Contexts for Learning Mathematics
@CFLM_math
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Contexts for Learning Mathematics is a K-5 inquiry-based math curriculum by Catherine Fosnot & Associates with an online classroom video teacher support system.
Joined February 2021
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Pinned Tweet
HUGE NEWS IF YOU LOVE PROBLEM STRINGS!
Cathy Fosnot and the New Perspectives on Learning team are writing a whole new K-5 series coming out this spring.
Problem Strings for Fluency and Beyond - one for each grade level.
Get on the list...NOW!
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Understanding the equal sign as equivalence and NOT as a symbol denoting the “answer is coming” is essential if children are to use algebraic expressions with meaning.
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“When I sit down to confer w/ a developing mathematician I have to constantly remind myself that my goal is to support the development of the young mathematician sitting next to me. It is not to fix the piece of mathematics.”
Conferring with Young Mathematicians at Work, Fosnot
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“Each time one prematurely teaches a child something he could have discovered himself, that child is kept from inventing it and consequently from understanding it completely.”
Piaget, 1970, p. 715
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Each time one prematurely teaches a child something he could have discovered himself, that child is kept from inventing it and consequently from understanding it completely.
― Jean Piaget, 1970
#iteachmath
#mtbos
#elemmathchat
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Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost.
- W.S. Anglin (1992)
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Memorizing facts without understanding the relationship between them does not support the development the cognitive network necessary for later fluency with computation or the foundations for algebra.
To get facts automatic, we need to work on the properties and relationships.
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Because memorizing procedures can often require kids to treat the numbers in each step as digits, they often lose sight of the quantities they are actually working with.
In this way, memorizing procedures get in the way of sense-making.
#mtbos
#iteachmath
#elemmathchat
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By abandoning the rote teaching of algorithms, we are asking children to learn more. We are teaching them to mathematize, to think like mathematicians, to look at the numbers before they calculate.
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Please do not tell kids that multiplication means repeated addition.
Multiplication is so much more than repeated addition.
Think about 1/2 x 3/4...where’s the repeated addition? It’s not there. In fact, you’re dividing by 2!
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Teaching mathematics is about facilitating mathematical development.
This means that we cannot get all learners to the same landmarks at the same time in the same way, any more than we can get all toddlers to walk at the same time in the same way.
Fosnot, Models of Intervention
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Sad truth...
The reason that some kids have trouble in school is because they are thinking about the problem in a different way than the teacher and nobody is letting them use their own way of thinking.
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Each time one prematurely teaches a child something he could have discovered himself, that child is kept from inventing it and consequently from understanding it completely.
- Jean Piaget, 1970
#mtbos
#iteachmath
#elemmathchat
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Algorithms have an important place in mathematics.
By abandoning ROTE teaching of algorithms, we are not asking children to learn less, we are asking them to learn more.
We are asking them to mathematize, think like mathematicians, to look at the numbers before they calculate.
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We cannot just hold up a model and assume that children see what we as adults see.
The math is not in the model.
The math is in the child’s head.
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Launching a “rich task” and having kids mathematize the situation only allows initial conceptions to emerge.
But that is not teaching.
Teaching means we need to get underneath kids’ strategies and support the development of the big ideas that allow the strategies to work.
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Number Talks vs Number Strings
Get ready to be blown away by this session hosted by Christina Tondevold
@BuildMathMinds
with Cathy Fosnot on the difference between number talks and number strings.
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The hallmark of numeracy is to look to the numbers first before deciding on a strategy.
It’s this kind of numeracy that we should be striving for, and a mathematics program needs to support teachers and kids in getting there.
#MTBOS
#iteachmath
#elemmathchat
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How to Question...
Are you looking for ways to ask better questions that don’t take over your students’ thinking, but still move them further along in their understanding?
Cathy Fosnot on how Ts can become more powerful questioners.
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“When I sit down to confer w/ a developing mathematician I have to constantly remind myself that my goal is to support the development of the young mathematician sitting next to me. It is not to fix the piece of mathematics.”
Conferring with Young Mathematicians at Work, Fosnot
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Please do
#numberstrings
!
Memorizing facts with flash cards or through drill and practice worksheets will not develop the important relationships needed for fluency.
Teaching facts for automaticity, in contrast, relies on thinking.
See some at
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The basics have shifted away from practicing algorithms, towards numeracy and deep understanding.
It’s not important that we perform all calculations ourselves but we better know what to tell digital tools to do and be able to recognize whether the answers we get are reasonable.
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“Children often confuse area and perimeter, primarily because the concepts are taught as formulas and the big ideas related to them are neglected.”
- Tabletops, Floors and Fields, page 7
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Teaching mathematics is about facilitating mathematical development.
This means that we cannot get all learners to the same landmarks at the same time in the same way, any more than we can get all toddlers to walk at the same time in the same way.
Fosnot, Models of Intervention
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If you have never done a number string with time on the
@mathrack
clock, you're missing out!
Check out the awesome Math Rack Clock and our new 1st and 2nd grade units on time at
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Developing number sense takes time.
Algorithms taught too early work against the development of good number sense.
#MTBOS
#iteachmath
#elemmathchat
#NCTMLA22
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What is numeracy and how do we develop it?
5 minute response...beautiful!
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Calculating with number sense as a mathematician means having many strategies available and looking to the numbers first before choosing a strategy.
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Mathematics should be thought of as a human activity of “mathematizing” - not as a discipline of structures to be transmitted, discovered or even constructed, but as schematizing, structuring and modeling the world mathematically.
- Hans Freudenthal
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Does your math curriculum support the development of the most important big ideas, strategies and models?
See Cathy build the landscape of multiplication that serves as the backbone of all of our multiplication units!
#elemmathchat
#mtbos
#iteachmath
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We are called to walk the edge, between the structures of mathematics & the development of the child.
We have to understand deeply the development of the mathematics by considering the progression of strategies, the big ideas & emergent models.
Fosnot - Models of Intervention
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By abandoning the rote teaching of algorithms, we are asking children to learn more.
We are teaching them to mathematize, to think like mathematicians, to look at the numbers before they calculate.
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Good math teachers are not just facilitators of learning. Good math teachers are mentors. They see young learners in their care as apprentices and welcome and initiate them into the community of praxis of the discipline - into the generating of mathematics.
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We cannot just hold up a model and assume that children see what we as adults see.
The math is not in the model.
The math is in the child’s head.
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With enough practice and feedback, humans can be taught to perform arithmetic algorithms.
The problem is that mathematics and numeracy (in contrast to arithmetic) require thinking, not just routinized behaviors.
#iteachmath
#mtbos
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Mathematics taught as an isolated set of skills and concepts rarely becomes part of an interconnected network of relations.
That's because structuring requires cognitive reorganization by the learner; the ideas usually cannot be transmitted alone.
Fosnot and Jacob, 2007
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Good math teachers are not just facilitators of learning.
Good math teachers are mentors.
They see young learners in their care as apprentices and welcome and initiate them into the community of praxis of the discipline - into the generating of mathematics.
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Using games in K-1 is a fantastic way to “assess” where kids are in development in the beginning of the year and all year long.
No need for written answer-getting exercises/tests that only show whether they got the right answer to those problems in that format.
I tend to start every year jumping back into
@ctfosnot
@CFLM_math
- exploring how I can tweak games, how I can play around with the games to suit the needs of my new students, & discovering new things within them that I may not have noticed previously.
#yrdsbmath
#earlyyears
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Calculating with number sense as a mathematician means having many strategies available and looking to the numbers first before choosing a strategy.
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Mathematics should be thought of as a human activity of “mathematizing” - not as a discipline of structures to be transmitted, discovered or even constructed, but as schematizing, structuring and modeling the world mathematically.
- Hans Freudenthal
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Do you love number strings?
All of our units use number strings to support automaticity and fluency.
Even better, teachers can observe other teachers facilitating dozens of strings like this one using our online classroom video library.
#numberstrings
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Launching a “rich task” and having kids mathematize the situation only allows initial conceptions to emerge.
But that is not teaching.
Teaching means we need to get underneath kids’ strategies and support the development of the big ideas that allow the strategies to work.
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We cannot just hold up a model and assume that children see what we as adults see.
The math is not in the model.
The math is in the child’s head.
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Many students who struggle to commit basic facts to memory believe that there are “hundreds” to be memorized because they have little or no understanding of the relationships among them.
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Computational fluency involves looking to the numbers first and deciding on a strategy for those particular numbers.
Computational fluency is not using the same strategy over and over for all numbers.
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When the humble reasoning of children is valued and nurtured in mathematics classrooms, doors open. When children are given the chance to structure number and operation in their own way, they see themselves as mathematicians and their understanding deepens.
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From Number Talks to Number Strings
In a number string, there is some talk happening. But there is also representation of strategies that children are offering and turn from one problem into another, using the model to progressively support kids.
Listen
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Launching a “rich task” and having kids mathematize the situation only allows initial conceptions to emerge. But that is not teaching. Teaching means we need to get underneath kids’ strategies and support the development of the big ideas that allow the strategies to work.
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Do your kids use 10+5 to solve 9+6?
Do you go after the algebra when they do or miss the moment?
You have got to hear how Cathy takes what a kid says about using 10+5 to get 9+6 and makes the moment into algebra!
Listen to the whole discussion at
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Big ideas in our units are not just big in the discipline of mathematics.
Our big ideas require cognitive reorganization in the kids' minds.
How are big ideas related to strategies and models...
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You can't just hold up a visual picture and expect that children see the mathematics in that model.
Mathematizing means modeling the world mathematically, that means we have to use contexts to generate the models that we know are the powerful ones.
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Developing number sense takes time. algorithms taught too early work against the development of good number sense.
Children who learn to think, rather than to apply the same procedures by rote regardless of the numbers, they will be empowered.
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By abandoning the rote teaching of algorithms, we are asking children to learn more.
We are taking them to mathematize, to think like mathematicians, to look at the numbers before they calculate.
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Conferring in math workshop is not about introducing new skills/concepts from a list of objectives, nor helping kids to get the right answer.
It’s about supporting the development of a young mathematician to become a better, more competent, mathematician.
#mtbos
#iteachmath
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Cognition does not start with concepts, but rather the other way around: concepts are the results of cognitive processes.
- Hans Freudenthal
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Calculating with number sense as a mathematician means having many strategies available and looking to the numbers first before choosing a strategy.
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We even take the topic of “time” seriously at
Check out the math rack clocks at
@mathrack20
and use them in our new units on time.
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In the Contexts for Learning Mathematics curriculum, we use math workshop every day.
Wanna know why?
Check out our new website to find out!
#iteachmath
#mtbos
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10
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By abandoning the rote teaching of algorithms, we are asking children to learn more.
We are teaching them to mathematize, to think like mathematicians, to look at the numbers before they calculate.
#mtbos
#iteachmath
#elemmathchat
1
9
19
Problem Strings for Measurement.
Problem Strings for Data.
Problem Strings for Geometry.
Problem Strings for Place Value.
Problem Strings for Algebra.
Problem Strings for Numeracy.
Problem Strings for each grade level.
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4
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Developing number sense takes time.
Algorithms taught too early work against the development of good number sense.
#MTBOS
#iteachmath
#elemmathchat
#nwmc22
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By abandoning the rote teaching of algorithms, we are asking children to learn more.
We are teaching them to mathematize, to think like mathematicians, to look at the numbers before they calculate.
#buildingthinkingclassrooms
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Whole-class discussions need to go beyond just a share.
We need to be asking how we can use student thinking to move the entire community forward.
Think - "How can we help kids see something new coming out of the discussion that they did not see going into the discussion?"
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ON LAUNCHING TASKS
As you develop contexts remember that you want to hook the children and help them imagine it.
DON’T TURN IT INTO A WORD PROBLEM OR TELL THEM A STRATEGY TO USE!
Give them an opportunity to find their own way to mathematize the situation.”
- Every CFLM unit
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By abandoning the rote teaching of algorithms, we are asking children to learn more.
We are teaching them to mathematize, to think like mathematicians, to look at the numbers before they calculate.
1
2
17
By abandoning the rote teaching of algorithms, we are asking children to learn more.
We are taking them to mathematize, to think like mathematicians, to look at the numbers before they calculate.
0
5
18
Teaching mathematics is about facilitating mathematical development.
This means that we cannot get all learners to the same landmarks at the same time in the same way, any more than we can get all toddlers to walk at the same time in the same way.
#mtbos
#iteachmath
0
2
17
During
#numberstrings
, it’s important to remember that...
We represent children’s thinking, not just their answers.
Different forms of representation support different noticing’s.
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8
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Heard about the Rekenrek or math rack, but not sure how/why to use it?
We love teaming up with
@mathrack20
to support automatization of the basic facts.
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Marinate in this for a little while.
Subtraction.
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PODCAST SERIES - Fractions!
"Math anxiety often comes when we don’t give ourselves permission to model the problem. If we would just do that, we often come up with new insights, new ways of thinking!"
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"Historically, time has been taught in our schools as a reading/drawing activity, rather than mathematics.
Reading hands on a clock makes little sense to a child who has yet to construct the meaning of measurement units and the durations they represent."
- The Timekeepers
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9
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Teaching mathematics is about facilitating mathematical development.
This means that we cannot get all learners to the same landmarks at the same time in the same way, any more than we can get all toddlers to walk at the same time in the same way.
Fosnot, Models of Intervention
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3
17
When kids are given the chance to compute in their own ways, to play w/ relationships and operations, they see themselves as mathematicians and their understanding deepens.
This also forms the basis for algebra and takes kids a long way in computing efficiently, and elegantly.
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5
16
Whenever we sit down to confer with a developing mathematician, we have to remind ourselves that the goal is to support the development of the young mathematicians sitting next to us.
It is not to fix the piece of mathematics.
#mtbos
#iteachmath
0
5
17
Are you using real sequences of contexts or simply word problems that camouflage school math?
In this podcast, you can consider 3 reasons why sequences of contexts, not just word problems, are essential in teaching and learning mathematics.
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11
16
Developing number sense takes time.
Algorithms taught too early work against the development of good number sense.
#iteachmath
1
2
15
What is numeracy?
Built a whole curriculum around it at
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7
16
What is numeracy and how do we develop it?
5 minutes of amazingness.
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When mathematicians write up their findings they do not merely reiterate everything they did. Instead, they focus on crafting a convincing and elegant argument or proof for other mathematicians to consider.
Elementary students should be given similar opportunities frequently.
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The purpose of models is not to fit the data, but to sharpen the questions.
- S. Karlin
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15
Students' strategies are our starting points, the moments we celebrate before upping the ante.
Understanding the progression of these strategies helps us also know just how to up the ante: we can see where we are in the journey and what is on the horizon for a given learner.
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5
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When we characterize mathematics as a human activity of mathematizing our lived worlds, & view the development of numeracy from the perspective of the learner rather than from that of a test designer, focusing on strategies, big ideas & emergent ways of modeling makes more sense.
0
4
15
Launching a “rich task” and having kids mathematize the situation only allows initial conceptions to emerge. But that is not teaching.
Teaching means we need to get underneath kids’ strategies and support the development of the big ideas that allow the strategies to work.
1
4
16
The first string in this unit is beautiful.
Modeled on the open double number line.
Try it! It's awesome.
1/2 x 36
1/4 x 36
1/8 x 36
5/8 x 36
7/8 x 36
5/8 x 48
#mtbos
#iteachmath
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3
16
"It is the story that matters, not just the ending."
Paul Lockhart
A Mathematician’s Lament: How School Cheats Us out of Our Most Fascinating and Imaginative Art Form
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3
16
There’s a difference between convenient ways to organize a textbook and how children think about mathematics.
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4
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Curious?
Check out
@mathrack20
and how we use the math rack clock in grades 1 and 2 units focused on the development of the Measurement of Time.
#iteachmath
#elemmathchat
0
8
14
We cannot just hold up a model and assume that children see what we as adults see.
The math is not in the model.
The math is in the child’s head.
0
6
15
Teaching mathematics is about facilitating mathematical development.
This means that we cannot get all learners to the same landmarks at the same time in the same way, any more than we can get all toddlers to walk at the same time in the same way.
0
5
14
Hey K-1 Teachers!
Fluency within 5 and 10 is critical!
Don't miss this awesome new book of number strings to support fluency with your young mathematicians at work.
From Fives and Tens to Automaticity
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6
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Have you ever seen Cathy Fosnot
@ctfosnot
build one of her landscapes?
Sit back and be amazed for 5 minutes.
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5
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Teaching mathematics is about facilitating mathematical development.
This means that we cannot get all learners to the same landmarks at the same time in the same way, any more than we can get all toddlers to walk at the same time in the same way.
0
8
15
Gallery walks...an important part of the process of developing a mathematical argument.
Critiquing the reasoning of others.
#iteachmath
#mtbos
#elemmathchat
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1
15
“There is no way you are going to get through this number string without a conversation on the distributive property."
10x13
2x13
12x13
20x13
22x13
19x13
Analyze
#numberstrings
with Cathy Fosnot on our online classroom video platform.
Here’s just a taste from the platform...
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