How do we know what they know?

The way teachers structure their grading sets a culture for the class and communicates what the teacher values.

As a teacher, It is important to think about the reasoning behind the grading decisions you make. If you do not accept assignments beyond a due date (or deduct points for late work), you may be communicating that speed and completion is more important than quality and understanding. If you do not allow (or require) students to re-take unsuccessful tests you may be unintentionally communicating a fixed mindset leading to students giving up and feeling that they will never be good enough. The best resource I’ve seen that moved my thinking forward on how to structure my grading is A Repair Kit for Grading: Fifteen Fixes for Broken Grades by Ken O’Connor.

In addition to how a teacher structures their grading is the type of feedback you provide. Dylan William’s book, Embedded Formative Assessment sited a study showing that as soon as a student receives a grade, learning stops. Even if a teacher painstakingly spends hours providing detailed feedback to each student, if there is also a grade on that assignment students won’t interact with the feedback. Ashli Black illustrates this point well here.

Here is what I have found to be effective:

<I’d prefer to eliminate grades altogether and focus only on growth and feedback, but this isn’t possible in today’s culture.>

  • Standards Based Grading makes student understanding of specific skills visible to the teacher for planning, and informing the student & parent. A weakness with standards based grading is that students and parents see the content as a checklist of skills to demonstrate and not a cohesive integrated body. In addition to listing “standards” in my gradebook, I added a “synthesis” category where I include grades for in depth modeling tasks which incorporate a variety of skills. Adding this category also reminds me as a teacher to be sure to develop or find and implement these types of tasks on a regular basis.
  • Regular class assignments get feedback only, no grade. The grade is posted on the online gradebook, which students can access anytime. Students are encouraged to respond to feedback and improve their work and I will adjust their grade accordingly.
  • Don’t grade everything! Sometimes practice is just practice. This also allows a teacher to spend more time providing feedback on key assignments.
  • Quizzes/Tests get 2 rounds:
    1. Students give the assessment their best effort. Then I write feedback on a few select questions that attempts to move their learning forward even if their work on the quiz is flawless.
    2. The next day students must respond to my feedback using a different color. Then I grade their demonstration of knowledge on each learning target using a 4 point rubric. If a student has shown that they do not understand a skill I mark this skill as “missing” or “incomplete” and they must schedule a time to work on this skill and re assess when they are ready.  When students get their quiz back they track their progress.Capture

Reasoning behind the 2-round quizzes:

  1. Test anxiety is a real thing. It impedes a students ability to access information. Students knowing that they will have another opportunity to review their work helps relieve some of this anxiety, allowing student to demonstrate their understanding.
  2. Students see me as their partner in learning. It changes our relationship. Students believe that I want them to be successful and that I believe in their ability to achieve at high levels.
  3. I’ve noticed that this system of assessing & grading builds a relationship with students over time and most students value the assignments you provide more and become less likely to turn in in low quality or late work. Motivated intrinsically instead of trying to avoid a consequence.
  4. Students develop a growth mindset though seeing the documented growth on each quiz. They see their faulty reasoning and their revisions and they recognize their growth that occurred during the assessment process.
  5. I get a better picture of their genuine understandings and misconceptions which better informs my teaching.

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This will be updated as we progress through the book:

Heavy Heart

I decided to start writing after reading this tweet and many others like it flooding my feed:


My heart is heavy. I can feel the stress of this situation so close to my home and being the second mass shooting in my town this month.

This, after I spent the morning thinking about the negative affects that stress has on thinking and learning. All of the people living in this city must feel it to some degree.

There is no blame to place, just a deep sadness that so frequently, individuals end up loosing touch with humanity. And its frequency seems to be increasing at an increasing rate.

Do you think that if schools more proactively taught and had students practice empathy that it would increase peace in our country? How can we teach empathy and help it to develop in our future adults?

We have to do something.

Notes and scattered thoughts from Learning & the Brain Conference

I attended the conference on Learning and the Brain: The Science of Character in Boston, MA on November 13-15 2015. Below are my notes and thoughts from sessions that really stood out to me during the conference.

Executive SkillsMargaret (Peg) Dawson and Richard Guare

Students need to be taught to practice executive skills which include (in order of development):

  • Response Inhibition
  • Working Memory
  • Emotional Control
  • Flexibility
  • Sustained Attention
  • Task Initiation
  • Planning/Prioritization
  • Organization
  • Time management
  • Goal directed persistence
  • Meta cognition

Executive skills, like any other skill can be improved through explicit practice and reflection. These skills are negatively impacted by fatigue and stress, which means when students are not getting enough sleep or are under stress at home or in school, they lose the ability to efficiently and successfully complete tasks – which then adds to the stress. I know by experience that this also is true of adults. A most basic example is that when you are running late for work you can’t find your keys, but if you are calm finding your keys is not difficult. Stress really hurts our thinking and development. More on that later.

There are two periods where children have a highly increased development of synapses in their brains (making many connections and developing executive skills): ages 3-5 and 10-12. From approximately ages 14-20 adolescents go through a synapse degenesis where brains basically complete a “pruning” of skills. Adolescents keep or lose synapses to make way for new learning and solidify important skills.  

Another interesting fact is that context affects executive functioning. For example, a student may be very good at planning and organizing at school but then in a social situation, they lose the ability to even be aware of the time of day. This is because their brain is still developing, not because the adolescent wants to make their life more challenging.

Strategies for teaching students executive functioning skills:

Include a time in the school schedule for daily coaching, setting and reflecting upon sub-goals is critical. Often students goals will not align to the goals that teachers may want for these children. This is ok, do not force them to adapt your goals for them but instead try to build in your goals using their interests where possible.

When coaching students:

  • Adapt common terminology when working with students around executive functioning.
  • Create a structure where students daily:
    • Review their goal
    • Evaluate their progress
    • Anticipate possible challenges
    • Plan strategies to achieve goal
  • Check out to see examples of an alternative school and their developed curriculum developing executive functioning skills with high school students.

How does the frequent use of cell phones & technology impact student self control? Interesting insight from Walter Mischel, author of The Marshmallow Test: Mastering Self Control:

With technology, temptation increases, which also causes an increase in the need for executive functioning. Students get more opportunities to practice planning, adaptability, flexibility, sustained attention and emotional control.

I’m sure teachers and coaches can leverage students ability to develop executive skills with technology  to help students translate it to other aspects of their life. This is another example of context as mentioned above. Students executive functioning skills are not easily translated between contexts (school, home, with peers, online) but through coaching and helping them to make these connections, adolescents can be use their strengths in different arenas to support their weaknesses.

Building ResilienceRick Hanson

Two conditions are important for any learning to occur:

  1. an activated state (we are good at this)
  2. Time for information to stabilize and take root. (This is very hard to do)

As teachers it is important to slow down from time to time in order to encourage and protect installation.

The brain has a negativity bias. When I heard this I realized I already knew this was true. For example, if I have a great day teaching and my students all were successful, then I am happy, but I don’t really think about it beyond that day. If I have a bad incident with a student or get a negative evaluation, it is all I can think about for weeks. The impact of the negative event is much more profound than the positive event. Think about how people developed surviving – if you failed at making a positive event, such as killing a rabbit for food, you  probably got to live another day whereas failing at a negative event – being chased by a lion, for example – your life ends. As a result, brains developed to scan for and focus on bad news & overreact to negative stimuli.

Here’s why this matters: Focusing on negative events increases stress. Prolonged exposure to stress negatively impacts our ability to learn, to think clearly and our physical health. It is critical for our health and growth as individuals and as a society that we sensitize ourselves to the positive and support this in our students: Here’s how –

  1. Notice when positive experiences are happening
  2. Help the experience last through thinking about the event, reflecting on its personal relevance, reliving its intensity in your mind, etc.
  3. Allow yourself to absorb its warmth
  4. Link it to any related negative experiences, while keeping the positive more prominent in your mind.

If as individuals and as a society we repeatedly internalize positive experiences then we will become harder to manipulate through fear, greed and consumerism.

Some related thoughts:

Educating Hearts & Minds – Dr. Kimberly Schonert-Reichl

She opened with this video:

  • acts of kindness improve happiness
  • rewards decrease motivation for altruism (I read this once before.. I think it was in Drive)

Practical Interventions for students with AnxietyJessica Minahan

Anxiety is often misdiagnosed as ADHD.

Behavior is a symptom of a different problem:

  • it is a way of communicating.
  • it has a function.
  • it occurs in patterns.
  • it can be changed.
  • We can only control our own behaviors.

Stop for a second and re-read those. Think about your students. This is deep stuff.

Negative behavior results in more predictable responses than positive behavior. This is comforting particularly to students with anxiety.

An example: Swearing or annoying tapping or fighting will generate a predictable response in 2-4 seconds. Doing the expected work may or may not come with some brief audible feedback from a teacher. Maybe a “nice” or a “good job” or no response at all. Students with anxiety like & seek predictability.

As teachers we need to make positive attention just as predictable, efficient, obvious and dramatic. Some students prefer private praise and it  is important that we as teachers understand and value this. we can also teach students how to wait by giving them strategies such as coloring or doodling.

Triggers for students with anxiety: (be prepared with a plan)

  • unstructured time
  • transitions
  • writing tasks
  • social demands
  • unexpected change

Support students at these times by making it predictable. This is a great reason to have a Warm up Routine. If you notice a student who finished their work sitting anxiously, make them busy ask them to help clean up, sharpen pencils, push in chairs…

Students with anxiety struggle with:

  • self regulating
  • positive thinking and stopping negative thoughts
  • social skills
  • executive functioning
  • Flexible thinking

Anxiety and homework: It is helpful to have homework include a preview or preteaching of the content  because student anxiety (and related behaviors) is reduced if they know what to expect next class.

Teaching executive functioning skills and growth mindset helps students learn to self monitor and identify their areas of stress. Help students build skills in order to manage behavior. ASCD article

Building ResilienceKen Ginsburg

“Stop looking at the child in front of you and start looking at the 35 year old that you are growing.”

  1. people matter
  2. young people need to feel valued
  3. young people are watching adults

Nothing discourages mastery more than an adult who steps in and says: “Let me do that for you.”

Resilience is a mindset.

Young people are most resilient if they know that adults genuinely believe in them, know them and have high expectations for them – but it has to be genuine.

Adverse childhood experiences (abuse, neglect, racism, violence) increase cortisol in their bodies which over prolonged periods increases likelihood to develop addictions, health problems, learning challenges.


If a child had adverse experiences, but there is a loving & caring adult in their lives, the adult protects the child’s from developing hyper cortisol affects.

Developing Resilience


Find competences and build confidence.

Adolescents who abuse substances are trying to become numb. This is because they are SENSITIVE PEOPLE.

Talk through what if’s when a child makes a bad decision. Allow them to conclude for themselves the best path to happiness.

Distinguish real problems from fake problems: Recognize when bad things are temporary and when good things are permanent.

General Themes

  • Be patient, kind, empathetic, accepting, tolerant and compassionate with everyone.
  • Promote kindness & teach empathy basic human needs
  • Students will learn more if they feel valued
  • Support student development of executive functioning skills through a comprehensive school-wide plan including daily coaching, goal setting & reflection.pft-0330

What ideas in these notes stand out to you?

How do you address them with your students?


Planning content for College Prep Math

I teach a single semester math class that students choose to take in an effort to avoid taking algebra 2. The class is primarily seniors whose highest level of math is geometry, and they plan to join the military or work and take a few classes at a time from our local community college.

With this in mind, I found out that our community college uses the Accuplacer as a placement test. From there, I looked at the College Boards Accuplacer Program Manual. This document lists math “skills” assessed under 3 categories: Arithmetic, Elementary Algebra & College Level Math. I put all of these skills in a spreadsheet and grouped them by content:

Based on this analysis, I should spend the semester focusing on the following topics:

  1. Proportional reasoning
  2. solving & graphing equations
  3. Systems of equations and inequalities
  4. Functions (Linear, Quadratic, Exponential)
  5. Solving, graphing, factoring Quadratic Equations

I am confident that the first 3 are skills my students need and I can support their understanding with great modeling tasks and real world applications (and mullets!).

I am struggling with #4 and #5 on the list above for students who are not planning on attending a four year college. This would mean spending weeks on quadratics just to prepare students for a test. I know I can find some good modeling problems, but in all honesty, in my 5 years working a professional civil engineer I did not need to factor any quadratic equations.

My gut instincts and knowledge of these students say to focus on application of exponential functions (credit, interest, population growth, etc) . I also believe that probability and statistics is missing from this list. I want to make curricular decisions based on research, data and experience, not just my “gut instincts”. I also don’t want the class to be a haphazard collection of unrelated tasks.

What would you do for the last six weeks of this class?


Based on Sandy’s comment below, I looked up our community college score requirements. Here is what I found. Why hadn’t I thought of doing this?!


They are only looking at the Elementary Algebra portion of the Accuplacer! I can filter the spreadsheet above to only show the Elematry Algebra skills and I end up with a much more manageable list. Looking back that the Accuplacer Program Manual (linked above) yeilds this information about the Elementary algebra scores:


Why are system of equations considered to be such an advanced skill?

Whelp, It looks like we will be considering some quadratics. Thank goodness for Algebra Tiles, the Math Assessment Project and Mathalicious!

Painless Proofs!

Last year I didn’t really teach proofs in geometry. It seemed so procedural and I knew it would take time away from more productive problem solving tasks.

I felt guilty for not including proofs in enough detail and I decided that this year I was going to teach it well.

Here is what I did:

  1. Our unit began with the Shell Center task: Evaluating Conditions for Congruency, I don’t always follow these lessons exactly as described, but we focused on the Must the Two Triangles be Congruent? part of this lesson. I found that this activity is much smoother and more effective when using AngLegs instead of trying to draw each triangle (described here and here).
  2. After this task we formalize our findings in our notebooks documenting which combinations guarantee congruent triangles [SAS, SSS, ASA, AAS] and which do not [SSA, AAA].
  3. Next students practice determining if pairs of triangles must be congruent based on the information given. I am careful to include a few with overlapping triangles and triangles who share a side or contain vertical angles to generate observations and class discussions. At this point I address the reflexive property and vertical angles as reasons sides or angles may be congruent. I follow up any student observations of congruency with “How do you know?” or “Explain why you decided these two angles must be the same.”
  4. Review definitions of midpoint, bisector, perpendicular, then a few images where I ask if you are given this information, what can you conclude is congruent. See the interactive notes here. This is the day students complete their first few **Really Basic** congruent triangle proofs.
  5. Prior to this class I printed Proof Blocks on colored paper and laminated them, then I used masking tape to affix them to  whiteboards. Once class began I randomly assign student pairs and had them work on a wall mounted white board (Vertical non-permanent surface) with a set of Proof Blocks at each work space.

.2015-10-15 12.34.06 2015-10-15 12.34.27

Next I project a proof and students copied the image onto the whiteboard, then marked the givens, saw what else they could determine was congruent, and decided if they had enough information to conclude and prove that the triangles must be congruent.

2015-10-07 10.38.232015-10-07 11.21.13

In order to make sure all students got timely feedback and to hold them accountable, I gave each group an index card and when they finished a proof I would check it. Once it was done well, they got a stamp on their card. At the end of class they turned in their index card with their names & stamps on it, similar to the review activity I described earlier.

I returned to this format with the proof blocks and different pairs about once per week after learning and incorporating new skills. The second half of the slides above were from a day after learning and practicing angles formed by parallel lines & a traversal. I intend to create new “blocks” as we advance through the school year. Next week students will prove that all triangles have a sum of 180 degrees and that base angles of isosceles triangles are congruent using the same format.

With students standing at white boards, they can glance around and see their peers work easily, they are more likely to collaborate, and the whiteboard & Proof Blocks make it much easier to adjust their work when there are any corrections required.

I just graded students quizzes over this unit and it is the first time in my 12 year teaching history that they did very well on the first attempt!

Constructions was MUCH better this year!

Last year I wrote a post about how I wanted to improve my teaching of geometric constructions this year.

This year, I re-read it and used it to help plan this unit better. I never provided the students with steps to make constructions! It was tempting, but I resisted the urge, and they did great!

Here is the sequence that I followed and plan to use next year:

Day 1 – Play with see how many of the challenges you can complete. I then had students submit a screen shot of their completed challenges through Google Classroom.

Day 2 – I assigned the construction design project described in my previous constructions post (and linked below). [credit for the designs: Mr. Baroody]

I gave the students compasses and paper to experiment and relate this to the computer task yesterday. They had to complete this at home. There was no discussion of “constructions” just making designs precise and accurate. Here are some students final designs:

[editable first page]


Days 3/4 – Working in pairs, students alternated roles for each level of Euclid, the game with one student operating the laptop and the other writing down the steps they used to complete each level. We only completed up to level 6. These steps became their notes for constructions.

2015-09-16 16.33.58 2015-09-16 16.33.47 2015-09-16 16.32.31

(Note: there is a geogebra constructions tutorial here.)

Day 5/6 – Construction station rotation – I randomly assigned groups and students rotated with their groups through different stations completing paper and compass constructions, using Popsicle sticks as their straightedges, so they couldn’t rely on measurements. They used their notes that they developed from playing Euclid the game. Sometimes they noticed that their notes were not sufficient, but since they had new partners, they would compare and add notes & diagrams as needed to improve their descriptions.

Day 6 – Constructions extensions. I asked them to apply their knowledge of constructions to new tasks, such as finding 1/8th of a segment, constructing a equilateral triangle and constructing a rhombus.

Students were so much more proficient at using a compass than at the end of this unit last year!

I think I need to develop more construction extension challenges to build on their basic skills next year. I may make a list of challenges and assign point values to them and then assign a total point value that they can get a variety of ways.

Please comment with any ideas to improve this sequence of tasks as an introduction to constructions that builds student confidence while supporting their creativity and problem solving.

First days of school: Mindset discussion/survey with Plickers

I didn’t use Plickers much last school year because I don’t ask a lot of multiple choice type questions in my classes. Most are more open ended to expose depths of student understanding. This school year, I found a way to use Plickers to facilitate group discussions and I am so excited to use them more this year!

While I am effective at supporting small groups working in my classes, I am not always the best at facilitating whole class discussions. I knew I wanted to hear students thoughts and dispel myths on how to learn math early in the school year. I have started this discussion in past years using Bowman Dickson’s survey found here.

Then I saw Julie Reulbach’s recent post on her plans for the first days of school. She mentions using Plickers as a brief survey.

I discovered that when setting up Plickers, you do not have to select a “correct” answer. Instead I decided it would be insightful to make the multiple choice options into a Likert scale. Capture

This way after each question I could project the response graph. It was perfect to be able to just show students their peers ideas and allow discussions to happen with little facilitation from me!

For example, when I projected this question, and then the response graph, a student said,


“How can you agree with this? We can always learn more and improve.”

Then students who agreed chimed in citing how their parents can’t do math, so they can’t do math.

To which the students who disagreed argued that you can change what you understand through learning and effort…

I let it continue occasionally asking probing questions. It was interesting to see how dispersed student responses were, and great to have this data to look at later.

I plan to complete this survey again later in the year to see if students move more towards a growth mindset with respect to learning math through the school year.

Another thought I had through this process was how Plickers could be used to facilitate Talking Point discussions too.