As I listened to examples of number talks in the math classroom, this stands out, teacher language matters.
“I think I heard you say.”
“How did you know you should have…..”
“Where do you think your mistake came from?”
“So you are saying…”
“How did you figure that out?”
“Do we see it another way?”
“How do we see this one?”
The teacher’s language conveys that effort, thinking processes and grappling with the ideas are what matters. Students have the freedom to explore the ideas. Relational equity develops as each student’s contribution is valued and analysed in an effort to come to a common conclusion.
And there is more … students see that math is a beautiful, creative and connected subject.
My brother nodded in agreement and chuckled, “Yes our mother was great with numbers.” She had the ability to use numbers fluently, a skill we we both easily acknowledged.
How well I remember standing next to my mother in a grocery store or bank as she quickly and easily calculated totals. I would line up the digits mentally in my head, carry or borrow if needed, and try to remember the numbers as I worked. What a cumbersome procedure! She, on the other hand, already had the answer and cheerfully let the clerk know the amount required. I recognized her skill and longed to have it. It wasn’t speed I wanted, it was the ease with which she worked. Why couldn’t I do that?
Was it the education my mother received during the 1920’s in the Netherlands? Or was it as a young, single woman when she worked for Unilever that she developed proficient number sense? It was not until later in my mathematical life that I learned her secret. Perhaps I would have benefited from a motherly ‘Number Talk‘.
A Number Talk? Yes, the kind of talks I plan to use with students this year. Parents certainly can engage their children in number talks. They’re fun! And teachers of math will find them a best practice for developing number sense in students.
I observed a number talk as part of my summer math course. Here are my observations.
How Number Talks Work
The instructor sets the students up for the number talk telling them they are to figure out a math question.
1.The question is written on the board and students are asked to solve it. No pencils, no paper, they are to calculate their answers mentally. No comment is given on how to approach the problem.
2. Students are given lots of thinking time and respond with a thumbs up once they have the solution. A low key way to respond so that everyone can take the time they need.
3. Students then share their answers and explain how they arrived at the solution.
4. Through the discussion students are led to see the interesting variety of approaches used to solve the question.
Open ended questions such as these spur the discussion on.
“Anyone try something different?”, “Anyone else do that?” These questions give everyone a chance to explain their process.
“I think I heard you say.” or “How did you know you should have…..” These statements give the student opportunity to clarify thinking and communication. The teacher does not add to the student’s explanation she only repeats what she hears. The onus is on the student to ensure that the explanation is clear.
“Where do you think your mistake came from?” Helps the student clarify logic and identify the error. Mistakes are part of the solution process both acceptable and interesting as part of the learning.
5. The process students use to solve the problem is written on the board so that it is easy for everyone to follow. This is important. Everyone needs to understand how the numbers are manipulated.
6. All solutions are represented on the board and students are asked to draw a picture of their solution and someone else’s solution. Another important step as it makes the learning more concrete and helps students see how they can work flexibly with numbers.
Number talks are one way to help students develop insight, ability and willingness to to break numbers apart and regroup them as they observe and discuss different ways to solve math problems.
Yes, the answer is important but it is not the most interesting part of a mathematical question. As we show students that problems can be solved in different ways we teach them the very, very important building block of number sense, a skill that is foundational for the rest of math.
Ahhh yes, a number talk would have given me insight into the skillful mind of my mathematician mother but I think I am on to her secret now.
Ugh! A didactic tale can fall flat. With little meaningful connection to the story the message is seldom remembered or lost to the reader. You may identify with this. I for one, recognize this didactic interaction not only in books but in my classroom.
Lesson 4 of How to Learn Math with Jo Boaler describes the didactic contract in the classroom. See if you recognize it.
The Didactic Contract identified by by Guy Brouseau, states that there are certain expectations for both the student and the teacher in a learning setting. Teachers are to demonstrate and guide their pupils, and students are to learn with ease.
Here is how it happens in my classroom. During a math class I quickly step in to clarify, add to a student’s math explanation, or demonstrate the next step. Students on the other hand, are quick to ask for help when faced with uncertainty because they do not necessarily believe that learning involves struggles or challenges. Together we fall into this unspoken contract. This way of interacting in the classroom becomes a barrier to learning.
Jo Boaler describes it well: We empty the interaction of learning and reduce the cognitive demand for the student.
As a teacher responsible for student learning, it is too easy to take ownership of something that belongs to the learners. Instead of this didactic interaction I need to allow mistakes, reflection, redirection, and meaning making, as students develop math skills. My goal as a teacher is to have an engaging classroom where students pursue questions and discuss their thinking about mathematical processes. This dynamic interaction would promote deeper thinking and meaningful problem solving.
It is critical in this setting to ask the kinds of questions that promote meaningful engagement. What is a good math question? Here is a challenge I love! I am building on what I know and will share more with you in a future post.
Current research helps us understand what high achieving math students do and it is most interesting, in fact Jo Boaler tells us it is worth shouting from the rooftops so here it is:
High achieving math students use flexible thinking, are able to easily decompose and recompose numbers and naturally compress ideas to move on to harder concepts.
What does all this mean? Let’s look her explanation.
Consider a simple computation such as this:
5 + 14
There are several possible strategies to use.
Counting all: First count all the blue counters to 5. Next, count the purple counters to confirm that there are indeed 14. Lastly, proceed to count all the counters from 1 to 19.
Counting on: Count the first set of blue counters to 5 and count on to 19.
Known facts: A student may simply know that 5 + 14 equals 19.
Derived facts: Students use what is known about numbers and facts to complete the computation. Fourteen is also 10 + 4 so, since 4 + 5 equals 9, add 10 and solve the problem to get 19. Although this may seem obvious to an adult reader it is important to note the process of decomposing and recomposing the number to make the problem easier.
Interestingly enough students who are low achieving at math use approaches that are more difficult. For example, imagine using a counting back strategy for subtraction.
25 – 14 = ___.
There are many steps involved in counting back from 25. This is a complex task and one where a student can easily become confused. Students who find math difficult often apply a ‘follow the rules’, problem solving process, lacking the understanding to make sense of numbers in flexible ways. How much easier this very problem would be, if the student worked flexibly with numbers as was done in the earlier example. Fourteen becomes ten and four. Now the problem becomes 25 – 10 = 15 and step two, 15-4 = 11.
Open Cones or Long Ladders?
High achieving students compress mathematical ideas. What does this mean? Think about learning multiplication. Initially students struggle, work through the process and practice examples. Once students understand what multiplication is, and how to use it, the concept is compressed and easily used in new settings.
Jo Boaler uses an image of an inverted cone to show what is meant by compression and how it helps students as they learn. Learning is compressed as students begin to apply understanding efficiently. New learning is built on compressed ideas and understanding grows.
Low achieving students who work at trying to remember rules, methods and procedures have a different view of mathematics, much like an endless ladder to be climbed and a long series of steps to be remembered. These students need to be guided to develop the skills of working flexibly with numbers, and to develop a deeper understanding of number sense.
All I can say is, “Thankfully we are developing a deeper understanding of how we learn.”
Once upon a time I lived in a world of the fixed mindset¹ and likely you did too. Some students were smart and others were not. Intelligence was viewed as fixed at birth and one of the roles of education was to sort students and direct each one to the correct vocation. And, if you think about it, too often we still organize learning in this way.
Let’s identify the qualities this fixed mindset promotes and you will recognize it right away.
Students working under a fixed mindset:
Are afraid to make mistakes
Give up more easily
Fear of constructive criticism
Feel threatened by the success of others
Think what this does to learning. Consider what happens when students see themselves in this way.
Contrast this with a growth mindset¹, a term which on its own sounds encouraging. A growth mindset states that intelligence can be developed and our true potential is unknown.
Students with a growth mindset are:
Not afraid of mistakes
Willing to take on a challenge
Inspired by the success of others
How can these ideas change instruction?
It is not the student who ‘knows’, that we should recognize rather the student who says, ‘hmmm I am trying to figure this out and I have not got it yet.‘ It is this student who demonstrates a growth mindset. As educators we need to communicate that everyone can get better if they work on it, which means that persistence becomes a key quality to encourage.
And for this reason our view of mistakes plays a critical role in our mindset. How do mistakes impact learning? The student who makes a mistake has multiple opportunities to learn. First from recognizing the mistake, and then from working through the process to correct it. All this creates more opportunities for brain synapses to fire and grow. Working through mistakes causes our brains change and develop.
What message shall we give to students?
Mistakes are fertile ground for learning.
As an educator with a growth mindset, I am motivated to create an environment where risk taking is safe and encouraged, and where learners at all levels are recognized for their effort.
¹Carol Dweck – Mindset: The New Psychology of Success Random House Publishing Group | December 26, 2007
Math, math, math, what are your thoughts on this subject?
Is it true that math is portrayed as a hard subject? As a student have you ever received the message that some people are math people and others are not?
Do we hold stereotypical messages about gender or race and ability to do math?
When you were in school what did you think about your own ability to do math?
You might be surprised to hear:
“All students can achieve at the highest levels in maths at all levels of school right up to the end of high school.”
Yes, there are countries in the world where this is the expected norm.
This summer I am using this blog to reflect on my learning in the course: How To Learn Math by Jo Boaler. This course is intended for teachers and parents and presents new research ideas on learning, the brain, and math that can change the way you think about math and how we learn.
The ideas on this blog will be a combination of my reflections and notes from the course. My hope is that along the way I’ll add clarity, and a deeper understanding to what I already know about math instruction and gain new ideas on how enlarge and enrich the world of math for my students. I hope you’ll join me in this adventure.
This summer I am taking an online math course from Jo Boaler, Professor of Mathematics Education, Stanford University.
It is fitting I think, that an online math teacher should take an online math course and it is logical then, that one of my interests is the very structure and organization of the course. How do I as a student interact with the course and with other students? In what way is this course engaging? How do I assess my progress? Does the course provide resources for further learning? And how on earth can an online course compete with summer in Alberta?
It must be engaging, and you may be happy to know it is, both in content and structure, because for a period of time each day I am passing up on warm sunshine and relaxing beach reading, to sit in front of my computer.
What are the features of this course that make it work for me? Each lesson consists of a series of short videos with accompanying text. I can view the videos several times if I wish and stop at any point to jot notes. The videos are short, from less then a minute to about 12 minutes and each video ends with a question that asks a response from me. In addition each video includes a forum where participants can reflect and comment on content. As a whole the course is easy to navigate, I can see my progress and understand what is next.
Already I can see these strengths:
– short chunks for learning
– immediate opportunity for response to the content
– interaction with other students
– flexibility to work through the material in any order.
– easy to navigate
– learn when I want ( and enjoy the sunshine too!)
Have you ever taken an online course? How was it structured? Did the course format work for you? What worked well? What would have improved the course for you? I would like to hear about it.
As I look back to the summer of 2013, countless memories remain from my experience as an English language teacher in the Czech Republic.
The beauty of Prague? Yes surely.
The challenge of daily communication? That too.
The warmth of people in the Czech Republic ? Absolutely!
All this and more….
What a crew we were, several accomplished ELL instructors and a group of enthusiastic, willing, professionals prepared to share part of their summer with English Language Learners in the Czech Republic. We were motivated by faith, striving in a meaningful way to share the life of faith in Christ, by leading a week long English Camp.
Could all the hours of preparation prepare us for the unknowns our team might experience as we conducted our week long English camp? Hmmmmm….. likely not.
Lessons for several levels of learners had been prepared with care, daily welcome and ice breaker activities were ready to go, evening programs to close the day with energy and good spirits were included in our arsenal of plans.
In the Czech Republic, students age 12 to 25 and several adults aged 55 and 75 had set aside a week of daily life to be learners. They were an eclectic and amazing bunch, from all walks of life. Our classes included adults who had experienced the Russian occupation and young people who were eager for a new future. As I think of them now, a year later, they were role models of students eager to learn. They were persistent, willing to learn from mistakes, and encouraged by each other’s success.
As a teacher I was happily immersed in the joy of learning for one short week with these lovely people. These students demonstrated that ability and ‘smartness’, grows with experience. These English language learners had a growth mindset when it came to developing new skills.
Learning involves risk taking and all of us, teachers and students, had to be willing to jump in and try something new. Each of us would evaluate our attempts at communication, revise, laugh to ourselves and forge ahead. How I applaud these students, who were willing to take the plunge. Sometimes it was with shyly spoken words and other times with full expression in a reader’s theatre.
Shared meals and conversation developed a learning community, as we laughed, cheered, problem solved, and persisted to share stories about ourselves. How could I forget, warm and gentle Pavell who came to learn this year because he heard about the class from others. Or wise and kindly George who approached my husband, Sid, on our last day, put his hand on his own heart, then reached out and put his hand on Sid’s heart, as he quietly said, “We are brothers. ” We reached beyond language barriers to truly communicate.
Our work was built on careful organization by our host church , First Baptist Church of Litomaurice. Every last detail, food, lodging, teaching facilities, and student registration was carefully thought out and arranged. How thankful we were for the love, care and commitment that went into this preparation.
We made connections across the globe and as I reflect on last summer I see no distinction between learners and teachers, we all came away changed.
I held my father’s large, calloused hand and matched his stride as we walked up the steps to the library and through the heavy wooden doors. We stood in the foyer, one long staircase before me led down to the children’s library and the other, up to the distant unreachable land of adult books. Perhaps someday I might venture there too.
After pausing, with some cautious uncertainty I slowly made my way down the stairs and opened the doors at the bottom. What wonder and delight filled my young heart as I stepped through those doors! In an instant I fell in love with books and reading. Books everywhere… colorful and inviting. Perhaps it was at that moment that the reader in me was born.
The memory of sharing that Saturday afternoon with my father is fixed in my brain and when the Edmonton Public Library and Vintage Edmonton displayed this image online it brought a smile. Good things have small beginnings.
This building was my happy introduction to libraries and the wonders they contain.
It demolished in 1966, to make way for more modern architecture.
That which remains is the happy heart of a lifelong reader.