# Guest Post: Math With Purpose

Chris Allen is a new addition to the AcerPlacer family. He is working towards a degree in engineering at Weber State University.

Often times the study of math is derided as a secondary subject, essentially a means to another subject such as physics or engineering. We sometimes see our math classes as merely an object to overcome to get to another goal. It’s somewhat rare for a student to stop and really consider, “Why even all the fuss about these?” In an age that we can simply take 15 seconds and look up a formula for anything we could ever want then magically we get an output that’s presumably correct, why bother with learning these archaic methods that can be thousands of years old (read: out of date)? Sure, the people who program the black boxes that feed us the right answers should/need to know this stuff, but I don’t.

The answer to that lies in part simply to build an intuition for when the black box might be feeding us a helping of bovine excrement. Take the story of an engineer who was running analysis on a trailer. This was a completely enclosed trailer and no part stuck outside of the trailer’s base. After putting a model of the trailer into the computer, it gave the center of mass about 3 feet outside of the trailer, which is physically impossible. However, this engineer trusted the computer all the way to the next project meeting, much to the engineer’s embarrassment.

Building this intuition for how the math outputs should look like is only part of what we learn as we learn different methods and approaches in math. The real gold in a strong mathematical education is not expecting 2+2=4, but by teaching us a diverse way of thinking about the world around us. How we can use a few basic rules and a bit of thinking out side of the box to solve nearly any puzzle. It’s a field that, despite appearances, creativity in mathematics is the most rewarded attribute.

We sometimes are told it’s important to learn math because it teaches us how to think, or some might say that that it teaches us different ways to think. But looking at the vast diversity in the mathematical fields of thought, I think the real reason why we should all study mathematics is that it teaches and reminds us that we can think.

# Guest Post: The Most Common Math Mistakes and How to Fix Them

Jodie Larsen has a BS in applied mathematics from BYU-Idaho with a minor in biology. In addition to teaching at AcerPlacer, she tutors students on several topics in math.

After many years of teaching / tutoring, I have helped and observed an enormous amount of students and have come to recognize the types of mistakes students most often make which has led me to realize what the most common ones are, and it’s not generally as simple as not knowing how to do a problem. Here are the most common mistakes I’ve observed and some suggested remedies.

## 1. Trying to memorize versus really understanding

I feel this is the most critical problem to correct in order to be accurate in all things math. Math isn’t a subject of memorization. It won’t be mastered simply by knowing verbiage and formulas. Many people have eidetic memories – a trait I always wish I possessed – however, that isn’t enough. Math is beautiful in so many ways, but in large part (in my opinion) because concepts can be combined in new, and endless, ways.

The key to fixing this problem is to ASK QUESTIONS. Students need to be voracious in wanting to know the WHYS in everything they are doing and in understanding the whys, students can then apply them in new situations with much more ease and accuracy. In order for this to be effective, though, the students need good teachers who are willing to teach them the reasons and meanings behind what is being done rather than just talking at them.

## 2. Not understanding the fundamentals before moving on

For the same reasons you shouldn’t build a house on sand, a person should make sure to learn the fundamental rules (learn and UNDERSTAND them) before trying to use said rules in more complicated applications. If students struggle with exponents, for example, but then try to do intense factoring, the factoring will be much more difficult. This mistake goes along with the mistake listed above because when learning the processes and rules which will continue to be used and applicable, a student must make sure they have a firm hold on the rules and will be able to apply them whenever needed.

The remedy to this problem is to be very vocal with your instructor / tutor and let them know that you’d like more practice and clarification until you feel you’ve mastered each rule or concept.

## 3. Poor handwriting and disorganization

This one seems perhaps a bit obvious, but it’s actually amazing how often this can throw people off and will cause major frustration when, for example, x’s look like y’s and 5’s look like s’s. Students will often say such things as, ‘I could get it right if I could read my own writing!’ and though they know it and can laugh about it, they don’t often work on it. This may be the hardest mistake to fix, but it just takes care and really slowing down.

Along this same topic, students sometimes aren’t properly taught how to organize their work, or they may just not take heed when they are. I often see cramming in tight spaces, overwriting to avoid rewriting, etc., and thus it becomes a sort of ‘Where’s Waldo?’ situation whenever I need to help students error-check a problem.

One of my biggest recommendations is to get a notebook of graph paper to do work in. There are built-in columns which make it very visually easy to place one number in each box and have everything nicely spaced out. As an added bonus, this can drastically help students who may suffer any form of dyscalculia – I witnessed this with a student years ago and it helped her (and her daughter) tremendously.

## 4. Using pen

I know, I know, some students are die-hard pen enthusiasts and though I understand the reasons why, it often really hinders them in math. If any mistakes are made, the student often tends to scribble them out or write over them instead of just doing the problem over.

Mistakes are made much more often with these types of actions and so I always recommend working in pencil. If students don’t like the scratching sound of a pencil, then I recommend a pen such as a FriXion pen which ‘erases’ with friction instead of an eraser (we all know how well those don’t work!) which can help avoid the mess.

## 5. Skipping steps (head math)

Though head math can be impressive, it’s also very error prone. I understand that students want to go as quickly as possible in order to get homework done sooner, but if mistakes are made, then the problem needs to be redone anyway, rendering the step-skipping rather useless.

I firmly believe that writing more steps out yields more correct answers and higher retention overall. A phrase I often use is, “When in doubt… write it out!” If you are working on a problem and you get it wrong, I will recommend you restart the problem and SHOW ME the steps while talking them out. Oftentimes, you will find your own mistake, and that’s empowering!

If I can summarize in one sentence, it would be this: Always ask WHY and practice extra material until concepts are mastered, use a pencil to write out ALL steps neatly, and graph paper is your friend. It’s as simple as that and will greatly increase your chance of success in math (and life) overall!

# Guest Post: My Path Through Math

There is an administrative assistant at AcerPlacer who loves sharing the story of her journey through math with students. Today, she was kind enough to write her experience so that everyone could see it.

If you think back to your very first day of college and the first classroom that you walked in to, how did you feel? Excited, overwhelmed,​ ​or amazed you made it to the right classroom? As you searched for what will be your unspoken assigned seat and looked around the classroom, what did you notice first? For me, I realized on the first day that there is nothing like the feeling of dread when you realize the class has 30+ students all in one space. How is the teacher going to help you if you have questions about the material, or just need some extra help?

While this may not have been your first college classroom experience, it was for me. I double-checked and even toured the campus before the semester started to make sure I knew where I was going (still got lost), I bought all my books early, and was ready to start my classes. I ended up in the engineering building and wandered into an advanced math class that made me run for the hills like the room was on fire. After what felt like an eternity, I eventually found my first class of the day — math. As if that wasn’t alarming enough, I walked in late and had to pick a random seat next to a stranger. At least my best friend was three seats down and looked just as panicked as I felt.

His expression and my feeling of alarm seemed justified. Just about everyone we knew had enrolled into a similar class or the class just one level higher. “Two-thirds of the students at community colleges, and 4 in 10 of those at four-year institutions take remedial courses. Math is a much bigger sand trap than English: Far more postsecondary students fall into remedial math than reading, and fewer move on to credit-bearing courses” (Gewertz, 2018).

Knowing that I wasn’t a mathematician, I thought to myself, “Here we go.” I was enrolled in the lowest level of math offered on campus. Could I do this? I could do this, right? As I sat down and unpacked my new school supplies, I looked around the room. I had an idea of what the college classrooms looked like, and it didn’t look like this room. What alarmed me the most about my new math class was that there were easily 35 students. All in the same room. With one teacher. In a math class.

Adelman (1999) states: “Of all pre-college curricula, the highest level of mathematics one studies in secondary school has the strongest continuing influence on bachelor’s degree completion. Finishing a course beyond the level of Algebra-2 (e.g. trigonometry or pre-calculus) more than doubles the odds that a student who enters postsecondary education will complete a bachelor’s degree” (p.vii).

Based on the study referenced above I was in big trouble. Math in high school was easy for me because I wasn’t required to take it during my junior or senior year. The last class that I was required to take was Algebra 2, and to be honest, I had no idea what was going on during the entire class. It was a miracle that I passed. Starting out in a new class, I felt that I could finish my math and avoid the so-called “sand trap”. I was going to do it. At least that was what I thought. I was in for a very rough semester.

Math was proving to be very difficult for me, and it was the class that I found myself dreading. In the upcoming semesters, I tried everything from traditional classes, computer-based classes, and even sought help from resources offered at my university. I just could not wrap my brain around this math thing. I had amazing professors who would help when they could, but I started to feel like a burden. I just could not understand what they were telling me or why we were moving “x” to the right side of the equation. What was factoring and why is this 3 all the sudden a negative number? I stopped asking questions. I came to the conclusion that I hated math. I hated all the rules, classes, material. All of it. It was the class I hated to attend, and even register for.

I learned that anytime I asked my family or friends for help that it only caused me more confusion and frustration. I found that not everyone who is wonderful at doing math can actually teach math. After a handful of math classes (13 to be exact), I found myself with a degree that was completed but out of reach because of my math requirements. How could I enjoy and pass higher level courses but not pass my math classes?! I felt defeated and hated to admit that math was again a class I had to repeat.

“Large numbers of students have been prevented from pursuing careers they’re interested in because of the math,” said Briars, a math consultant who was the president of the National Council of Teachers of Mathematics from 2014 to 2016. “They’re underprepared, but they’re put into the typical course sequence anyway. And we’ve done this at the expense of other mathematics, like quantitative literacy, or statistics, that is vitally important, and maybe more important for some careers” (Gewertz, 2018).

When you reflect on your previous classes, what made the class enjoyable? What made you successful in the class? Was there something in particular that stood out? For me, that answer is simple. I needed a small class that allowed me to ask questions and receive personalized help. I needed to be one of a handful of students, not one of 30+. I needed a class that had an uplifting, positive feel to it that encouraged mistakes and provided hands-on learning with an instructor who was invested not only in the topic, but also my success.

After what seemed like a never-ending nightmare of failed math classes, I had a degree that was one class away from being 100% completed and a job that only offered advancement if you possessed a degree. I had no idea what to do. I felt that I was out of options. I did the best thing I could have ever done for my math education. I discovered a new way of learning and really understanding math! So long, YouTube tutorials!

I was able to jump into a class that offered small classes, personalized help, out of class resources, and teachers who had the time to invest that had a real interest in my personal success and struggles. It is amazing how my view of math changed because I was finally able to get a grasp of what was actually happening. Why “x” moved to the right side of the equation, why that 3 becomes negative, and even how to read the trig wheel. Commonly I hear from students looking in to AcerPlacer, “Now I know that you work there, so you have to tell me that this program works, but will this program really help me test out and understand math?” I love that I can say, “Believe me, I know first hand that math can be a very difficult educational hurdle, but you are in the right place!”

AcerPlacer instructors have math-loving hearts of gold. They take the time and are truly invested in getting to know your learning style, your educational goals, and are always a great math support. They provide encouragement, comfort, math jokes, and bring not only their math experience, but also teaching methods that can unlock and help students grasp concepts. Each class is capped at 8 students per room so that it was easy to get the help I needed while in class. I could ask my instructor to repeat the material, say it a different way, and associate it with a story. The best part was that I never felt like a burden and I never felt out of place asking questions. I was completely comfortable admitting my wrong answers and thought process. For myself, it was the invested staff of instructors and the small personalized classes that helped me unlock so many math doors.

This program was the change that myself and many struggling students need! AcerPlacer was a game changer for me, and I love that I get to be a part of a team that helps students finish their college math requirements! So as the AcerPlacer t-shirts say… “Math is nothing to b² of”!

References:

• Gewertz, Catherine. (2018). Avoiding a Remedial-Math Roadblock to a Degree. Education Week, 37(32), 14–15.

# Guest Post: Valid Grading Measurements Continued

Today’s guest post comes from Madison Hodson, an instructor at AcerPlacer who studies mathematics/statistics education at Utah State University. This is a continuation of her previous article, which can be found here.

As described in my last article, I discussed the two principles necessary to produce a valid knowledge measurement (question on assignment or exam). This article will go into more detail and will provide examples for reference.

The validity of a measurement is broken up into two parts; one of those parts is measurement relevance. For a measurement to be relevant, it must reflect the unit goal or objective. That means for a question to be relevant it must pertain to the correct mathematical content and learning levels for each specific goal. If a measurement is relevant, then educated decisions and evaluations can take place based on the results.

In this example, let’s assume that the unit objective is, “When confronted with a real-life problem, the student determines whether or not computing the area of a surface will help solve the problem.” Listed below are three separate questions that could be included in the assignment. We need to determine which question is the most relevant.

Question 1.
Computing a surface area will help you solve one of the following three problems. Which one is it? (Circle the letter in front of your answer.)

1. We have a large bookcase we want to bring into our classroom. Our problem is to determine if the bookcase can fit through the doorway.
2. As part of a project to fix up our classroom, we want to put stripping along the crack where the walls meet the floor. Our problem is to decide how much stripping to buy.
3. As part of a project to fix up our classroom, we want to install new carpet on the floor. Our problem is to decide how much carpet to buy.

Question 2.
What is the surface area of one side of the sheet of paper from which you are now reading? Use your ruler and calculator to help answer the question. (Circle the letter in front of your answer.)

1. 93.5 square inches
2. 93.5 inches
3. 20.5 square inches
4. 20.5 inches
5. 41.0 square inches
6. 41.0 inches

Question 3.
As part of our project for fixing up the classroom, we need to buy some paint for the walls. The paint we want comes in two different size cans. A 5-liter can costs \$16.85, and a 2-liter can costs \$6.55. Which one of the following would help us decide which size can is the better buy? (Circle the letter in front of your answer.)

1. Compare 5 x \$16.85 to 2 x \$6.55
2. Compare \$16.85/5 to \$6.55/2
3. Compare \$16.85 – \$6.55 to 2/5

In the example given only the first question is relevant to the objective of, “When confronted with a real-life problem, the student determines whether or not computing the area of a surface will help solve the problem.” It is relevant because the students were faced with various real-life problems and had to determine when finding the surface area would actually be helpful. Based on the class’ answers to this question, the teacher could make an informed decision on whether or not the students achieved the objective.

The other two questions were focused on calculations that were not relevant to surface area. The results from these measurements would provide no feedback on whether or not the students had achieved the objective regarding when to calculate surface area.

As math instructors it is important for us to not only teach how compute numbers and algorithms but to teach logic and reasoning. Many individuals today struggle with the application of mathematics in real life. Perhaps had they experienced relevant math questions (to unit objectives and to life), these feelings would decrease. Creating mathematic contact that is both relevant and reliable takes a large time investment— however, as educators, it is worth our time to properly educate and evaluate our students.

# Guest Post: Ownership in Education

Kramer McCausland is an instructor at AcerPlacer. He is working on a double bachelors in mathematics and philosophy at Weber State University.

Student success is often on my mind. My hope is to find some simple equation that I can then offer up as the easy solution to the question of, “How can I learn this? How can I succeed in this class?” I often push myself to learn better techniques, to find better strategies, to practice being more clear. In short, I think that if I can be the perfect teacher, then every student I teach will succeed. Now, it is useful for us teachers to improve, but that’s only half of the puzzle, the other half is the student. I’m currently a student as well as a teacher and what I want to write about today is how to be a better student. What follows are the things I tell myself on the first day when I start a new class. And the tips and tricks I use to be more successful at learning.

The class I’m about to take is my class.

The knowledge I’m about to learn is my knowledge.

The responsibility to succeed is mine.

These are what I remind myself every time I take on the challenge to learn something new. The truth is that education is not passive. Too often, the student is portrayed as this empty bucket that the professor is there to pour knowledge in to. The sage on the stage. But, I’m afraid, that’s not how we learn. Education through osmosis is a nice fantasy, but the reality is different. It takes work, it takes patience, but most importantly, it takes ownership. It takes a firm conviction that this education is yours. The truth is that there are going to be bad teachers, there are going to be good teachers that have bad days, and there are going to be days where good students aren’t feeling like themselves. So how can we as students insulate ourselves against these misses in our education? By being truly responsible for what we’re learning. What I’ve compiled are a few tips for the proactive student.

1. Take good notes, review those notes, revise those notes: Make sure you’re jotting down the key points when learning something new. Then, check those points against a secondary source. In the day we live in, every part of education can have a corollary online (for instance, I just double-checked on google how to spell “corollary”). Very few people in this world can “learn” something after just hearing it once. Learn it in class, learn it again later, revise your notes on the subject as your learn more.
2. Communicate with your instructor: It may be surprising, but teachers are people too. They may gloss over, overlook, or entirely forget to mention something in class. If you’re unsure of what something means find an opportunity to meet with your instructor and talk about it. In the perfect world your teacher would clearly and concisely explain exactly what’s troubling you, but in this world it may take some leg-work on your part to get the best education.
3. Be patient with yourself: As cliched as it sounds, we all learn at our own pace. If I have one major qualm with the education system most of us find ourselves in, it’s the ideas of deadlines. I’ve met students who can understand everything they need to know about percentages in one hour of instruction, and others where the same material might take them 5 hours. Now, I’m not a total idealist here, and it is probably important that we learn how to learn quickly. But do your best not to become discouraged. Know yourself. Know how much you can retain in one sitting. And find steady study habits that work to your strengths (but that’s a topic for another day).

This education is yours. Take it seriously. Take ownership.

# Guest Post: Literacy in Mathematics

Our guest author today is Stacie Leavitt, an instructor at AcerPlacer who recently got a degree in math education from Weber State University. She will begin teaching for the Weber County School District this Fall.

A few years back there was a standard set in education that literacy should not be taught in our English classes alone, but that it should be taught in every single subject matter. Now for the History or Spanish teachers in a school, that may not feel like too high of a demand, but for math teachers it came as an abrupt surprise and a rather daunting task. “Now we’re not only teaching them math, but we have to teach them how to read too?! Those are almost entirely unrelated subjects!” As such, literacy is still highly overlooked when it comes to most math classrooms. However, if we take a deeper look at what literacy really is, maybe we can find more connection there than we thought.

Literacy in its most basic definition is the ability to read and write, but The National Literacy Trust includes [that], ‘A literate person is able to communicate effectively with others and to understand written information.’ So let’s dive a little bit deeper into these definitions. What exactly do we read and write? Our language is a mixture of symbols that when put in a certain order then mean a certain thing. We then need to be able to decode these symbols, use them to communicate, and be able to write about them. Similarly, much of math is being able to read the symbols to grasp their meaning, communicate about them, and then use those same symbols to write down your response. In fact the techniques used to decode and comprehend a paragraph are very similar to those used to decode and comprehend an equation. So how are literacy and math any different? They’re not really, it’s just teaching a new language within our own language. This is the idea that if we were to emphasize in our classrooms, we would not only be able to teach literacy but we would actually be able to teach math better and connect it more to skills that many of our students already have.

Now as a teacher myself, some of the main things that I have observed in different classrooms that separate literacy and mathematics are the absence of real world texts, few to no story problems, and the emphasis on the procedures instead of actual comprehension. In many of these classrooms, I can understand why a teacher would feel to build their curriculum this way due to the demographics of the school where maybe the majority don’t have high levels of literacy or math skills, are ESL learners, or their family situation can make it almost impossible to assign homework to take home. However this is exactly the classroom situation where there needs to be more focus on decoding and comprehension of text, especially the symbols and their meaning. Authentic math texts would be great for students to be exposed to in order to help them realize that math is more than just a process. It’s something people have wondered about, written about, built, discovered, and created. It’s both true and fallible and it’s ok to make mistakes in. Similarly real world story problems (not ones about buying 60 watermelons) can help them see how they can use these decoding and comprehension tools in a work or real world setting. But the biggest problem of them all is the the focus on the procedure. When this is overemphasized in a classroom it cuts off the need for students to become literate in math. There’s no need to decode the equations, comprehend what they’re meaning and what they do end up writing isn’t actually being understood. Their ability to tell you what they wrote and what it means is completely gone. Teachers wonder why, and honestly it’s because we don’t teach enough literacy in math.

# Guest Post: The Validity of Knowledge Measurements and Grading

Today’s guest post comes from Madison Hodson, an instructor at AcerPlacer who studies mathematics/statistics education at Utah State University. Here she exams what makes a “good” math question.

The validity of a knowledge measurement (a question on an assignment or exam) is broken up into two parts— relevance and reliability. For a question to be relevant, it must reflect the unit goal or objective and it must contain the correct mathematical content and learning levels. Reliable questions are those that when answered would give non-contradictory results.

If a measurement is relevant, then, once answered, proper evaluation can take place regarding the students knowledge and application of learned concepts. It is important for educators to clearly define their learning objectives ahead of time. This way, they can make sure their lessons cover all aspects of the objective. Having stated the objectives of each lesson also allows educators to draft relevant questions to homework and tests that cover and reinforce learned principles.

To aid in reliability, one of the most important qualities for each question is for it to be stated clearly. There must be no confusion as to what each question is asking and there are no ambiguous answers. This ensures that there are no discrepancies in the results from each question. Students either grasp the question and answer correctly or they don’t. The author personally thinks this aspect is important because she has felt confused by questions or answers on tests before.

Grading rubrics are imperative to ensure that scores are recorded based on fulfillment of the learning objective. For each question, there should be a rubric assigned that clearly designates the quantity of points that will be awarded for each answer. The rubric must be designed so that no matter who is scoring the question— there should be no controversy as to what answer(s) merit any specific amount of points. Having a rubric of this type will not allow any discrepancies between scores and will aid in the validity of each test score.

The purpose of knowledge measurements is stated in it’s name. Questions are posed to test the students knowledge and measure what they have learned and retained. By having clear objectives, relevant and reliable questions, and a precise grading rubric— educators, specially mathematics and statistics educators, are able to accurately determine their students knowledge, understanding, and application of concepts that have been taught. By taking time to generate relevant questions and watching for discrepancies within the students responses, educators will have valid results to base evaluations off of.

# Guest Post: F.A.T. City

Our guest author today is Stacie Leavitt, an instructor at AcerPlacer who recently got a degree in math education from Weber State University. She will begin teaching for the Weber County School District this Fall.

It’s a bit of a weird title, isn’t it? Let me explain. F.A.T. City is a workshop done by Richard Lavoie in order for adults to understand what having a learning disability feels like. F.A.T stands for Frustration, Anxiety, and Tension, three things that are extremely common among students with learning disabilities. Though we may think that in today’s world we are well aware of disabilities and that the fight has already been fought to gain rights for students with disabilities, we still sorely lack education for teachers in how to best help these students. This is especially important for the not-so-apparent learning disabilities. There are many students with dyslexia or processing deficits who either never get diagnosed or struggle getting the help they need. As teachers, this will be what we unknowingly need the most help on. In the words of Richard Lavoie himself, “I came to recognize — for the first time — the great irony of the teaching profession: Those of us who teach school usually did well in school ourselves and enjoyed the experience — why else would we return to the classroom to make our living? Therefore, the kid whom we can best understand — to whom we can relate most — is the one who does well in school and enjoys being there… Conversely, the kids whom we understand the least are the kids who need us the most. The struggler, the special-education student, the failure.”

Now as teachers in education, we need to look at what aspects of our behavior can be causing students with learning disabilities to experience frustration, anxiety, and tension and then avoid these as much as we can. It has been showed in many studies that if the negative affect is high in among learners then their ability to learn diminishes. So we as teachers need to make sure we are not instigating situations in our classroom that could cause unneeded anxiety or tension. Here are some common situations we create in our classrooms that we may not be aware of.

Other things to avoid are telling students to “try harder”. How does one just try harder? And teachers often state this as if students with learning disabilities are not already trying their hardest. We as teachers often see things as being easy because we’re coming from a higher vantage point of already knowing the answer. Take the picture below; do you see the image in the picture?

Why don’t you see it? Well look harder! Now, it doesn’t make much sense to tell you to look harder, does it? Rather you might need me to give some structure to help you see where the eyes, ears, and nose of the animal are in the photo and then you’ll be able to see that it’s a cow.

Similarly, it doesn’t help to tell students to try harder. Instead, if they have a learning disability and are not understanding we need to give more structure and guidelines to help them to see the overall picture we’re trying to show.
All in all, we must be careful to try and make our classrooms low in frustration, anxiety, and tension especially for students with learning disabilities. While some nerves can help heighten ability, too much can inhibit learning in many students. This responsibility lies mostly on us as teachers and we need to be educating ourselves in how we can best help students with learning disabilities in order to give them just as many opportunities to learn as everyone else.

# Guest Post: Reformers in Math Education

Today’s guest post comes from Madison Hodson, an instructor at AcerPlacer who studies mathematics/statistics education at Utah State University.

In this article I would like to highlight a few educational reformers. I would also like to discuss a brief history of education and the impact that it had on mathematics and mathematics education.

Johann Pestalozzi was an educational reformer of his time. He believed that all children deserved a fair chance to attend school and be educated. He tailored the curriculum to meet the needs of each of his students, especially the underprivileged and poor children. Johann’s influence was not limited to the 1700’s alone; today we have education and teaching criteria similar to Johann’s approach to education. Examples of Johann’s influence in today’s education include: an emphasis in children taking an active role in their learning, having a well-rounded education system, and a student-centered teaching approach. Johann also emphasized that school and home are both places for learning to take place and that parents and teachers must work together to make that happen. He also introduced the focus on not only what subject is being taught, but how that subject is being taught. These are just of few of the valuable contributions made by Johann that influence our educational system today.

Another reformer that I would like to mention is Franz Gall. Prior to Franz Gall’s work, the world believed that the intellect and all learning came through the heart and the soul. Franz Gall changed that belief to the knowledge that is still held today- that learning and intellect stems from the brain. With this newfound discovery, Gall divided and categorized the brain into separate sections that each serve a specific purpose. Gall’s findings are helpful in teaching mathematics because we can target specific concepts or learning ideas that engage various parts of the brain. We can also recognize that everyone’s brain works a little differently, meaning that individuals learn differently. As math teachers, there are many methods and examples that we can use to convey the same concept to our students. We have the resources and knowledge available to us to target specific teaching methods that will yield the best long term results.

As educators, we have the responsibility to not only help our students understand how to solve complex math problems, but also to teach them how these concepts can be useful and applied to real-life problems. Critical thinking and problem solving skills are indispensable assets that all individuals must learn in order to succeed. Math teachers are fortunate because we can combine our curriculum with applicable examples that will give students useful skills for life.