That’s All Folks!

After two semesters and an innumerable amount of lessons learned, teaching strategies acquired, methods developed, and so much more, my time of learning from Professor Golden will be done. I have learned so much more about teaching, content, and learning than I expected to. In my two classes with Professor Golden I have gained a deeper understanding and a stronger passion for being a math teacher.

Throughout this semester we have discussed several content areas that intimidate me, mostly because of my experiences learning them in elementary school. When we covered an area of concern for me I always learned new methods for approaching and solving the problems. Coming away from this class I feel much more comfortable and confident in my teaching abilities because I feel more comfortable and confident in my understanding and ability to solve the problems.

A significant take-away that I have from this class is the exposure to and learning of new teaching methods. My experience with teachers is that most of them teach you the material and then the students practice what they are taught. I believe that it is much more meaningful when the students are given the ability and support to be able to investigate themselves and find a use for the material.

I’ve had the opportunity to teach a class before, but I haven’t had the opportunity to teach the same class more than once. Being able to teach three times in the same classroom has really boosted my confidence and improved my teaching abilities and effectiveness. Rather than always feeling like a guest, I felt the respect of the students and was able to see them grow. It was also easier to see the areas that I need to improve.

I believe that I am a better math student and therefore will be a better math teacher one day because of the opportunities and experiences given to me in my two semesters with Professor Golden, and for that I thank him!

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Build It Up

While looking around on the web for a problem that is accessible for students and interesting for me, I came upon the Build It Up problem. The directions are very easy, it’s the execution that provides a challenge.

Directions from the site:

At first, I must not have paid too close of attention to the example solution because I had it in my mind that I needed to use all different numbers, it didn’t take long to figure out that it is impossible to get to 15 with all different numbers.


I then started playing around with different combinations, and after MANY failed attempts I started to get the hang of it and found a little bit of success. When I was working through my failed attempts I was reminded of the SMP.1 standard, “Make sense of problems and persevere in solving them,” as it would have been very easy to say this is difficult and I’m not doing it right so I give up, so to get to the place where you can find multiple correct answers takes perseverance.

One big thing that I noticed was how important the placement of the numbers is, I became very intentional with their order. Using the same four numbers, they would both result in 15, or not 15, it all depended on how I ordered them.

Another big noticing was, with all the combinations that added up to 15, the original 4 numbers added up to an odd number. Yes, some combinations that added up to an odd number didn’t result in 15. I didn’t come across any 4 original numbers that added up to an even number that also resulted in 15; this makes sense to me as: an even number+ an even number= an even number, and 15 is odd.

After having some success (it was tedious though) with building up, I decided it would be easier and would involve less guess and check and more meaningful actions if I try to build it down. I started with writing out the factors of 15 that would withstand being decomposed twice, (12,3) won’t work because 3 can only be decomposed once before it won’t work with our set-up. After I made the list of factors finding solutions was a lot easier and faster.

Saying that this way was easier and faster worries me though, I don’t want my students to always try to find the easiest and/or fastest way to do something. I am appreciative that I went through the struggle of figuring out how to build up to 15. If I didn’t go through the struggle I would not have made the observations I did and wouldn’t have come away from the problem with as much regard for factors.

I think this problem is accessible for students, they will go through the same process as I did of struggling and then making noticings and then testing their wonders to find solutions. I am interested in a few things about this problem. First, I am interested in how the students go about solving it, what they notice and wonder, what solutions they come up with, and what their “take away” is. I am also interested in all the possible solutions. While reading the comments about the puzzle, I noticed some people came up with equations for it. I would like to test myself and see what I can come up with.

Cheez-It Math

I was first introduced to using Cheez-Itz in a Math lesson when I was creating an area and perimeter unit. I was on Pinterest trying to find ideas and came across a post that, you guessed it, involved Cheez-Itz. Using whole crackers (no broken pieces), create different shapes with differing amounts of crackers. If each cracker has a side length of 1, then it is the students’ job to find the area and perimeter of the shapes. I was fortunate enough to do this with a fourth grade class and they loved it. They especially enjoyed being able to create their own funky shapes and use an object that was common but never associated with math before to practice and create with.

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When I sat down to write a lesson idea blog post Cheez-Itz popped into my head. Wanting to be able to talk about more than just what I have done with Cheez-It crackers I searched “Cheez-It math” and I am so glad I did. The first (and only) thing I looked at was a collection of activities for grades 2-4, available on The collection is free to download, just use this link! I am amazed with how versatile the scrabble Cheez-It crackers are.

The first activity relates to nutrition as the students are told to look at the nutrition label and discuss what they see. I think it’s a nice health tie-in.

The second activity was prediction, the teacher showed one cracker and the students were to predict the number of crackers inside their unopened bag. After they made their prediction they are able to open the bag. They can count how many crackers they see on the top, and alter their prediction, hopefully becoming more accurate since they have more information.

They are then able to pour out the contents of their bag and count each letters frequency. Once they know how many of each letter they have they create a graph, and then come together to create a whole class frequency chart.

After their graphs are created the students write fractions and sentences describing their results.

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After fractions the students go into area and perimeter , they do exactly the same things that I had my class do.

Next, the students made arrays with their crackers. With their arrays they solved using repeated addition and multiplication.

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After the arrays, students move into a brief spelling lesson where they are to create as many words as they can with their crackers. They are to record their words and place them into three categories, 2 letter words, 3-4 letter words, and 5-6 letter words. This is the final activity in the collection.

There is also a worksheet that goes along with all the activities that the students can fill out instead of recording everything in their notebooks. I was very impressed with how many areas were touched upon with one bag of Cheez-It crackers, it motivates me to find new ways to incorporate Cheez-Itz and other foods into learning time.

Decimal Multiplication

The math concept that was the most difficult for me when I was younger, and still challenges me is multiplication. I don’t remember my exact thoughts and feelings when my teacher covered multiplying decimals, but I’m sure they were not good. I want to make sure my students feel comfortable with the topics that are being covered. A few ways that I can do that is by making sure the progression of material is logical, paced, and at the correct level, as well as catering to the learning styles of my students.

I was procrastinating/ doing research on Pinterest and came across a few activities, posters, etc. that I believe would be useful for students when learning how to multiply decimals.

I think that the following poster is an excellent tool as it shows the different ways to represent a decimal. Being aware of these and being able to use them correctly will help with understanding. It is one thing for a student to see 3.89 and mentally say ” three point eight nine” while reading it, but by being able to read it and say ” three and eighty nine hundredths will be very beneficial.

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There was a wonderful resource that covers common decimal place value errors. I think this would be a very worthwhile thing to go over as a class. Some students may be doing these errors themselves so to show that they are common errors and the students aren’t the only one who makes the error will help with their confidence. It also provides good conversation about errors and the “rules” of decimals.

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Being honest I didn’t remember the steps involved in solving a decimal multiplication problem so the following was very helpful. I appreciate that within the steps to solving the decimal problem the concept of estimation is also used.

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I was never a student that enjoyed doing worksheets all day. With that in mind I tried to find activities that cover this content in an enjoyable and interactive way. That being said, the first activity I looked at was a worksheet, but I have to admit that I like it. It provides great practice for getting the steps down, but doesn’t challenge the students past that.

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I’m interested in this method due to the fact that I am a visual learner and I assume some of my students will be too.

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I wish I could say that I found an amazing activity that gets students up and out of their seats and focusing on the content, but sadly I wasn’t able to find one. This disappointed me as I believe that getting students up and working together on solving problems is very beneficial. I don’t know if I wasn’t looking in the right places or if I didn’t look hard enough, but I do know that my search for quality activities is not over.

Learning By Playing

Recently, after an in class discussion on how to solve for area, I played a game called Area Block. It is a simple, yet effective game that is a good tool for practicing your area computational skills. It is a two player game , and is played on a grid. Players take turns making a single shape on the board that has an area of 10 or less. After the first shape is made, the rule is that each subsequent shape must share a side with the previously drawn shape. The game is over when the board is completely filled in. The winner of the game is the player who has the most area filled in.

The first time I played the game, my opponent ( who was also playing it for the first time) and I didn’t have any strategies, we were simply trying to figure out the game and trying to get as many points as possible. Our board for the first game was nicely divided and our score was perfectly tied.

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After the first game we started trying to develop strategies. We tried starting different points and using different shapes. We noticed that by trying to think of strategies we had to think beyond our current turn. We thought of our area, and what shape our opponent would use, and so on. We were thinking of the shapes that would give us the most area, and that would be difficult for our opponent to work around. Despite our strategies our scores only differed by one point.


After the second round we thought we had a pretty good grasp on the game and played a round on a board that gave us more to think about. This board was not just a grid, but also had shaded in shapes already that we had to work around. This encouraged us to think about using our space wisely and what shapes would give us the most area. We still tried to use the strategies we developed in our second round. This round we also ended up with a perfectly tied score again.


If I were teaching are to my students I would incorporate this game or one like it. I like that it makes students think about area in multiple ways. They are able to have fun while practicing and are not just solving computations. They are forced to think about creating and solving for area with unique shapes, rather than taking the easy way out and only using rectangles. The open grid allows the students to get a feel for the game and the grids with obstacles already in it is a more advanced option for students who are further along with their understanding of area.


Relearning How to Multiply

I remember learning multiplication in third grade and the memories of that time are stressful ones. We were taught only one way to solve multiplication,

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Recently in a math education class we talked about multiplication and this was the only method I could remember learning. What makes that even worse is that when asked why do you solve the problem the way you do? What’s the reason behind the steps? I could only answer with “because that’s how I learned it thirteen years ago.” Before we discussed this method in class I did not remember the reasoning behind solving problems with this method. Now I realize that it is all about place value multiplying throughout the top number by one place value at a time, then using a zero in the answer row as a place holder when you move up to the next place value.

Within class discussions, I learned of a few new ways to solve a multiplication problems, such as using an array to break down the problem and make it more visual, such as this example,

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When using an array to solve the problem, students (and myself) can visually see the reasoning behind what they are doing. They can break the larger numbers 13 and 16 into smaller numbers that they are more comfortable with. They can then solve the smaller problems, 10×10 can be solved by a memorized fact or the student can physically draw ten rows and ten columns and then count the squares to reach the same answer. Once the student has solved each of the smaller problems they add the answers together to reach their final answer. This method makes sense to me because you can break up the numbers into combinations that make more sense to you and that you are more comfortable with. You can also see the layout and the grid, if needed you can count out the squares to double check your answer.

Another way to solve a multiplication problem is by grouping.

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Since the problem is 3×2, you make three groups and then place two stars, dots, tallies, etc. in each group. When you have done that for every group you count the total number of objects in all the circles.

A method that shows the very definition of multiplication, repeated addition.

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If you are using larger numbers (or smaller ones), you can break them down into smaller pieces. For example, 15×7 can be broken down by replacing 15 with 10 and 5, and 7 with 5 and 2. So 15×7 turns into 10×2+10×5+5×2+5×5.

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I am a proponent of teaching various ways of solving problems and having my students choose the way that they understand the most and are the most comfortable and competent with. I would rather give myself more work in teaching the methods and grading assignments and assessments and have my students actually learn than assume they will learn with cookie cutter methods.


Teaching Reflection

I was very nervous going into the experience as the lesson we were presenting was only created the previous day. Talking with the art teachers was comforting but also interesting. After giving them a run down of our lesson they had some questions that I had never thought of. I said the students would need construction paper to create their project and the teachers asked what size of paper the students needed. I just assumed there would be normal letter size; it never occurred to me that construction paper came in more than one size.

After hearing the students that taught the day before I was under the impression that the students understood and did well with working with fractions. I was incorrect in thinking that our lesson would go the same way. I didn’t do a good job with explaining what they were going to be doing for the day. One girl wasn’t in class the previous day when they used circles and she made it very clear that she did not know what was going on and was coarse when I tried to help her. I tried to explain it a few ways and she always responded with I don’t know what’s going on. I tried asking if there was a different way I could help her and she wanted nothing to do with me or my help, this was the first time I’ve ever encountered a student like that, it was unsettling but something I need to get used to.

I noticed that after explaining that they had the freedom to create their own stencil, the majority of them did not create a stencil and went on to create their project. When going around to the tables, I asked some of the students who did not have a stencil what fractions of the whole their pieces were. Many had to go back and we figured it out together. I told the students that they had to keep track of the fractions that they used, but didn’t tell them how to organize it. I liked the variety of organization that the students used, from a map with the fractions on it to a table with them.

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When it was time to add the fractions together majority of the students had challenges. It was nice to be able to talk it out with some of them and help them, but I wish I had the rime to go around to all the students and hear their understanding rather than just seeing their math. I think it would be good to have more time dedicated to this lesson. The 50 minutes we used could easily be doubled in order to provide adequate time for creating, analyzing, and reflecting. It was very clear that the students needed more time working with the concept of fractions in regards to a whole, and with adding fractions.



A few weeks ago, as a homework assignment we were asses student work on transformations. The activity that the students completed is called Aaron’s Designs, which gave students the chance to draw reflections and rotations of a given figure on a grid, and to describe transformations needed to make a given pattern. We were given this sample of student’s work and this rubric to help with grading.

I was surprised by the amount of time that it took me to grade the work. I kept finding myself wanting to make changes to the rubric clearer. I found myself in a grey area where if a student didn’t answer completely correctly what type of partial credit should they get? Do I give them the benefit of the doubt that I might not be understanding what they were trying to say, or do I grade a little more harshly and follow the rubric letter by letter? More often than not I found myself wishing that I could talk to the student and ask them questions to get more insight into their thinking and reasoning. The following image is a sample of a student’s answer that made me want to have a conversation with them.

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I wanted to ask this student questions such as, How did he use the same design? What did he do to it? Both times of what? Could you describe what he did with pictures or arrows to help strengthen your answer? What thoughts did you have while thinking about and answering this question? That’s what I wanted to know for one student on one question, and there were 15 students, no wonder it took me so long to get through them!

I was pleasantly surprised by the fact that every student got the first question correct. The students were given one shape already drawn on a grid and were asked to reflect the shape over the y-axis, and then reflect those two shapes over the x-axis. The second question gave some students problems, only about half of the students earned full credit for this problem. The students were given one shape already drawn in and were asked to rotate the shape 1/4 of a turn 3 times. According to the rubric, each of the three shapes that was correctly drawn earned them one point. The problem with this was that the students who reflected the shape like they did in question one still earned one point because reflecting happened to put one shape in the correct spot. Although they earned a point, they did not show understanding. The third and final question posed to the students had all shapes already drawn in, and asked the students to describe the transformations that may have been used to draw the design. This is the question that produced the majority of my questions for the students. Out of 15 students only 6 received full credit for this question, many students used the wrong terminology or didn’t describe the transformations in a way that made sense. for many students I wrote down questions along the lines of could you describe it any other way? With pictures or using arrows? For the students who received full credit , I asked as an extension if there was more than one sequence of transformations, and how many there could be.

This exercise made it very clear that you can’t get a full picture of a student’s understanding based off of a short paper and pencil test. It also shows how valuable it is to go over test corrections with students is, that way you have dialogue happening and they might be able to better or further explain their thinking. One of the most important take-aways that I got from the activity was how important a well written rubric is. It is important for you to help while grading and to grade fairly and not go crazy, and also for the students to know hat is expected/ how they will be graded, and also to ensure that they are being graded fairly.


No Child Left Behind


This week a video of President Obama was released, in it he talks about limiting the amount of time students and teachers spend in the classroom to to no more than 2%. “The administration now believes all the time spent memorizing facts has taken the joy out of learning–not just for kids, but for the teachers who are restricted in their creativity.” “And, tests should be just one source of information. We should use classroom work, surveys, and other factors to give us an all-around look at how our students and schools are doing.” the White House has laid out a plan urging legislators to follow the lead of many states by enacting new guidelines.

The Elementary and Secondary Education Act, which is supposed to be updated every few years, yet has not been amended since 2001 when President Bush renamed it to No child Left Behind. His update included “spend[ing] more money, more resources [on] methods that work. Not feel-good methods. Not sound-good methods. But methods that actually work.” The methods included a sweeping new federal system of testing and accountability, with its emphasis on standardized testing. Even though the law expired in 2007, it will remain on the books until it is replaced.

I think it is past time that we make changes to the No Child Left Behind Act. Just as there are various learning styles, there are different ways in which we excel at communicating what we know, therefore measuring student knowledge solely based on standardized tests is not a fair nor complete means of assessing and measuring. Some students get test anxiety and the pressure to do well on a test is too much, others don’t think in terms of multiple choice questions, but give them a more open-ended activity and they will show their understanding.

We need a complete look at the students’ progress throughout the year, not just looking at the endpoint. We need to take into account where each student started the year and compare their ending point to this, that way we can get a true representation for their progress throughout the year. By broadening the ways we assess children we can get a more complete look at their knowledge.

While the majority of those who are in the field of education are against No Child Left Behind, there are some who support it. They make claims like “standardized tests are reliable and objective measures of student achievement.” I have to agree that yes, they are reliable in the sense that they are multiple choice tests being scored by a machine and there is no human subjectivity or bias. I also have to disagree because while the scoring of the tests may be reliable, the results and what they mean are not. I have also heard before that “teaching to the test can be a good thing because it focuses on essential content and skills, eliminates time-wasting activities that don’t produce learning gains, and motivates students to excel.”Some of the “time-wasting activities” that they are referring to may be how some students learn best. Not every child has the ability to, or learns the best by lecturing and listening. With all that teachers are expected to teach to their students and the time frame they are expected to do it in before the weeks of testing makes for impractical teaching and learning situations.

No Child Left Behind may have been implemented with good intentions but its result was not a good one. It is time to take a look at what our students need and how we can help them and their teachers. There is no question that changes need to be made, the sooner they happen the better.

President Obama Article

No Child Left Behind Article

Standardized Tests- Pro & Con Arguments

Coding in Class

Recently I was challenged to experience coding. My first thoughts were ones of terror as I thought of having to understand and use long, complicated, and confusing lines of code, thankfully what I played around with was the exact opposite. I started out with Hour of Code (which didn’t take an hour). It was an easy and fun introduction to coding and getting a feel for how different commands fit together. The goal of Hour of Code is to safely get your character through the different mazes by putting different commands together. I also spent some time doing Coding with Anna and Elsa which is more coding but it is a review of angles. Both of these were fun introductions, after the introduction phase of my coding experience was completed I used the website Scratch to create my own game Zork & The Rocket by putting the coding practice to work. This experience provoked the thoughts and conversation of should we be exposing our students to this in math class.

I’ll be honest, I have mixed feelings about incorporating coding into math classes. I think it would be a great thing to expose your students to, but I worry about how much time it would take up. It also depends a lot on your students, if they are up to the challenge and if they can appreciate what they’re doing. If I was teaching lower el I wouldn’t even have my students do the hour of code or the frozen coding based on their level of difficulty. In upper el I might consider having them do the hour of code or playing some of the scratch games. It’s my fear that if these were brought up with elementary students you would be running around answering questions and helping students and that the activity and the intended outcome would be wasted. It wouldn’t be until middle school that I would have them do the hour of code and then play some scratch games and look at the coding behind the game. If middle schoolers want to try and create their own game they should be supported, but based on my experience with creating my game, I would think of having high schoolers try it.

Unless there was a new standard on coding I believe taking time away from teaching what is required to teach would not be a good use of time. Teachers are already faced with the dilemma of too many standards to teach and not enough time to teach them in. Having the hour of code or Scratch games would be nice to have as an option for students to be aware of as an option to do during an indoor recess or free time in the computer lab.

I think this was an eyeopening experience and I think it’s something students should be exposed to, but I can’t justify taking time away from required topics to teach such a time intensive topic.