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: The Fall Factor

Ryan Brown is an instructor at AcerPlacer and is working on a BS in mathematics with a minor in secondary education. He is also an avid rock climber.

Mathematics can be used for an infinite amount of reasons. I specifically am going to discuss how math can be used to find the safety of a fall for a rock climber. There is a simple equation that one can solve to find out the danger in each possible fall. The equation is: (Fall Factor) = (Height of the fall before the climbers rope begins to stretch)/( Length available to absorb the energy of the fall). It can also be written as F= H/L. A fall factor of 2 is the greatest that one could have while lead climbing; this would mean that the climber would fall past the belayer or hit the ground. The smaller the fall factor, the softer the fall for the climber. Knowing this equation can help climbers place gear safely so they can have the safest and most optimal climbing experience.

Guest Post: Math Stigma and a Study

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

“When will I ever use use this?” A question we, as mathematics instructors and connoisseurs, hear on a near-daily basis. You may have heard your past math teachers say such things as, “You’ll use this every day!” or simply look at you like they just cannot comprehend how anyone would NOT use math every day of their lives.

The truth is, math is all around us and we all use it, to varying degrees, more often than we may even consciously realize. I know people such as welders, electricians, and musicians who upon first consideration may not seem to need math, but it is shocking (especially in the case of the electrician) how often they utilize different types of math in their careers. Even if someone isn’t in any sort of math-related field, they will inevitably end up using such math as taxes, tips, and discounts. That’s not to say the average person will likely use properties of logarithms and population models on a daily basis, but it is quite helpful to know where those types of things are applicable. In my studies of Applied Mathematics and Biology, I was constantly surprised and pleased about just how many correlations could be made. Following, you will find one of my favorite such studies which I was a part of.

For this particular study, we headed out to a region which grew sagebrush and various other scrubby types of bushes and weeds. We started at a certain point and would measure out pre-decided lengths, say 10 feet, in random directions (if it is actually possible to be random which many studies claim isn’t – but that’s an entirely different story). At each 10 foot length, we would measure and take note of different diagnostics of the area such as the height and density (which we estimated visually) of the surrounding plant cover. The idea is that we could then create models for whatever type of animal (be it rabbits, voles, prairie dogs, etc.) and state how much cover said animal would have when needing to hide from predators. We created models which related the height of the animal to how protected that animal would be in that particular region. We also analyzed how visible those animals would be through certain vegetative densities, which of course would also show how visible they would be to predators. Animals which were too tall or too large wouldn’t have as high of a population density in that region due to the inability to hide properly. There are, of course, many variables and things to consider but having a lot of data allowed us to analyze a plethora of correlations and hypotheses.

As stated in the opening paragraph, one misconception about math is that “I’ll never use this.” I, along with my colleagues, aim to break this stigma by finding many practical applications and perhaps, just maybe, finding a niche in which each student can and will apply this beautiful thing we call math (or ‘maths’ in some parts of the world) and think of us whenever they calculate and realize…. we were right…. and that they’re actually enjoying it.


Guest Post: An Origin of Numbers

Drew Peterson recently earned a BS in physics from Weber State University. He is currently an instructor at AcerPlacer.

Start counting: 1…2…3…4…5… hopefully you can take it from there. Did you start from one like me? Maybe zero? Why didn’t we start at negative one, or even negative one thousand? It’s estimated that humans have been writing down numbers for at least the past 40,000 years (Ifrah, 2000). It’s really impossible for us to grasp how old this really us, but it really leads me to ask: Why have we been counting and writing numbers down for so long?

Let’s think back to the ancient world – I’m trading in a market place, and I need to know how many bushels of wheat to buy. I know how much I need to make a loaf of bread, it’s… that much. Visually, then, I can determine this from experience. Then the person I’m selling to needs to figure out how to charge me. She can see how many bushels I’ve taken but needs more rigidity — she needs to count how much I’ve taken. So the seller counts, maybe with her fingers, certainly not using the number system we think of now (one, two, three). Of course I don’t have zero bushels, in fact in that time I would ask how you could even see or think of zero of anything – let alone the wheat.

So by necessity we count, and by lack of necessity we didn’t need zero. Think of how bizarre an experience it would be to attempt an explanation of negative numbers to an ancient person, who only counts the things in front of them. I find it hard to put those shoes on, so I’ll produce an analogy: Imagine you’re building a shed, and you need to figure out how wide to make it. Your neighbor, who’s helping you, thinks he knows how long the shed should be. “Negative 10 feet!” he says. Of course, you stare in confusion as this answer makes no sense. How could you possibly have negative length?

Eventually, with the rise of currency, humans gained the need to measure nothing and negative of something, specifically when dealing with loans or any sort of deficits. For example, I’m back in my ancient trading market buying seeds to plant for my farm. I really need some seeds today to plant them in time. Unfortunately, I don’t have enough money, but the seller is kind enough to give them to me anyway. Now I owe her some money; I have a debt that needs to be paid. This is the idea of debt that we are commonly used to today, although our ancient people may not have thought of those as negative numbers.


Ifrah, G. (2000). The universal history of numbers: From prehistory to the invention of the computer (D. Bellos, EF Harding, S. Wood & I. Monk, Trans.).

Guest Post: Math Anxiety

Kramer McCausland is an instructor at AcerPlacer. He is working on a double bachelors in math and philosophy.

Math anxiety is wildly prevalent. Official studies vary a lot in their reporting of math anxiety, but in my personal work with students, I’ve found as much as 50% report some degree of math anxiety. Everything from mild dread when faced with a math problem to near terror at the sight of numbers. The art and practice of manipulating numbers is often portrayed as dull work, but for those with math anxiety, it can feel like an adrenaline-pumping fight for their lives. OK, maybe I’m exaggerating a bit, but I do have a lot of respect for students who, even though math is a source of major anxiety, choose to fight that good fight every day. I’ve compiled a list of some of the best tips I’ve found for combating math anxiety:

1) Stay organized, stay calm

Doing a complicated math problem can sometimes feel like heading down a rabbit hole. It can be full of twists and turns and dead-ends, and the sheer complexity of looking at that on a piece of paper can worry us. It is hugely important, as you head into these more complicated math problems, that you develop a step-by-step way of writing out your work. If you can’t look back at what you’ve written down and describe what you’re doing between each line of work, then you’re not being organized enough. Don’t be afraid to use scratch paper if you need it. Number your steps so you can see them clearly. Use colored pens or pencils to tell the difference between each new step. Clarity is your friend.

2) Keep your notes handy

First, take good notes. Well-written and organized notes are your greatest homework ally. As you work through your math homework, think of your notes as a leg up. Keep them nearby, but not visible. You want to challenge yourself to complete the hard work of mathematics without looking at your notes, but never feel ashamed if you have to pull them out every few problems (or even on every problem). As we learn math (especially if we’re preparing for a no-notes test), we need to train our brain to find the right answer without help, but it will help us avoid anxiety if we know we have somewhere to turn to when we’re stuck. If you get anxious about taking notes in class, ask your instructor if they have a printed copy of the notes. This way, you can dedicate yourself fully to paying attention instead of worrying about keeping up with what the instructor writes on the board.

3) Know your limit

In a perfect world, we would study math (and all of the beautiful things) simply because we enjoy it. But, for many of us, we’re studying math as a requirement for a high school or college level class. That means that we probably have a time constraint and will feel pressured to work ourselves into exhaustion. Now, a little bit of pressure is good, it will keep you motivated and focused, but know that you are going to have an upper limit. At some point, continuing to study will not yield additional knowledge. Do some amount of math every day as you prepare, but don’t burn yourself out by doing an outrageous cram session that you can’t remember the next day.

4) Test-taking: A breathing technique

A big part of math anxiety boils down to math test anxiety. There is no denying that taking a big math test can be scary, but keeping in mind a few test-taking tips can help us relax. Other than being prepared (which I hope you are), you should also learn a couple breathing and relaxation techniques to avoid psyching yourself out. To help you relax, close your eyes and try breathing in through your nose for 5 counts, holding for 2 counts, breathing out through your nose for 5 counts, holding for 2 counts, and repeat. As you breathe, just take a moment to notice where in your body you feel the breath moving in and out of your body. Maybe you feel the air traveling past your nostrils, expanding/contracting your chest, or as an up and down motion in your abdomen. The goal here is to just give your mind and body a break right at the beginning or in the middle of the test if you start to feel out of sorts. If you run into a hard problem, try to do 3 or 4 cycles of this breathing technique and then come back to it.

5) Test-taking: Relaxation technique

If the breathing trick isn’t your cup of tea, you can also find some peace of mind by trying some visualization. This takes some practice, and it may feel silly the first time you do it, so you should practice this a bit at home before test day. The goal is to be able to travel in your mind to your own personal “happy place”. I use this all the time as a mini-vacation I give myself during difficult tests or while doing my homework to relax. Start by closing your eyes and picturing a place, either real or imagined, that is calming to you. For me, I close my eyes and see the shores of a lake I used to go to as a kid. The details don’t have to be perfect, but throw in some small details to draw you in. For me, I’m sitting on a canvas chair, looking out at the blue water. I’m alone, and I have a book in my hand. But I’m not reading, I’m just looking out on the water and feeling the sand under my feet. Your happy place might be leagues different than mine, or might be pretty similar. Give this a try and see if it helps you keep a cool head when the going gets tough.

“Almost everything in life will work again if you unplug it for a few minutes, including you.”

— Anne Lamott

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.

For the sake of time teachers will often demand answers quickly, and since they may not be giving sufficient time to students, this can cause them to freeze or become anxious. In essence, the student becomes a deer in the headlights and we as teachers may assume a number of things from them being stupid to thinking that they just haven’t been listening. Now teachers may not always be conscious that they’re doing this. Oftentimes teachers get caught up in the need to cover all the material, they feel rushed, and then they consequently rush the students which really just makes things worse overall. Another tactic that teachers use both consciously and subconsciously is using sarcasm to demean the student when they don’t answer correctly or quick enough. Imagine this, say student Jimmy has dyslexia and it takes him twice as long as the other students to figure out an answer. You’ve asked the class what 2+2 is, they’ve answered and then you ask Jimmy what 2+3 is and he says 4. You tartly reply, “why yes Jimmy, 2+3 is 4”, the class laughs and you move on not thinking much of it. But what does it feel like to be Jimmy? He had just barely figured out your first question and bravely put forward his answer and in return he got laughed at. Now rarely do teachers intend to tear students down, but they can do it unintentionally if they aren’t careful. Haven’t we all let out a quick retort without thinking? As such, sarcasm should never be used to negate wrong answers given by a student who took a risk by answering.
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?

FAT City 1

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.

FAT City 2

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.