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## 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
• 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.

## Technology in Education

In an article by Tim Hickey called “Technology for Learning, not for Technology’s Sake: Toolbelt Theory and the SAMR Model” posted on nctm.org, the author covers two types of incorporating different styles of technology into math education. As stated in the title, these two ways are called the Toolbelt Theory and the SAMR Model. Each of these is its own unique way to involve technology in the mathematics classroom.

I’ll start with the Toolbelt Theory. The Toolbelt Theory implies that teachers should be giving students different ‘tools’ to add to their toolbelt. Once they are shown how to use a different type of technology they then have added another ‘tool’ to their lifespan toolbelt, or types of technology they will be able to use for the rest of their life. By looking past the standard paper and pencil approach, we are able to provide students with lifelong learning tools.

The second method, the SAMR Model, is an acronym standing for Substitution, Augmentation, Modification, and Redefinition. By using these for terms, you can incorporate technology into a lesson in many different ways. Substituting technology can help teach the same lesson but in a more engaging way. Augmentation will help teachers analyze class data. Modifications will require students to step out of their comfort zone and use technology as an assignment. And redefining a task allows teachers to be more creative in lesson planning.

I believe both of these methods can have a good impact in the classroom. I will certainly do my best to teach students how to use different types of programs and gadgets to add to their toolbelt. One way to achieve this is to use the SAMR Model or one similar in my classroom.

The next article is titled “Using Technology as a Learning Tool, Not Just the Cool New Thing” is written by Ben McNeely. In this article, he talks about technology and some of its problems along with its positives. The problems with technology are funding, the urge to cheat using the technology, and teaching people how to use it. The positives are the interaction that technology can spark, the skills that students can learn from it, and using it to connect with other generations.

## 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:

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.

## Increasing Student Participation

Student discussion and input can greatly impact learning in a positive way. The issue is that there are always students who are more vocal and have higher status. A few student will throw their hand up every time a question is asked to the class. Meanwhile, there are students who are constantly concerned that their question or answer is wrong. A major question we have to answer is: How do we get all our students to provide their input?

The article I read, “Increasing Student Participation” from The Teaching Center, which is part of the University of Washington in St. Louis. The article gave several ideas, but the ones I found noteworthy, along with some of my own ideas, are the following:

• Set up ideal physical setup for discussion
• Set expectations
• Establish environment of caring and respect
• Have class schedule set up for time for discussion
• Respond positively to student ideas

### Set Up Ideal Physical Set-Up

According to the article, there are many things that a teacher can do for the class to prepare them to participate in discussions. One thing is just the way the classroom is set up. The article suggests creating a U shape with the desks, which helps, since the students are facing each other and not just the teacher.

### Set Expectations

Also, set expectations for how much each student should participate. If the intent is to have every student contribute, let them know that every one of them has to contribute. One option is to have participation be part of the grade, but if we do our job as educators, getting students to contribute to a discussion should not have to be something that we have to force them to do, but eventually something that they will do on their own with our support.

### Establish Environment of Caring and Respect

One idea that the article did not address but I’d like to is establishing an environment of respect and caring. If we as teachers make it clear that every student’s input is valued, no matter their status in the class, then students will respect what other students are saying and not degrade them for it. As a result, students will not be afraid to speak their minds as much as they would otherwise.

### Have Class Schedule Set Up for Time for Discussion

One way to promote discussion in a classroom is by allowing plenty of time for it in the proper way. If a teacher only asks for questions and students’ thoughts at the end of a lecture, they may not remember what they thought throughout the lecture. On the other hand, if a teacher stops after every fifteen minutes of a lesson and asks a question to the students and asks for their questions and comments, the students may better remember what they just learned. Also, asking a question and taking responses immediately is not always the most effective way for students to formulate ideas and give a response. Giving students 10-15 seconds to think allows everyone to come up with an idea before anyone else responds. Mixing up discussion methods can make it so every student can give a response. Moving from partner groups to small groups to large groups to a whole class discussion can make it so every student feels comfortable sharing at some level.

### Respond Positively to Student Ideas

For students that lack confidence, pointing out their good ideas is very important. Even if their answer is not totally correct, pointing out what is correct or paraphrasing so it is correct can give them a confidence boost and think of their response as valuable. As stated in the last section, having students submit responses online or in some other private way and responding positively to that will give them the confidence to participate more in partner groups, where positive feedback will give them the confidence to participate in small groups, and so on.

The article “Increasing Student Participation” gave some intriguing ideas for how to get more students involved in discussions. I think that this is a very important subject to bring up, because all through my education, there were always people who would contribute every day and people that wouldn’t contribute at all. Sharing ideas can be a great learning experience for everyone. Hopefully we can use these ideas to have all our students contribute in meaningful ways.

http://teachingcenter.wustl.edu/resources/teaching-methods/participation/increasing-student-participation/

## 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.

It 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.

## Equity in Education: The Relationship Between Race, Class, and Gender in Mathematics for Diverse Learners

Equity in Education

Reading about a particular case study for class caught my interest. It was the case study in which a particular school was placing the majority of non-white students in below grade level courses, while just over one-tenth of white students were placed in these same classes. I was curious to learn what could possibly make such a disparity occur. I found the article “Equity in Education: The Relationship Between Race, Class, and Gender in Mathematics for Diverse Learners,” which was written Debra Rohn from the University of North Carolina at Charlotte for The Urban Education Collaborative. The article addresses how race, class, and gender affect learning outcomes and gives some theories on how to fix it.

The first topic discussed is “race and mathematics.” There is huge differences in average test scores between African Americans and Whites and Hispanic Americans and Whites, to put it plainly. This could be the result of direct discrimination, such as placing students of other cultures in classes below grade level just because the teachers do not expect them to do well based on their race. The article seems to suggest, though, that the more likely scenario is the expectations put on minorities by society as a whole. They are likely not expected to do well, and they realize this. The article suggests that many of them probably give up on their math abilities, since mathematics is widely viewed as a “White subject” in America. The article makes the claim that even when African Americans succeed in mathematics, some would describe themselves as “acting White.” Now, at first, this may seem to be more about equality than equity, but the solution that I gathered from the article is to give extra psychological help, of sorts, to children that are minorities. In an equitable classroom environment, students would be taught that a student’s race does not take them out of the running for a career in mathematics. One major issue is that the students are believing that they cannot pursue success in mathematics simply because of their race, and in an equitable classroom, they would be taught to think otherwise.

The second topic of discussion is “class and mathematics.” This has less to do with the idea that someone is behind someone else and more to do with the idea that students from lower socioeconomic status are set behind. When a child is raised by a single mother who may not have even graduated from high school, that mother is probably going to work long hours, which leads to her not being able to be with her children as much as parents from higher socioeconomic status. This leads to less development for the children of the lower socioeconomic status family, since they had less chance to communicate and do learning activities such as reading. This does put them behind from the beginning. A solution to this is for teachers to quickly realize this, and, rather than put them on a lower track for the whole education career, give them added supports and multiple modalities to work with.

The third topic was “gender and mathematics,” which focused on how many people think that men are naturally better at mathematics. There have been studies that show that this is not the case. As with race, the differences in test scores most likely comes from them thinking that they are less able to do mathematics, so they give up on being as successful as they could be. Also similarly to race, the solution is to eliminate these ideas in our classrooms, so that everyone gets a clean slate and can pursue whatever career choice they would like.

My view of this is that the solutions to all of these are to either reinforce positive thinking so that everyone believes that they can learn mathematics or help those that are behind so they can catch up to everyone else. Of course, one shouldn’t assume that a person that is African American or a woman isn’t at a disadvantage, because they could be someone who has a difficult past. Even if a classroom was all white males, we as teachers should know what past they come from, so we can potentially help them come out of it with hope of a better future. I can use this information to help my students overcome racism and sexism to pursue their dreams. I can use this information to help students who come from difficult beginnings get to a place where they can learn more information easily. The point is, teachers should learn as much about their students as possible, and cater to that. That is the point of equity. That is how it can be used with race, class, and gender.

## Addressing the Equity Principle in the Classroom

I discovered the article “Addressing the Equity Principle in the Classroom” in Vol. 101, No. 8 of The Mathematics Teacher.

The article describes a research project performed by the University of Wisconsin-Madison involving the math teachers of a chosen school district that participated in defining equity in the classroom. The following four responses emerged from the project:

1. Equity is about instruction.
2. Equity is about creating a specific classroom environment.
3. Equity is about equal opportunity.
4. Equity is about appropriate curriculum.

This study also asked teachers to identify one student that was performing below proficiency level or a student whom they had little personal connection. The teachers were given questions to facilitate getting to know the student and reflect on how getting to know students on a personal level correlated with achievement. Teachers reported in multiple cases that just getting to know these students better that achievement began to rise for each individual student.

In one of the reflections, the teacher realizes the stress one student has with English being a second language and working to provide for his family outside of school. Another teacher shared how a student simply talks to her about her interest outside of the classroom and have found commonalities in those interests, creating a positive environment in the classroom. This student has displayed a higher level of achievement since these interactions. This appears to have been accomplished without a distinct change in instruction, simply from more positive interaction and relationship building with the individual student.

After reading about this study, I determined that the methods math teachers use in the classroom that promote relationships and interaction among peers and teachers alike are very important to providing equity of access to mathematics. Teaching is a form of communication. By creating more mediums through which we can communicate concepts in which students and teachers can understand, we will have more pathways to communicate understanding.

So how? How do we build relationships that promote mathematical understanding? Even though this appears to be less specific than mathematical instruction, building relationships is very important to general learning. One of the things I will be doing in my classroom involves developing values of our community (classroom) through the students’ eyes, giving them ownership of what the environment should be. I want to break students into small groups, mixing them up periodically in the first period of time that they spend in the classroom. I want students to discuss what standards we want to live by in OUR classroom. There will be a couple that I will have at the beginning such as Follow the Code (school handbook), Look at the Person You are Speaking To. Everything else will be the product of the students. However, finding a way so every student is heard and validated is very important. This sets the tone for students to have ownership of the culture as well as places myself as a teacher as serving the students’ needs that they identified themselves. Once we establish these things, we can move into mathematical instruction with a foundation built on the needs of the students.

The one thing I think is missing from this study is how the students interacted with each other in these math classrooms. There are far more relationships in the classroom than just teacher-student. I believe the dynamics of peer relationships, especially in secondary education, can impact equity in a math classroom. Facilitating positive, productive interactions of students with their peers can break some of the barriers of equity in the classroom.

I discovered another blog titled “8 Characteristics of an Equitable Mathematics Classroom”. https://mathisforeveryone.wordpress.com/2015/03/05/8-characteristics-of-an-equitable-mathematics-classroom/

It promotes conversation for problem-solving which opens communication pathways for students to access understanding. The blog also states that “achieving equity is meaningless if it sacrifices excellence”. This should be the challenge of all math teachers to connect with students’ needs to achieve both of these things.