Throwback Thursday – Bar Model Solutions – by Students

Over the summer, we thought it would be fun to run some of the most popular posts from the past. It’s always interesting to see how students’ minds work. When I re-read a post from the past I always take away something different because I am in a different place with my own experience. Perhaps you are as well!


Bar Model Solutions – by Students!

Originally published 4/12/2016

After the post on Assessing Bar Model Solutions went up, Beth Curran sent a message: “We just did that problem!” She agreed to share some student work:


boys and girls 2

boys and girls 3

boys and girls 5

And when the students didn’t draw a model:

boys and girls 4

I see this as a comparison problem:

thinking blocks

5 units -> 125 students
1 unit -> 125 ÷ 5 = 25
7 units for boys -> 7 x 25 = 175 boys in all

(That’s the Thinking Blocks Model Drawing tool that allows you to insert your own word problems and solve – or you can use the pre-made questions!)

 

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Throwback Thursday – It can’t all be Singapore Math…

Over the summer, we thought it would be fun to run some of the most popular posts from the past. Here’s a look at some misconceptions around Singapore Math and Common Core Standards. When I re-read a post from the past I always take away something different because I am in a different place with my own experience. Perhaps you are as well!


It can’t all be Singapore Math…

Originally published 12/29/2014

This tweet posted by the National Council on Teacher Quality (@NCTQ) caught my eye:

NCTQ_Tweet

Now, I’ve heard decomposing called “branching” but can’t remember ever seeing this in a Singapore textbook. Where did this problem come from?

It’s nice that NCTQ recognizes Singapore’s Math as “tops in the world.” But it’s discouraging to see methods and terminology that are not a part of the Singapore curriculum attributed to it. Especially in the context of the nasty debate about CCSS. And especially since Singapore’s math curriculum–with its rigor, coherence, and focus–is often cited as a basis for more rigorous standards, including CCSS.

The problem posted is based on the concept of “Number Bonds,” which calls for students to decompose numbers (this is the term used in Singapore and in all major Singapore Math® textbooks distributed in the U.S.). Below, I’ve posted some examples of how this concept is presented in Singapore Math® series available in both the U.S. and Singapore.

This matter points to my BIG concern: As publishers and others adapt Singapore’s Math for the American market, new approaches creep in. These often are not based on the curriculum that helped Singapore’s students go from mediocre to best in the world in a dozen years. I’ve written about this in my comparison of Singapore math textbook series available in the United States.

So my plea to NCTQ: please use examples from an actual Singapore mathematics text when citing the components that make it so successful. And feel free to ask if I can help you find those examples.

Number Bonds problems in Singapore Math® textbooks

Here are some materials covering Number Bonds and “decomposing” numbers from actual Singapore textbooks:

From My Pals are Here, the most-used materials in Singapore:

MPAH 3A Mental Addition

From the U.S. Edition of Primary Mathematics, available in North America since 2003:

PM US 3A Mental Addition

From the Common Core Edition of Primary Mathematics, released in the U.S. market in 2014:

PM CC 3A Mental Addition_0001

And finally, from Math in Focus:

MiF_3a_mental_math

 

UPDATE:

Ugh! One more similar tweet from NCTQ.

NCTQ_tweet_#2

 

 

 

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Throwback Thursday – Journaling in the Singapore Math Classroom

Over the summer, we thought it would be fun to run some of the most popular posts from the past. Journaling in math class has become quite popular. We’d love to hear your successes or challenges with math journals!


Journaling in the Singapore Math Classroom

Originally published 10/13/2016

Communicating mathematically is a critical skill and goal for all of our students to reach by the end of middle school. In fact, Common Core Standards for Mathematical Practices, MP3, states that students will, “Construct viable arguments and critique the reasoning of other.”

Singapore’s Ministry of Education would tell you that there’s nothing Singaporean about Singapore math.  When developing their highly successful math curriculum, they took theory and ideas from mathematicians and educational theorists around the world and put them into action.

What should a math journal look like?

I have attended many workshops and make-and-take sessions on planning and preparing for student math journals.  Many have focused on setting up the student journal with a contents page and tabs to divide the journal into “notes,” “vocabulary” and “practice problem” sections.  While this will create a journal that looks really nice, what I have found to be most effective (and one that I actually use in the classroom) is taking a simple composition or spiral bound notebook and beginning on the first page.  Students make their first journal entry of the school year on page one and continue with entries on subsequent pages. Less is more!

Here’s what a journal entry page might look like:

journal-photo

The journal entry number just grows as the year progresses.  We might come up with the title as a class, or students can create their own.  The problem in the problem box can be copied by students or printed out for students to paste in their journals.

What should students put into journals?

There are four basic types of journal entries; investigative, descriptive, evaluative and creative.

Investigative: Students explore a new concept, solve a problem and make connections to prior learning.

  • Example: Three friends share a sleeve of cookies.  Each sleeve holds 32 cookies.  If each friend eats ¼ of the sleeve, how many cookies do they eat altogether?

Descriptive: Students describe or explain a concept or mathematical vocabulary.

  • Example: Use pictures, numbers and/or words to explain a polygon.

Evaluative: Students argue for or against a strategy or solution to explain why they think an answer is right or wrong, explain their choice of strategies or justify the most efficient strategy.

  • Example: Which of the strategies discussed in class today would you use to solve 245 – 97?  Why?

Creative: Students write their own word problem or create their own number puzzle.

  • Example:  The answer is 465 lbs.  What’s the question?

Here’s a sample student  journal page (click on image to enlarge):

scan0018

When should I ask students to make journal entries?

Journaling can be a very effective tool to develop communication skills in your students.  Depending on the type of entry, you could incorporate journaling into many parts of your math day.  Open a class with an investigative entry to engage students.  Consolidate learning and reflect on thinking with a mid-lesson descriptive or evaluative entry.  Enrich students with a creative entry for early finishers of independent practice.

The benefit of journaling for the teacher is it provides a concrete formative assessment.  By evaluating student responses, you can determine their readiness to handle a new task and check for understanding of concepts.  Student journals also provide a great launching point for discussion at parent-teacher conferences.

_____________

Check out a resource from a previous post: Singapore Math and Math Journal Writing

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Throwback Thursday – “Summer Math” Suggestions to Boost Student Understanding

Over the summer, we thought it would be fun to run some of the most popular posts from the past. All of these options are still on offer. Let us know if you have used one of these or something else to counteract the typical “summer brain drain”.

When I re-read a post from the past I always take away something different because I am in a different place with my own experience. Perhaps you are as well!


“Summer Math” Suggestions to Boost Student Understanding

Originally published 6/23/2016

School is out and summer is calling, but for many teachers and administrators, summer is a time to take stock and plan and budget for next year.

As a teacher, this is a glorious time of year, but also one of worry. I worry about my students.  I worry about those who needed extra support throughout the year understanding and retaining math concepts.  How will they fare next school year? Will they regress over the summer months if they don’t do any math work?

There are three categories of students who benefit most from summer math work:

  • Those who have struggled all year and maybe never quite achieved mastery on those critical grade level concepts,
  • those who easily forget concepts, and
  • those whose math confidence could use a boost.

With a Singapore Math program, there aren’t many ready-made options to pick up at the local bookstore.  Books that are available focus heavily on procedural understanding rather than underlying math concepts. So what’s a teacher to do?

Aside from recommending tutoring, I have found a couple of options that seem to meet my needs as a teacher and the needs of my students.

Workbook Work

Primary Mathematics Common Core Extra Prac 3

For those looking for a paper and pencil option, I recommend the Extra Practice books from Singapore Math’s Primary Mathematics series. Students should work at the grade level just completed (a rising 3rd-grade student should do summer work in the 2nd grade Extra Practice book).

The Extra Practice books offer parents and/or tutors “Friendly Notes” at the beginning of each unit that explain how to re-teach concepts in a way that is familiar to the student.  The notes are followed by practice pages that give parents sample problems appropriate for practicing the concepts and the student an option of working through problems independently.  Best of all, they include an answer key in the back so parents can check work and students can re-work problems, if necessary.

These books are written to cover a year’s worth of concepts; I am by no means suggesting that a child complete the entire book over the summer.  Teachers recommending this book will need to tailor the tasks to meet each student’s needs.  This can be as simple as highlighting the contents page to include units or pages that you would like the student to complete over the summer keeping those critical concepts in mind.

Another option for summer work can be found in online programs.  I have come across three online options for concept practice; Primary Math Digital, it’s twin Math Buddies and a program new to the US market, Matholia.

Online Options

Primary_Digital_Coming_Soon_Home_SchoolPrimary Math Digital (Free 15-day trial) and Math Buddies (Also a free trial) are backed by Singapore Math’s Primary Mathematics and Math in Focus series. Both offer students video tutorials that can be viewed by the student (and parent) an unlimited number of times.  These videos are scaffolded to follow the pictorial and abstract progression of learning.

Teachers can assign videos, practice and assessment tasks fMath Buddiesor students to complete over the summer at their own pace.  The practice pages can be a little challenging to navigate, but with some initial guidance, students should be able to complete the tasks independently.

Both programs require the school to purchase annual student and/or teacher accounts to gain access to the library of lessons. There are Homeschool accounts available. Expect a price tag of around $30 per student depending on the number of accounts purchased.

matholia logoAnother, more affordable option new to the US market is Matholia. Matholia was developed by two teachers from Singapore and has been used by teachers and students in Singapore as well as several other countries. This program also includes a library of video tutorials, practice and assessment tasks as well as fact fluency tasks and games.

The videos are easy to understand and are also strategically scaffolded for student understanding. The practice and assessment tasks are intuitive and easy for students to navigate. As with the other programs, teachers can assign tasks for students to complete over the summer.

Matholia also requires the school to purchase annual student accounts (teacher accounts are free) but is much more affordable at just $8 per student.

Don’t forget the concrete…

I can’t go without saying that any of these options will give students practice, but struggling students need more than just extra practice working through math problems.  They need more time in the concrete phase of learning using manipulatives; base-ten blocks, place value chips, model building with connecting cubes or paper strips, fraction strips or circles, etc.  So, please, consider not only sending these students home with books and login IDs but also with a bag of manipulatives for hands-on learning and practice.

Beach_of_Dreams_BeautifulNow…back to dreams of lazy mornings and time to relax and recharge.  Have a great summer and rest assured that your students will be prepared for the next grade with a little summer math work.

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Throwback Thursday – Anchor Tasks Demystified

Over the summer, we thought it would be fun to run some of the most popular posts from the past. It’s mid-July and teachers are already getting ready to go back to school. Here’s an article on planning for the concrete component of a lesson.


Anchor Tasks Demystified

Originally published 2/15/2016

With performance-based standards and 21st-century skill sets teachers are asked to teach mathematics with an emphasis on problems solving and inquiry learning, but how?  The answer is simple, with anchor tasks, of course, BUT HOW?

I have attended several seminars and sessions that have done a great job of explaining what an anchor task is and how using anchor tasks can transform my instruction while meeting the needs of all learners. Few, however, have explained how to implement them into my daily lessons.  I have been told anchor tasks are right there in the materials, but I have yet to come across a section labeled, “Anchor Task.”

In a recent seminar, hosted by Dr. Yeap Ban Har, I finally got the explanation I had been searching for… I had been looking in the wrong place!  Anchor tasks are not found in the Primary Mathematics Teacher’s Guides, but rather in the textbooks.

Dr. Yeap described the evolution of the term on his Facebook page:

Basically, an Anchor Task is the concrete component of any lesson!

How do I find an Anchor Task?

In Primary Mathematics 4A, Lesson 3.6c (Standards Edition) students will learn to interpret the fraction of a set as a whole number times a fraction.  The Teacher’s Guide leads teachers through an effective lesson where the teacher demonstrates how to find 1/3  of 24 using a couple of different methods.

TG - 4A - 3.6c_Page_1


I’ve included links to this same lesson in:


4A Standards TB p100To approach this lesson with more of an emphasis on inquiry learning, look to the textbook.

To create an anchor task, I took the example at the top of the page, find 1/4 of 20, and rewrote it as a word problem.  Students worked in partner groups to solve the following: There are 20 M&Ms in a bag. Three friends each eat  1/4 of the bag of M&Ms.  How many M&Ms did they eat altogether? Students were asked to find multiple ways of solving the problem and were given 20 chips to use if needed. Because our school has several Math Teachers that teach multiple grades, we devised a lesson planning document. (<-Click for a copy if you’d like to use it to plan your lessons)

Planning Sheet 4A - 3.6c Top

As students worked, I circulated around the room and quickly determined which students had mastered how to find  1/4 of 20, which students still needed support with this concept and which students were able to apply that concept to find  3/4 of 20.  Were they in the concrete, representational or abstract phase?

Planning Sheet 4A 3.6c MiddleAfter about five minutes, I gathered the students to share their methods of solving the problem.   This is where my direct instruction came in.  As students shared their strategies, I organized their independent learning into three methods.

I anticipated their strategies in my planning document and during my direct instruction I was sure to include any methods not discovered by my students on their own.

Planning Sheet 4A - 3.6c BottomStudents were then given the task of applying their newly discovered knowledge to solve the problems from the textbook, with my support, if needed.
The lesson ended with a journal prompt that was closely related to the concept learned.

A well-designed anchor task will engage students in the concrete and representational phases of learning a new concept.

Students will make connections with prior knowledge, reason and think logically to apply what they know to solve a problem with a partner or small group.  All students will be given time to work in the concrete phase to develop and hone their conceptual understanding.   As students are ready, they will naturally explore the representational or abstract phases of learning and discover strategies, or methods, for solving the given problem.  Sharing methods also allows students to communicate mathematically to explain and defend their thinking and consolidate their learning.

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