# Difficult Concepts

My students have found the following concepts/habits difficult:
–Modeling/showing their thinking consistently.
–Explain your work or explain the pattern.
–Discussing their strategies.

–Modeling/showing their thinking consistently.
In order to prevent “answer-getting,” I encourage students to always “show their work” and add a component that makes their thinking transparent. When I grade work, I always attribute points to the transparent thinking whether or not it leads to the right answer, because at least I can see and respond to what they tried. I also encourage students to cross out rather than erase things that don’t work so that I and they can see what they tried and address misconceptions. I also don’t introduce calculators until the first 2 chapters are complete, so students know they have to use a number line, an addition or multiplication table, a graph, etc. to justify their answer. Students also need to be regularly introduced to different ways to model the algebra as well as be mindful that they should model/show their thinking whether or not is is explicitly asked for in the problem or question.

–Explain your work or explain the pattern.
The first thing I do is make sure students have probelms that use or create patterns. Then I explicitly ask them to notice and explain it algebraically or in writing. The goal is for students to begin to intrinsically look for and notice patterns, but it is quite a daunting task at first. Most student skip that part or write ," IDK." So, when I see a pattern in students opting out, I bring the problem sup during math talk so we can attempt explanations and pattens as a group. I also start the year by looking for patterns in the addition and multiplication tables and by identifying and exploring the basic properties.

–Discussing their strategies.
When students respond to problems, they have opportunities to dicuss what they tried with a peer or peers in the groups they choose to work with. However, sometime the conversation turns into, “Here’s what I got. What did you get? Why’d you get that?” After circulating the room and hearing rather limited discussion, I realize that students need discussion modelled by my own consistent questioning when we share our work. Questions like, “How do you know? What do you mean? Why does that make sense? Can you show me in another way? How do we know that works too?” help because then I start to hear students pushing each other with those same types of questions. We also read/act out scripts of student dialogue in our CME curriculum that reflect student discourse over a problem. We initially do the problem as a class without reading the script and then the script acts almost as a summary. The script of student dialogue also models precise language and what we would ideally like hear from students discussing the problem (not the answers, but the different things they tried and how they figure out what strategy is the most efficient or how to self-correct a misconception).

As a middle school math teacher, I have found the following that students find difficult:

• Make sense of problems and persevere in solving them
• Solving complex problems that involves abstract thought
• Working in small groups

For the first item, make sense of problems and persevere in solving them, many students come into middle school with little ability to work through problems that require more than a couple of minutes. They want instant gratification in their work. I have used a method of giving small chunks of each problem, but it’s still too much for them to handle.
For the second item, using abstract thought, many students come in with very concrete and fixed mindset thinking. Having them work on a problem that has multiple solutions has worked for some, but not all in this category.
For the last item, working in small groups, this is more of an issue with personalities. Many students who are in an Algebra class tend to the top of their class and more often than not there are power struggles. Knowing my students is key when doing group work as I am able to group students in a more productive environment.

In my classroom, my students struggle with:

1. Being able to explain in words clearly what difficulty they are having with the task on hand. In other words, when a student raises a hand with a question, I approach and ask the student what help they need. But often they start by explaining what they have done so far. So I will repsond, “OK, but please try and define a clear question that I might be able to answer to assist you?” They know they need help, but they have not identified exactly what is holding them back from moving forward on their won.

2. Being able start with a clear and precise plan or approach to a problem. Students often want to jump right in and start working on a problem, but like a recipe, I try to teach students that they can be much more efficient and successful if they start by identifying the important values (in writing) and outlining the algorithm or process that they intend to follow from the start. I will often require students to “show” me a plan before allowing them to proceed past a certain point, like a check-point, on project work.

3. Being able to work collaboratively in a way that promotes listening and speaking from all group members. Some students are more prone to speak or listen, and I often have to actively intervene to suggest that perhaps “someone else” might have an idea when the group conversations are being dominated by an individual, or vice versa.

This will be my second year teaching and we are getting a new math series, so I’m interested to see more vividly where the struggle lies with my students. Upon reflecting on this last year, three areas that I view my students struggling the most are in -

1. Fractions - the operations of and, even further, does my answer make sense. We need to have more opportunities for modeling and visualizing what they are doing. That is something that I need to work on more as a teacher.
2. Perseverence - some students prefer to give up almost right away instead of working through the struggle, looking at their notes for assistance, past problems, the lesson outlined in the book. Some give up too easily the moment things get hard. Setting up a method of, “what have you done before coming to the teacher” will help them learn and utilize their resources more.
3. Math confidence - some are SO good at math and some are better at it than they think. Having the confidence in what they are doing and being accepting of the fact that they didn’t solve something correctly, yet they are still rewarded for trying and not getting frustrated. Building their self-confidence goes hand-in-hand with their math confidence.

I find students have trouble with the following three things:
measurement
place value
HOM (Applying Past Knowledge to Current Situations)

I think the solution is to help our students understand these conceptually vs. a rote process.
measurement - use “bits” of measure to divide larger “bits” to develop both a conceptional and relational understanding of length. Don’t start teaching measurement at zero!
place value - look for repeated patterns. Understanding decimals start at one not one tenth will allow students to see the pattern.
HOM (Applying Past Knowledge…) Again I think this is overcome by creating conceptual based opportunities which allow students to view problems through frameworks.

Sorry in advance, everyone, but my perspective is still from the content area of Science. However as I read the article, I see issues that I experience in my Science classes that I dont necessarily attribute to Mathematics goals. This article made this clear. A scientific hypothesis is a possible function that needs to be tested and verified.
I have often spoken of the language of mathematics that we use in science to express relationships, now i see beyond this to how relationships and functions are different.

My students struggle with moving from a data table that we create in science class to a set of ordered pairs and a possible linear expression of our tested hypothesis.

2. Choosing a different strategy when the first did not work out
3. Checking their answer. Does it make sense to the problem and does it work for all values or just the first.

My classroom has three big letters W H Y I use this this word more times a day than I can count, and after a few weeks, my student do too. Many students ‘just know’ the answers but can not explain how they got them or why they took the steps they did to get them. Getting kids to ask other kids why helps them with this. It also leads into the second issue of strategies. Often kids are stuck because they have not learned to try different things. During my math talks, students focus on strategies and sharing those strategies with others in the room. Finally, checking their answers, well, I just nick points if they don’t and they figure out they could do so much better by taking the extra minute to check their work!

• explain why they did what they did to get their answer
• check work
• connecting previous content

Similar to what others have said, one of the things, as my first year wrapped up, was how explicit I had to be when teaching students new material. I always want and try to have my students figure it out for themselves or work together to come up with an answer, however, often times I feel like I have provide them with explicit steps and have them practice it then.

1. Students have difficulty extending their mathematical thinking…extending what they know to new situations.
2. Students are satisfied with finding one way to solve a problem instead of using multiple methods.
3. Students are hesitant to explain and justify their thinking in classroom discussions.

To develop these important habits, I model and share student examples. I also try to group students after allowing them to complete an initial problem independently. The grouping is based on exposing the students to a variety of methods and to encourage them to justify their reasoning.

I don’t usually teach functions except as part of the design recipe. The math teachers usually handle that idea.

Three concepts that I find that my students have difficulty with are

1. applying the concepts that they have learned in their other classes to the Computer Science. I don’t think they can always see the connections. I am hoping to collaborate with the 7th and 8th grade math teacher to use the CS for Algebra to reinforce the concepts so they can see the connections.
2. Students don’t read the directions completely and just moved right into the puzzle.
This is a process that I am going to really work on this school.
3)Students working in small groups.

Three concepts my students have diffuculty with are…
Checking their work to see if it coherent and follows a logical strand.
Identifying minor details and understanding why that detail is a flaw.
Planning strategies to solve challenging problems (Persistence)

Functions are very difficult for students to grasp, especially as when they are introduced in the intermediate grades. The students I teach are diverse learners, many of whom, are English Learners. Functions are very foreign and abstract, students find them difficult because of the linguistic barrier that exists between English and their native language. Students struggle with persevering with problem solving, they want to be done. I often find modeling my own mistakes and growth from them helps the students with this practice, but it is very difficult. I would like to focus the first quarter on this practice and hopefully tackle a new mathematical practice every quarter.

*My students struggle with basic number sense, especially fractions, but decimals, percents, and even muliplication and division are holding them back from understanding more abstract concepts in Alegebra. I try to do some rotations that differentiate their tasks and give me a smaller, more leveled group to discuss concepts.

• My students struggle with perserverence and will often quite on problems that are definately within their skill level. They have been so “math injured” by the time they get to me that they do not try their best at math anymore. My plan this year is to scaffold the problem solving by removing any distractors, making the numbers easier (no fractions to begin with) and write the answer into the text of the problem. I do this to create an entry point for my struggling students that can lead to some much more valueable math discussions than simply shouting out the answer. We instead write equations to model the relationships listed in the problem. Eventually if all goes well, I can gradually get the students to solve more and more challenging problems.

Teaching math is a challenge because I often feel like the students are not developmentally prepared for the standards I am required to teach. I am always looking for more tools to help close the gaps and engage students in mathematics.

1. Students have difficulty graphing.
2. Word problems prove particularly difficult for many students.
3, Many students are resistant to or don’t see the need to show work.

Modeling as well as setting up steps students can follow helps students experience success in at least part of the problem and thus leads to perseverance that may eventually result in a correct answer,

My students want the answer to be right there in one step. If it takes more than two, they throw their hands up and demand help. They have very little perseverance. When they solve word problems the majority will not check to see if their solution made sense or was reasonable. We do a lot of solving word problems, and the students struggle and struggle.

My students lack the skill to connect the lesson opener examples with the skill practice, probably due to lack of visual and/or hands on examples.
Most of them seem to be below grade level, and new concepts and skill are hard to absorb and retain.
Most lessons are introduced with demonstration of examples, and students have to follow the examples but the questions sometimes are not directly linked to the given examples.
Very few students are able to connect, retain information and apply but these methods do not produce fruit for most of the students.

I find that students have difficulty with the following:

1. What does “slope” really mean?
2. What does the “y-intercept” really mean?
3. How do all the ways to represent a function relate to each other (i.e. equation, table, graph)?

I introduce slope and y-intercept by giving real world examples and talk about rates and initial values. Students typically relate to these and understand them. For some reason, they have a hard time seeing these same numbers embedded within the equation.

I also show many examples of multiple ways to show functions. We practice each one and compare them to see which is easier to use in which circumstances. Students respond well and see it within our examples but again have a hard time pulling it together on their own.

1. Like many others the concept of struggling as a learning process presents a huge challenge. I try to share personal experience of struggling and ultimate success to let students see it is a natural process with intrinsic benefits.
2. The concept of domain and range is typically introduced to students in the context of input and output. The seemingly simple idea of one output for every input is so confusing to kids.
3. Graphing could help students understand what a display of a function would look like. The attention to detail is critical when setting up the scale on a coordinate grid along with accurate plotting of ordered pairs.

In algebra, in particular, I find that my students struggle with connecting the abstract pieces and algorithms to a real-world application. Almost all of them, for example, can solve a multi-step algebra equation, but they don’t know how to connect that problem or its solution to a story problem. They don’t understand how to use algebra to solve a problem. It was interesting to read this article because I was just working yesterday on planning for the start of the year. I was thinking a lot about how to teach this idea of “function” to students. I appreciated the emphasis in the article on breaking down the language. I would like to use some SIOP/ELL learning strategies to examine the word “function”, both as a verb and a noun, as discussed in the article. I also would like to replicate the ball bouncing experiment, along with a couple other experiments, to potentially help students experience a function before they try to define it. I wonder how the coding activities could support this?