Some students think they are not very good at math, and that can be a challenging mindset. By setting the stage that everyone can be successful, and giving lots of examples students can relate to is a big help.
Caution needs to be taken because students don’t always easily see relationships. Functions deal with relationships.
I do not spend any time working with students on writing and understanding functions. After reading this article, maybe I should. In the coding that we do in our computer science class, relationships between objects, concept, etc. are important. Studying relationships (functions) might increase the student’s understanding of the overall concepts.
Word problems in this sense are never practiced. We do build all of our programs based on scenarios (stories, screenplays, etc.)
In my computer class I collaborate with the math, science, and LA teachers. I have noticed that students want to get through problems as fast as possible. When working through word problems with them I often have to read it with them and draw or write out the equation because they rush through and miss vital information. Getting them to take their time to analyze problems is my biggest challenge.
What little computer programming I have them work on in class, makes them think about the functions they need to complete the program. Again I face those students that don’t want to analyze the information if they have made a mistake. They would rather have someone look at it for them and find the problem.
Why does the author argue for caution when teaching students about functions?
When introducing functions to Algebra 1 students, I think we do too much, too fast. As an example, my textbook puts all of this in one lesson and my District curriculum allows for 1-2 class periods to teach this concept. (I find the allotment of time to lessons is about the same throughout the Algebra 1 course, regardless of the importance of the concepts in the lesson.) If this concept is important, shouldn’t we allow more time for student learning to occur?
I also think we focus too much on the minutia associated with this topic and lose site of the big idea. (for example: The Vertical Line Test is nice, but is that really what we want kids to take away from the lesson?)
How much time do you spend working with your students on writing and understanding functions?
As long as it takes. William Schmidt (University of Michigan) once described the secondary mathematics curriculum as “a mile wide and an inch deep”. This might explain why students don’t develop mastery in mathematics. Recent revisions to mathematics curriculum, including the Common Core Standards, haven’t changed this picture. I try to keep the focus of every lesson I teach on the “BIG IDEA” that underlies the lesson. Finally, sometimes I will skim over less important topics to make more time for the important ideas.
How much time do you spend practicing word problems?
*I try to avoid textbook “word problems”. They are shallow, contrived problems that have little to do with real mathematics. Instead of using those, I will present students with a contextual situation, ask them to learn something about it, start looking at numerical quantities associated with the situation and then see what math they can apply to the situation.
My students struggle with asking questions when they do not understand. They lack the communication skills to express their thinking. I usually present new concepts with plenty of wait time to allow students to ask questions, but find that certain students still do not feel comfortable. Also, my students struggle with showing their work and discussing their work.
Like everyone here, I too am having trouble getting kids to write out their work. Our solution, as so many teachers before us, was “No work shown. No credit!” They eventually see that writing out the work helps them stay on track and not miss steps. What Ponce made me think about was what their lack of writing out the solution meant. Is it simple laziness? I’m sure all of us know some students for whom this is the case. But in others I see that they get part of the concept, just enough to come up with an answer–sometimes right, sometimes wrong-- but they are afraid to write it all out because that lack of understanding will be exposed. Think of those kids who can find the value of y given x but can’t find x given y. They don’t fully get the idea of the idea of function as a relationship in both directions. A majority of our students are, or once were, ELL and I think focusing on the language of functions would be extremely helpful, if only to eliminate the tiniest bit of uncertainty that adds to their fear of functions that clouds their thinking.
My biggest issue is getting my students to see the importance of mathematics in their lives. Because of this they don’t want to put the effort needed to meet the new demands of the Common Core. They don’t want to learn the how or why mathematics work. They just want a calculator to get an answer.
Teaching math in general is hard for students because of the vocabulary. Usually I have a memory game the day or two after I have taught a new concept to make sure that the vocabulary we are talking about it is getting into their brains. When I first teach a concept I pay close attention to the way they use prior knowledge to make those connections. Then I use those prior knowledge connections when we do the memory game. That way it is related to them and things they know.
Many times the biggest trouble for my students is creating their own function from given inputs and outputs. I liked the one example about letting the students work through them and then having students go to the front and present their ideas to the class to talk about if they work or not. One caution I have is to make sure that I chose the students (who are wrong) the ones that can handle seeing their mistakes and not get offended or embarrassed.
My students have traditionally struggled identifying relationships of mathematical nature. A significant portion of their mathematical experiences have been, and continue to be, focused on answer-getting and rule following. Knowing that there are relationships with, through, and around numbers is a foreign idea. Indeed most of my students are very adept at categorizing things and terrible at seeing and identifying relationships. In general students have tended to resist the energy it takes to find a relationship.
A second struggle for students is identifying purpose of the mathematics, relevance. Junior high students have little capacity to even entertain a thought not directly impacting them in the moment, so to sustain brain power examining a situation that has nothing at all to do with them is largely an exercise in futility.
Finally, sharing individual thoughts and ideas with the class (and even in small groups occasionally) can be a daunting task. We try to establish ground rules and operating norms in class, but for some breaking down that barrier set in place after years of struggle in mathematics is a challenge.
The students find it difficult to correctly present their work. They can give the answer, but they find it hard to follow the correct steps to represent how they got the answer. Others find it difficult to understand word problems, so modeling and introducing vocabulary words are very important for them to be successful in the task. Also, visual representation/concept map will facilitate understanding of concepts. Next, students find it difficult to identify the patterns or relationship in a given word problem. Checking for understanding is very important, give students enough time to practice how to identify the core idea of the lesson, or scaffold the lesson to help the students understand. Most students don’t want to read so that is another challenge.
A few things I struggle with are getting students to check their work before they turn in an assignment or quiz, keeping students on task during cooperative work, and getting students to use appropriate math vocabulary when they share out. Regardless of how many conversations I have with students about simple mistakes, particularly arithmetic mistakes, there is still a contingent that I just can’t get to see the benefits of quickly checking answers. I have used Class Dojo to help track on task behavior during collaborative work, but would real like to not have to use an external monitoring tool. Students do a great job sharing their math thinking during math talks or during share outs after collaborative work, but I cringe when I hear them not using the simplest of math vocabulary that I know they know and understand. There seems to be no amount of modeling or begging I can do to avoid this–I have even tried having them make edits to a video of their explanation.
The first unit taught in Pre-Algebra in my district is functions. The way the unit was written was to begin by revisiting the previous year’s unit on proportional equations graphically with the use of ratio tables. The first struggle I noticed my students had was being taught the procedure of ratio tables, but no conceptually understanding; in other words, students had procedural understanding - use a ratio table to show if one cookie cost 3 dollars, how much 5 cookies cost - but they would increment by ones, rather than multiplying by five because they had no conceptual understanding of what a ratio table was, they simply used it as a T-Table.
The second struggle comes from the over-usage of ratio tables in 7th grade, where all linear equations were proportional, then moving into 8th grade where there is a y-intercept introduced. Students can identify patterns - 5, 10, 15, 20 - as multiples of 5’s, even when written in a T-Table of values, but when there is an initial or starting amount which causes the outputs to change to 3, 8, 13, 18 for example, students have lost their ability to find the pattern, even though the outputs change by 5’s still. Only a few students when given easy input/output relationships (times 2 plus 1, times 3 minus 1, etc.) have the basic math facts to help them realize the pattern is the multiples of 2/3 minus/plus one.
The third struggle I see in students is the difference between when something has a multiplicative relationship versus an additive relationship. For example, when writing an equation for a function, students don’t know whether the relationship between the input and output is additive or multiplicative nor whether the rate of change is additive or multiplicative. Because of this misunderstanding students struggle with slope as they can’t remember if it is input over output or output over input - and for that matter WHY is it output over input - because for all they know the relationship is additive and not multiplicative.
I see the same struggles with my students regularly. I do weekly assignments with my students where I recycle concepts that have been taught throughout the year, but I still have the same struggle, that after the “two” week time period they forget whatever was previously taught. Or even math concepts they “know” - all my students can tell me PEMDAS - yet only a few can actually use it correctly.
One of my goals is to encourage students to discuss mathematical topics and theories, and through discussion to actively participate in healthy mathematical debates. I welcome students to take risks and push their thinking to higher levels, ask questions, argue solutions and processes, offer constructive feedback, and critique peers. Through such collaboration and involvement, I believe students not only explore larger perimeters of knowledge and information, but also learn from each other, and apply the newly acquired knowledge.
One concept my students struggle with is vocabulary. When introducing new vocabulary, I try to break down what the word parts me or ways to help easily decode the words. I also introduce concepts first and then give them the vocabulary word so it is not completely foreign.
A second concept my students struggle with is appropriate “help” to another student. When starting the year off, I discuss with the class as a group the expectations for group and partner work. If I see work just being copied, I first go over and discuss with the students how I help them vs how they are “helping” and compare And contrast the ideas. They seem to catch on after that.
A third concept students coming to me struggle with is the importance of homework completion. This year, I tried to defeat this by giving homework quizzes every other week on specific homework problems. If a student was attentive while going over the homework, they were easily able to complete the quiz.
I have some students that struggle with this as well. Have you tried sitting with them one on one and asking them to show you exactly what they are doing to solve the problem?
In my computer class, students struggle using functions. As many times as I tell them that they must start off with input that gives them a known output to check for logic errors, they try to run with the unknown. They then turn in the wrong answer.
I have a hard time getting students to show their work on problems and to check their work as well.They want to find the easy way to an answer and once that answer is obtained they most often move on to the next problem rather than taking the time to check their work and see if their answer makes sense. Another problem I have is withe students trying to let one person in a group do all them work and then everyone else in the group just copying the answer.
After they have solved a problem, few of my students slow down to
ask “Does my answer make sense?” As I comment on their solutions, I
try to (gently) point out unreasonable responses. I have also found
that focusing on the units involved in the problem helps to focus
the students’ efforts. There are students who grasp the idea, and I
notice that their future responses seem to be more in line with the
question being asked.
As the article stated, a common problem I face is helping students
understand functions and, more specifically, function notation. I
can completely relate to these students, since as a teenager, I
struggled with this same concept! I love the idea of talking about
non-mathematical contexts of the word function! I may use that in
my lessons in the future! Currently, I use the idea of a function
being a machine. I name the machine using the letter of the
function, and talk about f(x) meaning, “the output of machine f when
x is the input.”
I teach online classes, and the biggest challenge I face is getting
them to read the feedback that I leave on their assignments! In the
first email I send out, I always include instructions on how to see
the feedback. I rarely see students responding to the questions I
pose in the feedback. When students are one quarter of the way
through the course, they have a required discussion with me. I
always take that opportunity to guide them through the steps to
viewing the feedback. It is always discouraging to realize that
I’ve been spending so much time typing them detailed and personal
messages about their assignments, and they skipped over the email
about seeing feedback. Following the discussion, I have seen a much greater response to my feedback.
knwhite46, I have found similar issues with encouraging group work. We’ve all been in the group where one person completes everything. Very little learning is happening! I love your idea of comparing how you work with students. Being the exemplar is certainly a best-teaching method!