How Should We Grade Students’ Math?

How should we evaluate and grade students’ solutions and answers to Math problems? 

a)    A correct final answer obtains a perfect score.
b)    Points for showing the solution, but more points for the correct final answer.
c)    Most points for the solution; minor deduction if the final answer is wrong.
d)    Most points for explanations and justifications for procedures (and its errors); major point deductions for inadequate reasoning, regardless if the procedure was perfectly executed and the calculations were correct.

Ideally, option (d) would be the best way to grade students’ maths. Students who can solve math problems and rationally explain how they come up with their solutions and answers, I think, are the best critical thinkers. On the other hand, students who can perfectly execute problem-solving procedures and calculations but cannot explain how they come up with such solutions and answers exhibits and practices rote-learning. Students who subscribe to this type of skills do not develop critical thinking and will fall short when variations in math problems were introduced and changes were made to the problems they are familiar with.

Based on my experience as a student, most of my teachers in math usually check my solutions and my answers. Some give more points to my solutions (c) while some give more points for the correct final answer (d). As much as enhancing critical thinking skills is concerned, I think option (a) is the least nurturing one because a student might just guess the answer without thinking critically, or worse, copy his/her answer from his/her classmates.

Math is an exact science. Thus, precision is of utmost importance. Correct solutions will most likely result to correct answers. Therefore, I think most points should be awarded for a solution, with a minor deduction for an incorrect final answer.

As the famous travel quote says:

“It’s not the destination… it’s the journey that matters.”

How about you? What are your thoughts on this?

Negative Reinforcement: Cancelling the Lowest

“Cancelling a quiz if students turn in all of their homework for the week increases the likelihood of the learner’s response.”

As mentioned in Ross Tan‘s  “Double Negative is Negative“, I agree that this practice will deprive students of the opportunity to gain knowledge and experience activities and tasks so they can perform well in school.

On the other hand, I think this negative reinforcement might also be effective if:

(1) the quiz is as challenging as the homework given. Most quizzes are usually given at the end of the week or at the end of a topic/lesson; thus, they summarize and assess students’ learning for that particular period.

(2) students show desirable performance in all of the homework given and not just “turn them all in”.

I think one of the best ways to modify this practice is by applying another type of negative reinforcement which was introduced to us during one of my geology classes in college. It is known to us as “cancelling the lowest”. We are given five (5) long exams throughout the semester. But instead of just averaging our scores in all of the five exams, the one with the lowest grade will be disregarded, averaging only the four remaining exams which will be part of our final grades.

I personally think that this type of reinforcement is effective as much as exam performance and motivation to study is concerned.  First, this reinforcement will not deprive and prevent me from experiencing the challenges of all five exams, as I am still required to take all of them. More importantly, this reinforcement will motivate me to study well, and get high scores in each exam, and relatively higher scores compared to the previous ones. If I fail or get a lower score in one of the exams, I will not lose hope and be motivated to do better on the next exam so this failed or low-scored exam will be disregarded as a reward.

I am a Math Teacher, Not a Math Expert

I used to think that to be able to teach Math effectively, I just have to be an “expert” in it. For almost a decade of teaching and coaching students, I spent most of my teaching life improving my “expertise” in Math. While I can say that I have encountered more than enough experience teaching students how to deal with and solve Math problems, I still cannot say that I am a Math expert.

What am I missing then? I spent so much time and too much effort devising and formulating strategies and techniques, honing my skills to be an expert in Math problem solving so I can help my students learn how to deal with and answer Math problems with ease, that I overlooked the art of teaching.  I forgot that teaching is more than just letting my students understand Math concepts and teaching them how to solve drills and word problems with ease using specific techniques and strategies that I prefer and find very effective. Rather, teaching is making ways for learning and providing guided opportunities for them to think, discover and solve problems on their own.

More than sharing knowledge or skills and focusing on preparing what to teach and what students must learn, it is important to come up with well-designed and well-thought teaching strategies to enable my students to actively participate and develop critical thinking skills. I have to know my students and identify what they already know prior to the new knowledge or skill they are learning. Focusing on what my students have and what they need to ignite learning will make them feel that my intention to teach them is personal. Showing and proving my students that I am an expert in Math will only intimidate and demotivate them to learn.

Expertise in a particular domain does not guarantee that one is good at helping others learn it. In fact, expertise can sometimes hurt teaching because many experts forget what is easy and what is difficult for students. (Bransford, Brown & Cocking, 2000, pp. 44-45)

“Expertise in an area does not guarantee that one can effectively teach others about that area. Expert teachers know the kinds of difficulties that students are likely to face, and they know how to tap into their students’ existing knowledge in order to make new information meaningful plus assess their students’ progress.”

(Bransford, Brown & Cocking, How People Learn: Brain, Mind, Experience and School, 2000, pp. 49-50)

I think I can dance (and more…)

Understanding Gardner’s Multiple Intelligences makes me realize that each individual excel in different fields. Most importantly, it makes me understand myself better. It gave me the opportunity to assess and evaluate my strengths and weaknesses, and think of ways to accept them and improve them if necessary.

Coming across Fritzie‘s So you think you can dance? entry made me realize that I have a different set of talents and skills. Except for being a good dancer, we are very different (and opposite, at some extent) in other skills and abilities she mentioned in her entry.

Everyone in the family is a born dancer, from my grandparents down to my youngest nephew. Simply put, it runs in the family. It is the talent that each family member enjoys as it becomes part of our quality time together. Personally, it is a passion.

I also love music and singing. It is one of the talents unique to me among my siblings. I, together with my parents, am part of the music ministry in our church. While I am not an expert, I can also play piano.

I am an introvert. I think that explains why most of the time, I enjoy alone moments, don’t talk too much and why I am frequently misunderstood as being shy and not a “people-person”.

I prefer interpreting graphs, maps, charts and illustrations over reading books, newspapers and other reading materials and any long prose. It would take me days, or even weeks, to finish our modules in this course. On the other hand, I can interpret illustrations and three-dimensional models with ease and enjoy designing them using scientific laws and theories. This might be the reason why I studied civil engineering for five years.

I consider reading novels and writing essays my weakest abilities. I am definitely not a wide reader. While most readers say that novels are more accurate and are always better than their movie adaptations, I would rather watch them in cinemas.

A lot of friends insist that I can write and encourage me to develop and improve my writing skills. However, as I always say, I would rather answer hundreds of Math exercises rather than write a one-page essay. I’ve been teaching Math and continuously learning and designing instructional materials for almost a decade now. Needless to say, Math is the subject I enjoy the most.

These differences don’t mean that I am better or more intelligent than other people.  Instead, this shows that each and every individual is unique. As teachers, educators and instructional designers, it is very important for us to identify and understand these differences among our students. This will help us come up with different approaches, techniques and strategies so we can teach and transfer learning effectively.

Introverted Learner and Teacher

Not just once did someone told me that I cannot become an effective teacher because I am introvert or if I stay being introvert. The problem is, introverts are always misunderstood.

So here are a few common misconceptions about Introverts (not taken directly from the book, but based on life experiences):

Myth #1 – Introverts don’t like to talk.

This is not true. Introverts just don’t talk unless they have something to say. They hate small talk. Get an introvert talking about something they are interested in, and they won’t shut up for days.

Myth #2 – Introverts are shy.

Shyness has nothing to do with being an Introvert. Introverts are not necessarily afraid of people. What they need is a reason to interact. They don’t interact for the sake of interacting. If you want to talk to an Introvert, just start talking. Don’t worry about being polite.

Myth #3 – Introverts are rude.

Introverts often don’t see a reason for beating around the bush with social pleasantries. They want everyone to just be real and honest. Unfortunately, this is not acceptable in most settings, so Introverts can feel a lot of pressure to fit in, which they find exhausting.

Myth #4 – Introverts don’t like people.

On the contrary, Introverts intensely value the few friends they have. They can count their close friends on one hand. If you are lucky enough for an introvert to consider you a friend, you probably have a loyal ally for life. Once you have earned their respect as being a person of substance, you’re in.

Myth #5 – Introverts don’t like to go out in public.

Nonsense. Introverts just don’t like to go out in public FOR AS LONG. They also like to avoid the complications that are involved in public activities. They take in data and experiences very quickly, and as a result, don’t need to be there for long to “get it.” They’re ready to go home, recharge, and process it all. In fact, recharging is absolutely crucial for Introverts.

Myth #6 – Introverts always want to be alone.

Introverts are perfectly comfortable with their own thoughts. They think a lot. They daydream. They like to have problems to work on, puzzles to solve. But they can also get incredibly lonely if they don’t have anyone to share their discoveries with. They crave an authentic and sincere connection with ONE PERSON at a time.

Myth #7 – Introverts are weird.

Introverts are often individualists. They don’t follow the crowd. They’d prefer to be valued for their novel ways of living. They think for themselves and because of that, they often challenge the norm. They don’t make most decisions based on what is popular or trendy.

Myth #8 – Introverts are aloof nerds.

Introverts are people who primarily look inward, paying close attention to their thoughts and emotions. It’s not that they are incapable of paying attention to what is going on around them; it’s just that their inner world is much more stimulating and rewarding to them.

Myth #9 – Introverts don’t know how to relax and have fun.

Introverts typically relax at home or in nature, not in busy public places. Introverts are not thrill seekers and adrenaline junkies. If there is too much talking and noise going on, they shut down. Their brains are too sensitive to the neurotransmitter called Dopamine. Introverts and Extroverts have different dominant neuro-pathways. Just look it up.

Myth #10 – Introverts can fix themselves and become Extroverts.

A world without Introverts would be a world with few scientists, musicians, artists, poets, filmmakers, doctors, mathematicians, writers, and philosophers. That being said, there are still plenty of techniques an Extrovert can learn in order to interact with Introverts. (Yes, I reversed these two terms on purpose to show you how biased our society is.) Introverts cannot “fix themselves” and deserve respect for their natural temperament and contributions to the human race. In fact, one study showed that the percentage of Introverts increases with IQ.

“Our culture made a virtue of living only as extroverts. We discouraged the inner journey, the quest for a center. So we lost our center and have to find it again.”

― Anaïs Nin

Source: 10 Myths About Introverts (by Carl King)

The Solitary Learner in Me

“What I find interesting in the description about solitary learners is that they “prefer to learn alone using self-study“. Ever since a kid I was not the type who would ask the teacher about different things. I would only do that if I did not understand something. I’m still like that today. My classmates might have wondered why I rarely approached our teacher to consult him/her…”

“The descriptors for this learning style are all “agree” in my case. It says:

1. Spends time on self-analysis – Agree!
2. Likes to spend time alone – Agree!
3. Feels that you know yourself – Agree!
4. Prefers to work on problems by retreating to somewhere quiet – Agree!
5. Like to make plans and set goals – Agree!”

(Excerpts from Winrich Beltran‘s Learning the Solitary Way)

While I couldn’t agree more with this, I still took this Learning Style Inventory, just to be sure.

Here’s the result: