Math Spotlight: Levels of Understanding

This week's post focuses on "Levels of Understanding."

Typically, before introducing students to the traditional way of solving math problems, I use manipulatives (ones cubes, tens sticks, hundreds blocks, etc) or printed models (pictures) to help students think about and understand what the computation really means. Using hands-on activities and guiding students through different types of questions helps me assess the depth of their understanding. My goal is not to simply teach students to memorize steps and procedures, but to ensure that they understand why they are doing what they are doing in each step.

When we think about learning skills and concepts, we should imagine those skills and concepts on a continuum of learning with children at different levels of readiness. Given a specific concept, your child may be at different levels at different times. I found the following descriptors from Kathy Richardson (she's a math guru that develops materials for assessing students' true understanding and misconceptions):

Ready to Apply (A) – The student can already do a particular task and is ready to use this skill in other settings. (This student receives more challenging work).
Needs Practice (P) – The student can do a particular task with some level of effort but still needs more experiences to develop facility and consistency. (This student typically receives work at grade level that increases in difficulty as his/her readiness increases.)
Needs Instruction (I) – The student has some idea of what a task is about but needs support. (This student receives direct instruction that begins with the lowest level of their individual understanding and builds up to problems with increased difficulty to help meet grade-level expectations.)
Needs Prerequisite (N) – The student does not yet understand the concept and needs to work with
mathematical ideas that precede the concept being assessed.

When I discovered these descriptors, I really wanted to jump up and down--I was excited because these categories of learning really capture how I think about individual students' understanding of a math concept and how I choose materials and create groups for focused instruction.

It is also important to consider the size of the number(s) when placing students at these levels for a given concept--the size of the number with which they are independently successful needs to be taken into consideration. I often find that concepts that students may seem to have "mastered" are merely in the process of truly being understood when they are presented with larger numbers. As we know, accuracy also becomes a larger issue as the size of numbers increase. To instruct students having difficulty (say learning the procedures for how to multiply 23 x 67), I begin by taking a step back and instructing them on how we would solve a simpler problem, like 23 x 7. Once the student has consistently demonstrated that they are able to complete problems at this level, we move on to adding a digit in the tens place for the second factor.

Next up...
I'm going to share examples of problems that are below, on, and above grade level based on NC state standards for math. When you see your child's math work, you can use these levels to have a better understanding of what they have accomplished. If they are successfully completing below and on grade level problems, but having difficulty with above grade level, this means that they are where they are supposed to be and that they are being challenged to push beyond the average 4th grade expectations. If they are having difficulty with "on grade level" problems, rest-assured that they are being served in a small group that meets them where they are and works to help them build up to solid grade-level abilities. 

By the way, I'm ready to reveal a secret...I'm a bit of a "math nerd" and I think that's cool!

Math Spotlight: Helping with Homework

How Can You Help with Math Homework?
I have recently had conversations with a few parents about math homework and would like to spend some time addressing math homework routines at home and school.

I have chosen to use spiraled math homework that covers many concepts instead of homework that only focuses on the concept that we are learning on a given day. This means that your child will see concepts that they have learned months ago, weeks ago, and possibly even concepts that are coming up. While I believe that this type of homework is really beneficial for me (and the child) as I hope to keep students' previous learning sharp, assess their ability to independently complete concepts we are currently focusing on, and see who can solve problem-types that I have not yet discussed, I know this may also be a source of confusion and frustration at times. How do you know what your child should have mastered already? (I'll address more about this in an upcoming post). And, most importantly, how do you help?

Can you help your child with math homework? Absolutely! It seems that resisting a parent's help with math homework is a typical "coming of age" behavior for 4th and 5th grade students. However, many of you feel that it is an important role for you to play in your child's education and I would agree. Looking over your child's math homework is definitely a way to keep yourself in the loop of how they are progressing and what the expectations are for 4th grade math.

Often, resistance comes from parents showing the child how to solve the problem using a different method than what was taught in the classroom. This can be confusing for a child when they are having difficulty understanding (or remembering) the method the teacher is using in class; however, for a child who has a good grasp on one method, introducing another method can help develop their math understanding. I often think we--students, teachers, parents--are looking for the "easier way" to solve math problems, but the truth is the only "easy" way to multiply or divide large numbers is to use a calculator! :) Regardless of the method students are taught or choose to use, learning how to do something new takes time. I teach at least two methods for solving multiplication and division problems because it allows students to develop deeper understanding and it gives them an additional way to check their work. Developing flexibility with different methods is also an expectation of our math curriculum.

It is truly my goal to do a better job of helping you help your child in the upcoming months. I will share videos that show the methods I have taught for multiplication, division, and other concepts. I will also share videos that can be used to help your child independently review concepts and additional websites that they can use to practice.

How do I use Math Homework?

Each morning, math homework is checked as students turn it in. Usually, it is returned immediately for them to make corrections on as many problems as possible before we leave for specials at 8:15. Students who have questions also ask for assistance at this time. If a student misses most of the problems, I may choose to work with them one on one during math time instead of having them return to try again on his/her own. This is so that I can gain a better understanding of what is not connecting for them--are they making accuracy mistakes or truly not understanding concepts? When problems are completed correctly, I usually assume that the child has a good grasp on the concept. When helping your child with homework, I only ask that I am aware of any assistance that was necessary so that I can keep that in mind as I plan for your child.

So, how can you really help with homework?
* Use the new stuff I am posting on the blog in the next few days to remind your child of how to solve the problems using the methods we are learning
* Initial beside of any problems that you help your child solve or write a quick note on the paper letting me know that your child did not understand. Some students even jot me a note that they didn't understand how to do something and need my help.
* Let your child know that Ms. Russell has given you permission to work on homework with them and even APPRECIATES it when you check over their work!
* If your child is doing fine with understanding math, but has accuracy issues, remind them that accuracy is a big focus in 4th grade and say something like "Hmmm...I see a few problems on here that are incorrect. I think you know what you are doing but some of your facts are wrong. Can you find your mistakes?" If your child is quick to become frustrated with homework, you may identify the point of inaccuracy (as I often do this in class to show the student in a positive manner, "You were good up until...").
* One practical tip for helping your child understand how to do a problem is to show them how to solve a similar problem using different numbers. After a few practice problems, your child can return to the homework problem and give it a try independently.
* If your child is missing (or misunderstanding) word problems, encourage them to jot down "the mathematical information the problem provides" in a list format. This is the method we use for understanding and solving word problems in the classroom so that language should help them.
* Practice multiplication facts! Multiplication fact mastery (speed and accuracy) is a precursor to successfully mastering nearly everything in 4th grade math--and beyond! You can use a set of flash cards to quiz your child and remove cards that are super-easy for them. Research suggests that students only practice 8-10 minutes in one sitting, otherwise the facts start to jumble in their mind.

Please let me know if you have any general questions about math homework that I have not addressed in this post! Thanks for all you do!

Introducing: Math Spotlights!

One of my goals for the New Year is to share more information with you all about math instruction. I hope to share the methods students are learning, details about our math class routines, and examples of below-, on-, and above-grade level expectations. This post will serve as an overview of our math routines and my approach to math.

What does math look like in our classroom?

• A typical week in 4th grade math includes 2-3 days of stations with students completing activities at different levels of difficulty. Students may be working independently or in a guided group with me, Mrs. Morris (our teaching assistant), or Mrs. Kuhl (our AIG teacher). In stations, students solve computation problems where we focus on being more accurate and making sure we understand the steps to different methods. Students also have stations where they review concepts, work with word problems, and use dreambox. 

• 1-2 days a week, I introduce new concepts while reviewing old concepts. For example, in the upcoming week, I will be introducing fractions concepts with some picture and number sorts (to gauge students current understanding of fractions). In addition, we will spend half of our math time reviewing and extending our understanding of how to divide larger numbers. 

• On Fridays, students complete some form of assessment. This may be review questions from concepts we learned earlier in the year or directly related to what we are currently studying. Often, this information is used to create leveled groups in stations for the following week. The difference between regular math days and "assessment Fridays" is the level of support that I am willing to give students. On assessment Friday assignments, I offer as little support as possible and really try to encourage students to show what they know. When I offer support to help a student get the answer, I write directly on their paper and/or initial it so that I know they did not complete the problem independently. 

Hopefully, this post has shed light on my approach to math instruction and my strong belief that learning is a process. While your child needs to learn given math concepts, it's not my belief that they must (or even should) master the concept on the day I introduce it. Coming up in our next Math Spotlight, how you can help your child with math homework!

As always, I appreciate everything you do to prepare your child to be ready to learn each day!