Using Calculators in Math Class...

I have recently started working as a tutor for math and science. Most of my math students are in grades 4 through 8. Although I haven’t been working with them for that long, I am already frustrated and appalled at their (in my opinion) over-reliance on calculators to do basic math operations. Although my boss told me not to worry too much about it, I felt determined to make them do mental math as much as possible.

As a science teacher, it frustrated me during my placements to see students miss “easy” questions in chemistry that required them to calculate the number of neutrons in an atom (atomic mass subtract the number of protons). I didn’t allow calculators during that test because I honestly did not foresee 15 year olds not being able to subtract 26 from 56.

If I had allowed them to use calculators, the benefit would have been for the students, mainly. They wouldn’t have missed a question because they made a math error. After all, I wasn’t testing their math, I was testing their knowledge of the composition of an atom. Unfortunately, not knowing how to do those basic operations cost them marks in science class, so I do see the benefit of using calculators.

However, I also felt that if I had allowed students to use calculators, I was providing a convenient way for them to cheat (they could easily hide a slip of paper under the calculator cover, or store words into its memory). More importantly, however, I felt that by allowing calculators for addition and subtraction questions, I would be undermining the importance of knowing those basic math skills.

Bethany Rittle-Johnson of Vanderbilt’s Peabody college of education and human development conducted a study published in the Journal of Experimental Child Psychology that looked deeper into how calculators affected grade three students’ math performance. Her team found that a student’s prior knowledge of math was the determining factor in whether or not a calculator made a difference in their education. Rittle-Johnson found that students who did not know their multiplication tables did not really benefit from the use of a calculator, but students who had a strong foundation benefitted immensely when given word problems to solve. "I think that the evidence suggests there are good uses of calculators, even in elementary school”, she stated.

Although this may be true, wouldn’t allowing students who do not know their multiplication tables to use calculators further make their math skills worse? What message does that send the student? “I know I told you to memorize the two times table, but I’ll let you use a calculator on the test anyway.” If a student knew they would be getting a calculator on the day of the test, why would they try to learn multiplication on their own?

What do you guys think?

Sources: http://news.vanderbilt.edu/2008/08/calculators-okay-in-math-class-if-students-know-the-facts-first-62879/

An Exceptionality to the Rule

Special education has become an integral part of the teaching profession.  What was once an area of teaching reserved only for the Special Education teacher is now a presence in almost every classroom.  Educational policy dictates that every school board in Ontario must “ensure that educational programs are designed to accommodate [special needs] and to facilitate the child’s development” (PPM no. 11).  Such programs are to consider the students’ individual strengths and needs as well as their individual instructional level.  Some teachers have the privilege of actually assisting in the development of their student’s IEP, and even have a chance to sit in during their IPRC meetings.  I believe that in doing this, these teachers not only have better understanding of the students they are working with, but they also now have a vested interest in the process since they assisted with its development.

Unfortunately, this is not always the case.  Some schools have a Special Education or Resource teacher to take on this role, and there are many teachers that have not received any training and do not know what to do in these situations.  The blame cannot be on the teacher.  Yet, it is imperative that teachers become familiar with their student’s IEP and what it means for them.  Many documents now explain this process and can assist teachers in developing these specific programs. 

In mathematics, this is especially important.  In many cases, students with exceptionalities will struggle with math.  Therefore, there is a greater need to understand student needs, and also to provide accommodations that will truly help students succeed. 

 

If you look at a standard Individual Education Plan, there are any number of accommodations you might find listed:

 

-          strategic seating

-          peer tutoring

-          alternative/quiet setting

-          chunking tests and assignment

-          extra time for processing

-          extra time for test and assignment completion

-          organizational coaching

-          access and use of assistive technology

 

There are many others as well.  Each of these types of accommodations are valuable for our students with exceptionalities.  However, when it comes to the one about assistive technology, this is where students can benefit the most.  Students can be assigned laptops, iPads, scanners, printers, almost every piece of educational technology you can think of, to the point where they need their own corner of the room to house all of this equipment.  When it comes to assistive technology, it can make a big difference how the students actually use it.  Sometimes this technology can be an iPad with hundreds of dollars worth of specialized mathematics apps or it can be something as simple as a calculator.  In some math classrooms, teachers want students to complete their work without calculators, in fear that they depend on them too much.  The reality is that for some of our students with exceptionalities, they do indeed depend on them.  They are a tool, and one that is needed.  We have due dates, seating plans and structured lessons, as well as general expectations for our classrooms, but in order to accommodate our students, math teachers sometimes need to make exceptions to the rules.

http://www.additudemag.com/adhd/article/7309.html

http://www.ncsall.net/index.html@id=325.html

https://www4.uwm.edu/Org/mmp/PPTs/Schefelker-Pruske-WMC08.ppt#257,2,Session Goals

Balancing Learning with Fun in Math

Balancing Learning with Fun in Math

In my other teachable, history, I find it much easier to find interesting and applicable activities to do at the senior level. I also found that many senior students like chemistry classes because they enjoy performing lab activities and experiments. However, when I was in high school, almost all of the students in the senior math classes with me were taking them because they were required for their science/engineering programs in university. This poses the difficult scenario of students taking a class because they have to complete it to get where they want to in university, rather than to learn about math.

I believe that I will find it very difficult to strike a proper balance between ensuring students understand the material and are prepared for higher learning and creating a class where students have fun, work in groups, and generally enjoy learning math.

I have looked at a few educators in an attempt to see how they make math more interesting for senior students. One of the top examples is Jonathan Winn, who teaches math in San Diego. However, he mentioned that he taught math to students who would otherwise not be taking math, instead of students who are taking math because they have to as a prerequisite. 

Math can be more more applicable, and hopefully enjoyable, in the college and workplace level courses because they are dealing with more concrete concepts and activities, such as calculating interest and realistic equations. If students see the validity in learning different math concepts, they will be more likely to participate in the learning process. 

University level math courses are a different topic though. Some of the concepts are difficult for students to understand and the courses are designed for entering university. For example, the Ontario curriculum states that "the Grade 12 university preparation course Advanced Functions and Introductory Calculus is designed for students intending to study university programs that will involve calculus." I figure that the best way for teachers to make this type of math more applicable for senior students is to show them a glimpse of how it is necessary for university level physics, chemistry, and biology programs. This may take a lot of effort, but I believe that if students can see where they will be needing their math skills in the future they will be more likely to appreciate and learn about it now.

If anyone has any tips for making math more life-applicable for senior students I would love to hear your suggestions.

Alexis Watson

Learning Through ... Videogames?

What if you could combine something that kids are obsessed with in their free time, with learning math?  This is exactly what Educade has done by providing in depth lesson plans for teachers which are focused on using Minecraft in order to teach Math.
Minecraft is a sandbox type game that allows the user to be creative and create structures in an online world. Unexpectedly, Minecraft became a cultural phenomenon after being released as a free to play game on PC.
Educade has created lessons for all grades, with areas such as arithmetic, geometry, probability, trigonometry, algebra, and calculus. It seems like an amazing way to make math fun and engage students as its a game that all ages have embraced. The one drawback though, is you can expect parents to raise an eyebrow when they see their child playing videogames and claiming they're just doing their homework!

Programming Mathematics

Currently in Canadian high schools, computer studies classes typically have low enrolment rates and no student is ever forced to take any courses. Conrad Wolfram is a British technologist who preaches a radical reform of how math is taught in schools in the Ted Talk below.

He talks about how school math doesn't look like real world math. This is prevalent when we hear our students say "when are we ever going to need this", and as adults who have been through the education system, we know they are probably right. They won't need the content, but it is more about acquiring skills like logical thinking. Wolfram argues there's other ways we can teach this, without having to teach something that is in a lot of ways useless when it comes to real life application.
He argues that we spend a lot of time in school teaching computation, but this is something that in the real world is all done by computers. His solution is to essentially teach students how to program a calculation, as this is more similar to what we see in the real world. What better way to thoroughly understand a calculation then to design a program to do it for us.
I really like the idea of this and find it interesting that in our education system as currently set up, there is very little emphasis on programming and I think its a tragedy (not just because my degree is in computer science). Students can't even choose to start learning programming until grade 10. In England, many schools are choosing to make computing mandatary for grade 6/7/8, and the government is really pushing the subject because it sees how valuable the skills are in any job, in any field. Also for some mathematics programs at the university level, students must take at least one course in computer science, so perhaps these two subjects can be tied together much more closely earlier in education.

If they were to make this radical change though, there would need to be some pretty huge changes to educate math teachers seeing as most wouldn't have the qualifications to be able to teach programming. The economic impact alone would be huge. Nevertheless, it shouldn't be an argument to not change just because that is the way its always been done. If you've read this far, check out the video below for a more in depth look by Conrad Wolfram.

Is "I'm just not good at math" a legitimate excuse?

As teachers, we will no doubt hear some students complain about not doing well in math class. They may blame us for not teaching them in a way that engages them. They may come up with excuses as to why they did not do well on a test (a party the night before, breaking up with their significant other, etc). Something that I often hear from my friends and classmates who often struggled with math is that it just simply isn't their strength and that only a certain kind of person could do well in math. But how true is this claim?

We are taught that as teachers, it’s our job to differentiate our lessons to suit all kinds of learners. Is there a type of student, however, that won’t excel at math no matter what we do simply because they can’t? Is it fair to evaluate all students using the same test when some have a natural advantage? Scientists are determined to find out and many feel this will change the face of education.

First off, it is a proven fact that we are born with a form of mathematical intelligence known as number sense. Number sense is an innate characteristic belonging to all creatures, not just humans. Without number sense, we wouldn’t know how to tell if there was enough room for all our family members in a minivan, or if there are enough seats at the table for all your dinner guests. It allows us to make sense of amounts and quantities in general terms.

Melissa Libertus, a post-doctoral fellow in the Department of Psychological and Brain Sciences at John Hopkins University went about measuring number sense in children who have not had any formal schooling yet. These kids were mainly preschoolers who didn't even know how to perform basic arithmetic. Their entire wealth of mathematical knowledge comes back to that intrinsic, animalistic quality of number sense. Well, that and whatever Dora the Explorer taught them about counting. What better place to begin to explore the possibility of a math “gene” or ability that nature supposedly endows to some and not others?

Two hundred four year olds participated in Libertus’ research. The children were given basic tasks to perform that involved counting, number recognition, etc. Children were also asked to estimate relative quantities. For example, various coloured dots would flash on a computer screen and each tyke was asked to name which colour appeared more often. This last skill was the best test of their inborn number sense.

In short, the researchers found that students who performed best at this estimation task also performed much better at the tasks that involved counting and other forms of “classic math”. When that natural mathematical tendency was strong, so was their performance in formal, structured math. Could this suggest that, yes, indeed, some students are just born destined to become better mathematicians than others?

"The relationship between 'number sense' and math ability is important and intriguing because we believe that 'number sense' is universal, whereas math ability has been thought to be highly dependent on culture and language and takes many years to learn," Libertus states. "Thus, a link between the two is surprising and raises many important questions and issues, including one of the most important ones, which is whether we can train a child's number sense with an eye to improving his future math ability."

In other words, while you can teach a student to memorize their multiplication tables and the quadratic formula, the skills required for mastery of the core, conceptual part of math (understanding concepts like less, or more, for instance), is inborn. However, this ability may mean more to students who don’t do as well in math.

A study published in the Journal of Experimental Child Psychology by Bonny and Laurenco indicates that this number sense ability and math competence had a stronger correlation when the students had lower scores in math. A lower number sense ability was more predictive of a low math test score than a high number sense ability could predict a high test score. This begs the question: what implication does this have for us as teachers?

Teachers can modify their teaching by focusing on core concepts and fundamentals and emphasize their importance. This points to the importance of elementary school math. While these students are still young, they should focus on strengthening their number sense skills so that in high school, they will be better prepared for more advanced math. Without a strong foundation, these students won't be able to thrive. 

Just because a student has a stronger innate ability that may suggest they are more mathematically inclined does not mean we should sit back and assume they are the future Einstein. They need guidance and coaxing as well. On the other hand, we should not be quick to dismiss a student as lazy simply because they did not fare well on a math test. Their intrinsic number sense abilities may not be strong as Little Johnny who gets straight A’s without studying. It’s kind of funny, in a way. Although there is documented scientific evidence to suggest that yes, some students just may not have what it takes to excel in math, that will not make a significant change in our roles as teachers.

On my first day of teacher’s college, I heard this wonderful analogy about a teacher and a gardener and it fits no situation more perfectly than this one. A teacher is a like a gardener. They are given a seed that is meant to be a tree. They don’t sit and glue together the sticks and branches to form that tree. Rather, they plant that seed, water it, give it whatever it needs to thrive, and stand back and watch.

That innate number sense ability is that seed.

Sources:
http://www.eurekalert.org/pub_releases/2011-08/jhu-ycc080811.php
http://www.sciencedirect.com/science/article/pii/S0022096512001798

The Importance of Understanding the Exponential Equation

Sometimes we, as teachers, teach concepts to our students that we understand are important and we understand would be quite beneficial for them to learn. We feel that, even if they do not use this particular skill in their future, they will at least understand the concept and have knowledge of how it applies to the world they live in. Essentially, this is what we strive for in developing life-long learners. But all too often, with a strict time schedule, we jump into skills and abilities without giving a non-generic real world connection, back ground or goal to the concepts and skills that are being developed.

Today, I would like to give an example of the importance of the exponential function and exponential growth. It is easy to explain half-life and depreciation through an exponential function. Albeit, far from the life experiences (and thus importance/interest) of our students. But why is exponential growth so important to us as individuals and as a species on this planet?

First off, as a teacher, listen to this short talk by Dr. Albert Bartlett from the University of Boulder Colorado. It will be your call whether or not to show this students and this will largely depend on the dynamics of your class. Dr. Bartlett is famous for stating that:

"The greatest shortcoming of the human race is our inability to understand the exponential function."

Now, for the classroom, here is a video from National Geographic that is an excellent visual representation of the current exponential growth that we as humans are in on Earth (Note: this is also good for Populations Dynamics in Grade 12):

The problem here is that exponential growth never ends and assumes (in the case of population) that resources are unlimited. This is not the case with our planet and we are currently beginning to face the problem of our inability to control (and understand) the exponential growth of humankind.

But here's the issue: this mis-understanding of limitless exponential growth does not only apply to population. Look at the growth of debt in the United States since 1940:

Finally, this series of graphs shows just how often this type of growth arises:

This is a dangerous trend which the vast majority of the population simply does not understand the basics behind and it arises in many important aspects of our species' existence on the planet. As we can see and much as Dr. Bartlett explains, it can be detrimental for us to not have any knowledge of exponential growth. One important area that exponential growth has arisen and that can be addressed by individuals on an individual basis is consumption.

Here's an example to run with students out of Washington that can easily be adapted for your area (for example, Lake St. Clair/Lake Erie or Lake Superior):

Resource Consumption Activity - the Problem with Exponential Growth 

The activity is a very clear example of water consumption that runs through three scenarios. The resource limit is the entirety of Lake Washington (770 billion gallons of water). In the first scenario, a gardener takes 1 gallon of water from the lake to water her plants each day. The water source will last 2.1 billion years. In the second scenario, she gains one client each day that requires an additional gallon of water each day (consumption grows linearly) it takes 3,400 years to deplete the resource. Finally, in scenario three, the gardener is a true business mogul and doubles the amount of clients each day (consumption grows exponentially). This time, the entire lake is drained within 40 days.

This is an excellent activity that can be done with your class to show the severity of exponential growth. As this concepts arises in all grade 11 and 12 courses, having students understand why it is so important by giving it some real-world background can be very beneficial to students developing their mathematical understanding and ability. 

Question Starters for Accountable Talk

Here are some questions that I was given by the math coach at our school. These questions are design to promote accountable talk in the our Math Classrooms.

How have you shown your thinking? (picture, model, number, sentence)	Which way (picture, model, number, sentence) best shows what you know?
How have you used math words to describe your experiences?	How did you show it?
How would you explain ________ to a student in Grade ___?	What mathematics were you investigating?
What questions arose as you worked?	What were you thinking when you made decisions or selected strategies to solve the problem?
What changes did you make to solve the problem?	What was the most challenging part of the task? And why?
How do you know?	How does knowing __________ help you to answer the question _______________?
What does this make you think of?	What other math can you connect with this?
When do you see this math at home? At school? In other places?	Where do you see _______ at school? At home? Outside?
How is this like something you have done before?	What else would you like to find out about _____?
How do you feel about math?	What does this math remind you of?
How did you solve the problem?	What did you do?
What strategy did you use?	What math words did you use or learn?
What were the steps involved?	What did you learn today?
What would happen if…?	What decisions can you make from the pattern that you discovered?
How else might you have solved the problem	Will it be the same if we use different numbers?
Prove that there is only one possible answer to this problem!	Convince me!
Tell me what is the same? Different?	How do you know?

How Can We Reach Out to Kinesthetic Learners

One of the main issues that I see in math classes and am extremely guilty of myself, is only tailoring math lessons to two of the three types of learners. Almost every math lesson that I have seen, or taught, is geared towards audio and visual learners. We are missing out on those students who learn kinesthetically, and in my experience these students make up the majority of the class. I feel as if we get caught up in doing the same old thing and teaching the same way as what we are used to because it is easy. We cannot allow ourselves to slip into this cycle, and deprive our students of a more effective way of teaching a lesson, that with a little imagination and creativity we would surely be able to come up with. However, I am  not speaking for all teachers. I know that some teachers are excellent at differentiated instruction and I hope that others will be inspired by their effort to engage their students.

 I have seen many hands on assignments used in math classes that are engaging for students, although I do feel that assignments such as this are under utilized. During my time in high school I can only remember doing one hands on activity in one of the five math classes that I took while attending. Not only was the assignment more enjoyable for me than doing seat work day after day but I fell like it was the most effective way for me to learn the material. 

It is difficult for me as a tutor and supply teacher to take advantage of differentiated instruction because of time constraints but I do try to use it whenever I can. I can relate to the kinesthetic learners because I was once in the same spot as them and I am motivated to teach lessons in a way that will grab every students' attention. As the push for differentiated instruction increases I hope that more and more teachers will become motivated as well.