Note: I am currently unable to access my lap top/ ipad or wifi. I have attempted to complete the readings and blog post on my phone using my data plan. It hasn’t been the easiest thing to accomplish. When life returns to normal I will fix up the errors and properly cite material. Sorry for the inconvenience.

Catherine

This unit on Conceptual Challenges really synthesized for me my role not only as a teacher but also as a student. Until recently I believe I have been the antithesis of Piaget’s statement “Not how fast but how far”. Educationally I believe I have always been on the fast track, not in terms of being educationally superior but rather looking at a volume of work I needed to conquer and setting about conquering it, not learning it. Homework and assignments were a check list of activities that I tackled and prided myself if I got through it (honestly never considering if I understood it or could explain it but rather could I do what was asked as a robot would).

Reflecting over the past few days on Heathers experience and the other students, as well as, Harvard Grads and Faculty, I realized that early on in my education I was considered ahead of my peers. By the time I was in grade 5 I was two years younger than my peers. It was around this time also I began to feel like a fraud. My confidence slipped and I would say I became a very average student until grade 11. In grade 11 something finally clicked. Did my brain catch up with the material? Did I become more confident? Did I just learn how to play the system and know what I needed to do to get good grades?

From grade 11 through until about eight years ago, I kept on that track I learned how to do what needed to be done to get a job done “well”. At least in the eyes of others. I taught curriculum, got through units, students produced work that they could be proud of. But what were they really learning? Had I really not just taught them how to play the system the same way I had.

I probably would have continued right on that path if a slap in the face moment had not occurred. I had to face my misconceptions head on. LIke Heather’s teacher in the video I believed that students arrived at my door with the background knowledge to proceed from where their last curriculum left off. Never once did I question the teaching that was going on in those rooms, rather if the students arrived not knowing something they just were not good at it.

I would review if needed (lecture style), and dispense new information (lecture style and perhaps with a model I demonstrated with) and often found myself thinking, that went really well, these kids have to understand this I did a great job. What a fantastic sage on the stage I was (notice I did not say teacher).

The slap in the face moment came when the first week of classes with grade sevens they were struggling with the most basic of concepts. Frustrated and decidedly sarcastically, at the time, I reverted to a primary teacher reviewing math concepts. It was then I became dumbfounded. None of the students had any understanding of WHY they did things in math. They perhaps knew the how’s of computation but application and understanding were sorely lacking. Later that day, to make myself feel better I walked in to the class that my students from the previous year were in. I asked the same question, and I got the same dumbfounding answers. They had no clue. HOw could this be?

If this was true in Mathematics it had to be true in other subjects as well. I sat down that night deciding how to map out my future as a teacher. My plan was to change my lessons from students listening to a chalk and talk to me listening to what they knew, talking to them about why they did something and trying to get them to apply that knowledge to new situations in whatever way possible.

Goal setting became important. As Blanchett (1977) stated “a good experiential situation must permit the child to establish plans to reach a distant goal,while leaving him wide freedom to follow his own route (p 37).” This led to my understanding of how I was rushing through the curriculum to check off units I had completed. I needed to slow down. In 1987, Duckworth stated that “learners need time to explore phenomenon (Chapter 6).”

Exploration became a large part of my classroom time. Allowing students to manipulate and create their own models. Give them an opportunity to try ideas and learn from the results.

In the Confrey (1990) article there is a very poignant section on arithmetic that discusses the difference between rote learning and meaningful learning. The following portion stood out for me “We label students as wrong, but do not delve into the preconceptions that may have led to this”.

Fosnot’s (2013) book delves deeply into how children can benefit from constructing their knowledge. Taking what they know (or think they know) and expanding on that. Allowing this will help them see if what they previously believed was true or if they had a misconception. Without the opportunity construct their knowledge students may never understand how to move forward and deepen their knowledge base.

After reading the articles and watching the video I began to wonder how, in mathematics specifically I could improve my own understanding of what my students knew and what misconceptions they may have.

I found a very helpful article by An and Wu (2012) entitled: Enhancing Mathematics Teachers Knowledge of Student Thinking from Assessing and Analyzing Misconceptions in HOmework.

First of all I have not been a big fan of homework for about the past 8 years, as well. During my epiphany, mentioned above, I realized that homework seemed to be busy work. Also that I assigned “busy” homework and did not really use the results to any end, other than marking it as done or not done. An and Wu (2012) bring up this point as well. Their research focuses on how we can use the grading of homework as a way to understand what our students know and what misconceptions they may have. If we assign fewer, more meaningful questions and take the time to evaluate that work we will have a much better picture of that students knowledge. We will be able to identify misconceptions and have the opportunity to allow the student time (with teacher direction and assistance to understand and correct these misconceptions).

This leads directly to my thoughts about how technology can help in this area. I envision my students choosing three of their “assigned work questions” one from each of the three sections to complete “on line”. Students could access a variety of programs that would enable them to show and talk about how they solved the problem. Why they did, what they did, why it made sense to them, as well as, if they believe they have solved the problem properly. This would allow the teacher to not only see the work the student has done, but also allows them to hear the rationale. Having this valueable information to refer back to would not only aide in understanding the students misconceptions but also be an excellent marker to refer back to once the student has progressed past this problem.

References: (Not in proper citation format to be fixed later)

Confrey, et al. Article from class notes list 1990

Fosnot, C. Chapter 1 and 2 from class notes list, 2013

An,S. and Wu, Z. Enhancing Mathematics Teacher’s Knowledge of Student Thinking from Assessing and Analyzing Misconceptions in HOmework. International Journal of Science and Math Education (2012) 10: 717

Hi Catherine,

I, too, had a similar epiphany a few years ago while teaching grade 7. I assumed that they came to me knowing and understanding most of the basic concepts required for them to continue in grade 7. This was in all subjects, but particularly math and science. It took me awhile to figure out that, not only did they not have some of the basic knowledge required for more rigorous math and problem solving, they also had no idea why they did some of the things they did in math. I took a huge step back a couple of years ago and decided to focus on the why as well as the how. My principal at the time told us that we should concentrate on a mile deep, and not worry about the mile wide, in other words, don’t worry about covering all the curriculum expectations, but make sure that the basics they need are solid. This has been my philosophy ever since and find my students are much more motivated in math class and have a better understanding of how and why they do things.

Anne

Hi Catherine,

Thanks for your post, especially regarding aspects of homework and the benefits (or lack thereof) of assigning it. It’s something I definitely continue to struggle with in my practice as I feel homework is expected by students and parents. I question their role in actually help students learn and what benefits they can derive from them. Without question, I have made an deliberate process of reducing the quantity of homework with hopes of focusing on the important questions or key conceptual material. As an alternative to homework, I have tried to utilize other forms of assessment in the classroom such as concept questions or white boards. Through these methods, I feel that we have better collaboration and communication in the classroom to fully delve into topics. This is perhaps an area where digital technology can assist with an increasing number of outlets that allow for student assessment and learning (like Kahoot, Quizlet, Socrative).

Thanks for posting!

Your discussion point about understanding the “why” behind math struck me. I (now in my practice) always discuss with the students why we are learning a certain topic/idea. For example, we are currently studying the concept of time in my grade 2 class and we had a brainstorming session about why we need to know how to tell time and in what instances we would need to know the time. In this way, students can make real life connections between their own lives and what we are learning and hopefully become more engaged because they can see a real life application to their learning.

Catherine, your post raises a number of points. I like the autobiographical standpoint from which these points are raised. In reference to the article, “We label students as wrong, but do not delve into the preconceptions that may have led to this.” Assessment, done meaningfully, can sometimes reveal misconceptions. How was Heather’s alternative conception or preconception revealed in the video? For the class and Catherine, is there a math question that could be developed or you have tried that has revealed an alternative conceptions? Thank you for your review of the article, Samia

HI Samia,

I love the questions you posed because it really hits at the heart of all assessment. While I do believe a teacher can pick up blatant misconceptions in paper and pencil tasks (for example I had a student once who, when learning long division always mixed up the number placement. When looking at his written work it was not difficult to see the misconception he had) but correcting this misconception was not as simple as a paper and pencil response of “the numbers go the other way around”. Fixing this misconception required discussion, demonstration and an understanding of why the numbers in those places did not make sense.

For less blatant misconceptions, like Heather’s understanding of the seasons in the video the teacher thought Heather, and likely others understood the concept. It was only by the interviewers talking to Heather and having her explain her understanding that the misconceptions became apparent. As Heather continued to talk to the interviewer you could tell that she became less certain of her correctness.

If the interviewer had never verbally questioned Heather and had her demonstrate her knowledge and/or understanding it is likely she would have carried on with the same misconceptions. For an assessment to be truly valuable I believe there needs to be a verbal component. Having students “talk through” their understanding, and providing feedback as to why they believe this and finally demonstrating with proof and hopefully an opportunity to construct knowledge on their own through hands on activities or simulations would be, in my opinion, the most effective method of accurately assessing understanding. The teacher in the video made a similar comparison when Heather picked up manipulatives to help her explain her point.

The problem with the type of assessment I am advocating is TIME! Where does the teacher find the time to verbally assess or interview each student? Are we not likely to assume the students who are doing well on homework assignments, seat work, and projects understand the concept and look to assist the student who is struggling? This is where I see blended learning classrooms being beneficial. Teachers can work with small groups and discuss understanding, while other groups of students work on other modules.

Catherine