You’ve probably heard it – a professor holds up a jar with several large rocks in it. “Is it full?” the professor asks. Her students say sure. She then adds gravel, which fills in the space around the rocks. “Is it full?” she asks again. Her students, beginning to understand, say no. The professor then proceeds to pour in some sand, filling the spaces between the gravel. “Is it full?” she asks again. The students are uncertain, until the professor pours water into the jar, filling the jar to the top. “What is the moral of this lesson?” asks the professor. A student yells out, “You can always fit more in the jar!” The professor shakes her head. “No,” she says. “The moral is that if you don’t put the big rocks in first, they’ll never fit.”

Reflecting on this parable, I decided to select four big “rocks” – one for each quarter – that are the foundational ideas I want to cover this year. Building lessons while keeping the four big concepts in mind would help ensure that they wouldn’t get lost in the day-to-day gravel and sand of the school year.

This year I will be teaching Math 8 and Integrated Math 1. So, with the California Common Core State Standards for Mathematics in hand, I picked my four rocks.

For Math 8, I picked these four:

- Linear relationships, including their graphs and equations
- Exponential relationships, including roots
- Rates of change, including slope
- Right triangle congruence and similarity, including the Pythagorean Theorem

For Integrated Math 1, I picked these four:

- Arithmetic (linear) vs. Geometric (exponential) growth
- Systems of Equations
- Represent and interpret data sets, including bivariate data
- Rigid motion in the coordinate plane

What do you think? Do you agree with my “rocks”?

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Boaler, J. (2008). Promoting ‘relational equity’ and high mathematics achievement through an innovative mixed-ability approach. *British Educational Research Journal, 34* (2), 167-194.

**Summary of Research**

This study followed students placed in mixed ability collaborative groups in math and compared them with students in traditional math classes over four years. Through lesson observations and student interviews and assessments, Boaler found that the students placed in the heterogeneous collaborative groups not only outperformed the students in traditional math classes in mathematics achievement but also showed increased relational equity, treating each other with more respect and offering each other more support.

**Design of study:**

Quantitative and qualitative data on 700 students were collected over four years at three California high schools. Two high schools used traditional teaching methods, which incorporated 21% lecture, 15% teacher questioning, 48% individual work, and 11% group work in ability-grouped math classes. The third school used reform methods, with 4% lecture, 9% teacher questioning, 72% group work, and 9% student presentations in heterogeneously grouped math classes.

**Data Collection:**

Over 600 hours of lessons at all three schools were observed and analyzed. Sixty students were interviewed every year, sampling from high and low achievers and different cultural and ethnic groups. All students were given questionnaires in the first three years of the study asking about their experiences and perceptions of math. All students were assessed for mathematics understanding, and selected focus groups were given applied mathematics assessments that were videotaped.

**Emergent Themes:**

- Students who work in heterogeneous groups learn to take responsibility for their own learning and the learning of others in the group resulting in higher mathematics achievement for all members.
- Students who work in heterogeneous groups learn to value other people’s perspectives and respect diverse cultures and backgrounds, which has positive implications for their future participation in a global economy.
- Students who work in heterogeneous groups show improved communication and collaboration, as well as enhanced critical thinking and problem solving.
- Teachers need to conscientiously support and model group work in order for it to be successful.

** ****Results:**

Despite starting out at lower levels, students at the reform school outperformed students at the traditional school within two years. More students (41% vs. 27%) at the reform school went on to advanced math classes by the fourth year of the study. Achievement differences between ethnic groups were reduced. The students at the reform school showed increased relational equity: helping others learn, taking responsibility (59% vs. 5%), and respecting other’s viewpoints.

** ****Needs:**

Teachers require training to instruct students in the practices that make group work effective. Teachers need to help groups avoid unbalanced workloads, and make collaborative norms explicit. Teachers must also act as advocates for respect and responsibility, paying attention to how students are working together and not just to the mathematical outcomes.

** ****Implication of Study for Stakeholders:**

The traditional teaching method in which teachers dispense information and individual students process it is not the most effective way to improve academic outcomes, and it does not address social justice, social and emotional learning, or citizenship goals at all. Grouping students heterogeneously by race, ethnic group, gender, and ability, and allowing them to work together to process a given topic improves outcomes in all of these areas, as well as engaging students in the development of twenty-first century skills such as collaboration, critical thinking, and communication. Teachers need support and training to enact these radical changes to their teaching methods so that they are implemented effectively in their classrooms.

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Math is all about finding patterns. One of the most famous patterns was discovered by Fibonacci. He found that if we start with 1, 1, and then add the two previous numbers, we get a pattern that looks like this: 1, 1, 2, 3, 5, 8, 13, 21,…. If this pattern is drawn as squares, it creates a spiral that is found in art and nature.

Snowflakes have a pattern, too. Even though every snowflake is different, every snowflake has something in common: they all have six points. This is because water freezes in a hexagon, so snowflakes are also hexagonal.

Animals intuitively use math. Take the spider: its web is created using a logarithmic pattern to stabilize it, and then the spider completes the web using an arithmetic pattern. It would take an engineer quite a while to figure that out, but a spider does it intuitively. Dolphins use math, too. They use it to echolocate.

Athletes use math intuitively as well. And dancers are extremely good at using physics and math to create complex patterns. Jugglers use tons of math!

The work world uses amazing amounts of math. It’s not about being fast (we have computers for that) but problem solving and thinking about numbers creatively. Math in the real world is hands on and collaborative.

The modern world gives us math games that are really fun. Here are some of Jo’s favorites: Wuzzit Trouble, Motion Math, DragonBox, and Mathbreakers.

I really loved this course. I’m going to save up the $125 to take her teacher-focused class.

The world is mathematical!

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It is up to us to make sense of math. One way is by using intuition, which is a general sense or feeling about what works. According to Sebastian Thrun, founder of Udacity, “Every time you see a math problem it is useful to stop and think intuitively, what do I think is going on here?”

A great aid to intuitive thinking is visual representations through drawing. That takes the math out of the realm of formulas and algorithms and puts in into a concrete context.

There are “Big Ideas” in math that the formulas and algorithms are about exploring and using. However, sometimes in math class, those big ideas get lost, and consequently, students get confused. For example, the Big Idea of Pi: In every circle and every sphere the circumference is always a little bit more than three times the diameter. If students can learn the big ideas and hold onto them, they will find math really clear and easy to use.

In other words, formulas are only useful if you understand the big idea that they are illustrating, so students should always start by trying to understand the big idea. Drawing a picture helps a lot. Keeping in mind that math is about relationships between quantities helps too. So asking questions about how the quantities relate (like diameter and circumference) can help facilitate understanding of the big idea (pi is the same for all circles).

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She starts by discussing some more math myths: That math is just a lot of rules that need to be memorized, that it is all about right or wrong answers, and that it is not a creative subject. But studies show that high achievers think of numbers flexibly and low achievers believe there is only one correct way to do any problem.

For example, there are many ways to work out 18 x 5:

Multiply 10 x 5 = 50 and 8 x 5 = 40 then add the results, so 50 + 40 = 90

Multiply 20 x 5 = 100 and then subtract 2 x 5 = 10, so 100 – 10 = 90.

Since 5 = 2+2+1, multiply 18 x 2 = 36 and double that 36 + 36 = 72 and then add the last 18, so 72 + 18 =90

18 x 5 is the same as 9 x10 because 18 = 9 x 2 so 18 x 5 = 9 x 2 x 5 and 2 x 5 is 10, so 9 x 10 = 90

That was four different methods – there may be more!

But some students believe that it is not allowed to change numbers in this way, but this type of flexible thinking is really important. Students can get in the habit of thinking about multiple ways to solve any problem. One way to develop this habit is to work with others.

Math is not an individual, solitary activity. Talking about math can greatly increase a student’s level of understanding. In fact, mathematician Uri Treisman studied UC Berkeley students who were failing calculus classes and leaving the university. He found that there was only one difference between students who were failing and those that were not: The successful students worked together on their math and talked about it, where the unsuccessful students worked alone. So Treisman put the failing students in study groups, and within a year they started out-performing the other students. It works because discussing problems engages students in reasoning and metacognition.

Lastly, it is important to remember that all math is connected. It is not just discrete topics that need to be mastered. For example, proportions come up again and again in math over a lifetime of math learning – fractions, similar triangles, dilations, graphs, slope, rates of change, the Pythagorean Theorem and Pythagorean triples. Higher achieving students see these connections to big ideas, whereas lower achieving students just try to memorize methods.

The PISA study showed that students with a fixed mindset (who believed that people are either good at math or not) achieved at lower levels, and the kids with a growth mindset were the highest achieving. And students with a growth mindset who also used positive strategies (who made connections and didn’t memorize) were the highest achieving students in the world.

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This week, I tried the “First Day Graffiti” project. Only I did it on the second day of school. Because I’m a rebel like that.

My topics were:

It makes it EASIER to learn in a class when…

It makes it HARDER to learn in a class when…

I am most likely to participate in classes when…

Other students help me learn when they…

I learn best when the teacher does…

Here are my results (click on the picture for a larger view):

In a nutshell, students want:

A quiet classroom

Instruction at a slower pace

Group projects

Teachers with a positive attitude and mood

Help from their friends

Clear, step-by-step instruction with lots of examples

I’ve posted these in my classroom as a reminder when I plan my lessons. These students really care about their education, and they know what they need. I need to listen.

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Our brains grow in response to stimulus, and mistakes are stimulating! When we get an answer correct, there is not the struggle needed for brain growth. So it is important to try difficult problems and make mistakes in order to build a strong brain.

As Thomas Edison said, “Genius is one percent inspiration, and ninety-nine percent perspiration.”

The more mistakes we make, the more successful we are. So how do get this mistake-making experience? By trying different things, being okay with being wrongs, and by not judging ourselves or our ideas.

As we move in to Common Core, students may be frustrated by the open-ended nature of the problems they will begin to encounter. It is important to keep a growth mindset and be determined to struggle on, knowing that mistakes actually aid the learning process.

Fast math isn’t always good math. Math requires deep thought, and deep thought takes time. What is essential is making connections and asking (and answering!) questions such as why things work the way they do.

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Carol Dweck’s mindset model says there is a “Growth” mindset, where students believe they can improve with hard work, and a “Fixed” mindset that believes that people are either smart or not and that can’t be changed. It’s not uncommon to have a growth mindset in some areas (like sports) and a fixed mindset in others (like math).

When it comes to math, those students who have a growth mindset consistently outperform those with a fixed mindset. This is because mindset influences behavior. People with a growth mindset persist until they succeed, and see mistakes as feedback.

Ironically, praise for being “smart” can lead to developing a fixed mindset! It leads students to shy away from difficult work so that their self-image of being smart isn’t challenged. However, student that are praised for being hard workers had a growth mindset. They wanted to reinforce their self-image as “hard workers” and so picked more complicated problems to do. It is important that students choose demanding tasks, because their brains grow the most when they struggle.

The key to a growth mindset is to keep positive self-talk even in the face of negative messages. This could be “I can I can learn anything I choose to” or “I can achieve at the highest math levels if I want to.”

One study had teachers gave half their students a note with their feedback that read, “I am giving you this feedback because I believe in you.” Those students achieved at significantly higher levels than the students who didn’t get the note with their feedback. As teachers, it important to give our students the message that we believe in them and expect high things from them.

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To counteract the assertion that “some people are just better at math,” Boaler directed us to the article “Teaching the Brain to Learn”, which posits that the “distributed neuro functional systems,” which are the basis for later cognitive ability, can be shaped by the type of activities students do. It also stresses that emotions are connected to learning as well, influencing the affective filter in the brain. This is why emotionally meaningful experiences lead to students improving their cognitive abilities, while emotionally stressful situations can disrupt the learning process. Student engagement is the key to enhancing outcomes, which is why project based or discovery learning can have great results, as long as it is rigorously structured. Otherwise, the outcomes are no better than direct instruction methods.

We then were watched a series of videos. The first asked people “What do you think about Math? Why?” and then reflect on the people who didn’t like math – what turned them off? Some said they didn’t know how it related to the real, physical world or how it was applicable to their lives. Some said there were too many rules to memorize, or that it was boring and too repetitious.

Next was smashing math myths, and how the brain works. Math isn’t just in textbooks, but living, breathing math is all around us. There is no such thing as a “math person” – everyone is a math person, and everyone is born with the innate ability to do well in math. What we think of as “math people” are that way because of the experiences they have had. You can catch up to brainy people! The brain is like a muscle, the more you work it out the stronger it grows, but you have to keep increasing the “weight” which means the more you struggle with an idea, the more your brain grows.

The brain is very plastic. London taxi drivers have to learn so many streets – twenty thousand! – that their hippocampus actually grows bigger. Interestingly, once they retire and are not accessing this information any longer, the hippocampus shrinks back to its original size. A seven year old girl who had half her brain removed was able to teach the remaining half to take over many of the tasks of the missing hemisphere. In one study, students worked on a task for just a few minutes each day for 6 weeks. At the end of the study, they had changed the structure of their brains.

There is a stereotype that math is “for boys.” Unfortunately, that is reinforced by the kinds of toys that are marketed it for boys. These are the type of toys – building toys and strategy games – that help build the math part of the brain. Stereotypically “girl toys,” such as dolls, don’t do this. Always resist stereotyped thinking. Always. The ideas in the world are for everyone.

To sum up, we were asked to summarize the three main points in the lesson that we feel everyone our age should know. Mine would be:

1) The brain is plastic. It can grow based on how we work it out, and those things that we struggle with and are challenging build it more.

2) Certain activities and games, such as origami and backgammon, can help develop our math ability.

3) There aren’t “math people” – everyone is a math person. It is possible to catch up to the “brainy” people. Always resist stereotyped thinking – Always!

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