# Introducing Functions

## Article Review

• Why does the author argue for caution when teaching students about functions?
• How much time do you spend working with your students on writing and understanding functions?
• How much time do you spend practicing word problems?

## Discussion Topic

Pick three concepts or habits that students find difficult (for example, “I can’t get my students to check their work when solving word problems”). For each concept or habit, share the way you introduce the idea and talk about how students respond.

Post your thoughts to the forum below, and come back later to read other entries and comment on another teacher’s response.

This is a companion discussion topic for the original entry at http://studio.code.org/s/algebraPD/stage/3/puzzle/1

I can’t get my students to work well in partners and share the tasks. It’s usually one students trying to do all the work and everyone copies.
Also it’s hard to get students to go back and check their work.
And I also have a hard time getting students to remember formulas.

One of the biggest problems I see is my students don’t want to take the time to show work. It’s all about how fast they can get done with the work. With easier concepts they feel they don’t need to work the problems out and as we move into more difficult problems they try the same approach and it hurts many of them.
Along with this, many students become set in there ways and will have a certain way of solving problems in their head and it is tough to break them of it.
When working on group projects, the student in the group who is stronger in math tends to be the one who carries the majority of the load. This causes others in the group to miss key concepts through participating in the project.

In my computer science class my students are “forced” to check their work. If they hit the run button they know right away if they have made an error and they have to go back and make corrections. I guess the hardest part for them is dealing with the frustration of their program not working. However, when it does work they get very excited and have a great sense of satisfaction.

I can’t get my students to feel comfortable showing work in Math. If they ever write some calculations on the side, they always erase them before turning the assignment in.
I can’t get my students to check their work while test. They are asked to do so on classwork or homework, but it is not mandatory on test and quizzes. They often chose to skip this last step and end up not noticing their own mistakes.
It’s hard to get students to make real life connections even after I lay it out for them.

1. The obvious argument is how we confuse our students in our terminology and how this can lead to disconnect. For example, the word function can be used in a multitude of ways and be used as either a noun or a verb. This can obviously be very confusing to a lot of our students. Also the challenge of how can different mathematical terms build upon each other.
2. The next year school year will be the first time teaching Algebra in 6th grade so I don’t have a definite answer.
3. Once again I will teaching Algebra for the first time next school year.

I find my students are resistant to showing their work to all the problems. Nowadays, it’s all about how fast they can get it done. I have even taking points off for not showing work but most of them do not care and continue to work out the problems without work shown. It is so challenging to get them to do it. I try to incorporate word problems and real-life types of problems as much as possible as these are the ones that have more meaning to real-world types of problems. They need to know how to work them out and persevere through the problem to find a solution.

I am big on vocabulary. Traditionally I have had many ELL students in my Math classes, so the part in the article about how the word ‘function’ can be used in many ways (various nouns) was an eye opener for me. Vocabulary is key to introducing the core concepts.

He cautions the need to have students understand and relate what a function is rather than to only do a problem procedurally without comprehension.

Oooo oooohh!!! I have a new “thing” that may help.
Recently I found ( can’t remember where) a print out of “thinking bubbles” that simply said “jobs”… ex. one said Encourage Group, another said Ask A Question, another said Give Advice, etc… each student got one. My low language users did the “encourage group to keep trying” one. My “smartest” students may get “What if we tried this” card… I loved it; because I could do a participation grade on if everyone in group did their “job”.
Hope this helps give you an idea to try in your class. The kids seem to like having a job. And as they get better or more comfortable they can switch them.

I have been struggling with having my students write clear, complete, and accurate responses that include sufficient detail. Additionally I have had difficulty with having them write out all steps when solving their problems, even those that they think are obvious. Finally, I have had to constantly tell them make sure to read all parts of problems/tasks to make sure that they are answering not only all parts of the question, but even that they are answering the correct question.

I have a hard time getting my students to show all their work. Many of them want to take shortcuts and end up with an incorrect answer. I demonstrate how this should look as I explain and show how to solve the problems on the board.

I also have problems getting them to completely answer all questions. It seems like the majority is always looking for a quick and easy solution even though we discuss what it would take to completely answer the question as a whole group.

My biggest obstacle is in getting the students to actual finish a homework assignment.

I have the same problem. Most of my students are always looking for the quickest way to find an answer and by doing so, they end up with an incorrect answer. I deduct points for not showing work, but it doesn’t seem to matter.

Functions is the basis of Algebra. Fully understanding a function will help all students in understanding Algebra. Although I feel the definition of function is lost and forgotten in the daily or weekly conversations that contribute to my classroom.
I am always trying to focus on the correct vocabulary when discussing but after reading the article I am realizing that they may not fully understand the vocabulary they are using, (domain, range, independent variable etc.)
I try to use word problems in every lesson. I will admit that I do not necessarily have the students make the connection between vocabulary and word problem. That will definitely help their understanding of function and the vocabulary words.

I have a hard time getting students to show their work. They focus on getting the answer right and I want them to focus on the process. Although getting the answer right can re-confirm the process they will sometimes circumvent the process all together and try to guess the answer. My goal would be to instill in them an appreciation for hard work, not simply an appreciation for success.

My students have trouble with word problems, common core problems, multiple step problems, etc. They will get lost in the length of the problem and start to shut down. They feel overwhelmed and often do not know where to start. In the past I have used strategies like TTQA but those no lobger seem effective with the new common core questions. I have started to focus on showing the students how to break down the problems into steps and tackling it one step at a time. When I give the students basic problems they do great but as the problems become more wordy they struggle. I find my students have trouble understanding the concept of functions so I will try the strategy in the article that the pre-alegbra teacher did with focusing on vocab then moving on from there.

Like many others my students don’t like to show their work. I have them show their work in a various ways such as writing in sentences what their answer means or simply showing step by step what they did. Many of my students don’t like to do this; main reasons given to me are 1) I can do it better in my head, writing it down confuses me and 2) they don’t know exactly what to do so they don’t want to show me they don’t know.

Another topic that might seem super simple that my onlevel and special ed 7th graders have difficulty with are fractions. When it came time to figuring out unit rate or if two things were proportional they were bogged down thinking they were dealing with fractions. I saw it comes down to some not having a secure foundation with multiplication tables and others not having a firm number sense with fractions in general.

Lastly with word problems my students have difficulty picking out the important information or translating the words into a mathematical sentence to figure out what to do. We talk about the meanings of is and of and discuss terms for various operations.

This past year was my first year teaching math and I learned a ton! I’m hoping to get better ideas for introduction to help students connect these concepts rather than just memorizing what I say.

My students have the same problem. I’ve learned that most of mine are so used to just getting the correct multiple choice answer for a test that they don’t care about the process. My goal for my students is for them to be more independent learners and to value discussion/showing and critiquing work with their classmates.

General knowledge: In my accelerated algebra class, I am constantly looking for ways to make sure that students understand the why (conceptual) and not just the how (procedural). I believe that some of my students are just not ready for the level of math that the class involves and therefore the why is beyond them. For those that are ready as soon as they understand the why, the how is easy and they remember it because they know why it is the way to solve the problem.

Writing equations and solving them: Many times my students (especially when the task involves whole numbers or even integers) and solve word problems in their head and do not see the necessity in writing the equation, going through the mathematical steps on paper to solve and then checking their answers so I will give them problems that involve fractions or decimals that would be very difficult to do without writing down and then they realize that they have to go through the steps.

Exponential functions: Students have written, solved and used linear equations for a couple of years by the time they get to my class and are very familiar with them but when I introduce exponential functions they struggle to understand them. Because students are new to exponential functions as well as the fact that exponents in general are not in their core math understanding as are the four basic operations used in linear functions, students struggle to grasp exponential functions. By using real-world examples and activities, students can better relate to and understand exponential functions.

I believe that the author’s point of view is based on the fact there is interconnection between concepts in a given branch in Math. So, it is important for a learner to master the prerequisite concepts to move on to the next level with success. One week, including time for practicing word problems, would be required to students on writing and understanding functions.

There are students that are unable to distinguish dependent variables from independent variables. So, I usually present a related situation and ask to identify what could be considered as variables and which one that could affect the other one. Some students may respond very well while others may not.