Week Five: Teaching Strategies-Math for All

“We’re kids. Why are we learning about taxes and overdrawn accounts? Why don’t we do problems that talk about video games or something we care about?” asked a 6th grader in my class.

Now, he had a point. They are eleven year-old students, learning percents through questions about things they don’t quite understand. Yes, they will use it later on, but maybe there is a different way to present the problem where their interests allow them to become more engaged in these types of questions.

When we first encountered the problem, I asked them to read it on their own. They all looked at me like I was crazy. They all said they couldn’t do it. After explaining what the question said and what overdrawn accounts were, 90% of the class shouted the answer. The other 10% were still trying to work through the problem. So, it wasn’t about not knowing how to go about the problem, it was that they didn’t actually know what the question was saying because they didn’t have experience with those terms.

The article we read, “Teaching Strategies for ‘Algebra for All'”, by James R Choike, was very informative and made some great points. It also gave something we could use, as in, strategies we could implement in the classroom to address certain issues.

For example, in the “eliminating distracters” portion, it spoke about how numbers and words could prevent students from approaching the problem. They might give up before starting, even if they have the skills to complete the problem. Some students, especially young students, will get caught up in the words.

Another point I liked was the multiple representations idea and being able to connect a thread through them. As individuals, we take different approaches to solving things. We like to approach problems in a way that makes sense for us. So, expecting all students to approach a problem in the same way, or in one particular way, is limiting. You may not get through to every student and their learning styles. We can reach the same answer through different mediums, and letting students explore those mediums is important. Then, you will be able to make connections between the representations.

By showing multiple representations, it might be easier to root particular concepts with the students. You can expand a simple problem by showing different approaches.

Another, not-so-surprising, point he made was about not beginning the year with remediation. I actually agree with that. There are other means to revisit topics without wasting time in the beginning, not knowing what students do and do not know. Time is precious, why waste it?

Students need to be able to grow and make their own conclusions. They will never learn if we just give them answers. They need to be able to think for themselves and experiment, making their way through unexplored waters.

At the end of the article, it talks about establishing a safe classroom environment. I think that is one of the most important points. If a student does not feel comfortable expressing their thoughts in class for fear of being ridiculed, then they won’t be able to get anywhere. If they feel like they can communicate, ask questions, and offer their thoughts, there will be a greater opportunity for learning and growth.(At least, in my opinion.) I encourage my students to ask questions, try problems, and respect one another. If they can’t do that, then we can’t move on as a unit. We can all encourage each other’s learning if given the opportunity.

This week, I read an article about when a calculator should be used, and I thought it was pretty cool that they said to let students decide for themselves when it is okay to use one.

Here it is: Calculator

I retweeted the article from a math student/teacher that posted it. I also came across this favorite math quotes tweet on twitter, and I wanted to participate…..but the quote I liked was too long, and I forgot to @ the person doing it. In all, I’m still getting used to Twitter, but it’s getting easier to navigate. I also liked a few tweets, which I thought were interesting. And my post from last week was mentioned and tweeted as well.

That’s all for Week 5! Let me know if you have any questions or ideas!


Week 4: Why Teach Anything? The Point of Teaching Math

This week, we read an article and listened to a podcast that detailed much of what we read. It dealt with reason and sense making and why we need to focus on it. There were ideas of how to incorporate it into the curriculum, but the largest issue might be that reasoning is not something you can teach. It’s like the saying goes, “You can lead a horse to water…”. Making sense of problems and approaching them from a logical angle could work, but reasoning is a whole other creature.

In a perfect world, we  could insert reasoning and sense making without an issue. However, it is not a perfect world. They do bring up great points, and I think the main takeaway is that something has to change and the only way to kick-start that change would be for everyone to work together and address the issues at hand with regard to mathematics at the high school level.

There was one particular idea that I’m glad they brought up. High school students come from a variety of backgrounds. Every classroom is different, containing varying levels of abilities and students from different backgrounds. Some students, especially minorities, will be placed in lower level math courses under the assumption that they may not be attending college. Therefore, they do not receive the same opportunities to learn as another student in a higher level course. The idea of equality in the classroom in reference to obtaining the same learning opportunity is spot on. Otherwise, we will continue to perpetuate an idea of a “math type”. (Sort of like setting “math people” aside, continuing this idea that you are inherently good or bad at math.) Such attitudes need to be turned around, at least in my opinion. It is not fair to strip educational opportunities from anyone regardless of race, ethnicity, or socioeconomic status. Education is not an exclusive club.

I came across a post this week about why we should teach mathematics. Mostly, it was a conversation starter. The author gave their own reasons, but also inquired about other reasons from the readers. Mostly, I believe we should teach math for the sake of teaching it. I feel like we have taken a turn for usefulness. We only care about those things that are completely useful to us. If we cannot instantly benefit, then we don’t care. Learning for the sake of learning seems to be a thing of the past. For example, why do we read Shakespeare? Why do we read Huck Finn? Why do we read Frankenstein? Why do we read anything? For the pure enjoyment of it. For the culture that surrounds it. For understanding. The same thing goes for math. We should teach it because it offers another dimension to understanding the world around us. It doesn’t exist to torture unsuspecting students. We should teach it for the beauty of it. That may not be enough reason for some, but it is for me.

It really got me thinking about what other people might think about why we teach math. I think this question resides on more of an individual level.

Here is the link to the article, (with my comment toward the bottom):


Continuing with the equality idea, I saw a few things that I liked on Twitter dealing with gender bias in STEM classrooms. And, being a female in math, I agreed with many points. The most important part, though, was the fact that we think this bias is dwindling with our generation, but it is just as present. (And history repeats itself.) Though, I’m also inclined to be wary of the article, if only because it was a woman who wrote it. So, the article about bias is, potentially, biased. In any case, I present you with the article:


I believe that is all for this week! Let me know if you have any thoughts!



Week 3: Blog Comments & Tweets-“Would a Grade By Any Other Name Work as Well?”

Have we lost focus on the point of education? Series of tests, quizzes, papers, standardized tests weigh heavy on our minds. Many times we remember taking the test: “It was hard; it was easy; everything I studied wasn’t on the test; everything I studied was on the test.” An exchange of phrases whizzes by, the only remnants being a grade you file away in a folder that you may never see again.

Why is this important? When are we going to use this? Why do I need to know this formula? Why doesn’t ‘X’ solve its own problems? I have my own to worry about. Students are not the same. As individuals, we have different interests. Therefore, we value different things. We shouldn’t be expected to pursue interests that are not our own. Sure, being educated on basic concepts is a great idea, but ‘why’ knowing such concepts is a great idea is the question. Too often, the answer of why becomes getting good grades, because it’s on the test, you just have to know it, it’s on the MCAS or the SATs, and so on. There is no real purpose. There is no drive. Grades as motivation is a surefire way to burn out fairly quickly. Personally, I think it limits potential and creativity. Too often, we look for the instructor’s approval over our own thoughts and ideas. Reading people and giving them what they want is easier than thinking for oneself. I find that to be a huge issue.

What if we just trashed grades? ANARCHY?! No. Let’s actually think about it for a second. Maybe another minute. Grades are like trophies. Trophies serve as a ranking mechanism. Sonner or later, everyone will want a trophy. Trophies then become everything. Everyone gets one and the meaning of a trophy becomes tainted. What once meant to serve as a ranking system, a means of motivation, has become tainted like little league sports. Grades seem to be the only reason to do anything. It has become a threat. If you get a bad grade you will not get into a good college, unless, perhaps, you slip the interviewer a twenty under the table. Jokes aside, even if we don’t throw away grades, I think the whole system needs to be revised.

I read a few articles this week, but one in particular stood out to me. It was about grades and if colleges should stop giving them. I thought many points were valid and that it was a good idea to experiment with. There was a plan as well, and although it may have holes, I think it is a great outline for how we could work around and tailor a schedule to the student and their interests which they are allowed to explore with. Most students now do the bare minimum to pass, but what if they were given the opportunity to pursue their interests without being chained down by a major from day one. They might be able to find something that they love without being pressured to choose as soon as they are thrown into a college environment.

Anyway, before this gets too long, I think it is worth the read and offers some pretty good ideas that we could experiment with. No grades could offer an increase in productivity, creativity, and overall performance, given other motivators. (No grades does not mean no work. You still have to work hard to keep your place within a class. If you don’t want it, then you are in the wrong class.)

Link to Blog: Grades Post

Here is my post on the Blog: “Thank you for this post! I completely agree that grades need to be looked into. This seems like a great solution to something that I have lived through. Grades are the “be-all, end-all” for most students. Things would be much different if students, including myself, weren’t so focused on getting good grades over completely understanding a topic. I think this system is absolutely brilliant, and I would love to see it implemented at some point. It would be a great experiment. I think we have lost sight of what education is, and it would be a good idea to start revising the system to optimize learning. I truly believe that, if given the chance, students will find that motivation to explore and discover their interests on their own. It’s sort of like asking, “What would you do with a million dollars?”, except it’s, “What classes would you take if grades weren’t an issue?”. It is an extremely liberating notion for students. Granted, there may be some unforeseen issues with an ungraded system, but I see more merit in working hard for something and pursuing interests despite failure. In all, I think it would be a great idea to experiment with and see the results.”

Also, I had a few Tweets/ Retweets. I was mentioned in one about growth mindset, which you all should take a look at. Growth Mindset is a great concept. And I tweeted a few links about growth mindset as well.

I still haven’t quite figured out that copy and pasting deal with Twitter….so this might be the best I can do for now:

<blockquote class=”twitter-tweet” data-lang=”en”><p lang=”en” dir=”ltr”>More on Growth Mindset! Take a look! <a href=”https://t.co/b2yCSgQooP”>https://t.co/b2yCSgQooP</a></p>&mdash; Rebecca Pereira (@rpereira418) <a href=”https://twitter.com/rpereira418/status/698315809953869825″>February 13, 2016</a></blockquote>


<blockquote class=”twitter-tweet” data-lang=”en”><p lang=”en” dir=”ltr”>I keep soming across &quot;Growth Mindset&quot;, and I quite like the idea. <a href=”https://t.co/IGT8W9hoxn”>https://t.co/IGT8W9hoxn</a&gt; <a href=”https://t.co/piBNZdIQpd”>pic.twitter.com/piBNZdIQpd</a></p>&mdash; Rebecca Pereira (@rpereira418) <a href=”https://twitter.com/rpereira418/status/697798428185722880″>February 11, 2016</a></blockquote>


<blockquote class=”twitter-tweet” data-lang=”en”><p lang=”en” dir=”ltr”>Should colleges stop giving grades? <a href=”https://t.co/UyemyIbVj9″>https://t.co/UyemyIbVj9</a></p>&mdash; Steve Shea (@SteveShea33) <a href=”https://twitter.com/SteveShea33/status/696762880096600064″>February 8, 2016</a></blockquote>


Week 2: Blog Posts/Tweets


This week, I saw a few articles I really liked, but I want to share this one because I have seen and heard a lot of things related to grades lately. Also, in my experience, grades have been detrimental to learning. Personally, I just want to learn. I want to know everything. Realistically, that is not really possible, but I like learning for the purpose of learning. The first question in any class seems to be, “when is the test?”. When the test day nears, the question becomes, “what is on the test?”. Once answered, students are scrambling to cram only the essential test material into their brains.

In essence, they are not learning. They are studying those few problems only to forget them a day after the test. This is temporary knowledge.

Last semester, I had this really great professor for my course in Logic. He told us we would have three tests. He also told us that we didn’t need to worry about them. Not once did he discourage us. From week 2, there were help sessions. We did problems every class. Not only that, we had discussions on other topics related to philosophy and the breakthroughs that are occurring within the field, especially in relation to mathematics and physics. I enjoyed going to every class. Every topic built on the last. Memorizing the rules were not required, but we did them so often that they became second nature. On my own time, I would do the homework problems and familiarize myself with the operators. If I was unsure, I would ask the professor and he was always glad to answer the question. In a class with over 30 males and 3 females, I felt comfortable to answer questions and go to the board to answer my own question sometimes. (He would help if I got stuck.) Here’s the thing, the homework was not graded. When it came time for the test, I didn’t feel pressured. I knew everything I needed to know. He did not place emphasis on the test. He cared more about us learning and thinking for ourselves. He created the perfect environment for that, and I learned so much. Also, if you asked me to solve questions now, I could probably do them without much of a problem.

The whole point in what I’m trying to say is that grades should not be the first priority. Learning should be the most important part. Creating an environment in which learning can take place is very important. Practicing and making students want to practice on their own time might help facilitate learning. We should not emphasize grades as much as we do. I saw a post that gave mathematicians grades, and I couldn’t help but laugh. (I’ll find it and post the link below.) We need assessments, but not at the expense of grades and learning. Students will take the initiative if encouragement is thrown their way. I am willing to experiment and see how not placing grades on my students’ quizzes helps or hinders them.

Report Cards for Famous Mathematicians

I was also able to have a conversation over Twitter with another person that I re-tweeted, and I shared the article I read with him because it was related to what he posted. He tweeted the article out to his followers.

That’s about all for this week!

Week 2 Reading: What Makes it a Proof?

“The very function of proof is not to prove the obvious in order to meet some sophisticated and purely professional standards and goals, but to prove what is not obvious.”-Morris Kline

This is how most people are inclined to think, at least in my opinion. But, the question here lies in what is considered to be obvious. What assumptions are we allowed to make? Obvious varies from person to person, from audience to audience. What is obvious to a nuclear physicist might not be obvious to your everyday lawyer.

Most students, especially early on, are not typically strong with proof, (if I’m honest, I’m not strong with proofs either). This might be due to levels of exposure, or the fact that they might think certain things don’t require proving. Also, throughout the article, many of them believe proof is by example. They show an example, and there is the proof.

I think that might be a start. It’s like the basis step. For example, is what they are giving me even true for a few cases? If it is, then they can move on to generally applying what is given by using prior/existing knowledge and assumptions. I would have started by looking at a few examples and seeing some kind of pattern. Since the proof gave a hint, I would have started looking at the hint first, expanding it and seeing what it is telling me. Then I would have proceeded from there. These students all seem to have the same idea, but they don’t have a strong background in proof or even the basic concepts. (At least not enough to make the connections.)

As teachers, we should work on providing an exploratory environment where students can think and make connections. We need to give students more tools that they may be able to use in coming up with ideas. Create an open classroom focused on thought an interpretation and making the rules of the game clear. Explain what proof is. Explain different types of proof. Explain that math is connected and not this distorted group of topics that have no relation to one another. The goal is to make deeper connections, but the question is how, and I think more experimentation and research is needed on that front.

More focus needs to be on the student and their understanding and the article seems to fall short on explaining itself. Though, it does make some useful points and brings up an issue that students are having. Yes, we should analyze student work, but we also need to expand student knowledge and encourage deeper understanding through inquiry and study. With that being said, we might want to try to discourage teaching strictly for the test. Students will never learn if they study for the one grade and never go back to the material. Maybe we need a strong use of cycling within the classroom. So, go back to old topics and show how they might be connected to the new ones. Topics in the classroom are not isolated. There is a connection. And students need to see that. It’s like history. dates are chained together, and one event may be connected to another event resulting in this domino effect. Math is the same way. There are chains, links, between topics.

In all, analyzing students work is helpful and seeing where students need help, but applying that knowledge to helping students root their understanding in the best manner is the issue that needs more research. What ways can we help our students?


Week One: Blog Comments

This week, I read a post titled “The Secret Question (Are We Actually Good at Math?)”. I follow a math blog on my personal tumblr, and he posts some pretty interesting and cool things. So, after an initial search of many things that didn’t catch my eye, I decided to scroll through his blog. I clicked around and found a quote that caught my eye.

“Students are not allowed to make disparaging comments about themselves or their mathematical ability, at any time, for any reason. Here are example statements that are now banned, along with acceptable replacement phrases.

– I can’t do this –> I am still learning how to do this.
– That was stupid –> That was a productive mistake.
– This is impossible –> There is something interesting and subtle in this problem.
– I’m an idiot –> This is going to take careful thought.
– I’ll never understand this –> This might take me a long time and a lot of work to figure out.
– This is terrible –> I think I’ve done something incorrectly, let me check it again.

Please keep in mind the article we read by Carol Dweck. The banned phrases represent having a fixed view of your own intelligence, which does not reflect the reality that you are all capable of dynamic, continued learning. The suggested replacement phrases support and represent having a growth mindset regarding your abilities and your capacity for improvement.”

After seeing this quote, I was instantly curious and clicked the link to the article. The whole article addresses the fact that as mathematicians, some of us believe that we don’t deserve to be where we are. That whole mind-set is conditioned, and I like that someone put that into a cohesive piece. It’s well worth a read. I also think it would be a great idea to start off every year with these phrases and an intervention of sorts. We are not idiots, and I think that now is the time we start embracing that fact if we haven’t already. Math like a sport. You need to practice to get better, but the fun is in playing the game. The best part is that you don’t have to be born to play, you can still participate.

I commented my thanks to the author for writing the piece and shared my own experience with my students and how I always tell them that they are not bad at math. They are young children and their first thoughts shouldn’t be that they are bad at math. Instead, they should have the idea that math is fun and that it is for everyone.

Here is the link to the blog post:


Week One: Responses to Readings

“Compared to What?”

Several studies show that comparison is beneficial to students. However, the writers of the article saw a gap in literature involving school-aged children. As the title suggests, their main goal was to see what comparisons provided maximum benefit for learning in 7th and 8th grade students. The issue being brought to light is the fact that, although we know comparison is beneficial and implement comparisons in a teaching environment, we may not be effectively utilizing the strategy. In many cases, what is being compared misses the mark of what should be compared.

What exactly are we comparing? What should be compared? Which types of comparisons are most effective? The authors look into three types of comparison for their study which varied in solution methods and problem features. One type compared similar problems with the same solution. Another type compared moderately similar surface features with the same solution method. The last type compared different solution methods to the same problem. They hoped to find out what type, or types, would be most effective in order for teachers to use to improve their use of comparison. They wanted to hit on the “three critical components of mathematical competence” which involved “procedural knowledge, procedural flexibility, and conceptual knowledge”.

After completing the study, the researchers found that, “comparing solution methods generally led to greater conceptual knowledge and procedural flexibility than comparing equivalent or different problem types”, but it “did not lead to greater procedural knowledge”. Being able to solve a problem using different solution methods indicates a certain level of understanding beyond purely getting an answer. It creates connections to simpler levels of calculating expressions with variables. They also found that in order to compare, it might be better to have something known to compare to. Going into the study, the students had some experience with equations and the steps to solving them. They did not go into the lessons completely blind. When comparing, it seems that you need some prior knowledge, or at least something worth comparing to. For example, it might be beneficial to be somewhat comfortable with a certain solution method before being exposed to another one rather than being thrown two comparable methods without introduction.

They concluded that comparing two different solution methods seemed to be most beneficial to learning if carefully implemented by teachers. Problems and solution methods must be chosen carefully. Also, comparing problem types might be a good form of comparison if the goal is flexibility of solution methods. It seems that the two types combined could potentially expand benefits if properly executed. Another conclusion was that comparison is only effective with careful support. Questions must be designed to support comparisons; they must encourage certain responses. They show that comparison is beneficial, but teachers must be careful as to what exactly is being compared. You can’t just compare anything and hope it works. In all, future research should be geared toward what is being compared in order to maximize benefits to students.

As I read through the study, I kept thinking, “this is a good idea, but what about other groups?”. The limitations of this study were addressed, but I think it runs deeper than classroom conditions and mathematical ability. In either case, comparison wouldn’t harm learning, but the issue lies in the fact that this study cannot completely be generalized to many other students. The only thing this study says is that a particular group of 7th and 8th graders, the majority of which are white, benefited more from the methods approach of comparison. Other parts of the study suggest that the authors pushed the results in that direction in regard to how they formulated the packets. Other factors need to be taken into consideration. There needs to be more research in order for this to have a bigger impact, at least, in my opinion. The study is far too limiting, which they made mention of. Overall, there needs to be more research done. There are so many other questions to be answered in regard to benefits: who will benefit? Does the research vary cross culturally? Does it vary depending on age? Does it vary based on environment? And the list of questions goes on.

While I think this was a great study to conduct and that they made great points, it further shows that more information needs to be obtained. There are many different possible combinations of studies that can be run, and that may be where the problems arise in retrieving data.

Common Core: High School Algebra

Personally, after reading this, I couldn’t help but think, “this was how I learned algebra”. Overall, we hit upon everything on the list. However, there were some parts about theorems that I am certain were not covered. After having gone over them throughout many college courses, I think that those theorems would have been great additions to the curriculum, at least in more depth. (Such as the Binomial Theorem and methods of proof.) Also, since technology keeps improving and is a constant in mathematics, it might be beneficial to have a technological component to solving algebraic problems. (At least being able to graph more complex functions using certain programs. However, only after graphic representation is understood. Otherwise students will come to rely on such methods without understanding what it means.)

I like the fact that it seems like the curriculum is not just geared toward pure computation. It seems like it is geared toward understanding particular concepts as well by building on certain basics, such as with equations and inequalities. The focus seems to be on getting a firm grasp on what it being taught. However, it does not always work that way given other outside factors that have become the norm among students, such as studying for the test rather than for purposes of learning.

Another problem is that many kids are thrown into algebra without prologue. Many classrooms are not uniform in the material being taught or the sequence in which the material is presented. All of the material presented seems fair to follow… on paper. However, prior knowledge, or lack of knowledge, may get in the way of understanding. Many of these topics should be taught sooner as a way of easing some students and encouraging others. Also, students might benefit from more proof within algebra. There seems to be an introduction into it, but not much of a focus. I’m not saying to have a whole month dedicated to proof, but it should be interwoven and given a bit more attention as an introduction to what may be learned in geometry. I believe that conceptual focus might be key to rooting certain mathematical topics: more answers to why questions instead of how.