many administrators and elementary teachers who will tell their staff or their students that they hate teaching math or aren't math people, and we have generations of people who came through our education system who feel this way too. This is a societal problem, considering most jobs require a lot of mathematics, from skilled trades, to the medical field, to computer programming. We have to change the way that math has been traditionally taught, the way that we were taught, because it has led to far too many people not understanding it, and even worse, being scared of it. Growing up we learned procedurally and by rote memorization, many of us remember sitting at the kitchen table and being drilled on times tables to the point of tears, and when you ask someone to share their math stories, like what their experiences have been with math, we hear about a lot of negative, stressful and anxious situations. We are trying to shift the teaching of math because we want students to enjoy learning math, to experience in a way that mathematicians experience it, as a thinking and creative subject. We don't believe that students are only ready to solve a problem once "we've explained operations, algorithms, or rules" (Dr. Catherine Fosnot, p. 25) rather that a contextual problem can be used at the start of learning to construct and explore mathematical ideas, strategies and models (Fosnot, p. 25). Investigation is key to , however "learner's initial informal strategies are not the endpoint of instruction; they are the beginning. Teachers must transform these initial attempts into more formal and coherent mathematical strategies and models (Fosnot, p. 29). When students investigate concepts within a context, concretely, and make connections between the different strategies that they are using, they develop flexibility when thinking about numbers and a toolkit of strategies that they can transfer to new situations. Students who develop conceptual understanding at the same time as procedural understanding will be able to recreate a forgotten procedure or a fact, because they understand the big ideas and have a range of strategies to tap into. For example, if I don't have the fact for 8 x 6 memorized, but I know what 8 x 5 is, then I can use the strategy of using a known fact and just add another group of eight to that and I've got my answer. Or, I may want to think about factors of 8 and decompose it to 4 x 2, and because I understand the Associative Property it doesn't matter how I group the factors so I can do 4 x (2 x 6) , which is 4 X 12 and that may help me to see the 48. Or, knowing the distributive property, I could think about 8 as being (4 + 4) x 6, which would give me 6(4) + 6(4) or 24 + 24. This is a fairly easy example, but think about how these strategies could help a student to do double digit computations mentally. Check out this Jo Boaler video, it's an excellent example of students who have developed a conceptual understanding of multiplication, they don't need pencil and paper to do a tedious stacking or long multiplication algorithm, that is actually pretty easy to mess up because students lose track of numbers they are regrouping and forget the 0 place value holders, they can do it even more efficiently in their heads:
If students develop a conceptual understanding of math facts, algorithms, procedures/formulas, and rules, they can recognize when an answer doesn't make sense (like if I multiply two, two digit numbers I should get something that is at least in the hundreds, anything in the double digits or anything in the ten thousands wouldn't make sense), and they have strategies for mental math. Memorization helps us to be more efficient at problem solving, so we do want kids to know their facts, but we also want kids to understand a concept, because when you understand the big ideas behind them, you have strategies you can use can use to reconstruct facts when you forget them.
Beliefs are key, and we believe that students learn most effectively through problem solving, investigation, collaboration and accountable talk. We communicate our beliefs to our students by the tasks we choose, how we speak to them about math and mathematicians, and every time we value a correct answer over patient problem solving and thinking. Our beliefs impact how our students view and approach math. Shifting our thinking to a growth mindset, where we believe that everyone can learn math, is key. Carol Dweck has been the driving force behind this idea that everyone can learn math, that no one is born "smart" at math, that the brain is capable of growth and creating new synapses, and that this growth happens when we make mistakes and explore and learn from them.
Our beliefs drive all of our curricular decisions, and the creators of this wiki believe that solving problems, problems or tasks that are intentionally chosen because they loan themselves to particular Principles/Properties/Big Ideas, strategies and mathematical models, will foster students in becoming thinkers, innovators and the creative problem solvers of the future. We live in a world where students have access to technology, devices and sites that can do complex calculations or give accurate answers, so we believe that the emphasis in mathematics needs to be placed on thinking, reasoning, proving and communicating, since this is what will be needed for future jobs, jobs that don't even yet exist. Procedural knowledge and fluency will always be important, but simply memorizing procedures and rules without that conceptual mental velcro leads to students not knowing how and when to apply them, or how to reconstruct them when they have been forgotten, and sends the message that there is only one "correct" way of doing math. There needs to be a balance in math programmes, games and intentionally chosen or constructed strings are an excellent way to practice and consolidate facts and strategies that have been investigated through problem solving, and support students in a fun and non-threatening way. (http://www.theglobeandmail.com/news/national/education/making-a-profit-on-apples/article16856546/).
Numeracy, just like the development of literacy skills, is developmental, and much like learning to read, there are developmental markers or concepts that need to happen first, like knowing that there is only one counting number for each thing that I count, and the last number that is said tells me how many in total, needs to happen before kids can learn to skip count or count on, for example. There are learning trajectories for math concepts, strategies and models that we need to be on the look out for as students work through problems. These big ideas or enduring understandings need to be highlighted during the whole class discussion, and are the driving force behind our learning goal or reason for choosing a problem. Naming the strategies that students are using, like one to one correspondence, skip counting by 2s, skip counting by 5s, counting on--instead of starting the count again, etc.,..., helps us to know where the student is developmentally and where to push them from there, to a more efficient or abstract strategy. Understanding what strategies students are using, when to introduce visual models to support these strategies, what big ideas or concepts they have consolidated, or what misconceptions they still have, helps us to know where to go next in our instruction and how to differentiate for our students.
Students need to be challenged to see the patterns in the numbers, know which operation to use, check for reasonableness with mental math and estimation, communicate and make their thinking visible, and use a range of strategies, which will support them with persevering when faced with any problem. Math is thinking.
Our Beliefs
We are a group of instructional coaches and educators who work collaboratively together in classrooms, as well as in this digital space, to co-learn and deprivatize all of the work that we are doing to build our content knowledge and ignite student thinking and reasoning. We also believe, based on research and how our own students feel about math, that we need to change how we are teaching math. Teaching math the way that we were taught isn't really working and there is a growing body of evidence, that didn't exist when we were kids, to show that there is actually a developmental landscape or continuum to how we learn concepts, and that conceptual and procedural learning need to go hand in hand. When we were growing up the emphasis was on learning procedures and tirelessly practicing them until we perfected them. Traditionally, the emphasis was on teaching by telling (Dr. Catherine Fosnot, Young Mathematicians at Work: Constructing Fractions, Decimals and Percents, p. 25, 2002), teachers began lessons with a modelled example, explaining operations and rules first, before letting students try them out on a word problem. Many of us grew up being taught how to use procedures and rules, like when you divide fractions invert the fraction and multiply, without any real understanding of why we were doing that or why it worked. This led to a whole lot of people, in many generations, to not understanding math, and even hating it, proudly declaring, "I'm not a math person." You don't see many people wearing illiteracy like a badge of honour, it rarely evokes emotions of anxiety, unless being asked to read Shakespeare, but it has created something called "math anxiety". Have you seen this video of Patricia Heaton, an actress who stars in a TV show called "The Middle", on "Who Wants to be a Millionaire"?This is math anxiety at its finest, it's actually fairly common, and there are many administrators and elementary teachers who will tell their staff or their students that they hate teaching math or aren't math people, and we have generations of people who came through our education system who feel this way too. This is a societal problem, considering most jobs require a lot of mathematics, from skilled trades, to the medical field, to computer programming. We have to change the way that math has been traditionally taught, the way that we were taught, because it has led to far too many people not understanding it, and even worse, being scared of it. Growing up we learned procedurally and by rote memorization, many of us remember sitting at the kitchen table and being drilled on times tables to the point of tears, and when you ask someone to share their math stories, like what their experiences have been with math, we hear about a lot of negative, stressful and anxious situations.
We are trying to shift the teaching of math because we want students to enjoy learning math, to experience in a way that mathematicians experience it, as a thinking and creative subject. We don't believe that students are only ready to solve a problem once "we've explained operations, algorithms, or rules" (Dr. Catherine Fosnot, p. 25) rather that a contextual problem can be used at the start of learning to construct and explore mathematical ideas, strategies and models (Fosnot, p. 25). Investigation is key to , however "learner's initial informal strategies are not the endpoint of instruction; they are the beginning. Teachers must transform these initial attempts into more formal and coherent mathematical strategies and models (Fosnot, p. 29). When students investigate concepts within a context, concretely, and make connections between the different strategies that they are using, they develop flexibility when thinking about numbers and a toolkit of strategies that they can transfer to new situations. Students who develop conceptual understanding at the same time as procedural understanding will be able to recreate a forgotten procedure or a fact, because they understand the big ideas and have a range of strategies to tap into. For example, if I don't have the fact for 8 x 6 memorized, but I know what 8 x 5 is, then I can use the strategy of using a known fact and just add another group of eight to that and I've got my answer. Or, I may want to think about factors of 8 and decompose it to 4 x 2, and because I understand the Associative Property it doesn't matter how I group the factors so I can do 4 x (2 x 6) , which is 4 X 12 and that may help me to see the 48. Or, knowing the distributive property, I could think about 8 as being (4 + 4) x 6, which would give me 6(4) + 6(4) or 24 + 24. This is a fairly easy example, but think about how these strategies could help a student to do double digit computations mentally. Check out this Jo Boaler video, it's an excellent example of students who have developed a conceptual understanding of multiplication, they don't need pencil and paper to do a tedious stacking or long multiplication algorithm, that is actually pretty easy to mess up because students lose track of numbers they are regrouping and forget the 0 place value holders, they can do it even more efficiently in their heads:
If students develop a conceptual understanding of math facts, algorithms, procedures/formulas, and rules, they can recognize when an answer doesn't make sense (like if I multiply two, two digit numbers I should get something that is at least in the hundreds, anything in the double digits or anything in the ten thousands wouldn't make sense), and they have strategies for mental math. Memorization helps us to be more efficient at problem solving, so we do want kids to know their facts, but we also want kids to understand a concept, because when you understand the big ideas behind them, you have strategies you can use can use to reconstruct facts when you forget them.
Beliefs are key, and we believe that students learn most effectively through problem solving, investigation, collaboration and accountable talk. We communicate our beliefs to our students by the tasks we choose, how we speak to them about math and mathematicians, and every time we value a correct answer over patient problem solving and thinking. Our beliefs impact how our students view and approach math. Shifting our thinking to a growth mindset, where we believe that everyone can learn math, is key. Carol Dweck has been the driving force behind this idea that everyone can learn math, that no one is born "smart" at math, that the brain is capable of growth and creating new synapses, and that this growth happens when we make mistakes and explore and learn from them.
Our beliefs drive all of our curricular decisions, and the creators of this wiki believe that solving problems, problems or tasks that are intentionally chosen because they loan themselves to particular Principles/Properties/Big Ideas, strategies and mathematical models, will foster students in becoming thinkers, innovators and the creative problem solvers of the future. We live in a world where students have access to technology, devices and sites that can do complex calculations or give accurate answers, so we believe that the emphasis in mathematics needs to be placed on thinking, reasoning, proving and communicating, since this is what will be needed for future jobs, jobs that don't even yet exist. Procedural knowledge and fluency will always be important, but simply memorizing procedures and rules without that conceptual mental velcro leads to students not knowing how and when to apply them, or how to reconstruct them when they have been forgotten, and sends the message that there is only one "correct" way of doing math. There needs to be a balance in math programmes, games and intentionally chosen or constructed strings are an excellent way to practice and consolidate facts and strategies that have been investigated through problem solving, and support students in a fun and non-threatening way.
(http://www.theglobeandmail.com/news/national/education/making-a-profit-on-apples/article16856546/).
Numeracy, just like the development of literacy skills, is developmental, and much like learning to read, there are developmental markers or concepts that need to happen first, like knowing that there is only one counting number for each thing that I count, and the last number that is said tells me how many in total, needs to happen before kids can learn to skip count or count on, for example. There are learning trajectories for math concepts, strategies and models that we need to be on the look out for as students work through problems. These big ideas or enduring understandings need to be highlighted during the whole class discussion, and are the driving force behind our learning goal or reason for choosing a problem. Naming the strategies that students are using, like one to one correspondence, skip counting by 2s, skip counting by 5s, counting on--instead of starting the count again, etc.,..., helps us to know where the student is developmentally and where to push them from there, to a more efficient or abstract strategy. Understanding what strategies students are using, when to introduce visual models to support these strategies, what big ideas or concepts they have consolidated, or what misconceptions they still have, helps us to know where to go next in our instruction and how to differentiate for our students.
Students need to be challenged to see the patterns in the numbers, know which operation to use, check for reasonableness with mental math and estimation, communicate and make their thinking visible, and use a range of strategies, which will support them with persevering when faced with any problem. Math is thinking.