The Bigger Picture

In chapter 5 of Multiple Pathways to the Student Brain, Dr. Zadina discusses various ways that teachers can help students move information from working memory into long-term memory. She states that sometimes the brain doesn't need to move information into long-term memory, but school is a case where we do need to move the information. I love the idea of providing opportunities for students to look at the bigger picture. So often, my students only want to look at one topic at a time and they don't want to think about how it relates to what they have learned before or what they might learn next. I think that they find it overwhelming to look at so many details at one time. Week 2 of Dr. Zadina's workbook also offers suggestions on ways that we can help students see the bigger picture. One method could be helping the students relate the material to real life as much as possible. We are currently reviewing for a unit test on finding the solutions of quadratic functions. This was the perfect opportunity for me to use these methods to help the students see the bigger picture.

First, I helped my students make a list of the different major topics that led into solving quadratic functions. They discussed adding, subtracting, and multiplying polynomials, learning how to factor, learning how to graph quadratic functions, and interpreting the key parts of the graph of a quadratic function. We talked about how each one of these skills was absolutely necessary in order to find the solutions of a quadratic function. We also discussed how the solutions of a quadratic function could be found using one of four methods, each method would result in the same solutions. We talked about what the solutions looked like on a graph, and what it meant to have zero or only one solution. The students were amazed at how connected all of the material was and how much they were using the concepts they had previously learned. I think that this definitely helped the students to see the bigger picture.

In addition to discussing how the concepts fit together, we solved quadratic equations involving real world situations. For example, we solved problems about throwing an object or dropping an object from a building. The questions asked students how long it would take for the object to hit the ground. The students had to interpret the question and understand that hitting the ground meant the height would be zero. They were free to choose a method to find the solutions. Finally, they had to interpret each solution. I urged them to go back and read the problem each time so that they could make sure they were fully answering the question. If you were asked how long it takes to reach the ground, the positive solution would be the correct answer and it needs units. I think that this really helped the students see the real world application of finding the solutions of quadratic functions. It gave them the opportunity to see that there was a real application of this concept other than in the math classroom!

Critical Thinking

In week 2 of Dr. Zadina's workbook, she discusses the need to incorporate opportunities for students, especially older students, to develop the frontal cortex. She suggests giving students opportunities to analyze, synthesize, and evaluate information in order to aid in the development of the frontal cortex. Last week in my algebra class, we learned how to solve quadratic functions using the quadratic formula. It is very easy to make minor mistakes when using the quadratic formula, so I wanted my students to be careful when substituting values in for the corresponding variables. I created a "find the mistake" task in which I stated that two students each found the solutions of the same quadratic function. I wrote down their work, but the work had errors which made the solutions incorrect. It was up to the students to find and explain the errors, determine if one or both of the solutions were incorrect, and show how to find the correct solutions. After the students completed the "find the mistake" task, I asked them to switch with a classmate in order to evaluate someone else's work. We discussed the answers and I asked for verbal justification of the errors that they found. We then anonymously discussed some of the explanations that students wrote down. I wanted my students to discuss what answers seemed to go above and beyond what was asked, and which ones failed to answer the questions.

The activity went really well. The students appreciated receiving feedback from their classmates and they told me that it would make them work harder so that they could receive all good feedback next time. They also wanted their work to be used as a positive example to the class. This was definitely an activity that I would incorporate again. I think it gave my students the opportunity to engage in various levels of critical thinking and it also reinforced the need to provide justification for solutions.

"Wire What You've Fired!"

In week one of Dr. Zadina's workbook, she discusses the need to use repetition in the classroom. She explains that repetition is necessary in order to "wire what you've fired" (Zadina, 2008). She goes on to briefly discuss possible teaching techniques when asking students to memorize events or processes. Read it, write it, say it... practice the stages over and over so that they become wired in your brain. I definitely think that repetition and practice are necessary in order to wire what you've fired, but I think that this is a very tricky subject. When I teach my math students new content, it is 100% necessary that we practice and discuss misconceptions and make mistakes in order to form a deeper understanding of the material. But the point of practicing is not so that they can memorize a process, it is so they can experience problems and use logical reasoning to figure out the answers based on what they have learned. For example, if I am teaching my students how to solve multi-step equations and I use the following example: 3x - 4 = 8. According to the properties of equality, you add 4 to both sides, 3x = 12, and then divide both sides by 3, x = 4. A student who is memorizing information would think that, for every multi-step equation, you add a number to both sides and then divide each side by a number. When the student realizes this is not the case, memorization has become more cumbersome than trying to deeply understand why we are performing the steps.

Teachers are being urged to stop asking students to memorize information, instead the students should have such a deep understanding of the material that memorizing becomes unnecessary. Along these lines of memorization, I have heard so many arguments about the memorization of multiplication facts. When I was in elementary school, I had to do mad minutes in order to memorize the multiplication tables. In my opinion, this is a set of information worth memorizing. CCSS-M are encouraging teachers to help students develop a more conceptual understanding of multiplication. I think this is great; I think understanding why multiplication works the way it does is something that has been overlooked in elementary school for many years. However, I think this is a situation in which conceptual understanding does not and cannot take the place of memorizing multiplication facts. I have high school students who can't figure out the greatest common factor of 24 and 32 because they don't know their multiplication facts. It is such a hindrance in Algebra I that I have been tempted to do mad minutes with my students as their daily warm-up. Fluency is out of the window if the student can't multiply. Some argue that students shouldn't have to memorize these facts because they will always have a calculator available to them. I think that the calculator is an amazing tool and we should absolutely incorporate it into the math classroom, but when my students have to use the calculator to perform basic multiplication facts it becomes a crutch rather than a tool.

Here is a cool article about multiplication fact memorization:
Mastery of Multiplication

Emotions and Learning

After reading Chapter 3 of Zadina's book this week, I started thinking about how seriously negative emotions can affect the learning process. When I think of the times that I have been incredibly stressed out or anxious about a situation, all I can remember is constantly thinking of those negative feelings. They took over my mind and my thoughts, I couldn't concentrate on anything but trying to change the situation so I could feel less anxious or stressed. So when students feel anxious, stressed, threatened, or scared, I can totally understand why they can't focus their attention on learning. As stated by Zadina, "The effect we most often think about in regard to learning is that strong negative emotions impair thinking and, therefore, learning in general" (2014). Ultimately, students who are experiencing strong negative emotions are not engaging in their full learning potential. Most of their thought processes are being devoted to those negative emotions.

The Science of Positive Thinking
I found this really interesting article written by James Clear that discusses the benefits of positive emotions and positive thinking. Barbara Fredrickson, a leading researcher in the field of positive thinking, believes that positive emotions "broaden your sense of possibilities and open your mind, which in turn allows you to build new skills and resources that can provide value in other areas of your life" ( Clear, 2013). Unlike the narrow thought process that takes over your brain when you experience strong negative emotions, positive thoughts can help your brain see more possibilities than normal. This can ultimately lead to building new skills for success.

I think that creating an atmosphere in which my students feel safe is a top priority. I don't want my students to feel overly stressed or anxious in my classroom. I want everyone to be able to openly discuss ideas and possibilities without feeling intimidated and unsafe. I think that creating a safe space in which my students can share their thoughts will ultimately broaden their understanding and strengthen the connections that they form with the material. 

Mind Maps

At first glance, I wasn't a huge fan of mind maps. I like to organize my thoughts in a very logical list fashion, so I was immediately freaked out that a mind map should be a freely written somewhat hodge podge of thoughts. Secondly, I initially brushed off the idea of mind maps because I didn't think that it would be a helpful tool for my math students. I was so turned off by mind maps that I decided I must not be understanding the full idea or all of the ways they could be used. I found a few websites that made the idea of mind maps much more appealing to me:

Balancing your Mind Map
The first link shows how computer software can be used to make a more organized version of a mind map. I think that it would be helpful for me to jot down a mind map and then use some sort of software to organize the map on the computer.

Students Guide to Mind Mapping
The second link explains various ways that mind maps can be useful for students. I found the note-taking section interesting. As mentioned before, I tend to write notes and thoughts in a list fashion. However, as pointed out in the Mind Meister Blog, how often do teachers lecture in a perfectly logical and sequential fashion? Most times, we go back and add thoughts and bits of information along the way. Mind maps can easily allow students to add in those important bits of information in the correct places. Mind maps can also be a great tool to create a to do list.

Mind Maps - Why Have I Never Heard of These?
I still wasn't sold on the idea of using mind maps in math. The final link gave me some great ideas as to how I could incorporate these mind maps into my classroom. Mind maps can be used in two major ways in the math classroom. First, the maps could be used to organize broad ideas and explore relationships among various topics. For example, as mentioned in the blog, mind maps could be used to compare and contrast things such as ratios, fractions, and percents. In addition, mind maps could be used to solve a numerical problem, or organize the steps necessary to solve the problem. Mind maps could be a great tool for students to explore new material and begin forming those connections necessary to learn the material.

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