The How and Why of Math

A fun start-of-the-school-year team-builder!

Calm on down and divide.

At the beginning of the school year, it's the perfect time to establish routines and expectations for math. It's also a great time to activate students' prior knowledge (and do a little formative assessing), while also giving them some choices. It's OUR classroom, after all. Enjoy!

Successful Math Student= Math Literacy + English Language to Discuss, Explain, and Evaluate Learning

Below you will find my notes from the 2016 WIDA National Conference in Philadelphia, PA. which I attended as an English language (EL) teacher.

•View the history of WIDA as a national consortium for 39 U.S. states.
•To learn more about the experience, visit the WIDA conference website.

WIDA Mission

WIDA advances academic language development and academic achievement for children and youth who are culturally and linguistically diverse through high quality standards, assessments, research, and professional learning for educators.

Unique PD Tools for Advancing ELLs’ Mathematics and Language Development

Focus Group: 7 PD days, 25 mainstream and ESOL teachers, Show-and-Tell and small group AND whole group sharing:

-implementation of new learning/strategy in the classroom

-evidence through classroom artifacts, student work, videos

-professional discourse, all equals

-feedback from colleagues

-LOTS of research-based resources

-planning time

Mathematical pedagogy component- emphasizing TEACHING practices

NCTM: National Council of Teachers of Mathematics, created practices that teacher must do in teaching math

•Problem-solving based math

•CGI: cognitively guided instruction, language and number sense games

•math talk and discourse moves

-questioning patterns

-writing

•Using Children’s literature for teaching mathematics

Cognitively Guided Instruction

What’s most difficult and why?

How many more? (vocabulary hard) part + unknown part= whole-> whole-part=unknown part
How many left? Can be visualized (cookies being eaten) **easiest! Written as whole-part=part and had an action they could visualize (eating cookies)
There are 14 hats in the closet. 6 are red and the rest are green. How many green socks are in the closet? Most difficult because the whole was given first and very abstract (Whole-part=part unknown)

•Comprehensible Input: communicate context by talking, reading, watching about it

-students make sense of context, action/location, math problems (comes from context), computation, answer (in context)

Productive Classroom discussions:

-Selecting language-rich mathematics tasks

-Anticipating student responses

-Patterns of questioning

-talk Moves:

Revoicing
Repeating
Reasoning
Adding on
Waiting

Don’t just focus on vocabulary!

There is basic, everyday language that is involved in mathematics: sentence level, linguistic complexity

•Specific organizational styles of different methods of math

•Students need to use authentic language to communicate their ideas

•Mathematical literacy: teachers have to consider the principles of language development

•Integrate language and math: relevant, meaningful, interactive, and purposeful activities

Practices should relate to students’ languages, cultures, life experiences, allow students to construct mathematical meaning in a variety of ways, provide activities that allow for interaction and authentic use of language in different context

Exploration of instructional tools:

Cubing game: pocket dice, looking at a concept from different perspectives

Three-way tie: can be used in multiple content areas, can be used to assess background knowledge, necessitates discussion in the classroom, correcting misconceptions, practice using the terms authentically in written sentences

2x2 sentence builders: making connections between words, asking students to write
Problem-solution space: focus taken off the solution, focus on the process, stating the problem in your own words, sketch problem and solution

Discussion prompts:

make sense of the tool: explore and engage
Benefits for developing language skills: vocabulary usage, language forms and conventions, linguistic complexity
Benefits

Take-Aways:

•Instructional tools have explicit language focus (vocabulary, sentences, oral and writing skills, AND support development of mathematical ideas

•Classroom implementation of tools create opportunities for students to practice L/S/R/W

Contact with questions/requests: Galina (Halla) Jmourko: jmourko@pgcps.org

Rodrigo Gutiérrez: rodrigog@umd.edu

On a normal day, an "I wonder..." question can be solved with math!

In mid-October, I attended the National WIDA Conference in Philadelphia, Pennsylvania. My colleagues and I took a long walk from the conference center to the "Rocky Steps" in front of the Philadelphia Museum of Art.

After walking for about an hour in the lovely fall weather, we stopped at the statue before the steps. Two of my colleagues checked their pedometers...and they showed completely different distances! How is it possible that two people could have walked different distances while walking side-by-side.

All 5 of us discussed this. "Well, maybe one of your fit-bits is broken, " was one suggestion. Another said, "And you ran to the store back there. Maybe that accounts for how much further you walked," but a friend interjected "She didn't take that many more steps to get there!"

A colleague of mine who loves solving puzzles chimed in: "Maybe it's the difference in your strides!" My two colleagues with pedometers looked interested. I suggested that we could use math (specifically multiplication) to see if stride length x steps could = distance. But how? Well, they could start at the same point on the sidewalk and count steps until their strides met again: the lowest common multiple of their stride length (we didn't have a tape measure or yardstick with us:)

Well, my coworker in purple, although it's hard to tell from the angle of this photograph, is about 4 inches shorter than my red-shirted friend. When they reunited in step, it was a ratio of 4:3. We all said or thought "Of course! Two people can walk the same distance together and it might seem like they've walked different distances if we consider their steps of equal distance. But since their strides are different, the number of steps taken by "purple" was much great than those taken by "red:" 4 steps for every 3 taken, to be exact.

From research, here's data that shows how their strides might compare:

A widely quoted estimate of stride length is 42 percent of height, although further research shows that ratio is only moderately accurate. Rough estimates of steps per mile based on a stride to height ratio are:

Height	Steps per Mile
4 feet 10 inches	2,601 steps
4 feet 11 inches	2,557 steps
5 feet even	2,514 steps
5 feet 1 inch	2,473 steps
5 feet 2 inches	2,433 steps
5 feet 3 inches	2,395 steps
5 feet 4 inches	2,357 steps
5 feet 5 inches	2,321 steps
5 feet 6 inches	2,286 steps
5 feet 7 inches	2,252 steps
5 feet 8 inches	2,218 steps
5 feet 9 inches	2,186 steps
5 feet 10 inches	2,155 steps
5 feet 11 inches	2,125 steps
6 feet even	2,095 steps
6 feet 1 inch	2,067 steps
6 feet 2 inches	2,039 steps
6 feet 3 inches	2,011 steps
6 feet 4 inches	1,985 steps

In the moment of asking and answering an "I wonder why...?" question out of the classroom and on vacation, I exclaimed "This is authentic math!" and wanted to document this one moment when math is used in real life.

I would love to try to go on a walk with some of my students (of varying heights) wearing pedometers (without mentioning the purpose), and have them lead their own mathematic inquiry about how the distances-walked afterward were so different! With hope, their dialogue around this could lead them to their own hypotheses and I, as facilitator, could guide them to the same test and reasoning that I came to with my colleagues. Math is understanding life, and multiplication fluency is an everyday tool are two mantras I want my students to take away from a year with me as their teacher.

Want to see an exemplary teacher explain how to draw students in with authentic math? Watch this!

Multiple Meaning Math

For my 1st grade students, all English language learners, I've been focusing on how the meaning of known words can be applied in math. Concepts taught at this grade include addition, subtraction, counting coins, tally marks, telling time to the half-hour, and the attributes of shapes. Since they also must be able to count by 2's, 5's, 10's up to 100, the number line and number grid because valuable tools to visualize the problems we work with in a more concrete way.

Still, the way we talk about math can be confusing! For example, if you ask a 6-year-old to "count up from 2 to 5," processing the meaning is a challenge. "Up," "to," and "from" have multiple meanings, and none of them will have been heard or seen by ELLs before when talking about a number line. "Up" is above me, "to" is where I'm going (e.g. "to school") or what I'm going to do ("to eat lunch"), but rarely where I stop. Finally, "from" usually indicates the person who gave me a gift, not where I started from. So activities with a number line on the floor, with explicit instruction that defines the "math-meaning" of these words is helpful for all students.

Another example of multiple-meaning words challenging understanding are the words "more" and "less" when applied to numbers, not visual/tangible quantities. Consider: what would a child think when you ask "Which is more?" and the choices were two numerals. I would imagine the thought processes might go like this:

"Hmm...the numbers are the same size. They are close to each other. Let me get my unifix cubes...wait, what was the question?"

So for the past two months, I've been teaching small groups during math stations, about the questions we often discuss that contain the words "more" and "less" and how we can consider those questions using manipulatives to explain our thinking.

Here is the vocabulary card I created to explain the meaning of "more" and "less" in math:

We started by identifying, between two quantities, which was more and which was less. Then I extended the challenge into math questions using the comparatives "more than" and "less than" with kinesthetic learning. We said together "more" while moving our hands apart, as though around a balloon being inflated; and "less" with our hands starting apart and coming closer together. We also had a discussion about connecting the words "more" and "bigger," "less" and smaller.

In another lesson, we read this math text:

Reading this book, which focuses on seeing "[quantity] more" animals, helped students connect more with addition. They showed me how they could use their fingers or unifix cubes to find the answer. They practiced writing the number models on their whiteboards (i.e. if we saw two more tigers, how many did we see? 3+2=5) and justifying why they chose to add, "...because more means bigger"). We also practiced extending this understanding to the number line. I modeled, then they tried counting up or down (words I didn't use in these lessons; not yet) when we talk about more and less. They started to show me which direction (left or right) they would count on a number line if we heard the words "more" or "less," and could explain why.

After 3-4 lessons about the use of "more" and "less" in math, I chose to give a formative assessment to see which students needed support with answering questions containing this words and how. To do so, I included questions related to visuals, number lines, and simply words. I gave my students the choice to use their fingers, unifix cubes, and/or the number line to explain their answers.

In the following video, I talked with my student, who goes by AB, about the challenge question. It was important for me to see that he understands "more" and "less" better when he approaches a problem with the number line; using unifix cubes to show "more than" is still a strategy that I'll continue to address. Check it out and let me know what you think!

Mathematically Motivated

Conquering Math Anxiety - The Power of Yay Math: Robert Ahdoot at TEDxAJU

"It's become my life's work to understand how and why people arrive to that reaction ("Uhhg. I hate math!" expressed with anger and inadequacy) and tonight I'm going to share with you the solution to it." Let's listen in:

(See the link to his "Yay Math" website in my blog/website links below).

Growth mindset is a trending term amongst educators, with legitimate reason: sometimes feeling overwhelmed with the Common Core Standards and assessments, new curricula, and growing teacher turnover can lead to pessimism, which seeps from teachers' thoughts to their actions and then directly to their students' attitudes. Teachers and students both benefit, academically and emotionally, from maintaining a growth mindset and developing the teachable (and trainable) trait of optimism.

Developing GRIT takes time. Lots of time. However, teachers should start their students on the right foot by teaching, explicitly, that math is learned by making mistakes. Positive attitudes and growth mindset help us learn from our mistakes; focusing our feelings on the "right" answer is actually the wrong way to do math.

To demonstrate this, I would do some basic inquiry/discussion lessons where we, as a class, work together to solve a question whose type we may have never seen before. I'll time us, explaining to students that the time we spend to try, try, try, reflect, and learn from mistakes will show that we're smarter mathematicians. We should be the tortoise in this race, because it's not a race at all.

After each lesson, we would share how the lesson made us feel. Feeling "smart" due to speed of answering would be a marker of misconceptions, and I would review that grit is a mathematician's trait that we need to work on and that speed can lead to frustration if we don't have that positive personality yet.

The amount of time we spend on each day to solve a problem with mistakes would be charted. Here's an example of how it could look, as created on Kids' Zone: Learning with NCES.

I would explain to students that this isn't showing that we race to correct answers; instead, it demonstrates that the time we spend following Pólya's principles (Understanding the problem, making a plan, trying the plan, and reflecting on our successes or mistakes) focuses how we approach a problem and adds to our strategy tool box.

Every time we work slowly and learn from mistakes, our toolbox collects more tools and our smiles stay firmly on our faces. This type of behavior must be taught through modeling, practice, practice, practice, and also PRAISE, PRAISE, PRAISE while bringing the fun!

Curious George (Pólya)

I'm curious about math? How about you?

Principle 2: Devise a plan

The National Council of Teachers of Mathematics (NCTM) developed a standard for learning math: "Apply and adapt a variety of appropriate strategies to solve problems." When students are differently-abled or learning math literacy in English, spending time talking about problems they're looking to solve provides students with a classroom full of brainstormers! Through this method of instruction, students are exposed to more strategies they could possibly use and the scaffolding to formulate and explain their thinking.

This video shows how to monitor every student's thought process in choosing a strategy to solve a problem (although I would ask them to explain how they'll use the strategy they chose in addition to explaining why they chose that strategy);

Here is an example of the need for students to be taught how to talk in every content area (skip to minute 3:44 to see students talk about math)!

Check out this example of a 4rd graders using each of Pólya's principles (at their teacher's suggestion!)

Principle 3: Carry out the plan

This relies on the nurtured character traits of persistence and patience. Problems must be attacked in a slow, careful, measured, and precise way. What does it mean to be precise, though? Here's a video that teachers can use as a first step toward preparing their students for carrying out their strategies with precision.