misscalcul8

8.01.2016

Mathematical Mindsets: The Highlights {Part 3}

The task is challenging but accessible .
The boys saw the task as a puzzle
The visual thinking about the growth of the task gave the boys understanding of the way the pattern grew
They had all developed their own way of seeing the pattern growth
The classroom had been set up to encourage students to propose ideas without being afraid of making mistakes
We had taught the students to respect each other's thinking
The students were using their own ideas,
The boys were working together
The boys were working heterogeneously.

When we don't ask students to think visually, we miss an incredible opportunity to increase their understanding.

Additionally, students did not think they were finding a standard answer for us; they thought they were exploring methods and using their own ideas and thoughts, which included their own ways of seeing mathematical growth.

The researchers found that when students were given problems to solve, and they did not know methods to solve them, but they were given opportunity to explore the problems, they became curious, and their brains were primed to learn new methods, so that when teachers taught the methods, students paid greater attention to them and were more motivated to learn them.

The teacher taught them the methods when they were needed, rather than the usual approach of teaching a method that students then practiced.

When students are asked to think intuitively, many good things happen. First, they stop thinking narrowly about single methods and consider mathematics more broadly. Second, they realize they have to use their own minds—thinking, sense making, and reasoning. They stop thinking their task is just to repeat methods, and they realize their task is to think about the appropriateness of different methods. And third, as the Schwartz and Bransford research study showed, their brains become primed to learn new methods (Schwartz & Bransford, 1998).

When teachers are designers, creating and adapting tasks, they are the most powerful teachers they can be.

Making math tasks richer:
1. Can You Open the Task to Encourage Multiple Methods, Pathways, and Representations?
2. Can You Make It an Inquiry Task?
When students think their role is not to reproduce a method but to come up with an idea, everything changes (Duckworth, 1991).
The mathematics is more complex and exciting because students are using their ideas and thoughts.
3. Can You Ask the Problem Before Teaching the Method?
4. Can You Add a Visual Component?
5. Can You Make It Low Floor and High Ceiling?
When students are invited to ask a harder question, they often light up, totally engaged by the opportunity to use their own thinking and creativity.
6. Can You Add the Requirement to Convince and Reason?

In every math conversation, students were asked to reason, explaining why they had chosen particular methods and why they made sense. This opened up mathematical pathways and allowed students who had not understood to both gain understanding and ask questions, adding to the understanding of the original student.

She explains that there are three levels of being convincing (Boaler & Humphreys, 2005):

Convince yourself
Convince a friend
Convince a skeptic

It is fairly easy to convince yourself or a friend, but you need high levels of reasoning to convince a skeptic. Cathy tells her students that they need to be skeptics, pushing other students to always give full and convincing reasons.

When I ask students to play the role of being the skeptic, I explain that they need to demand to be fully convinced. Students really enjoy challenging each other for convincing reasons, and this helps them learn mathematical reasoning and proof.

Open up the task so that there are multiple methods, pathways, and representations. Include inquiry opportunities. Ask the problem before teaching the method. Add a visual component and ask students how they see the mathematics. Extend the task to make it lower floor and higher ceiling. Ask students to convince and reason; be skeptical.

Chapter 6: Mathematics and the Path to Equity

When we have gifted programs in schools we tell students that some of the students are genetically different; this message is not only very damaging but also incorrect.

Some people who have excelled in math choose not to be proud of the hard work and struggle they went through; they prefer to think they were born with a gift. There are many problems with this idea, one being that students who are successful through hard work often think that they are imposters because their achievement was not effortless.

The researchers went on to study the factors in the students' environment that led to different feelings of belonging, and they found that two factors worked against feelings of belonging. One was the message that math ability is a fixed trait; the other was the idea that women have less ability than men. These ideas shaped women's, but not men's, sense of belonging in math. The women's lowered sense of belonging meant that they pursued fewer math courses and received lower grades. Women who received the message that math ability is learned were protected from negative stereotypes—they maintained a high sense of belonging in math and remained intent on pursuing mathematics in the future.

We need all teachers to believe in all students, to reject the idea of some students being suitable for higher-level math and others not, and to work to make higher-level math available to all students, whatever their prior achievement, skin color, or gender.

Some teachers believe that some students cannot achieve at high levels of high school because they live in poverty or because of their previous preparation. In Chapter One I gave an example of high school teachers who made this argument to their school board, but teachers such as those at Life Academy are proving this wrong every day, through teaching high-level mathematics and positive messages to all students.

This is unfortunate, as we know that students who are advanced in math from an early age are more likely to drop math when they get the opportunity and achieve at lower levels.

Making math more equitable:
1. Offer all students high-level content
2. Work to change ideas about who can achieve in mathematics
The studies also show, encouragingly, that students who have a growth mindset are able to shrug off stereotyped messages and continue to success; this speaks again to the huge need for students, and teachers, to develop growth mindset beliefs about their own subjects and transmit growth mindset messages to students.
3. Encourage students to think deeply about mathematics
Unfortunately, the procedural nature of mathematics teaching in many classes means that deep understanding is often not available, and when girls cannot gain deep understanding they underachieve, turn away from mathematics, and often develop anxiety. Girls have much higher levels of anxiety about mathematics than boys do (Organisation for Economic Co-operation and Development [OECD], 2015), and the unavailability of deep understanding is one main reason for this (Boaler, 2014a).
4. Teach students to work together
When the Chinese American students found mathematics difficult, they were supported—first by knowing that everyone was struggling and then by working together to solve problems.
5. Give girls and students of color additional encouragement to learn math and science
The researchers found that the levels of anxiety held by women elementary teachers predicted the achievement of the girls in their classes, but not the boys (Beilock et al., 2009).
Researchers found that when mothers told their daughters “I was no good at math in school” their daughter's achievement immediately went down (Eccles & Jacobs, 1986). Teachers need to replace sympathetic messages such as “Don't worry, math isn't your thing” with positive messages such as “You can do this, I believe in you, math is all about effort and hard work.” Subsequent experiments showed that women underachieved when they simply marked their gender in a box before taking the test, compared to those who did not have to do that. Role models are extremely important to students—and one of the reasons it is so important to diversify the teaching force.
6. Eliminate (or at least change the nature of) homework
PISA, the international assessment group, with a data set of 13 million students, recently made a major announcement. After studying the relationships among homework, achievement, and equity, they announced that homework perpetuates inequities in education (Program for International Student Assessment [PISA], 2015).

Additionally, they questioned whether homework has any academic value at all, as it did not seem to raise achievement for students. This is not an isolated finding; academic research has consistently found homework to either negatively affect or not affect achievement. Baker and LeTendre (2005), for example, compared standardized math scores across different countries and found no positive link between frequency of math homework and students' math achievement.

Mikki (2006) found that countries that gave more math homework had lower overall test scores than those that gave less math homework (Mikki, 2006). Kitsantas, Cheema, and Ware (2011) examined 5,000 15-and 16-year-olds across different income levels and ethnic backgrounds and also found that the more time students spent on math homework, the lower their math achievement across all ethnic groups.

When we assign homework to students, we provide barriers to the students who most need our support. This fact, alone, makes homework indefensible to me.

It is unfair and unwise to give students difficult problems to do when they are tired, sometimes even exhausted, at the end of the day. I wonder if teachers who set homework think that children have afternoon hours to complete it, with a doting parent who does not work on hand. If they do not think this, then I do not understand why they feel they can dictate how children should spend family time in the evenings.

The value of most math homework across the United States is low, and the harm is significant.

Homework should be given only if the homework task is worthwhile and draws upon the opportunity for reflection or active investigation around the home.

7.31.2016

Mathematical Mindsets: The Highlights {Part 2}

Chapter 4: Creating Mathematical Mindsets: The Importance of Flexibility with Numbers

The best and most important start we can give our students is to encourage them to play with numbers and shapes, thinking about what patterns and ideas they can see.

Successful math users have an approach to math, as well as mathematical understanding, that sets them apart from less successful users. They approach math with the desire to understand it and to think about it, and with the confidence that they can make sense of it. Successful math users search for patterns and relationships and think about connections. They approach math with a mathematical mindset , knowing that math is a subject of growth and their role is to learn and think about new ideas. We need to instill this mathematical mindset in students from their first experiences of math.

When students see math as a broad landscape of unexplored puzzles in which they can wander around, asking questions and thinking about relationships, they understand that their role is thinking, sense making, and growing.

Instead of approaching numbers with flexibility and number sense, they seemed to cling to formal procedures they had learned, using them very precisely, not abandoning them even when it made sense to do so. The low achievers did not know less , they just did not use numbers flexibly—probably because they had been set on the wrong pathway, from an early age, of trying to memorize methods and number facts instead of interacting with numbers flexibly (Boaler, 2015a). The researchers pointed out something else important—the mathematics the low achievers were using was a harder mathematics. It is much easier to subtract 5 from 20 than to start at 21 and count down 16 numbers.

Notably, the brain can only compress concepts; it cannot compress rules and methods. Therefore students who do not engage in conceptual thinking and instead approach mathematics as a list of rules to remember are not engaging in the critical process of compression, so their brain is unable to organize and file away ideas; instead, it struggles to hold onto long lists of methods and rules. This is why it is so important to help students approach mathematics conceptually at all times.

The left side of the brain handles factual and technical information; the right side brain handles visual and spatial information. Researchers have found that mathematics learning and performance are optimized when the two sides of the brain are communicating (Park & Brannon, 2013).

The implications of this finding are extremely important for mathematics learning, as they tell us that learning the formal abstract mathematics that makes up a lot of the school curriculum is enhanced when students are using visual and intuitive mathematical thinking.

The antithesis of this approach is a focus on rote memorization and speed. The more we emphasize memorization to students, the less willing they become to think about numbers and their relations and to use and develop number sense.

The hippocampus, like other brain regions, is not fixed and can grow at any time, as illustrated by the London Black Cab studies (Woollett & Maguire, 2011), but it will always be the case that some students are faster or slower when memorizing, and this has nothing to do with mathematics potential.

All subjects require the memorization of some facts, but mathematics is the only subject in which teachers believe they should be tested under timed conditions. Why do we treat mathematics in this way? We have the research evidence that shows students can learn math facts much more powerfully with engaging activities; now is the time to use this evidence and liberate students from mathematics fear.

It is important to revisit mathematical ideas, but the “practice” of methods over and over again is unhelpful. When you learn a new idea in mathematics, it is helpful to reinforce that idea, and the best way to do this is by using it in different ways. We do students a great disservice when we pull out the most simple version of an idea and give students 40 questions that repeat it. Worksheets that repeat the same idea over and over turn students away from math, are unnecessary, and do not prepare them to use the idea in different situations.

First, practicing isolating methods induces boredom in students; many students simply turn off when they think their role is to passively accept a method (Boaler & Greeno, 2000) and repeat it over and over again.

Second, most practice examples give the most simplified and disconnected version of the method to be practiced, giving students no sense of when or how they might use the method.

When textbooks introduce only the simplest version of an idea, students are denied the opportunity to learn what the idea really is.

When learning a definition, it is helpful to offer different examples—some of which barely meet the definition and some of which do not meet it at all—instead of perfect examples each time.

Students are given uncomplicated situations that require the simple use of a procedure (or often, no situation at all). They learn the method, but when they are given realistic mathematics problems or when they need to use math in the world, they are unable to use the methods (Organisation for Economic Co-operation and Development, 2013). Real problems often require the choice and adaptation of methods that students have often never learned to use or even think about.

One significant problem the students from the traditional school faced in the national examination—a set of procedural questions—was that they did not know which method to choose to answer questions. They had practiced methods over and over but had never been asked to consider a situation and choose a method.

It is also part of the reason that students do not develop mathematical mindsets; they do not see their role as thinking and sense making; rather, they see it as taking methods and repeating them. Students are led to think there is no place for thinking in math class.

In a second study, conducted in the United States, we asked students in a similar practice model of math teaching what their role was in the math classroom (Boaler & Staples, 2005). A stunning 97% of students said the same thing: their role was to “pay careful attention.” This passive act of watching—not thinking, reasoning, or sense making—does not lead to understanding or the development of a mathematical mindset.

Large research studies have shown that the presence or absence of homework has minimal or no effects on achievement (Challenge Success, 2012) and that homework leads to significant inequities.

Research also shows that the only time homework is effective is when students are given a worthwhile learning experience, not worksheets of practice problems, and when homework is seen not as a norm but as an occasional opportunity to offer a meaningful task.

Two innovative teachers I work with in Vista Unified School District, Yekaterina Milvidskaia and Tiana Tebelman, developed a set of homework reflection questions that they choose from each day to help their students process and understand the mathematics they have met that day at a deeper level. They typically assign one reflection question for students to respond to each night and one to five mathematical questions to work on (depending on the complexity of the problems).

Questions that ask students to think about errors or confusions are particularly helpful in encouraging students' self-reflection, and they will often result in the students' understanding the mathematics for the first time.

Number talks are the best pedagogical method I know for developing number sense and helping students see the flexible and conceptual nature of math.

A growth mindset is important, but for this to inspire students to high levels of mathematics learning, they also need a mathematics mindset. We need students to have growth beliefs about themselves and accompany these with growth beliefs about the nature of mathematics and their role within it.

7.30.2016

Mathematical Mindsets: The Highlights {Part 1}

This book I would say has changed my thoughts on math, teaching, and teaching math more than any other I've read in my seven year career. I will recommend it and link it forever. I will have to post my highlighted notes from it in several posts because no one would ever scroll through all of it otherwise! There is just so much to process and that I will need to read over and over again- so many opportunities for growth and change!

It's only $10.71 for the paperback and $7.99 for the Kindle version. You NEED this book. But until you get your own, this should be enough to make you want more.

Enjoy!

Mathematical Mindsets: Unleashing Students' Potential through Creative Math, Inspiring Messages and Innovative Teaching
Jo Boaler

Introduction: The Power of Mindset
When students get the idea they cannot do math, they often maintain a negative relationship with mathematics throughout the rest of their lives.

Research studies have established that the more math classes students take, the higher their earnings ten years later.

Research has also found that students who take advanced math classes learn ways of working and thinking—especially learning to reason and be logical—that make them more productive in their jobs. Students taking advanced math learn how to approach mathematical situations so that once they are employed, they are promoted to more demanding and more highly paid positions than those who did not take mathematics to advanced levels (Rose & Betts, 2004).

That single belief—that math is a “gift” that some people have and others don't—is responsible for much of the widespread math failure in the world.

Math is conveyed as a really hard subject that is uninteresting, inaccessible, and only for “nerds”; it is not for cool, engaging people, and it is not for girls. It is no wonder that so many children in schools disengage from math and believe they cannot do well.

Part of the change we need to see in mathematics is acknowledgment of the creative and interpretive nature of mathematics. Mathematics is a very broad and multidimensional subject that requires reasoning, creativity, connection making, and interpretation of methods; it is a set of ideas that helps illuminate the world; and it is constantly changing. Math problems should encourage and acknowledge the different ways in which people see mathematics and the different pathways they take to solve problems. When these changes happen, students engage with math more deeply and well.

They believe that mathematics ability is a sign of intelligence and that math is a gift, and if they don't have that gift then they are not only bad at math but they are unintelligent and unlikely to ever do well in life.

Chapter 1: The Brain and Mathematics Learning

If you learn something deeply, the synaptic activity will create lasting connections in your brain, forming structural pathways, but if you visit an idea only once or in a superficial way, the synaptic connections can “wash away” like pathways made in the sand. Synapses fire when learning happens, but learning does not happen only in classrooms or when reading books; synapses fire when we have conversations, play games, or build with toys, and in the course of many, many other experiences.

If brains can change in three weeks, imagine what can happen in a year of math class if students are given the right math materials and they receive positive messages about their potential and ability.

The new evidence from brain research tells us that everyone, with the right teaching and messages, can be successful in math, and everyone can achieve at the highest levels in school.

What I am saying is that any brain differences children are born with are nowhere near as important as the brain growth experiences they have throughout life.

Every second of the day our brain synapses are firing, and students raised in stimulating environments with growth mindset messages are capable of anything.

A lot of scientific evidence suggests that the difference between those who succeed and those who don't is not the brains they were born with, but their approach to life, the messages they receive about their potential, and the opportunities they have to learn. The very best opportunities to learn come about when students believe in themselves.

In other studies, researchers have shown that students' (and adults') mindsets can change from fixed to growth, and when that happens their learning approach becomes significantly more positive and successful (Blackwell et al., 2007).

The highest-achieving students in the world are those with a growth mindset, and they outrank the other students by the equivalent of more than a year of mathematics (see Figure 1.6 ).

It turns out that even believing you are smart—one of the fixed mindset messages—is damaging, as students with this fixed mindset are less willing to try more challenging work or subjects because they are afraid of slipping up and no longer being seen as smart. Students with a growth mindset take on hard work, and they view mistakes as a challenge and motivation to do more.

When students are given fixed praise—for example, being told they are smart when they do something well—they may feel good at first, but when they fail later (and everyone does) they think that means they are not so smart after all.

Praise feels good, but when people are praised for who they are as a person (“You are so smart”) rather than what they did (“That is an amazing piece of work”), they get the idea that they have a fixed amount of ability.

Telling students they are smart sets them up for problems later. As students go through school and life, failing at many tasks—which, again, is perfectly natural—they evaluate themselves, deciding how smart or not smart this means they really are. Instead of praising students for being smart, or any other personal attribute, it's better to say things like: “It is great that you have learned that,” and “You have thought really deeply about this.”

Chapter 2: The Power of Mistakes and Struggle

“Every time a student makes a mistake in math, they grow a synapse.”

One reason it is so significant is that it speaks to the huge power and value of mistakes, although students everywhere think that when they make a mistake it means that they are not a “math person” or worse, that they are not smart.

When teachers ask me how this can be possible, I tell them that the best thinking we have on this now is that the brain sparks and grows when we make a mistake, even if we are not aware of it, because it is a time of struggle; the brain is challenged, and this is the time when the brain grows the most.

First, the researchers found that the students' brains reacted with greater ERN and Pe responses—electrical activity—when they made mistakes than when their answers were correct. Second, they found that the brain activity was greater following mistakes for individuals with a growth mindset than for individuals with a fixed mindset.

The study also found that individuals with a growth mindset had a greater awareness of errors than individuals with a fixed mindset, so they were more likely to go back and correct errors.

It tells us that the ideas we hold about ourselves—in particular, whether we believe in ourselves or not—change the workings of our brains. If we believe that we can learn, and that mistakes are valuable, our brains grow to a greater extent when we make a mistake.

He points out: “Imperfection is a part of any creative process and of life, yet for some reason we live in a culture that has a paralyzing fear of failure, which prevents action and hardens a rigid perfectionism. It's the single most disempowering state of mind you can have if you'd like to be more creative, inventive, or entrepreneurial.”

He also summarizes the habits of successful people in general, saying that successful people:

Feel comfortable being wrong
Try seemingly wild ideas
Are open to different experiences
Play with ideas without judging them
Are willing to go against traditional ideas
Keep going through difficulties

It's also a good time to reinforce important messages—that when the student made this mistake, it was good, because they were in a stage of cognitive struggle and their brain was sparking and growing.

I said “Do you know what just happened? When you got that answer wrong your brain grew, but when you got the answer right, nothing happened in your brain; there was no brain growth.”

If we want students to be making mistakes, we need to give them challenging work that will be difficult for them, that will prompt disequilibrium.

In workshops with Carol Dweck I often hear her tell parents to communicate to their children that it is not impressive to get work correct, as that shows they were not learning.

This is a radical message, but we need to give students strong messages to override an idea they often get in school—that it is most important to get everything correct, and that correctness is a sign of intelligence.

When mathematics is taught as an open and creative subject, all about connections, learning, and growth, and mistakes are encouraged, incredible things happen.

7.29.2016

How To...Teacher Moves

In my own personal effort to #ExpandMTBoS, I'm starting a new category of blog posts called 'How To' so I can share the strategies behind the resource. I hope new and veteran teachers alike can find something useful. Click on the tag to the right for more posts!

This is a collection of ideas and resources that I've read and wanted to use or have already used.

Get presenter’s to the front. One can only speak and the other can only point. They explain their thinking for one pair. Keep this light, safe and fun. If a student does not explain clearly enough or missing key elements, just let it go, they will most likely come out in later explanations.

Ask a student in the class to re-explain the presenter’s thinking.

"Get low" in the classroom so students don't look to you as the answer keeper.

Students write two truths and a lie about a function or math problem; see here for variations; include these type of questions on assessments.

Grade or give feedback with two or more highlighters.

When students say:

"What do I do next?" reply with "What do you think?"
"What do I do next?" reply with "How did you start?"
"Is this right?" reply with "What did you do?"
"Can you help me?" reply with "What should you do first?"
"I got the wrong answer." reply with "Can you find a mistake in your work?"

When asking students to share their responses with the class, say "Thank you" to acknowledge their answers without confirming if it's right or wrong. Practice that poker face!

Put self-assessment questions on quizzes and tests for students to reflect on what they think they know.

Include quadratic equations when teaching solving systems by substitution. {Great idea Meg!}

Help address gaps, spiral content, self-test, study, or review by giving students index card problems as they enter the room. You can make an answer key or have students line up or sit down based on the answers they get. {Thanks Nora!}

You can jump a few DOK levels by reversing the question....give students math problems and ask them what they solve instead of asking them to do the calculation. {Learn more from Fawn}

Here are three quick games to play when you have extra time in class that involve some strategy and logic. Always be prepared; this is a great back up. {Thanks to Julie!}

Sarah Carter shares four of her favorite review games. I know I always have my default games so it's great to mix it up with some new ideas. Here's another collection from Kim that I really loved and have gotten away from.

I am a huge fan of card sorts but this post really inspired me to kick mine up a notch. These ideas work great for INBs, individual studying, and pair practice. {Love these Brigid!}

7.28.2016

Bell Ringers 3.0

One disadvantage of teaching in a tiny school is that you can't just reuse everything because you have the same students for three years in a row. So every year I have to find new first day of school activities and change things like my daily bell ringers.

Every year I find a new obsession so I would probably just change it anyway.

Updates from last year:

Changed the colors to match better
Took off the week labels {how is it that we got to school for 180 days but it's more than 36 weeks....so confusing}
Changed "Weigh It Wednesday" balance bender puzzles to "Work It Wednesday" brain teasers {thanks to weekly KenKen pdfs for educators!}
Changed "Thoughtful Q's Thursday" to "Number Talks Thursday" {excited but completely unprepared for these}
Updated estimation180 and WODB photos with new ones {thanks everyone who submits those!}

Here is THE powerpoint!

Now I have some questions. Last year I printed out front and back handouts every week for students. I know some people use Google Classroom for warm ups but I just don't think I can rely on our Internet on a daily basis.

I know for sure I want students to write on Mental Math Mondays.

I'm thinking I could use Google Forms for Tough Guess Tuesday estimation180 photos. Most people did not or could not calculate the error and error percentage. Do I need them to do that? Do I need them to write a description and a reason? How would I display the information in a useful way?

Work It Wednesday are brain teasers that don't necessarily require writing...students could use dry erase markers on their desk. Do I need them to write anything?

For Number Talk Thursday, it's supposed to be mental so students could use marker again. But I also used some dot images so I could print those on paper for students to write on. I kind of like the idea of printing more than one of the same image so they can practice seeing different strategies.

I know for sure I want students to use Plickers for Freaky Friday WODB. They love Plickers and I only use them a couple times of year. But do I want them to write their reasons? Or just call on random people to share their answers? I obviously don't grade these so do I NEED them to write?

I guess what I'm truly struggling with is....will they do it if I don't make them write it down and turn it in?

It would be great if I could use less paper...maybe fit one week per side, cutting the amount of copies I need in half. But my favorite part of last year's handouts was the questions I asked every week. They were random and let me get to know the students so much better. I guess I could use Google Classroom for those too...how would you do that? Every Friday post a question?

How do you guys handle your warm ups in a way that makes your heart smile? :-)

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