# Mathematical Mindsets

So a few of us have started the book and are making our notes/comments in the provided document; my principal created a shared doc in Google. I appreciate this way as I can read what others have written and get a general sense of what’s to come. The most recent chapter, Chapter 3, really caught my attention. In it, the author talks about how students view math vs. mathematicians view math. Students tend to view it as procedures, rules, and/or calculations. However, mathematicians tend to see it as creative, beautiful, and full of patterns. It was the sentiment that math is a study of patterns that made me take notice.

I have, for years, told my students that math was about patterns. That it was like a puzzle one needed to solve. I have always viewed math as a series of patterns and puzzles. I remember when I was in Kindergarten, a friend and I were talking. She was bragging how she could count to 100 while I could only count to 20. This irritated me. I wanted to count to 100 too. I remember going home and working this problem out in my room. Why I didn’t ask my mother is beyond me. But I was a stubborn kid (and for those of you who know me now, I’m still pretty stubborn). I remember looking at the numbers and ‘analyzing’ them. I thought, “If I can count to 20, then I can figure out how to count to 100.” And as I looked at the numbers, I saw a pattern. The numbers repeated. I began to realize that once the numbers in the one’s place (although I didn’t know place value at the time) were done, they started over again. And the numbers in the ten’s place began at 1, then went to 2 when all the numbers in the one’s place had been used. I had figured out the problem and went to school the next day bragging that I too, could count to 100.

Part of my success with math has come from A) a reading disability, so I gravitated towards math, and B) the fact that I was able to play and manipulate numbers on my own, okay and C) just being a plain old stubborn kid! We need to help students view math as patterns. We need to get over our own fear of math. We need to explore and allow conversations to happen in math. This is where the learning happens and a love of math will develop.

This is such a great book! Full of inspiration. I’m so glad we are reading it as a staff.

One final note: From grades 2 – 6 I was convinced I was going to grow up to be a mathematician. While that’s not my occupation, I’d say that I am one! We all are!

# Place Value Basics & Mult./Div.

Last year, I began using Jon Corippo‘s 8 p*ARTS . I saw great success with the repetition. As a result, I thought I’d like to do something along similar lines with Math. Now, I will admit, what I came up with isn’t nearly as fun. However, the repetition is there. This is for 5th grade and can easily be modified for other grades. Here’s what I came up with.

Place Value Basics

The plan:

• Today’s Number – Have the student of the day decide on the day’s number anywhere from billion to thousandths place. However, the number must be at least to the tenths place.
• 10 times greater – Take the original number and make it ten times greater.
• 100 times greater – Take the original number and make it one hundred times greater.
• 1,000 times greater – Yup, take the original number and make it one thousand times greater.
• Add 10 times greater and 100 times greater – add the numbers.
• Write a number that is GREATER – Have students change ONLY a digit that is AFTER the decimal.
• 1/10 times less – Take the original number and make it ten times less.
• 1/100 times less – Take the original number and make it one hundred times less.
• Subtract 1/10 and 1/100 – subtract the numbers.
• Write a number that is LESS – Have students change ONLY a digit that is AFTER the decimal.
• Prime factors of the first 2 digits of the whole number – Only take the numbers in the ones and tens place and find the prime factors.

An example is given on the second slide. This should be done daily, with an assessment each week. The first week or two should be done as a group until the class understands what is expected. Once they ‘get the hang of it’ all that is needed is the number and the students can do this independently.

Update: Since the beginning of the year, I have added a new daily practice paper. Now that they can do the Mult/Div paper well, I switch back and forth. I will soon add a fractions practice paper to the mix.

# Google My Maps in Math

When one thinks of incorporating Google Maps into their curriculum, the first thought is Social Studies. While that’s quite natural, I have incorporated Maps into most subjects. My latest brainstorm came when I was teaching adding and subtracting fractions. Yup, you read that right, I have students adding and subtracting fractions using Google Maps.

One simple option is to plot a point and place a real-world problem in the description box. Good start, but what if the students used distances to find the sum or difference?

I created this map of our town and included lines (using the draw line tool) to various locations in our town. This is where it got #eduawesome! The distance (which is displayed once the line is chosen) is shown in decimals! This means they have to convert the decimal to a fraction or mixed number, find the common denominator, and THEN add or subtract! And just to make life a bit more fun, I wasn’t too precise on all my lines. This means they also had to ROUND to the nearest hundredth and in some cases simplify!

For example:

Find the difference between Route A and Route B.
Route A – From school to a local Mobile Home Park (0.753 mi)
Route B – Keefer’s Inn to the high school (0.599 mi)

In this case, Route A was rounded to the nearest hundredth (0.75) while Route B was rounded to the nearest tenth (0.6). Then students had to convert this to a fraction and simplify. Route A = 75/100 = 3/4. Route B = 6/10 = 3/5.

Finally, the students were tasked with finding the difference. There were a lot of steps in there, but it was so much more fun than writing out and solving problems in the workbook. This was more in depth than any workbook I’ve seen, more fun, easy to create, and used a variety of acquired skills. I will be doing more things like this in the future.

# Tweet The Author

I recently wrote for the CUE Blog on how to own a premade curriculum. I spoke about taking it and tweaking it so that we, as teachers, feel the ownership instead of feeling like we have no say. Part of the way I take ownership in my math class is to use Andrew Stadel‘s Estimation 180 site. It’s a nice warm-up for the students and a great way to practice several of the 8 Mathematical Practices found in CCSS.

Most recently, we have been going through a series that has us estimate the value of coins in a container. It started with pennies, then progressed through until the day we estimated the value of dimes. We went through our normal routine – three minutes to discuss and find an estimate that is too low, too high, the actual estimate, and how they arrived at that estimate. Then, as normal, we viewed the video answer. Upon finding the answer, the RSP co-teacher got a discussion started. She disagreed with the answer. We left it at that so that the students could either agree or disagree. After they reviewed the previous two days’ answers and compared the answer to the dimes, the class determined that they too disagreed. Following the Mathematical Practices, they had to justify their reasoning, which they did. They reasoned that the pennies and nickels were mounding up to the point of almost spilling over, whereas the dimes didn’t quite reach the top of the container – same container.

Fortunately, with modern technology, we didn’t have to let the discussion die there. So, I got on my phone, hooked it up to the screen so the students could be active participants, jumped on Twitter, and sent Mr. Stadel a private message. This is what they wrote, well, told me to type:

After they got over the initial shock that one could actually do this, they got excited. Now, I have only been a participant in a few of Mr. Stadel’s sessions at conferences but was fairly certain that he would be open to what we had to say and would most likely respond. And he didn’t disappoint! The kids were VERY excited that he did respond. Okay, I was pretty excited too. This was such a real and relevant experience.

We then talked about how many more dimes we thought could fit. They determined that 1/2 roll more – 25 dimes – would be needed.

THIS is why we, as educators, need to be connected. THIS is why we continuously grow our PLN.

I would like to thank Mr. Stadel and Ms. Luke (RSP teacher) for making this all possible.

# Shifts in Math

One of the bigger shifts in math, aside from the building blocks in the CCSS Framework, is the ever dreaded ‘Explain your answer’. When this first came out, I wasn’t sure what they meant, the frameworks hadn’t been written yet. I had students explaining that they ‘added the ones then regrouped to get the answer.’ And while they were technically correct, there was a lot missing.

# Better Understanding

Now, I have a better understanding of what is needed. The students need to break down each step and explain, using academic language, what their thought process is. I furthered my understanding when I went to a training on this. Honestly, it wasn’t the best training, but it got me thinking. I used some of the techniques to create a better lesson.

# The Lesson

First, I created a template that the students were going to be using. Then, as a group, we walked through each part of the template and filled it in. There was  A LOT of guidance this first time. I’m hoping with practice, they will become more independent. Students worked in table groups to solve their table problem. Finally, they were to film their process of solving the problem. Using their ‘scripts’ students explained the process for division. Here’s an example:

# Multiplication & Division Basics

In the beginning of the school year, I created Place Value Basics. This was meant as a daily review to get students thinking quickly about some of the basics we learn. It was a big hit! My students went from doing it in 40 minutes (I know, but they needed the time) to around 8 minutes. Pretty good, right!?

Well, they had been bugging me to change it up. THIS is a good sign. So I came up with Multiplication and Division Basics. As some were still having a bit of trouble with Prime Factors, I kept in on this version.

Each year I teach this before Winter Break. Then after this, we head into fractions. Fractions take up all of the 2nd Trimester and by the time 3rd Trimester and the State Test roll around, students have forgotten how to multiply and (deep sigh here) divide. The problem is they have a shaky footing on these concepts before hitting fractions. I know, I’m the teacher… I should go with what they know and base lessons around them. Yes, in an ideal world that is happening. However, the pressure to do Benchmark Assessments and my district’s pacing (don’t get me started on that), and prepping them for the next grade are all too much for me – and them I suspect.

I have seen a great success with my students and the Place Value Basics. I am hoping that they can have the same success with this. How long will it take us in the beginning? Ugh, I hope not the 40 minutes! It’ll take us a while the first week or so, but in time they will successfully complete it in 8 minutes or less! Again, I will start off doing this whole class.

# Which One Doesn’t Belong

Last year I was introduced to a great website called Which One Doesn’t Belong by my friend, Nancy Minicozzi. The site is full of visuals in which students must decide which of the 4 images doesn’t belong.

Looking at the following, which would you say doesn’t belong?

Based on your perception you may have said the heart because it is the only one that is NOT a polygon. Or perhaps you thought the star for the lack of the red outline. Possibly you were drawn to the green arrow as it is comprised of two different shapes. Or maybe it was the last shape because it lacked a color on the inside. No matter what you thought, you were right.

This is a fantastic, low-risk activity for students to participate in. I have done it a few different ways. I have had students study the image and then move to different parts of the room. While gathered there, they talk about why they chose the image they did. Other times I have used tools like Padlet for them to record their thoughts. No matter what method I use, they enjoy the activity.

This year I have 1 non-English speaker. At first, he was reluctant to participate, but now he regularly raises his hand to explain why he chose the image he did. I also have students moving from table to table because they can see how the different images don’t belong.

In addition to the low-risk appeal, I am able to teach my students how to have conversations. We begin sentences with, “I agree with…. because…”, or “I disagree with… because…”, or even better, “In addition to what … said…” Quality language, thought process, and communication skills that are being utilized daily.