"What three words come to mind when you think of school math?"

This post is a duplicate of a post I made to my classroom blog.

In a survey at the beginning of the year, I asked my 125 Math 6 students to tell me what three words came to mind when they thought of school math. Almost everyone answered, and this Wordle, which I also shared at Back to School Night, shows all the words that were listed by two or more people. The larger a word, the more students listed it. (The colors are random.)

To my teacher mind, some of these words describe associations I'm pleased to see incoming sixth graders having with math, some are merely neutral or factual, and some are associations I hope will change. I'd class them like this (listing the words in each category in order of frequency):

Positive (8): fun, challenging, interesting, learning, challenge, exciting, yay, smart
Neutral/Unclear (22): hard, multiplication, numbers, difficult, homework, easy, complicated, addition, subtraction, division, fractions, shapes, adding, equations, algebra, practice, math, complex, school, long, work, ok
Negative (9): boring, confusing, ugh, scary, irritating, annoying, weird, meh, stressful

I think it's great for students to find math challenging, as long as they're not discouraged, so I put "challenging" and "challenge" into the positive category. But I'm actually uncertain whether to classify "hard" and "difficult" as neutral or negative. Hard work can be intensely satisfying and can lead to great learning, but when "hard" is one of the three primary associations eleven- and twelve-year-olds have with math, well, I worry that they're feeling overwhelmed. At the other extreme, "easy" can be OK if it is a word used by a happy and confident student, but it might also be contributing to why "boring" appears so many times, so it reminds me I need to provide challenges in math class for all kids.

By the end of the year, I'm hoping "fun" and "challenging" replace both "hard" and "easy," and that the rest of the negative words are wiped out!

As for the students' description of the subject matter of math, arithmetic looms large, which is not unexpected for students coming out of elementary school. I do find it interesting that many mathematicians describe math as the study of patterns, yet not one student listed that word, even though they've undoubtedly looked many times at patterns in math class. Could it be that they believe "real" math is the symbols and arithmetic, not the patterns and relationships? If so, I'd like to change that so their view of math is more expansive.

Finally, I would love to see the word USEFUL showing up by the end of the year. Middle school math is arguably the most useful math students learn, but I hope they will realize how powerful it is now, not just later in life.

Back to School Night

Aaaaaaand, it's a wrap.

Back to school night felt much more chill this year than it has in recent memory. Probably because I talked a lot less and tried to run it more similarly to how I run class with students. We started with a brief intro of me because parents are really curious about that stuff, but then jumped right into a problem I've liked for a long time.



I asked parents to make a guess. Crickets. I made a guess, and that helped break the ice. Once the gridded rectangles and scissors came out, parents got into it. They were finding volumes, realizing that the cut corner had to be removed from both sides, looking for patterns, talking to each other, asking if the corner length had to be an integer, and in general, being great students. When we discussed the need to be convincing, it made sense that a general rule could help with that.

Dum, dum, dum... enter desmos:



We talked about different approaches to this problem and why it makes sense to talk, collaborate, and learn from each other in math class. I used the problem as a way to describe my most common structure for class (problem posing --> intuition --> strategies, collaboration, checking for reasonability, changing ideas --> class consensus --> formalization, showing other approaches --> application to new problem space) and made the pitch that learning that happens in this order is far superior to just skipping straight ahead to the formalization step before anyone's hands have gotten dirty, both in terms of engagement and in the depth and quality of learning that's going to take place.

We talked a bit about content and the sequence of math courses at the school since it's a weird one.

And finally, my favorite slide: what I need from parents (inspired by @fawnpnguyen's back to school night slides, available here)


I'm not sure if this what parents wanted or expected, but it was fun for me! Wish there was a way to get quick feedback from my audience, but formative assessment is severely lacking in the back to school night business. It certainly beats going through a list of content objectives and the grading policy (ahem, what I used to do).

First two weeks of school

It's been a really fun first two weeks of the school year. Yes, exhausting as well, but super exhilarating and exciting too. This year, I started the year a bit differently, focusing more on how I wanted students to work together and think mathematically than on specific content. Because it was the first year I was doing this, I could do the exact same problems with all of my classes. Here's how it went:

Day 1:

• Students came in and saw a seating chart with randomly assigned groups of 3 or 4 and were directed to one of the vertical whiteboards. I wanted to establish this as the norm from the get go.
• Students filled out Google form describing a class in the past they've enjoyed, a class they have not enjoyed, questions that they have about this class, and questions that they have about me. I used the last two prompts as ways to discuss my expectations and structure for the class and to start building some personal relationships.
• Students worked individually for a few minutes and then discussed this problem, which I stole from IMP Year 2. Our first unit will be Statistics for all classes so I thought it would be good to do a fun, but challenging problems, that related to probabilities and ways of counting events.
• Homework was to fill out a Google form asking them about themselves and to keep working on the problem above.

Day 2:

• Students were grouped randomly anew and shared their work on the Tying the Knots problem. We spent the last half of class with group presentations sharing out their progress and practicing how to present and interact with presenters.
• Homework was to write a reflection on themselves as a learner and to start writing up the process and solution for the Tying the Knots problem (I used the Problem of the Week standard categories).

Day 3:

• New random groups, and I used one of @sophgermain's activities for helping students get to know each other. Nothing huge, kids just shared one thing they did over the weekend with their group.
• New problem! This one was incredibly fun. I originally thought that we would take it into proof by induction, but after input from @woutgeo and @hpicciotto decided to stick with a more intuitive visualization of the sequence.
• I basically let students work in their groups without too much guidance from me. Most realized it gave the Fibonacci sequence pretty quickly, but were not able to explain why. Many tried to develop a closed form rule, without much success (surprise, that's actually pretty hard to do). Most groups started trying the extensions, but didn't get super far. I stopped the class a few times and asked various students to explain their group's work. One of my classes this year doesn't have as many whiteboards as I'm used to having, but our desks can be written on so I'm going with that for now.
• Homework was a reflection on their process and feelings when working on these problems and presenting/watching presentations.

Day 4:

• New groups and I answered some more questions about the class and about me. I continued to have them share out a few personal tidbits in their groups as they are still very much getting to know each other (especially the freshmen). Today's questions were about favorite ice cream flavors and favorite movie.
• This was a slightly more structured day. I pushed students to be able to explain why the pattern that was produced matched the Fibonacci sequence. It was helpful to project pictures of their written work and explanations and use that to get more precise and tight in our language. I felt okay adding on to their explanations as needed since there were more extensions to explore (2 by 2 by n case and 3 by n case).
• Homework was to work on the two extensions and to start an integrated review problem set.

Students' work on explaining the derivation of the recursive formula

Some work from the first day on developing a closed form. I was not sure as to whether I should discourage students from going in this direction as finding the closed form rule is extremely challenging. 

More fun student work at the beginning of the exploration.

P.S. @daveinstpaul shared a great follow-up programming project in which students need to write a program to generate all of the possible ways to tile a 2 by n rectangle and then extend it to an m by n rectangle. I'm going to check in with one of the programming teachers tomorrow to see if this might make sense as a posible extension in her class.

P.P.S. A new teacher who I think is going to be amazing visited my class today and I got completely turned around in what I was saying and did not do a great job of moderating the discussion. It's been too long since anyone has observed me, and I just didn't feel comfortable with the kids yet to laugh it off so awkwardness ensued. Bah. We need to be visiting each other's classrooms much more frequently.

Rethinking Grading: Ch. 5

Chapter 5: How to Reform Grading: Making Change Happen
Cathy Vatterott

Changes need not be grandiose to have a huge effect on student learning or to improve the accuracy and validity of student grades.

We must decide what we believe about the purpose of grading.

If they believe the purpose of grading is to accurately reflect achievement, then it becomes inconsistent to punish behaviors such as cheating, tardiness, or attendance with grades.

If an individual teacher believes the purpose of grading is to reflect academic achievement only, they could begin by removing nonacademic behaviors from the grade, by no longer grading practice work, and by giving more ungraded formative feedback.

When we agree on purpose, methods follow purpose.

Lesson learned:
One. Start small.
Two. Let it grow.
"Teachers need time to grieve the loss of what they thought was right."
Three. Include all stakeholders.
Four. Create a belief statement or guiding principles.
five. Have a comprehensive communication plan.
Six. Make students and teachers your allies.

When implementation is top-down with no teacher by-in, there's often a limited understanding of the changes and no commitment to the mission. Teachers notoriously find ways around policies they had nothing to do with creating.

Rethinking Grading: Ch. 4

Chapter 4: What, How, and When to Grade
Cathy Vatterott

Pre-tests set the stage, shave instruction for all, and guide individual learning. After the pre--testing process, formative assessment provides feedback to students while they are still learning; summative assessment shows the level of mastery at the end of the learning cycle.

Most teachers of you and formal feedback and formative assessment as two different things. It's easier to think of formative assessment as structured tasks designed by the teacher, results of which may be marked or documented in some fashion, so students and parents can have a record of the students progress toward the learning targets.

Feedback is a two-way recurring conversation between teacher and student.

For teachers to be able to give feedback to students, it is necessary to limit direct instruction enter create activity-based lessons.

All feedback does not have to come from the teacher; peer feedback can also be useful.

As we get targeted feedback to individual student and as they are empowered to learn in their own way, the differences in learners become smaller.

If, after repeated attempts, a student or group of students has failed to master a learning target we must take a fearless inventory of our instructional process and ask yourself these questions;
What's their level of learning properly diagnosed with pretesting?
What's the feedback about learning timely specific and helpful?
Did our differentiation move the student or group of students forward?

Using the result of a pre-test, feedback, or formative or summative assessment, teachers can identify patterns in the students work or clusters of student need. Students can then be organized into two or more groups for ungraded group learning the activities at each table are based on the errors that students made on the form of assessment.

In a purely standards-based grading system, only summative assessment counts in the final grade.

Typically formative assessments are evaluated and descriptive feedback is given to the learner, such as with practice tests.

Ungraded practice tests are especially beneficial to learn as they Activate "retrieval learning" and strengthen the connections in the brain.

One technique for practice test is called "find it and fix it." Rather than marking the answers that are incorrect, the teacher notes to the student, "five of these are incorrect; find them and fix them". This requires a student to reengage with the questions and precipitates a lot of learning.

Mastery checks: these assessments are written using three levels: green, yellow, and red. The green level questions are basic skill problems and didn't really require only one or two steps to solve. Yellow level questions require multiple steps and or multiple ideas to solve. The red level questions are generally questions of the students have never seen before, requiring them to go beyond knowledge they have obtained and\or apply the knowledge to a new situation. Students are expected to attend all three levels of questions. Their answers help the teacher to determine the students his level of mastery.

The current consensus is that homework should be formative assessment the checks for understanding or that helps prepare students for summative assessments. Therefore, and I truly standards-based system, homework should not be graded. Standards-based policies usually state that homework will be reviewed and feedback will be given, but not counted in the grade.

The final achievement of learning is more important than the steps it took to get there.

Formative assessment is assessment for learning and occurs when there is still time to improve. Summative assessments are assessment of learning that occur the end of a predetermined learning cycle, after learning has taken place.

How in assessment is used is what determines whether it is formative or summative.

Students who eventually achieve mastery should not be penalized for earlier struggles.

The most recent evidence of learning is the most accurate and grades should be replaced by the most recent evidence.

Student should never be allowed to retest without showing additional evidence that they have mastered the concept that caused him to do poorly on the original assessment.

Remember that our goal is to minimize the number of retakes a student needs to show mastery.

We want to hoops to result in additional learning, not just for students to complete missing work.

Feedback is free help-there is no grade or Mark associated with feedback.

Formative assessments give students multiple opportunities to improve, free from the threat of grades while they are still learning, and summative assessments verify and report their learning progress.

Rethinking Grading: Ch. 3

Chapter 3: What Grading Looks Like in the Standards-Based Classroom
Cathy Vatterott

The standard show us the results that we want students to achieve. We then work backwards from those results to create more specific learning targets. We synthesize or unpack the standards into learning targets, usually written as "I can" or "We can" statements.

But when we organize individual targets into lesson-sized tasks, keeps them separately, and assess them separately, students may fail to see the relevance and connection. A better method is to group targets together so that several targets may be addressed by the same activity.

Self assessment is formative assessment-it should always focus on improving the students progress toward the learning target, not I'm getting a better grade.

Learning is not so much instruction or a lesson to be taught, as an activity to be experienced.

I never heard of a student not doing *his* work; it's *our* work he's not doing.

If we want to encourage students to view mistakes as a necessary step in learning, we need to remove the threat of grading while they are learning.

Grades are not necessary for learning, but feedback is. In fact, feedback has been shown to be one of the most effective strategies to improve learning.

Rethinking Grading: Ch. 2

Chapter 2: Why We Need a New Grading Paradigm
Cathy Vatterott

Treating all students the same resulted in a certain percentage of students who failed.

Instead of teach, test, and MoveOn and one large group, learning is a series of mastery's for individual students-teach, check for understanding, apply learning, get feedback, revise learning, and get more feedback until mastery is achieved.

Unlike the old paradigm of one-shot learning, a feedback loop exist that makes learning dynamic-feedback to the students informed their learning and teachers change instruction as they see what individual students need.

Within the traditional grading paradigm, it's not safe to make mistakes. In a traditional paradigm, failure is a judgment and a validation of her students lack of ability.

Learning is hard and frustrating, but ultimately achievable and satisfying. Mistakes are a natural part of learning and mistakes or something you do, not something you are. Lack of understanding is a puzzle to be solved-not a validation of stupidity.

As grades are used to punish behaviors, they overshadow the grades students receive for learning.

In the traditional grading paradigm, when teachers grade everything, the grade means nothing.

When first attempts, including practice, are graded and went all grades are permanent, students are penalized while they are still learning. Mistakes are permanently recorded and there is no redemption.

If you have a bad week practicing, you don't show up on Friday night with -5 on the scoreboard. The only way to win the game is to get better at the learning.

Rethinking Grading: Ch. 1

Chapter 1: The Culture of Grading
Cathy Vatterott

But teachers intervene-they teach with the goal of having all students learn. "If the distribution of student learning after teaching resembles a normal bell-shaped curve, that, too, shows the degree to which our intervention failed. It made no difference."

Belief #1: Good Teachers Give Bad Grades
As teachers, we bought into the idea that a bell curve indicated rigor and misinterpreted it to be a rule to follow. We came to believe that of success were scarce and great spell into a bell curve then we were tough teachers.

Grade inflation is the arrive from the belief that rigor equals a scarcity of high grades and that the purpose of grading is to sort and rank.

Rigor and difficulty was often equated to the amount of work done by students rather than the complexity and challenge of the work.

Such practices reinforce the belief that some students could not learn and perpetuated a system that not only allowed four but actually expected failure. In many ways, sorting and ranking practices institutionalized failure and conveniently of dissolved teachers of the responsibility for student failure.

Belief #2: Not Everyone Deserves an A

Many people feel strongly that grades reflect more than learning. We review grades as a package deal; to succeed, seras must have it all-academic achievement and moral virtues.

Belief #3: Grades Motivate Learners

The first misconception is that learning is only a means to an end-to escape punishment or get a reward, the learning has no intrinsic value, and that students would not be interested in learning for its own sake.

The second misconception is that a single entity called motivation exists, the students either have it or don't have it, and it can be manipulated by external forces.

The third misconception is that the most effective method is the use of rewards and or punishment and that grades are in effect the reward and or punishment for all students.

Our believes have led to an abuse of grades.

Students have come to believe that effort however week, not learning, earns them the A.

And our relentless pursuit of the almighty A and the perfect GPA, something got lost-learning.

Reality Check #1: NCLB

This was a foreign concept to teachers-we had never been expected to ensure that all students were proficient. We didn't know how to do that. We were not even sure that it was possible.

NCL be exposed a dirty little secret-graves don't equate with performance on standardized test.

Accountability for learning demands grades that are reflective of learning.

Reality Check #2: Grades Are Misleading About Succeeding

A puzzling example is that good grades in high school and students cheaper car insurance. Why-because good students are safer drivers or because good grades mean you are an accomplished rule-follower who will follow the rules of the road?

We thought that we were rewarding the right thing-completion of tasks, compliance, promptness. But in that process if we devalued mastery of deep conceptual learning, we have hampered students his future success. Maybe the grading practices that we thought were preparing students for the future really weren't.

Reality Check #3: The Common Core State Standards Changed Everything

Although standards and standardized test has supposedly driven instruction for years, we now see that we have been focusing too much on low-level rote learning.

Too often, we have neither allowed nor expected students to think. We have filled her head with facts and formulas and reward them for restarting it. We have done the analyzing, synthesizing, and evaluating instead of expecting our students to do it. We have done too much of the work of learning, perhaps because we didn't trust him to want to do the work, or perhaps because we weren't sure they were able to do the work.

To successfully navigate the standards, student grades will need to reflect mastery of skills, not memory of content.

Today we must prepare them for a world in which they must know how to take initiative, self-advocate, solve problems, be creative, and accomplish tasks without minute-to-minute supervision.