Narrowing Participation Gaps

One thing that we touched upon this week in class was found in the chapters we read from ‘Strength in Numbers’ that participation in your classroom can be tricky. It can be difficult to make students who seemingly do not care answers questions and volunteer. This creates a “participation gap” where the students who participate, participate a lot and the students who do not participate, never participate. It is no surprise then that this correlates almost one-to-one with achievement as students who participate are the students who gain a deeper understanding material and know it (that is why they are volunteering answers).

The ‘Strength in Numbers’ story about the teacher who drew out participation from all of her students really intrigued me to find out more about how to get your students to participate. Especially when you have students that are all very different in terms of personality, race, backgrounds etc..

I found this article called, “Narrowing Participation Gaps” by Victoria Hand, Karmen Kirtley, and Michael Matassa that does just that. The url is: http://www.nctm.org/Publications/MathematicsTeacher/2015/Vol109/Issue4/Narrowing-Participation-Gaps/

The article discusses three specific ways to encourage and increase participation by all students which, in turn, will narrow the gap. First,

  1. Organize Mathematical “Contributions”
  • Participation is not only talking or answering questions
  • More than one way to get math answers
  • Prompt students unclear answers with directing questions

The first bullet point makes the point that teachers too often ask a question and evaluate the student’s knowledge based off of who answers and how they answer the question. It is important to remember that math is all about process and teachers need to evaluate how the students got to their answers. There are other ways of participating in class other than speaking as well. An important quote that goes along with this explanation is, “This orientation also prioritizes correct uses of academic language over students’ sense making”.

The third bullet point is one that was not said, but I observed from the example in the article. These directing questions can be uber-focused on the words that students use such as “length” and “width” as it may have a different meaning to the student than it does to the teacher. Thus, these clarifying questions help the teacher know what the student is saying and it helps the student understand what he/she is saying. This can also help narrow the participation gap when students aren’t afraid that they will be “wrong”, but rather they will get guided to the right answer. It is important not to simply give the answers to the students as they are learning absolutely nothing there. Confusion is the best way to learn.

2. Expand “Smart-ness”

  • Expand perception of who is “good” at math
  • Complex Instruction in groups
  • Assigning roles
  • Reward different ways of thinking

Group work, in general, causes more participation, as each member is responsible for their own work. This is very true for what roles are assigned and each student has a responsibility. In my opinion, group work sometimes creates less participation as the “smart student” does the work. Assigning roles changes this and allows students to see the way others think and it will expand what they deem as smart as well.

3. Engage Instead of Motivate

  • Takes away the blame on students
  • Change the classroom as opposed to the student
  • Don’t label students

While I agree it is important not to label students, I do not necessarily agree with “engage instead of motivate”. Yes, there is a time and place where you do the work along with the students, but giving motivating problems is good as well. I believe there must be a balance of both to really get the students to participate. The article talked about how teachers should allow foreign language speaking students to solve the problems in their first language and then to explain it in english and I think this is a great method. Bottom- line I believe you still have to motivate your students and it is not all about doing the work with your students because then they will start to bank on you for the answer and that will only hurt their learning.

Group Work Rules Rule

This week in class we were instructed to read this list of rules that would be good to choose from and implement into our expectations of group work in the classroom. The list is entitled Group Norms for Working in Groups (link: https://docs.google.com/drawings/d/1NCgFW54NTbAxi14Ui_O7zp01IRsmJtuqpBkQdHwlcpg/edit). I found the list to be very interesting, and I wanted to go through it and discuss which rules I really liked and ones would alter a little bit.

Here is the complete list of the Roles and Responsibilities of Learners:

  • Take turns talking
  • Listen to each others’ ideas
  • Believe your classmates have something of VALUE to contribute
  • Disagree with ideas, NOT people (be respectful)
  • Value WHAT is being said, not WHO is saying it
  • Know you have the right to ask anyone for help
  • Helping is not the same as giving answers or telling
  • Everyone has the duty to give help to anyone who asks
  • Say your ‘becauses’ – give justifications and explain assertions
  • NEVER say, I’m so bad at math! RATHER say, I don’t get this yet!
  • Remember, confusion is a part of learning
  • Stay focused on your group’s work; curb off-topic discussions
  • Take risks
  • Be persistent

Here is the complete list of the Roles and Responsibilities as Facilitator: 

  • Use random group assignment – we can all learn from each other
  • Respect individual think time first, then small-group talk, then whole class discussions
  • Regulate participation patterns
    • Let’s hear from someone who hasn’t spoken today.
    • No hands, just minds. Look up when you think you know and I will call on somebody.
    • Even if you only have a little idea, tell us so we can have a starting place.  It doesn’t need to be all worked out. (Rough-draft thinking is normal in PS’ing)
  • Know what to table and what to pursue
    • I’m not sure I follow. Could you please show me what you mean?
    • Could we please table that idea for now? I’m not sure that is the direction I would like to go for right now.
  • Catch students being ‘smart’;  use vocabulary to name it
    • posing interesting questions and problems
    • making astute connections, connecting two seemingly disparate ideas
    • representing ideas clearly
    • devising a useful representation
    • developing logical explanations
    • working systematically
    • extending ideas
  • Assign competence; praise them
    • in public, in the classroom
    • specific to the task – so they know what they did that is being valued
    • in an intellectually meaningful way – tied to the mathematics
  • Ensure that arguments should rest on mathematical justification, not on social position

The first rule that I really liked was to disagree with ideas, not people. I always felt in high school that there was one girl who always argues with everything that I would say, and it did feel very much so like targeting against me. I think that this rule would help reduce that and make kids feel more comfortable to share their ideas. It is also way more respectful to disagree with someone when you thing their idea may be wrong, but remember that everyone’s ideas have value. This way student’s can find what is good in an argument and what can be improved.

The second rule that I really liked was to say your “becauses”. I LOVE this.  I think a lot of the time when students are lost or confused it is because they do not understand the why or how behind a concept. This way if a student is confused they can ask someone who understand to say their becauses and hopefully the confused student will understand it better. Also, if a student is not quite getting something, the teacher can ask them to say their becauses for what they do have, and the teacher can see where the confusion is happening and try to correct the confusion.

The third rule, which is my ultimate rule, NEVER say, I’m so bad at math! RATHER say, I don’t get this yet! I am such a firm believer that everyone can do math. I never ever want to hear in my classroom the phrase “I am bad at math.” No, you are not. Maybe you have not gotten it yet. YET. Math is just another language to all my literature lovers, math supports scientific claims and research to all my scientists, math is the basis for statistics that we make decisions on my historians and politicians, math and music go hand in hand for counts and beats musicians, math is lines, shapes, and points my artists, and for my math lovers, continue to love it. Math is relevant in every subject, even if you do not openly realize it.

Something that I would alter would be how I regulate the participation patterns. I would really like to do roles so each time someone has a different role. There would be a leader, a calculator, a writer, and a fact checker (or something along those lines). That way each student could see themselves in different roles of mathematics. Also, I would allow students time after I posed a question to think about it or solve it then try to call on people randomly. This way everyone has a chance to share their ideas.

What this list of rules really sparked in me was how creative I could be with displaying them and how to introduce them. I love to do projects and have “cool” things around me. Some people may call me “extra” but that is just the way I am.

So, I took to Google to find how other teachers have used decorations to display their classroom rules and expectations. I found this which I love so much:

Image result for fun ways to display classroom rules

I love Mario games. This is so cute to me, and I think students would enjoy it. Plus, there is a lot of space to put your rules on the board.

I also found this article: (link: http://blogs.edweek.org/teachers/classroom_qa_with_larry_ferlazzo/2016/06/response_classroom_rules_-_ways_to_create_introduce_enforce_them.html) that gave some great ideas on how to introduce the rules to the class.

The list includes:

  • Me and We Rules: Involve students in writing the rules with you. In small groups have each student in the group write down 2 or 3 rules are expectations they would like to live by. Have each group sort them into up to 5 rules. Write group contributions on the board. Come up with a creative title for the activity. Winnow the rules down to NO MORE than 5! Make sure they are written behaviorally.

     

  • Eternity Rule: Write one rule down that is important to you-an Eternity Rule. That means that from now until you retire you will teach, reinforce, and live by that rule. Now have students come up with at least 3 more Eternity Rules for the rest of the school year. These will be the rules they will live by as well.

     

  • Roll the Die Rules: Create cardboard six sided cubes with rules that you value on each side of cube. Now have students in small groups roll the die for approximately 2 minutes. Each student gets a turn. Have students calculate the number of times a rule showed up. The rule that shows up most is shared with the class. All groups share the rule that received the most hits and then the teacher leads a discussion regarding those identified rules.

     

  • Rule My World: To teach the rule “entering the classroom politely-no pushing, shoving, yelling, etc.” have a pair of students’ role-play pushing, shoving, and yelling prior to the entering the room. In small groups the teacher has three written prompts-what went wrong with the way the students entered, what’s a better way to enter, and why is pushing and shoving impolite? The teacher conducts a sharing activity and writes down responses on the board. The teacher reinforces the rule by asking students to remember the role-play and the group activity.

I really like the idea of involving the students in writing the rules, that way they feel they have a say in the class. Also, eternal rules that I always abide by would be good, because I can include the three I previously talked about that I really liked. My biggest take away of how to introduce and display rules is to be creative and do it in a way that I like so students can get a sense of who I am. Also, allow students to have a say and have fun with writing and learning rules.

Modeling in the Mathematics Classroom

 

The primary focus of the chapters that we read from Strength in Numbers was creating equity and access to mathematics in the secondary education classroom. In the theme of making mathematics accessible to all students, I found an article about using modeling in the high school classroom, called “Mathematical modeling in the high school curriculum” (link: http://www.nctm.org/Publications/Mathematics-Teacher/2016/Vol110/Issue5/Mathematical-Modeling-in-the-High-School-Curriculum/). Mathematical modeling is centered around “using mathematical approaches to understand and make decisions about real-world phenomena.” Utilizing this type of instruction, the teacher will give students a real-world problem that they will come up with multiple solutions to. In the article, the example given is centered around comparing different prices of gas at gas stations, and if it is more economical for a driver to drive outside of their “usual” region to find gas. Problems such as the example given are what the concept of modeling focuses on. Using this type of instruction in the mathematics classroom reminded me of the chapters from Strength in Numbers in the essence that utilizing modeling can help to make mathematics more accessible to students. Modeling focuses on making connections from mathematics to the outside world, making it more meaningful to the students. Furthermore, it aligns with the principle of asking students to see themselves in the mathematics that they are learning. In modeling, students are asked to use and develop problem solving skills to investigate a given scenario that applies to their day-to-day lives. The concept is reminiscent of project-based learning (PBL) but on a much smaller scale.  

Screen Shot 2018-01-16 at 8.19.33 PMIt asks students to perform tasks similar to that of PBL, but from the description in the article, modeling should take place over one to two class periods. The concept of modeling in itself asks students to apply their learning to real-world situations, deepening their understanding of the material. A portion of the article focuses on the teacher’s role in modeling. It addresses questions that the teacher should be asking him/herself before the lesson begins, such as what other resources students may need access to in order to properly address the question that the model gives the students. This indicates the preparation that should go into preparing a modeling activity for the students in the classroom. The article prompted me to consider how much of an influence giving students the opportunity to integrate their learning into real world problems can have on their learning. At the end of the article, a is quoted who describes her appreciation for having the opportunity to model in her mathematics classroom because it helped her to “remember the math.” Modeling gives students to apply what they have learned in their classes outside of the classroom, as they will eventually do as adults.

In my own classroom, I can use the information gathered in this article about how to model, and the benefits of modeling in the mathematics classroom to integrate modeling into my curriculum as a teacher. As we discussed in class, I would be sure to give students the opportunity to work in small, random groups to exploit the skills of each individual student. Giving students an opportunity to apply their thinking is a common theme in recent articles read, and in the assigned reading for class, as well as what research has supported in the past. It gives students real-world applications to what they are learning, answering the perpetual question “when are we going to use this in real life?” This article was further support for me to ensure that I create a classroom centered around applications for the mathematics that students will learn. Doing this will not only give them an opportunity to apply their learning, but will help them gain a deeper understanding of the mathematics that they are learning, and thus retain the information gained for a longer period of time. Altogether, this makes mathematics more accessible to the students, as it aligns with the ideal in Strength in Numbers. Modeling in itself can be changed to fit what the teacher utilizing it needs for their classroom (i.e. a model can be made shorter or longer, what the model is will depend on what is being learned in the classroom, and how frequently the teacher uses models to apply student learning).

Paired with ideas that I noticed in the article that I wrote about last week, as well as what I learned in my Curriculum and Instruction (C&I) class, I began to wonder about the impacts of modeling, or PBL on students in classes. In my C&I class we visited New Tech in Sioux Falls, and those students had significantly lower standardized test scores than other schools in Sioux Falls in the mathematics subject area. Modeling gives students an opportunity to use problem solving, but not to the extent that PBL does. It also inherently employs aspects of an equitable classroom, aiming to make the mathematics more accessible to all students. Thus, I questioned how deeply modeling affects students in the classroom. I found a study through an online database where modeling was utilized in one differential equations course while another professor used a traditional lecture technique in his differential equations course. The study found that on the same final exam, students in the class that used modeling as a instruction technique had a mean score 12.4% higher than the students in the traditional classroom. Although the study admits that it was “quasi-experimental,” it still gives serious implications to the usefulness of modeling in the classroom. Link: https://www.sciencedirect.com/science/article/pii/S073231231630147X

References

Hernández, M., Levy, R., Felton-Koestler, M., & Zbiek, R. M. (2016). Mathematical modeling in the high school curriculum. Mathematics Teacher, 110(5), 336-342.

Promoting Equity in Mathematics

From the little bit that we talked about equity in class, I realized that I did not know exactly what this word meant or how it applied in schools. I thought that equity was creating same lessons and teachings and giving all students the same opportunity; however, after reading Carly’s blog, I understood that it is way more than that, so I set out to find an article to better inform me about what equity was and how to promote it in our classrooms. The article I found was

Promoting Equity in Mathematics: One Teacher’s Journey

Alan Tennison
        The first thing in this article that stood out to me was the definition of educational equity as “the concept that all students, regardless of their personal characteristics, backgrounds, or physical challenges, must have opportunities to learn mathematics. Equity does not mean that every student should receive identical instruction” (page 28).  It was nice to see a definition laid out for me so that I could actually learn what equity is.
        One huge problem that schools have today is that they place students into classrooms based on how the teachers perceive their abilities. This is hard because there is no right or wrong way to do it and students get placed into the wrong groups because they are not test takers or they are very quiet or other outside factors that do not deal with their intelligence level. Then these students are in these “low-track classrooms” where the curriculum is easier, the teachers are less experienced and the other students in the classroom are less-engaged. This creates a great divide among students in the low-track and the high-track and the low-track students have little to no hope of ever catching up with the high-track students.
        The overwhelming issue that this article talked about was that once students are tracked into lower achieving classes, the expectations go way down and the material seems useless to the students. Personally, it makes sense. If I was a student that the school deemed “low-achieving” and I was put into a math class where I had to re-learn the material that I did not master well enough the prior years, it would be a lot easier to give up and not try as hard.
        The author then discusses how he tried to change this by ending tracking in his school and making it so that all students had the same expectations. The result of this was that the students attitudes changed and they cared more about mathematics. He began to use problem-based learning that incorporated many different math concepts and students could now make connections as to when they would ever use the topics in real life. The percent of students enrolled in a math course their fourth year increased from 10% to 60% and the average math ACT increased from 16.9 to 19.9. As we are all math people, we know that numbers do not lie.
        In my opinion, I think it is important to create classrooms with students all along the spectrum in terms of “perceived mathematics ability” as every student has something to learn form one another. Pairing a student who completely gets a topic with a student that is lost helps them both. The confident student explains the topic to the other and this helps him/her understand their thinking and helps him/her remember how to do the problem. The student that does not understand gets the information explained from someone who has also just learned the material, so they are able to better explain the material student to student.

Making a Math Classroom Equitable

This week in class the concept of mathematical classroom equity was introduced, which is a concept that immediately elicited my attention. I have often contemplated the idea of equity, but in the concept of equity vs. equality. It was a topic introduced to me a while ago when I found this image:

Image result for equity vs equality

At the time I was considering the argument of equity vs. equality in a political sense, because in our current political climate many groups campaign for equality when they really wan equity. I had not thought about it in an educational sense.

So, when the topic came up in class that we would be looking at Case Studies and deciding whether they were equitable or not, I was immediately interested. My main item of discussion and knowledge for equity in the classroom comes from our assigned reading of chapter two Equitable Mathematics Teaching from Strength in Numbers Collaborative Learning in Secondary Mathematics by Ilana Seidel Horn. URL: https://drive.google.com/file/d/0B1zVoWMFl08-TGN2NWxQMXVPVGc/view

I found the chapter intriguing to read and took out many good points and concepts from it.  The first thing is how the book defines equity in math as “equitable mathematics teaching involves using models of instruction that optimally support meaningful mathematical learning for all students.” Meaning that teachers should be using a variety of methods and techniques in order to reach students of various learning styles.

The second thing I found most helpful was the three practices they listed for collaborative learning environments that influenced equitable math teaching.

  1. What counts as math involves how mathematics is presented to students and the messages about what success means.
  2. Pedagogical practices focus on the work of teaching.
  3. Relational practices address the relationships that students build with others in the school and classroom.

Finally, the four principles for equitable math teaching where:

  1. Learning is not the same as achievement.
  2. Achievement gaps often reflect gaps in opportunities to learn.
  3. All students can be pushed to learn mathematics more deeply.
  4. Students need to see themselves in mathematics.

There are many things here that I would love to implement into my own classroom. Like, using group based work in order to help build the classroom as a community of learners so that way they feel part of a collaborative effort. They could take on a role in their team that meets their strong suit. Also, having across classes activities. That is to say, the algebra students work on the calculations to some 3-dimensional shapes the geometry students are making. That way they feel more connected as a school.

Having students see themselves as mathematicians is also so important. I am a firm believer that everyone can do math, because it is a universal language that can be taught in many different ways. If one way is not working for a student, then it should be I as a teacher to make my classroom equitable so that that student can find a way that helps them learn math. Every student deserves a fair opportunity to learn such that they can be at the same level as all of their peers. Constantly berating students with quizzes, homework, and tests when they are doing poorly does not mean they are going to learn math. Students learn math in many different ways, but they can all learn math.

After learning more about equability in education I like to see equity more like this:

Image result for equity vs equality

After learning all this information about how I can make my classroom equitable, I wanted to be able to see it in action. What are the different ways equity can be incorporated? So, I went to the National Council of Teachers of Mathematics and found this video that showed me a lesson of how a teacher used equity in her math classroom. URL: http://www.nctm.org/Conferences-and-Professional-Development/Principles-to-Actions-Toolkit/Equitable-Pedagogy/

In the video the students are learning about finding the area of a square, and they all have to go to the board to present how they found their area. They are all in partners by the looks of it, and the partner groups have varying degrees of difficulty in their square. What I mean by this is some students have a regular square that has a flat side on the bottom, but other students have their square skewed so a corner is touching the bottom. Like this:

The students who may struggle more would be given the square on the left so they could count the grids from top to bottom and left to right then multiply to find the area. While the students who understand it more get the square on the left where they have to combine the areas of a square and triangle to find the total area. This way both groups of students are given materials at their level of understanding.

Finally, in the video you can see a great example of peer collaboration. When two students with the harder square are at the board presenting, one student asks them how they got the area of their triangle. This forces the two students presenting to explain their rational, allowing the teacher to check their thinking, and the student who asked the question gets to learn something new.

One adjustment I would make to an equitable classroom would be to find a better way to mix students at the upper end and lower end of the class to see how further growth and understanding could be resulted from that. Right now, to me, it seems like equity is giving the “smarter” kids material for deeper understanding and other students material to try to reach the “smarter” kids. Equity has been an interest of mine for a while now, and I am excited to learn more about how to implement it into my future classroom.

What are some best practices for teaching high school mathematics?

By Kevin Reins

This week I read the responses of a two part series in Ed Week by Larry Ferlazzo. The question of the week was, “What are some best practices for teaching high school mathematics?” This intrigued me as I was preparing the 18th revision of my secondary mathematics methods course here at USD.

“…there are a zillion different instructional strategies and practices that math teachers can use in high school.”

The focus of part 1 was on these ‘instructional strategies and practices,’

  • teach to big ideas (see image above), it allows students to have interconnected schema
  • focus on the processes and connections between different processes
  • use instructional routines (see Why instructional routines?)
  • keep a record of conversations when you orchestrate full group discussions
  • be selective and cautious in your use of technology
  • incorporate high leverage long-term strategies
    • -David Wees
  • embrace mistakes, normalizing mistakes, safe space for discussion and correction, utilize error analysis
  • formal error analysis through test corrections
    • -Jillian Henry
  •  provide relevance and contexts for the mathematics
  • engage students in a variety of practices and strategies
  • provide scaffolds for those who need the extra support when working with challenging content
  • plan intentionally and deliberately so your instruction is impactful, consistent, and effective
  • develop a community of learners where group participation and interaction is expected
  • employ student-centered teaching and learning
  • provide opportunities for students to develop and strengthen their skills of mathematical communication (including vocabulary)
  • make the development of a variety of problem-solving techniques a priority
  • eliminate the blank paper; require students to write (1) determine a strategy that could be used to solve the problem, (2) write a question that you have about the problem, (3) record everything you know about the content related to the problem.
  • develop their ability to ask good questions during problem solving phases, Entry (getting started), Moving (when stuck), Reflection (thinking about thinking), and Extension (deeper thinking).
  • utilize graphic organizers to help them employ processes independently
    • -Tammy Jones & Leslie Texas
  •  What works at elementary or middle level works for high school
  • pose interesting problems or set the stage for students to pose interesting questions/problems about the situation
  • encourage investigations, experiments, collaboration, and discourse as students explore problems
  • expect representations or models for the problems being investigated
  • engage students in discourse, creating mathematical arguments and critiquing the reasoning of others
  • proving their work with both formal and informal proofs 
    • Anne Collins

Part 2: Students must ‘engage in math problem-solving’ and not just ‘follow procedures.’

The acquisition of best practices for teaching high school mathematics is necessary for student academic success.

The focus of part 2 was on engaging students in problem-solving. The following was said by the experts interviewed,

  • you must have as your guiding philosophical principle the belief that all students can learn
  • you must provide opportunities for them to fall in love with learning
  • Standards for Mathematical Practices can serve as a guide for the ways students need to be engaged in mathematics
  • choose open-ended problems
  • focus more on the process rather than the correct answers
  • challenge them with mathematically rigorous tasks, choosing a Higher-Level Demand Task
  • learn how to anticipate student responses and misconceptions for tasks
  • ask students to find multiple strategies to the tasks you present
  • learn how to help students learn from mistakes
    • -Wendy Monroy, LA math coach
  • math learning should be developing conceptual understandings of the mathematics
  • focus on the conceptual relationships
  • create a synergy between the lower levels and higher levels of thinking through inquiry
  • create a social environment that promotes team work and collaboration
  • provide an open, secure environment that allows for mistakes as a part of the learning process
  •  use an inductive teaching approach (vs deductive)
  • reduce teacher talk time (increase productive mathematical discourse)
  • differentiate by content, product, and affect (Tomlinson)
  • use all types of assessment; visible thinking routines, “I use to think… Now I think…” (Harvard University’s Project Zero)
  • use a flexible fronts layout of the classroom which encourages more collaboration
    • -Jennifer Chang Wathall, educational consultant in concept-based mathematics/curriculum
  •  give challenging problems that build patience and persistence in their maturing problem solving skills
  • then spend ample time in joyful struggle
  • create rich mathematical dialogue that leaves the building
  • 12 challenging problems that 5 of which will appear on the final, and give them time in class to work on them (e.g., A point P, inside a square, was 3, 4, and 5 units away from three of the corners. Find the length of the side of the square.)
    • -Sunil Singh, author of Pi of Life: The Hidden Happiness of Mathematics
  • sufficient time to make sure that students know how to solve problems using different methods
  • look for opportunities for students to have multiple entry points or strategies for solving a problem
  • take time to discuss strategic choices
  • find flaws in short cuts and when certain methods won’t work
  • open their mind to new and different approaches
    • -Matthew Beyranevand, author of Teach Math Like This, Not Like That: Four Critical Areas to Improve Student Learning.

So after bolding all of the big ideas of the laundry list of instructional strategies and practices that were provided I compared it to the content that I normally would teach in my secondary mathematics methods course. The result was two ideas, one new, and one that could use a deeper focus. I would like to incorporate more ideas on how to utilize math mistakes in the classroom as learning opportunities. I also would like to explore instructional routines a bit more.

To think more deeply about embracing mistakes, normalizing mistakes, and creating a safe space for discussion and correction, I think it is important to start off knowing what some common math mistakes in high school are. I found Math Mistakes website that does just that. This should be a good start for a discussion on how to utilize some of these mistakes when they pop up as a learning opportunity.

With respect to instructional routines, I read Why instructional routines? It turns out I know what they are and how one should utilize tasks in teaching. One instructional routine that David Wees talks about is, Contemplate then Calculate, as a tool for learning how to use the 5 Practices for Orchestrating Productive Mathematical Discussions. The high level goals of Contemplate then Calculate are to support students in surfacing and naming mathematical structure, more broadly in pausing to think about the mathematics present in a task before attempting a solution strategy, and even more broadly in learning from other students’ different strategies for solving the same problem.

“Instructional [routines] are tasks enacted in classrooms that structure the relationship between the teacher and the students around content in ways that consistently maintain high expectations of student learning while adapting to the contingencies of particular instructional interactions.” Kazemi, E., Franke, M., & Lampert, M. (2009)

I’m looking forward to expanding my teaching and learning opportunities to include both instructional routines and normalizing mistakes.