number sense in secondary education

Usually, when I think of building number sense, I think about primary math education.  I know, though, number sense is something we all must work on all the time!  I'm curating ideas for secondary math and building number sense.

1. Knowing how precisely a high school freshman can estimate the number of objects in a group gives you a good idea of how well he has done in math as far back as kindergarten, researchers at Johns Hopkins University found.  This Science Daily article is quite interesting.


2. This is a go-to book for ideas: Building Powerful Numeracy for Middle and High School Students, written by Harris and published by Heinemann.  


3. This site says it's a community for number string design!


4. The state of Texas has a well-developed academic competition developed by the University of Texas - one of which is number sense.  Practice tests are available online here.
   


5. The San Francisco Unified School District Math Department has put together a page full of resources on number talks. Check it out!


6. Fawn Nguyen put together this set of 90-number talks for middle schoolers — enough for a full school year if you do this routine every other day.

What tools, routines, or materials do you use to build number sense in secondary students? How often do you work on number sense? How important is it in your math program?

Factors and Multiples Game - prep for Factoring!

I saw this tweet this morning ...

Low entry, high exit.https://t.co/ruIXZm809F#mtbos pic.twitter.com/eSutEl0FfY

— Matt Vaudrey (@MrVaudrey) September 23, 2016

I decided to check out the game!  LOVE it for a bit of fun with factors and multiples!  Here's my first result ...

It would be fun to do this with students ... then post different chains around the room, have students analyze moves, suggest different patterns. What could I have done differently in the chain above to increase the number of links?

Use the activity BEFORE factoring polynomials to improve thinking about possible factors!

Game links ... here and here.

#70Days Number strings

Math Talk
Number Talk
Strategy Sharing
Number Strings
Cluster Problems
Problem Strings

This morning I checked out the "youcubed" site to see what had been uploaded recently.  On the teachers' page, the first video/article is about number talks.  Since I plan to start the year with these, I've been reading as much as possible on the topic.  I took part in Jo Boaler's course, How to Learn Math last summer and was overwhelmed with making connections to advanced algebra 2 (a new/different curriculum for me).  Now that I have experienced the curriculum and the "typical" student, I am ready to infuse the course with more numeracy activities for sure!

What caught my eye on this site was in the "comments" section of this first article.  Someone shared a site new to me on "Number Strings."  I read Building Powerful Numeracy for Middle and High School Students this summer so I was somewhat familiar with the idea of a number string.  This new site on strings is an excellent resource.  Number strings start with a helper problem - a basic problem in which students share their answer and explain their strategy.  Typically in a string there are additional problems related to the original helper problem.  It might look like this:

3 is 100% of what number?

3 is 50% of what number?

6 is 50% of what number?

3 is 25% of what number?

6 is 25% of what number?

3 is 10% of what number?

6 is 20% of what number?

3.8 is 10% of what number?

This string is longer than some.  It is inspired from a multiple choice test question: 3.84 is 12% of what number?  The choices of answers included:  0.03125, 3.125, 32, or 46.08.  In the discussion online,  a teacher remarked that she didn't want her students to use a calculator or set up an equation.  Instead she wanted to prepare them to answer the test question by working on number strings.

I know I want to work on fraction and decimal skills with my students.  I'm debating about whether to start number talks with whole numbers.  I'm guessing that my 9th and 10th graders have never participated in a number talk ... unless they have participated in our Texas UIL number sense competition training.

Where do I begin?  Will it be OK to jump in with fractions - or will that stymie conversation too much?  Thoughts?

#70Days Lead from the middle!

I read an excellent article today ... Why Do Americans Stink at Math?  Please, read it!

Several years ago I read the book, Innumeracy: Mathematical Illiteracy and Its Consequences by John Allen Paulos.  It was interesting to see documented the very thing I had heard too many times.  Parents would say, "I was never good at math either.  I can't even balance my checkbook."  Not many people would admit to not being able to read the newspaper, but parent after parent would say they could not do simple arithmetic. Now my guess is those parents could do basic operations but they lacked confidence in mathematical understanding.

As teachers we tend to teach the way we were taught.  Parents are often quite comfortable when we do that.  In many ways our math instruction has not changed in the last 10, 20, 50 years.  Teaching rote algorithms, memorizing rules, applying them to routine exercises - these are standard practices in math classes.

So what is the concern?  Math is so much more than rote algorithms!  And our students need to understand numbers conceptually, able to manipulate them easily, make decisions involving number comfortably.

The author of the article today points back to the 1960s and "new math."  That was a prime opportunity for math instruction to make a shift but it did not.  Today it's the Common Core State Standards.  There is such furor over the CCSS that it makes one wonder if opponents have even read them.  Why wouldn't we want our children to be proficient in mathematical practices as described here:

1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.

These eight practices are in all grade levels; they are the backbone of the Common Core State Standards in mathematics.

The author points to teacher training as a primary culprit.  As I mentioned before, we teach as we have been taught.  Teachers are asked to implement new standards, to emphasize critical thinking and deep understanding of concepts when many of them admit - "math is not my thing."  In my early college days I told my advisor I wanted to teach elementary school.  At that time, the degree I would earn was in elementary education.  We also had to declare a minor.  When I said I wanted my minor in mathematics, she shook her head and said all elementary teachers need to minor in reading.  I was stubborn and insisted on studying math.  While I did take quite a few math classes, I took only one "methods" course in teaching math.  To be honest I don't remember much about that course. Today I hope pre-service teachers discuss number routines, number talks, making meaning using manipulatives, how to engage young students in math discourse and problem solving.

So what can be done?  As teacher leaders, we need to lead from the middle.  Every teacher needs a PLN - personal learning network.  Blogging and Twitter are the go-to platforms for just in time training!  Not only that they provide a safe place to write about the realities of the classroom, to ask for feedback, to share teaching ideas, and to learn new strategies.

If you are not already involved in the Math Twitter Blog-o-Sphere (MTBoS), I encourage you to visit this site, set up a Twitter account, and start a blog for the new school year. As you "meet" new teachers online, follow them on Twitter, read their blogs, even make comments.  Join a math chat!  Reach out ... ask for feedback.  You are not alone in your math teaching journey! Together we can change the headline ... "Why Do Americans EXCEL in Math!"

Numeracy - formulating my plan of attack!

I just ordered this book for summer reading and planning: Building Powerful Numeracy for Middle and High School Students by Pamela Weber Harris.

This year I have had the privilege of working with advanced students taking a pre-AP Algebra 2 course in ninth and tenth grades.  I enjoy working with them tremendously.  And I am shocked at the apparent lack of number sense.  Fractions, in particular, have been troubling for my students.  They don't understand the concept of a common denominator, and especially don't understand how it is related to the factorization of the given denominators.

Next year I want to address number sense in a strategic way.  

Already in our math department we read this article  recently and discussed it briefly.  I also submitted the article to Read, Chat, Reflect so I hope to discuss numeracy at the secondary level sometime soon on the Twitter chat, #EduRead.

Our PLC also discussed numeracy.  Next year every test will have a non-calculator portion as well as a calculator portion to force the use of mental arithmetic.

I wrote about numeracy last year in a blog - but now I need to re-read it, digest it, and work with the resources I found then!

Then later this summer, I am presenting in our district's summer professional development extravaganza on the topic of "warm-ups."  I submitted this proposal to share the wonderful resources I've learned about through #MTBoS but also to give myself a contextual reason to build systematic warm-ups addressing numeracy for the fall.  (Planning to share Estimation 180, Visual Patterns, 101 Questions, Would You Rather, and more!)

I have noticed in my search that little is written on numeracy at the secondary level.  If you have materials, suggestions, an outline of expectations in numeracy in high school math, I'd love to see those.  AND if you are interested in collaborating on such an outline, let me know!  You can reach me at educator {dot} ch {at} gmail {dot} com.

#MTBoS30

Primary Dicey Numbers!

I had fun tutoring elementary students this summer.  We used tiles, cards, pattern blocks, dice, and poker chips to build number sense.

One of the activities we did was to roll 3, 4, or 5 dice to create the largest possible number, then the smallest possible number.  We added them, subtracted them, graphed them on a number line, and talked about the patterns we saw.

Do you need extra practice for your young mathematicians?  If so, check out this free center activity, Dicey Numbers!

How to Learn Math #5: Number Sense!

I worked on Session 5 of How to Learn Math yesterday.  This course both affirms my knowledge and practice ... AND it challenges me to grow and learn!  I have long believed that building number sense is huge and that we don't do enough of it in schools.  I have been tutoring young students this summer - students going into grades 2, 3, and 5.  These students in particular needed help with composing and decomposing numbers and in using "friendly" numbers to assist in problem solving.



As I thought about number sense I was remembered that I posted about that topic several months ago.  I am repeating the post below because it has some useful links in it!

~~

Usually when I think of building number sense, I think about primary math education.  I know, though, number sense is something we all must work on all the time!  So I did some Internet research to find resources and ideas for middle and high school.

1. Knowing how precisely a high school freshman can estimate the number of objects in a group gives you a good idea how well he has done in math as far back as kindergarten, researchers at The Johns Hopkins University found.  This Science Daily article is quite interesting.


2. One of my first "hits" in my search revealed this book, Building Powerful Numeracy for Middle and High School Students written by Harris, published by Heinemann.  The Heinemann website has a brief description.  I plan to put this book on my summer reading list!


3. If you are looking for number sense strategies and tricks, this website looks awesome!


4. The state of Texas has a well-developed academic competition developed by the University of Texas - one of which is number sense.  Practice tests are available online here.


5. NCTM has published several number sense tasks - with explanation in their Reasoning and Sense Making Task Library


6. The state of Washington has posted sample number sense writing prompts for all grade levels.


7. San Diego school system has posted ideas for number sense routines.


8. University of California also published a workbook on number sense.


9. Minnesota has published an interesting paper showing the progession of number sense through primary years to high school.

I enjoyed this research ... now ... how shall I build the recommendations and examples into my daily practice?  My students would definitely benefit!

I'm working on a plan for warm-ups ... for when students first enter.  I know I want to incorporate number sense activities, as well as problem solving and review of the daily curriculum topics.  Once a week I want to use Math Munch in some way.  There are so many ways to go ... so much to do ... How do you structure those first 5 - 10 minutes of your class?

I look forward to hearing about what others do to build number sense!

Primary Math Number Mysteries

Do you love a good mystery?

I am less than 100.I am an odd number.
My tens digit is the same as the number of days in a week.
My ones digit is the number of fingers on one hand.

What number am I?

Our math resources sometimes lack enough practice for students to build independent math skills. 

Number Mysteries is a free resource with eight place value mysteries that could be used as extra practice for seat work or a math center.

Download the pdf here or here!

In Number Mysteries, students identify a 2-digit number using clues. Clues involve common numbers such as “half of a dozen” or the “value of a nickel.”

Students could be asked to represent the mystery number using concrete materials, drawings, or in standard form.

After working through these exercises, students should create their own puzzles.  Student-created puzzles with illustrations could be put together in puzzle books and/or hung in math centers to challenge classmates.

Tutoring this summer!

I'm going to do some tutoring this summer with elementary students - grades 2, 3, and 5. The school they attend uses the Singapore math books. I'm putting together my manipulative kit ... found

most of what I need at the dollar store!

This is what I have on my list so far ... Sharpie marker, whiteboard, whiteboard markers, dice, poker chips – 4 colors to make place value chips, flat square tiles, index cards to make number bonding cards, and sticky notes. I'd love to find one set of base 10 blocks.

In addition I've been scouring the Internet for engaging learning activities that focus on number sense. I've collected several thanks to the many math bloggers!

I also created an activity - a simple hundred board counting activity where pieces of the board are broken and numbers are missing. Students simply fill in the missing numbers. The focus is on counting by ones but also +10 and -10.



(Download free copy of Hoppy Hundred Board Activity)

What are your favorite manipulatives for helping children with number sense?  Do you have an activity to share?

I am thinking of a number ... eureka!

Students love to unravel “I’m thinking of a number” puzzles! 

Use these puzzles to introduce solving equations intuitively. Or use these puzzles after students have learned how to translate words into equations. Either way, your students will enjoy the work while reinforcing basic solving skills.The puzzles relate to these common core math standards:

• CCSS.Math.Content.5.OA.A.2 Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product.
• CCSS.Math.Content.6.EE.B.6 Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.
• CCSS.Math.Content.7.EE.B.4 Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
• CCSS.Math.Content.8.EE.C.7 Solve linear equations in one variable.