Transitioning to the Standard Algorithm

 

When should the standard algorithm be introduced?

When should the standard algorithm be introduced?

 

Many of my second grade classrooms are in the middle of their addition units. The classes often teach place value and addition strategies during the months of September and October. When introducing addition strategies, teachers rarely start using the standard addition algorithm (see # 4). Manipulatives and visual representations are heavily used during the first month of school.  The process below differs per school, but I’m finding that this is often the case in many of the second grade classes that I’ve observed. Keep in mind that I’m missing other approaches, so perceive the following as a few highlighted strategies that are used during the first few months of school.


counters-01

Finding the sum of objects

Students are introduced to some from of unifix cubes or counters. Students are asked to compile the groups of counters to find the sum. For the most part students find this task quickly and are ready to move onto the next portion.  This is also a first grade skill that’s reviewed at the beginning of second grade.

circle11-01

Identifying numbers on the number line

The number line usually follows the counters. Sometimes the number line makes an appearance before the counters, but it’s usually afterwards. The number line is used extensively. Students are asked to find numbers on the number line. This builds number sense and an understanding of the reasonableness of an answer. Eventually students are asked to add numbers with hops showing the addition involved.

Find the sum of 25 and 15.

Find the sum of 25 and 20.

Base-ten blocks are then introduced to emphasize place value. Students are asked to combine base-ten blocks to find the sum. They are asked to find the sum of the base ten blocks and place them on the number line.

basetenblocks-01

What is the sum?


The four processes above aren’t necessarily mandates, but it’s found in the sequence of the textbook.  I should mention that the same process is used with subtraction later in the year. What I’m finding is that there’s rarely a mention of using the standard addition algorithm to find the sum. I don’t necessarily think that’s a problem, but it raises the question of when should the algorithm be introduced? In what cases should the algorithm be introduced and is it only used in certain circumstances.

Think of 200 + 198. Would your students use the standard algorithm for this?  Some might, other might prefer to use another method. Regardless of when the standard algorithm is introduced, there will still be students that would prefer to use the algorithm.  Is that the most efficient method?

The topic of this post was tackled during last Thursday’s #ElemMathChat. Most of the questions revolved around when the standard algorithm should be introduced and mistakes that occur when students focus on the steps.  There were many useful answers, but I’m not positive if one right answer climbed its way to the top.  There are many factors at play here. Some teachers feel pressured to move through the curriculum at a high pace because of testing.  They might teach the algorithm sooner, while others might not mention it and scroll through the prescribed lesson sequence.  Most teachers would like students to have a conceptual understanding of numbers and systems before moving towards a standard algorithm. How much time is spent developing that truly depends on the teacher and student. I’m not judging any teacher is these situations as we are all in this together, but I believe having this discussion is important.  I think this also plays a role in how other algorithms are introduced.

 

Exploring Rules and Patterns

exploringrulesandpatterns

This past week my upper elementary classes started their equations, patterns, and rules units.  The units are composed of patterns, special cases, student-created rules, and solving equations.  To be honest this is one of my favorite units and involves a good amount of pattern exploration.  Through exploration, students construct their own understanding of how mathematical rules can be developed by analyzing patterns.   Many of these activities involve manipulatives or visual representations of various patterns.  I’m going to highlight three specific activities that seemed to work well this past week.

Analyzing the Perimeter

What's the Rule?

What’s the Rule?

Students were given a handful of square geometry blocks.  They were asked to find the perimeter of one block.  This was quick as students just needed to count the sides of the block.  Four!  Students then put together two blocks and found that the perimeter didn’t double, instead it was six. Students continued the patterns and discussed with their group what the rule could possible be.  Some groups used the whiteboards to write possible solutions.  Throughout this activity students struggled at first and then came to an understanding that the rule just didn’t include one operation. After the rule was discovered the students found the perimeter of 100, 200, and even 1,000 squares put together in a horizontal row.  I believe this activity also helped establish the reason for having mathematical rules.

Rule Tables

Students used four dice, a whiteboard, iPad, and dry erase marker to complete this activity. Two of the dice were operation and they had + and – on the sides.  The other two were typical six-sided 1-6 dice.   Students rolled all four dice and created a rule.  For example, if a student rolled a 6, 2, +, and – then he/she could say the rule is + 6 – 2.  Students wrote the rule on top of the whiteboard and used one of the die to roll five numbers that would be included in the in column.  Afterwards, students were asked to find the out column using the rule that was created.  A few examples are below.

studentrules

The students then took a picture of their product and sent it to Showbie.  Later on that day the class discussed how to combine rules.  So instead of + 6 – 1 this rule could be + 5.  The students were then combining all of their rules.  This activity led to some productive discussions on how to simplify or expand rules.

Visual Patterns

visualpatterns

I came across Fawn’s Visualpatterns site a couple years ago.  This is a fantastic resource that I introduced this past week.  I printed out some of the patterns and placed them in manilla file folders.  The picture of that is located near the top of this post.  The six folders were placed around the classroom.  Student groups visited each folder and determined the rule. While in the group students worked together and filled out the sheet below.

Screen Shot 2014-10-11 at 8.44.25 AM

Modified from this site.

Students took whiteboards and started to build possible rules for the pattern. Once they accomplished this they filled out the table and graphed the relationship.   I appreciate that students are asked to graph their findings.  This could lead into so many other math topics. Students only rotated through two folder stations so we’ll continue this activity next week.  By the way, the students were stoked when I showed them the visual patterns site and not because it has the answers.  A few students even said they were going to check out the other patterns on the site.  I’m looking forward to utilizing this resource a bit more next week.


How do you introduce patterns, rules, and equations?

 

Math Genius Hour Research

titleresearch-02
About two weeks ago my classes started their math genius hour projects. Students started with a  wonderwall, created multiple questions, and then decided on one question that they wanted to emphasize.  The students generated and picked the question.  I signed-off on the question and the students chose what math strand they wanted to highlight. Eventually the math strand will dictate what will be included in the presentation.

questions

If needed, students can refine their questions to show more detail.  Some students may do just that.  Some of the students won’t really know until the research process starts. As you can see, students definitely were wondering how ____ is made. I can’t describe in words the amount of curiosity that this type of project brings to the table. Instead of asking the teacher for their answer (they still do though) I’m asking students to research for themselves.

Most of the students will begin researching their questions using a variety of online resources next week.  What I found out last week was that students weren’t aware of how to search, gather, or cite information. Before the students begin their research I wanted to review how to use online sources correctly.  A few students raised their eyebrows and asked why we’re learning about research skills in math class.  Many of the students have never heard of the term digital literacy before.  I thought this was a great opportunity to discuss the importance of being able to find and use online resources effectively.  The class explored (1) (2) a few different resources on how to search for information online.

We also looked at the following questions:

- Who created the website ?
- How does the site address end ? (.gov .edu .net .org)
-Does the page contain any type of advertisement ?  

So where are we now? Some students have already started to research their topics.  Students are asked to find at least three sources before picking a presentation tool.  They are filling out the sheet below to compile their information.

Math Genius Source Sheet

Click for pdf

Once the research is complete students will pick their presentation tool.  I’m also looking at having students reflect on their math genius project journey in their journals.  Now, I’m looking at what type of presentations tools they can use … that’s my research for the weekend.

Estimating in the Elementary Classroom

Bv6El2hIYAAS03r.jpg-large

Using Estimation180

My school finished its ninth day of school yesterday. It’s been a journey as students are understanding class routines better.  At this point in the year, students and teachers are starting to become more solid in their processes.  Many of my students arrive to class at different times. Some students are at an elective or leave class a bit earlier/later than the rest of their peers. Regardless of the arrival time, when students enter the room they follow a flow chart. Students have their own folder and materials inside that are ready to go. I usually have some type of visual brainteaser for the week and a grade specific Scholastic math magazine. In the past I’ve used different types of math warm-up activities to start class.

This year I adapted my warm-up strategy. I wanted to individualize the type of responses within that warm-up time slot. After researching a few different tools, I decided to try Andrew Stadel’s Estimation180 this year. I think of Estimation180 as an opportunity for students to develop a stronger sense of numbers and practice estimation skills in the process. Initially, I thought that the site would be great for middle or high school students. I then found the below sheet and site that seemed helpful. This is one way in which student can document their thinking.  The template also includes lessons that could link to Fawn’s Visual Patterns site.

Click to download template

This template inspired me to adapt the sheet to fit an elementary classroom. I changed the template a bit to work with a third grade math class.  A few colleagues and I will be using this sheet early next week.

So now, students enter the classroom, pick up their folder and begin to work on their daily estimation challenge warm-up  sheet.  The estimation is displayed on the whiteboard. Students pick a high, low and exact estimate. I ask the students to prepare to tell me about the reasoning that they used to come to the concluding estimate. The class then completes the online portion of the site and submits a response. We then look at other responses and reasoning.

Photo Sep 05, 12 42 44 PM

After a brief discussion the result is revealed. Students write in the correct numbers and find the + / – . The entire activity takes about 5 – 10 minutes.

Photo Sep 05, 12 40 19 PM

I’m planning on using Estimation180 a few days a week and incorporate Visual Patterns for the rest of the days. The template also includes a few different reflection pieces.  I feel like these activities provide students opportunities to produce a product and reflect on the results. At some point I’d like to add a journaling component to encourage more reflection and possible goal setting.

Thoughts on Questioning Techniques in the Classroom

photo credit: mag3737 via photopin cc

photo credit: mag3737  cc

Every year I find that pairing the right math activity while asking specific questions can yield some amazing student learning experiences. I would assume that most math teachers would agree that only giving a specific solution to a student doesn’t necessarily help them understanding concepts. Offering solutions without feedback or questions can encourage students to care only about finding the answer. The act of “answer finding” limits understanding and diminishes curiosity.

When I started teaching I spoke constantly. I would give examples and statements that I thought would help all my students. Looking back, I spoke more than I should. As I progressed in my career I found that constructing a mathematical understanding doesn’t always ignite from just listening to the speaker.  There’s a time and place for listening, but being engaged in the learning process is vital.  I soon found that a balanced instructional approach was needed so I decreased the amount of talking and started to ask math related questions instead.

Although statements are beneficial, effective questioning techniques can provoke a response from the student. Offering guiding questions, or questions that encourage students to delve deeper in their explanation benefits the student. I feel like part of my job is to create an environment where students are able construct mathematical understanding. When students struggle with that understanding, questioning techniques can be another tool that teachers utilize. Questioning also helps students think more independently and explain their mathematical reasoning in a verbal or written form. Students need to be able to explain why and how they find solutions.  This type of communication is an important skill to develop.  Before planning on using questioning techniques in the classroom there are some important points to consider.

The environment

Students have to be open to answering the questions that are posed. In order for questioning techniques to work, students need to feel comfortable enough in the classroom to offer their ideas and explain their mathematical thinking. This environment is often intentionally built by creating a positive classroom learning community early in the school year.  Students will often participate less if they feel as though their input isn’t valued.

The timing 

Teachers can spend extensive time planning, but I find the best times to use effective questioning techniques are in the moment. Learning can be messy and teachers need to be able to have questions available depending on where students are in their mathematical understanding.   I’ve seen great question techniques used in whole class and small group settings.

 The questions 

The questions that are posed truly matter. When I started teaching my questioning techniques were less than stellar. Through time I’ve learned to expect more from my students. When given a chance, students are fully capable of expressing their thinking. Teachers need to allow students opportunities to do just that. The questions should prompt a response from the student beyond yes or no.  I want to get the students talking about their math process and learning.

 

mathquestions

 

Other classroom questioning resources are below.

Effective Questioning Techiques
Asking Questions
Using Questioning to Stimulate Mathematical Thinking
Leveled Math Questions

 

 

 

 

 

 

 

 

 

 

 

Bridging Procedural and Conceptual Understanding

Yesterday I was putting together a few math projects when a Tweet caught my eye. The Tweet below started a short conversation that I thought was interesting.

David’s Tweet had many responses.  Most responses revealed that educators tend to side with solving one problem ten different ways rather than having students solve ten similar problems.  I started to reflect on how teachers give assignments that ask students to complete repetitive problems that often reinforce procedural mathematical thinking.  I also started to think how in an effort to provide practice, teachers may focus on procedural aspects first and then move towards practical application.  I find this happens frequently with math concepts at the elementary level.  What I don’t find often is the viewpoint that practicing procedural aspects can be embedded in solving specific problems multiple ways.  This type of thinking reminds me of number collection boxes.

Regardless of the assignment I want to be able to give specific feedback.  A larger problem that involves multiple steps can provide opportunities for teachers to pinpoint where misconceptions are and give direct feedback.  This isn’t always possible with ten similar shorter problems.  Below is an example of a few problems that you may find in a fifth grade classroom.  I don’t condone using these types of problems as they are definitely utlized, but I think we need to ask what’s being assessed when students complete this type of problem?  Students are simply asked to find the volume and show a number model.  I appreciate how the problems ask students to show their number model, but these types of problems seem to measure procedural understanding.  Do students know the formula?  Yes, well then they can answer many of these problems, even 10 in a row.

Procedural

 

I think the above problems have a place in the classroom, but shouldn’t necessarily be the norm.  Usually these types of problems are found on homework sheets.  The problem below which was adapted from a recent fifth grade test is more challenging, but gives students opportunities to showcase their own mathematical understanding and persevere.  Some would say that these two problems are completely different.  I would agree, but similar concepts are being assessed.  They do look different and the second requires more skills to complete.  Students need to be able to use their procedural understanding and apply it to the situation.  Also, one key element that’s missing from the first problem is the student explanation.  Students are required to show their mathematical thinking in the second problem.  This is big shift and can reveal student misconceptions more clearly than the first problem.  I struggled with the decision, but eventually had students work in groups to complete the problem below.  Students were allowed to use any of the tools in the classroom to find a solution.

newadvanced

 

At first, all groups struggled with this problem.  Near the end of class all the groups presented their findings.  What’s interesting is that all the groups had different answers and ways in which they came to their conclusions.  I was able to offer opportunities for students to see and ask questions about different math strategies.  During the next class I was able to pull each group and give feedback.  This activity took a good amount of time to complete, but I feel like it was worth the commitment.

Through this experience and others I’m continuing to find that it takes a “bridge” to connect the procedural and application pieces.  At times I feel like there’s an assumption that if students are able to answer 10 similar procedural problems that they will be able to simply apply that knowledge in a multi-step problem.  This isn’t always the case and sometimes the bridge doesn’t fully form immediately.  Performance tasks, similar to the problem above can be one way in which teachers can help the transition from procedural understanding to practical application.  Being able to apply that knowledge to a math performance task can be a challenge for some students.  When teachers focus so much on the procedural, that’s the only context that students see and practice.  A blend between procedural and application needs to be established within the classroom.  I feel like activities like this help bridge this gap.


How do you bridge mechanical and conceptual understanding?

Math and M.C. Escher

 

Math and M.C. Escher

Math and M.C. Escher


During the last week of school my students started to explore topography concepts. Topography usually isn’t the first thing that is thought of when someone mentions the word math. That’s why I find it so interesting.  I truly enjoy teaching this topic because it often brings out the best from my students.  I find that most upper elementary students tend to thrive when given geometric shapes and asked to explore, rotate, translate or even turn them inside out.

I generally introduce the unit with M.C. Escher.  The class learns a bit about the life of Escher and his contributions to the world of art.  Moreover, we discuss how art and math are related. This is often a deeper conversations as students start to expand on the notion that mathematics can be found throughout our world.  Topics like the golden ratio and Pi often get brought up during this time.

After learning about Escher’s life and his influencers, the class looked at his different artistic creations. Usually my students recognize at least a few different creations.  Students seem to gravitate towards his optical illusion pieces or the famous Waterfall work.  As each work of art was discussed the more students found mathematics as an integral part of Escher’s work. After reviewing the different pieces of lithograph art, the class watched a short video on how Escher’s design and math are connected.

After the video the students were asked to have a conversation about how math can be found in most art.  The words symmetry, rotations, slides, translations, reversals, surfaces, and perspective were all brought up during the discussion.  What’s nice is that the vocabulary was brought up naturally as students spoke to one another.   I was able to highlight the words and facilitate the discussion as needed.

Eventually the discussion ended and the class moved to the next activity.  I planned to have the students create their own Escher-like artwork.  The students reviewed how to have “Escher-like eyes” when creating their own pieces.  I was proud of the student responses and the imagination that came forth during this discussion.  The class then reviewed the directions to create their own Escher-like creations.

The students went through the directions and asked questions.  Once the expectations were clear I passed out a 8 inch by 8 inch square to each student.  Students created their own tessellation template.  In the future I’m probably going to cut the square dimensions in half so the patterns become more evident.

This slideshow requires JavaScript.

Eventually the students used the template to create an Escher-like creation.  Students showcased their work to the class using the vocabulary mentioned above.  The students were able to bring their work home on the last day of school.  All in all, this is a lesson I’m intending on using next year and a definite #eduwin in my book.

 


How do you incorporate art and math?