Good Morning Blog

Good Morning. This is the second post in a series about topics that have traditionally not been handled honestly in Mathematics classroom. The first post was about adding and subtracting fractions .  In this post I would like to address two aspects of square roots that are often treated with less precision than they ought to be; square roots of negative numbers and simplifying square roots. To better understand both of these issues, it is important to have a clear understanding of what is meant when we say that a is the square root of b . We mean that a times a is equal to b . More precisely, . Square Roots of Negative Numbers When students first learn about square roots, they generally have only worked with the set of Real numbers; natural numbers, whole numbers, integers, rational numbers, and irrational numbers. The collection of all of these numbers is called the Real numbers. But "real" in this sense has nothing to do with these numbers existing and other numbers not ...

Good Morning! When I ask my students how they feel about fractions, I don't really need to listen to what they say. I just need to pay attention to their faces. They're generally not fans. When I ask them why they don't like fractions, the responses are not very diverse; they hated adding and subtracting fractions and they hated having to find the lowest common denominator. And I get it. If the fractions have a common denominator, you can just add the numerators. But if they do not have a common denominator, you have to find one first and then rewrite the original fractions as new, equivalent fractions with the new, common denominator. Perhaps you may remember your teacher saying something along the lines of, "You need to find the lowest common denominator first. If you can't find it you can multiply the two denominators." We will come back to that statement later. First, I want to discuss the topics of Greatest Common Factor (GCF) and Lowest Common Multiple (...

Over the past 5 years I have taken a total of 10 (ten) online graduate courses through Purdue University Northwest, Indiana Wesleyan, and Indiana University all in an effort to be meet the Higher Learning Commission's criteria to teach dual credit courses. The course design has been similar for each. Students login to some learning management system, get the assignments, work on them, submit them, etc. There is one element of course design, however, that has been the most beneficial for both myself and the other students in the class; online discussion. Not every course I took had online discussions, but the ones that did were the most engaging. Inserted from GIPHY The way the online discussions would work is each week the students and the professor would engage in online discussion about the assignments for that week. The questions would be posted, someone would reply with an idea of how to start, others would offer criticism or support, or continue with the next step. Everyone wo...

Every parallelogram is a trapezoid. Remember in school when your teacher told you that every square is a rectangle but not every rectangle is a square? Do you? DO YOU? Because it's literally the same thing. Every parallelogram is a trapezoid but not every trapezoid is a parallelogram. For some reason this is a debate that periodically rages in many Mathematics teacher circles. Why? Because textbooks disagree on the definition of a trapezoid. The definition of a parallelogram is basically standard. A parallelogram is a quadrilateral with two pairs of parallel sides. The definition of a trapezoid, however, is not as standard. Some textbooks define a trapezoid as a quadrilateral with at least one pair of parallel sides. While other textbooks define a trapezoid as a quadrilateral with exactly one pair of parallel sides.  It is this second definition that poses the problem for parallelograms. If a trapezoid is defined to have exactly one pair of parallel sides, then a p...

So here we are. Well, here I am. In this period of social distancing, unless you are my wife or one of my children, we are probably not in the same place. On Friday, March 13, 2020, the school I teach at closed due to the COVID 19 pandemic. The closure was originally supposed to last until April, 10. Since then, the governor has closed schools for the duration of the school year. Most, if not all schools nationwide have also closed. To continue educating students, schools are transitioning to online learning, or eLearning. This has put schools, teachers, students, and parents in a new a stressful situation. Schools are empty. Teachers teach from home using some form of a blended, online learning model. Parents, even those deemed essential workers, are left to be supervisors for their kids. Teachers who are also parents have to do both. In a regular classroom environment, everyone is working on roughly the same things at the same time. The teacher is only steps away. When there is...

One of the "revelations" I had over the past few years that has helped me change my instruction was that having someone else waste your time sucks. I enjoy having my time wasted as much as the next person. Which is to say, I do not enjoy having my time wasted at all. It is one of the things that irritates me the most. When I realized that students often feel that way in school, I knew I had to be different. I didn't want to lecture any more than necessary. And when I did, I wanted to err on the side of talking too little. If a student can learn something without me, I want them to do so. If I could assign 15 problems instead of 30, I'd assign 15. I was reminded again recently of what it feels like to have your time wasted by someone else. I am in a graduate class and one of the requirements for the class is that we post solutions to homework problems on a discussion board and others respond to them. You can't see the solutions others post until after you pos...

There's a saying that good teachers meet their students halfway. I'm not sure that is true. Don't get me wrong. I think it is critical for teachers to meet their students academically...somewhere. I just don't believe the meeting point needs to be defined with that much specificity. If I have a student who is capable of going 80% of the way toward learning a concept, meeting that student halfway is a disservice to the student. I should only meet them at the 80-20 mark. If I have a student who is only capable of going 25% of the way toward learning that same concept, meeting that student halfway isn't sufficient. I should meet that student at the 25-75 mark. My responsibility to each student is different for the exact same concept. So why should it look the same for each student in my classroom? It shouldn't. Our classrooms are filled with students who come in with varying ability levels, various levels of self-worth, various levels of self-doubt, and...