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Volume on!

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For anyone wondering, this is how it goes down…👀😂 from one of @Louis C.K. ‘s specials. 👍🏼☺️#teacher #school #teachersoftiktok #fyp #foryou 

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@teachwithtadgh
For anyone wondering, this is how it goes down…👀😂 from one of @Louis C.K. ‘s specials. 👍🏼☺️#teacher #school #teachersoftiktok #fyp #foryou
♬ original sound - Brian Atkinson

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1 Describe in one phrase your understanding of how children best learn

2 Describe in one phrase your understanding of inquiry learning

3 Record the area of mathematics teaching you are most confident about

4 Record the area of mathematics teaching you are least confident about

5. Describe in one phrase why math instruction is different than other subjects.

Write your response to each question on one sticky note or index card. Include the number of the question.

Groups choose one of the questions, synthesize the answers and represent the information in some visual format (pie graph, tally marks, bar graph, venn diagram) that can be shared with the rest of the group.

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Looking at the Why?
Prior to becoming a PYP teacher, when I reflect upon my teaching I think I was a so-so maths primary teacher. I understood the skills and concepts children in my class should be learning and tried to make maths learning fun mixed with the all-important skill rote learning routines I had also grown up with whilst in school which still seemed to be the expectations of the day.
Joining my first PYP school 7 years ago, I started experimenting with how we could use the PYP framework with maths learning.
I began playing around with central ideas for my students to explore and lines of inquiry that matched PYP key concepts. That became a big game changer for me in how I'd help guide my students towards their maths learning and importantly, allow them to take ownership of their learning.
When we create a successful maths planner using a strong central idea married with supporting lines of inquiry (matched with key concepts) some of the benefits are:
° students have a clear understanding of what the unit is about
° it gives them freedom to explore areas of maths thinking that interests them
° it transforms the learning to become inquiry-based learning in a natural way
° it creates a shift from skills-oriented to more meaningful concept-oriented learning
° it provides them with opportunities to take ownership of their learning
° by generating their own questions to explore they are doing what real mathematicians do and their engagement increases
° they gain a sense of pride in what they have decided to enquire into

Central Idea:
The tricky part, which I still stumble over occasionally, is getting that great central idea. It takes time and a lot of thought to create a central idea that allows children to:
° foster a curiosity towards the concepts
° instill a need to explore skills & strategies
° to be broad enough to allow students to take ownership and inquire into avenues that they find interesting in that particular maths strand. How?
° should be able to cater to all the different inquiries when they reflect during their summative (see below)

(I wish the IB would create some 'think tanks' and share possible central ideas with possible lines of inquiry out to PYP schools as a starting block for teachers.)
Present or Don't Present a Central Idea?
There are two camps it seems amongst PYP teachers: those that present a central idea towards the beginning of a unit and have it displayed for constant reference and discussion and those that don't reveal the central idea till the end of a unit or have children determine what the central idea could be as a summative.
Personally, I'm in the former camp and so I feel it is important to discuss a maths central idea early on in a unit. By presenting the maths central idea, kids are encouraged to generate their own questions in small groups of things they feel they need to know and what they need to do to gain a deep understanding of it. This provokes curiosity towards the maths topic and also allows them to take ownership of their learning because they will be given opportunities to explore their own questions during the unit.
Quite often we are in the midst of a maths unit and when I see all the different avenues the children are taking their learning, I can find faults in the central idea we are discussing. The central idea can sometimes feel limiting to where they have actually gone with it so at the end of the unit, we discuss how the central idea could be improved. This serves as a in interesting reflection activity for the children and it helps me create better central ideas for next year. My students enjoy when I share with them: "This was a central idea students last year created. Let's explore it and see what changes we think we should make to it over the next few weeks." They appreciate the idea that there new central idea they create will then be used in the following year for the next students to explore and possibly debate over its effectiveness.
Lines of Inquiry
Using the PYP key concepts to form lines of inquiry, I feel, is an easier process. You can look at the maths concepts and skills you want the children in the unit to explore and so use this to create the lines of inquiry.
PYP Maths Scope & Sequence, the conceptual understandings can be a good place to start creating a central idea. Tweaking them to add some grit for students to inquire into rather than just being told the fact needs to happen.

An example maths unit:
Central Idea: When angles co-exist, connections and relationships form.
° FORM: What different types of angles, triangles and quadrilaterals are like
° FUNCTION: How we can estimate and measure the size of angles
° CONNECTION: Patterns and relationships that exist between angles
° CONNECTION: How we use angles in our daily lives
Because the maths unit becomes student-led due to them generating their own questions to explore from the central idea, sometimes areas outside the planned lines of inquiry are examined. I think this is great.
The children needn't be constrained within the lines of inquiry I had planned for the unit. Usually I don't share the lines of inquiry with the children anyway. I use them as a personal skeleton guide of where I feel the learning should go based on the scope & sequence. If particular students or the class as a whole,aren't taking their learning to broad enough avenues, then I can refer to these to help guide the class.
Everytime I create a maths unit, I always create a line of inquiry using 'connection'.
Eg, CONNECTION: How we use angles in our daily lives
CONNECTION: How we use mental maths strategies in our daily lives
CONNECTION: How we use 3D shapes in our daily lives
This serves as a reminder to me that I need to guide the children to understanding why we are learning about this maths topic. Children need to know the relevance of what they are learning and by finding a connection that what they are learning does or will relate to their own life experiences, they will become far more engaged. An easy, but powerful way for the children to inquire into this is they will often ask their parents how they use this particular maths in their lives. We collate all the parents answers and discuss together.
Since the unit becomes student-led, the summative assessment needs to accommodate all the diverse learning that took place. It would be ridiculous and grossly unfair to have my class all inquire into different avenues of a maths concept only to then hit them with a traditional maths test which wouldn't serve any real purpose. So, what I have found to be a powerful assessment (and reflection) tool is to simply give the children the central idea and they write /draw what they understand about it.

Here are some blogspots explaining how the summative assessment works and the advantages I have found in using them: ° °

Measuring Time Summative Assessment
Probability Summative Assessment

Blogpost with some example central ideas & lines of inquiry I've experimented with:
Example Central Ideas & LInes of Inquiry

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Looking at the Why? Prior to becoming a PYP teacher, when I reflect upon my teaching I think I was a so-so maths primary teacher. I understood the skills and concepts children in my class should be learning and tried to make maths learning fun mixed with the all-important skill rote learning routines I had also grown up with whilst in school which still seemed to be the expectations of the day.

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How is mathematics taught using an inquiry approach?  How is mathematics addressed within and outside of the context of the programme of inquiry?  How do learners construct and transfer meaning and then apply with understanding?

The aim of this workshop is to support you with the skills, knowledge and ideas needed to successfully address the teaching and learning of mathematics within the context of their educational setting.

You will have an opportunity to discuss the teaching and learning of mathematics, share resources and work on relevant curriculum documentation.

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camsbaugh6@gmail.com

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8 Habits of Teachers Who Support Productive Math Struggle

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8 Habits of Teachers Who Support Productive Math Struggle

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Productive struggle is the kind of effortful learning that develops grit and creative problem solving. It results in students understanding content at a deeper level and applying that learning to more difficult and complex problems. It's what we want for all students. But did you know that some teaching methods could block students' potential for developing productive struggle in math?

Here's a roundup of information on some common teaching habits and their unintended consequences:

1. Calling on students who know the right answer

It may be tempting to call on the Hermione Grangers in class, but by doing so we miss an opportunity to show students how to respond when they get stuck. Consider that the kind of modeling you want to share with the whole class may be to watch Ron Weasley answer incorrectly. Then you can take the opportunity to help students ask questions to work through the problem. When you use an "I don't know" moment to trigger questions rather than a feeling of failure or disappointment, you're helping all students. It shows that you value the process of making mistakes and figuring things out.

Helpful resources:

2. Praising students for their smarts

Researcher Carol Dweck shares that praising students for their intelligence can have negative consequences on their motivation to achieve. Students are actually more motivated to face challenging tasks when teachers congratulate them for hard work, rather than for getting the correct answers. However, this does not mean that learning is just about effort, says Dweck.

"Students need to try new strategies and seek input from others when they're stuck. They need this repertoire of approaches-not just sheer effort-to learn and improve" ( Dweck 2015).

Try praising for creativity in figuring out a problem, playing with ideas, thinking out of the box or persevering in the face of challenge.

3. Creating bulletin boards to display high achievement

Turns out students show more engagement and better learning outcomes when they see examples of perseverance in learning rather than great feats of intellectual achievement, according to this study on motivation. So instead of making an example of high scores, try sharing examples of creative problem-solving or progress toward a learning goal.

Check out these ideas from other educators on creating custom goals for your classroom that foster deeper learning and intrinsic motivation.

4. Focusing on teaching procedures and formulas

Memorization and drills don't develop effortful, deeper-level thinking. For this goal, try presenting students with non-routine problems. Allow them to figure out what math they need to use and their own method of solving the problem. Letting students visualize the problem and find non-formulaic solutions increases engagement and promotes creative problem-solving.

Explore examples of visual math activities:

5. Marking student responses as right or wrong

"Right or wrong" feedback can actually lower student learning outcomes, according to studies. Students need informative feedback in order to course correct toward a solution. Informative feedback goes beyond getting the right answer and helps students keep questioning. It provides context to why the answer was right or wrong. Teacher Linda Himes, with 36 years of classroom experience, shares this example:

"I watched a student struggle with creating equivalent fractions. Through his own diligence, he finally figured it out. Instead of saying, 'Good job!' or 'See, I told you you were smart,' I said, 'Wow, you just figured out how to make equivalent fractions by seeing the relationships between the numbers in the fractions." The grin that spread across his face was priceless. He knew he had done something challenging. His attitude towards math changed in that moment!"

6. Giving easier work to struggling students

This sends the message that some students can't handle challenging work. Instead, look for instructional activities that allow students at all levels to access the learning and then progress to each individual's edge. Check out these examples provided by youcubed, an organization co-founded by education author and Standford mathematics professor Jo Boaler.

7. Following a strict schedule for covering new material

Allowing students to struggle in the process of learning new material takes time. Just because students didn't reach the right answer in the allotted time doesn't mean learning didn't happen. When students are encouraged to engage in the process of productive struggle, they generate questions and ideas around the problem. This process is critical to reaching deeper levels of understanding a topic.

"Conditions of instruction that appear to create difficulties for the learner, slowing down the rate of apparent learning, often optimize long-term retention and transfer." - Robert A. Bjork, from Making Things Hard on Yourself, But in a Good Way: Creating Desirable Difficulties to Enhance Learning

8. Making students feel ok about not being a "math person"

Research shows that everyone is capable of mathematical thinking and that embracing a growth mindset can have powerful effects on learning. On the flip side, soothing students' math frustration in this way can result in resignation and blocking productive struggle. Whether it's math anxiety or a preconception that there are people who are naturally good or bad at math, remember that attitudes about math and self-perceptions affect student learning.

For a reminder of the kind of teaching habits that develop productive struggle in all students, not just in math, download this printable poster:
MIND Research Institute is a social benefit organization working to ensure that all students are mathematically equipped to solve the world's biggest challenges. MIND also created ST Math, a visual instructional program that builds a deep conceptual understanding of math through rigorous learning and creative problem solving to engage, motivate and challenge PreK-8 students toward higher achievement. 8 Habits of Teachers Who Value Productive Struggle PDF

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8 Teaching Habits that Block Productive Struggle in Math Students

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Which common teaching methods might be hindering productive struggle in math? Check out this free poster to help students embrace productive struggle!

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Graham Fletcher Progressions Video Series

Open Middle

Math Proofs

Youcubed

Inside Mathematics - problems of the month

Nrich maths

Illustrative mathematics

Robert Kaplinsky Lessons

Which One Doesn’t Belong

Would You Rather Math

Three Act Tasks

Number Talks

Counting Collections

Numberless Word Problems

High Yield Routines

High Yield Routines - illustrated slides

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Open Middle

Math Proofs

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Beliefs About Teaching and Learning Mathematics

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Beliefs About Teaching and Learning Mathematics

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by Terry Heick
This is part 1 in a 3 part "Return To The Classroom" series, so-named because we, somehow, loathe the phrase "Back to School." Today's post is on assessment, with a look at rubrics for the Depth of Knowledge framework.
My first three years as a teacher, my district used Norman Webb's Depth of Knowledge framework.

Bloom's Taxonomy gets most of the press in education, and UbD 6 Facets of Understanding, and Marzano's New Taxonomy are also useful frameworks for dispersing "cognitive actions," but it was the "DOK" model that I used first, and so it has a soft spot in my heart. I found the numbers more useful-as a new teacher-than the abstractions of already complex language. Making sense of a taxonomy requires a teacher to understand the structure and language of the thing, internalize it, contextualize it within a classroom, and then further understand how to use it as a teaching tool.

Taxonomies can be used to create assessment, map big ideas in a curriculum, structure discussions-in fact, this should probably be a post of its own. But back to Webb & DOK-Grant Wiggins recently shared a post on assessment design-or rather, the reality of "power verbs" as they relate to rigor (always a fun word) and complexity. He shared the DOK rubric in text form, and I thought it might be useful to create a graphic for each one individually for easier sharing and curation.

The basic organization here is higher numbers = higher levels of thinking. In the same way educators value student thinking in the highest levels of Bloom's, thinking-and requisite work-designed at DOK levels 3 and 4 represent said "higher level thought."

As with all things in education, thinking is a matter of trend. DOK is seemingly trending down, while Bloom's is always in fashion. Whether or not you use it as an instructional design and teaching tool or not, the language in the descriptions below might be useful to you-if for no other reason, than to see what makes certain tasks in certain content areas increasingly complex.
A Depth Of Knowledge Rubric For Reading, Writing, And Math
Level 1: Recall of Information Level 1 requires students to receive or recite facts or to use simple skills or abilities. Oral reading that does not include analysis of the text as well as basic comprehension of a text is included. Items require only a shallow understanding of text presented and often consist of verbatim recall from text or simple understanding of a single word or phrase.

Level 2: Basic Reasoning about text Level 2 includes the engagement of some mental processing beyond recalling or reproducing a response; it requires both comprehension and subsequent processing of text or portions of text. Inter-sentence analysis of inference is required. Some important concepts are covered, but not in a complex way. Standards and items at this level may include words and phrases such as "summarize," "interpret," "infer," "classify," "organize," "collect," "display," "compare," and "determine whether fact or opinion." Literal main ideas are stressed. A Level 2 assessment item may require students to apply some of the skills and concepts that are covered at Level 1.

Level 3: Complex Reasoning about text. Deep knowledge becomes more of a focus at Level 3. Students are encouraged to go beyond the text; however, they are still required to show understanding of the ideas in the text. Students may be encouraged to explain, generalize, or connect ideas. Standards and items at Level 3 involve reasoning and planning. Students must be able to support their thinking. Items may involve abstract theme identification, inference across an entire passage, or students' application of prior knowledge. Items may also involve more superficial connections between texts.

Level 4: Extended Reasoning. Higher-order thinking is central and knowledge is deep at Level 4. The standard or assessment item at this level will probably be an extended activity, with extended time provided. The extended time period is not a distinguishing factor if the required work is only repetitive and does not require applying significant conceptual understanding and higher-order thinking. Students take information from at least one passage and are asked to apply this information to a new task. They may also be asked to develop hypotheses and perform complex analyses of the connections among texts.

Writing Level 1: Recall of Information Level 1 requires the student to write or recite simple facts. This writing or recitation does not include complex synthesis or analysis, only basic ideas. The students are engaged in listing ideas or words as in a brainstorming activity prior to written composition, are engaged in a simple spelling or vocabulary assessment, or are asked to write simple sentences. Students are expected to write and speak using standard English conventions. This includes using appropriate grammar, punctuation, capitalization, and spelling.

Level 2: Basic Reasoning Level 2 requires some mental processing. At this level, students are engaged in first-draft writing or brief extemporaneous speaking for a limited number of purposes and audiences. Students are beginning to connect ideas using a simple organizational structure. For example, students may be engaged in note-taking, outlining, or simple summaries. Texts may be limited to one paragraph. Students demonstrate a basic understanding and appropriate use of such reference materials as a dictionary, thesaurus, or Web site.

Level 3: Complex Reasoning Level 3 requires some higher-level mental processing. Students are engaged in developing compositions that include multiple paragraphs. These compositions may include complex sentence structure and may demonstrate some synthesis and analysis. Students show awareness of their audience and purpose through focus, organization, and the use of appropriate compositional elements. The use of appropriate compositional elements includes such things as addressing chronological order in a narrative or including supporting facts and details in an informational report. At this stage, students are engaged in editing and revising to improve the quality of the composition.

Level 4: Extended Reasoning Higher-level thinking is central to Level 4. The standard at this level is a multi-paragraph composition that demonstrates synthesis and analysis of complex ideas or themes. There is evidence of a deep awareness of purpose and audience. For example, informational papers include hypotheses and supporting evidence. Students are expected to create compositions that demonstrate a distinct voice and that stimulate the reader or listener to consider new perspectives on the addressed ideas and themes.

Level 1: Recall Level 1 includes the recall of information such as a fact, definition, term, or simple procedure, as well as performing a simple algorithm or applying a formula. That is, in mathematics a one-step, well-defined, and straight algorithmic procedure should be included at this lowest level. Other key words that signify a Level 1 activity include "identify," "recall," "recognize," "use," and "measure." Verbs such as "describe" and "explain" could be classified at different levels depending on what is to be described and explained.

Level 2: Skill/Concept Level 2 includes the engagement of some mental processing beyond a habitual response. A Level 2 assessment item requires students to make some decisions as to how to approach the problem or activity, whereas a Level 1 item requires students to demonstrate a rote response, perform a well-known algorithm, follow a set procedure (like a recipe), or perform a clearly defined series of steps. Key words and phrases that generally distinguish a Level 2 item include "classify," "organize," "estimate," "make observations," "collect and display data," and "compare data." These actions imply more than one step. For example, to compare data may require first identifying characteristics of the objects and then grouping or ordering the objects.

Level 3: Strategic Thinking Level 3 requires reasoning, planning, using evidence, and a higher level of thinking than the previous two levels. In most instances, requiring students to explain their thinking is a Level 3 activity. Activities that require students to make conjectures are also at this level. The cognitive demands at Level 3 are complex and abstract. The complexity does not result from the fact that there are multiple answers, a possibility at both Levels 1 and 2, but because the task requires more demanding reasoning. An activity, however, that has more than one possible answer and requires students to justify the response they give would most likely be a Level 3 activity. Other Level 3 activities include drawing conclusions from observations, citing evidence and developing a logical argument for concepts, explaining phenomena in terms of concepts, and using concepts to solve problems.

Level 4: Extended Thinking Level 4 requires complex reasoning, planning, developing, and thinking-most likely over an extended period of time. The extended time period is not a distinguishing factor if the required work is only repetitive and does not require applying significant conceptual understanding and higher-order thinking. For example, if a student has to take the water temperature from a river each day for a month and then construct a graph, this would be classified as a Level 2 activity. However, if the student is to conduct a river study that requires taking into consideration a number of variables, this would be a Level 4 activity.

At Level 4, the cognitive demands of the task should be high and the work should be very complex. Students should be required to make several connections-relate ideas within the content area or among content areas-and have to select one approach among many alternatives on how the situation should be solved. Level 4 activities include designing and conducting experiments; making connections between a finding and related concepts and phenomena; combining and synthesizing ideas into new concepts; and critiquing experimental designs.
A Depth Of Knowledge Rubric For Reading, Writing, And Math
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A Depth Of Knowledge Rubric For Reading, Writing, And Math

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by Terry Heick This is part 1 in a 3 part "Return To The Classroom" series, so-named because we, somehow, loathe the phrase "Back to School." Today's post is on assessment, with a look at rubrics for the Depth of Knowledge framework. My first three years as a teacher, my district used Norman Webb's Depth of Knowledge framework.

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Lesson Part 1
Lesson Part 2

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1. NCTM Position
2. What Does Fluency Look Like?
3. Fluency without Fear - go to "Math Fluency and the Curriculum"

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Five Principles of Extraordinary Math Teaching

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Five Principles of Extraordinary Math Teaching

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Five Principles of Extraordinary Math Teaching | Dan Finkel | TEDxRainier

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Questioning and Vocabulary Supports That Inspire Language-Rich Mathematics

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Questioning and Vocabulary Supports That Inspire Language-Rich Mathematics

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Questioning and Vocabulary Supports That Inspire Language-Rich Mathematics

Sue O'Connell

Most of us remember learning mathematics in quiet classrooms. Teachers demonstrated procedures, students silently practiced with worksheets and workbooks, and answers were quickly assessed as right or wrong. In contrast, today's vision of mathematical proficiency blends computational fluency, conceptual understanding, and application, moving math beyond memorization and computations into reasoning and problem solving. As we work to adjust our teaching practice to align with our goal of developing mathematical thinkers, we recognize the importance of talk and writing. Through math talk, our students are able to process their ideas, hear others' thinking, and ultimately revise and refine their own understanding.

Creating language-rich classrooms can be challenging for teachers. Figuring out what questions to ask, determining how to cultivate productive math talk, and finding ways to support precision in communication challenge us as we rethink math instruction.

Questions That Promote Math Talk

What questions do you ask to promote talk and writing in math class and how do your questions foster diverse thinking? Each type of question challenges students to think about mathematics in a different way.

What is the big idea? questions ask students to use their observations and insights to generalize as they discover rules or shortcuts. For example:

What keeps happening? Why?

What is the rule?

Will it always work? Explain.

What did you learn? questions ask students to rethink lessons and summarize their learning. They reflect on what they learned and ask questions about what remains unclear. For example:

What is something new you learned today?

What questions do you have?

What is easy or hard about this skill? Explain.

With careful attention to the questions we ask, we prompt diverse thinking and stimulate rich class discussions that support our students as they test their thinking, refine their skills, build conceptual understanding, and solve math problems.

Cultivating Math Discussions

And who answers these questions? Simple instructional shifts, like asking students to reflect silently followed by partner talk, rather than calling on one student to answer a question, allow all students to process the question and voice their thinking, which yields rich classroom discourse.

Specific questions that encourage student-to-student talk serve to extend discussions and draw more students into the classroom conversations. For example:

Did anyone solve it in a different way than Liam did?

What do you think about what Colin said?

Does anyone want to add to what Bailey shared?

Some students have difficulty diving into conversations, unsure how to insert their ideas into the discussion. Supporting students by discussing and posting possible sentence starters helps them find ways to insert their thinking into ongoing conversations. For example:

I agree with Blake because ...

That reminds me of ...

I think Molly said ...

I would like to add ...

I don't understand why you ...

Could you explain what you mean when you said ...?

And creating a classroom environment that allows students to feel safe to share their thinking, even when it is in a developmental stage, relieves anxiety and leads to more engaging conversations. Some pointers for creating a safe environment for math talk include

Acknowledge the value of mistakes. Not only are they OK, but they are expected in math class.

Allow students time to share their ideas with partners and work out their thinking prior to revealing that thinking to the whole class.

Provide opportunities for students to change their minds as they hear and process others' comments.

Praise effort, perseverance, and collaboration.

When possible, deemphasize right or wrong answers and emphasize reasonable procedures, conceptual understanding, and mathematical insights.

Precision

The language of mathematics is extensive and specific, and students often lack the words to adequately express their ideas. Attention to math vocabulary, using any of the strategies below, helps students internalize this technical language and allows them to more precisely share their thinking.

Introduce new words in context, using pictures, examples, synonyms, and connections to past math ideas. Many teachers record new words on math talk charts, similar to anchor charts. The word is recorded along with pictures, examples, or numbers that help explain its meaning.

Display words for reference. Regardless of grade level, students benefit from seeing new words as they are learning them. Classroom word walls allow students to locate and use precise language as they are talking and writing.

Create ongoing opportunities to talk and write using math vocabulary. Classroom experiences like sorting math vocabulary, webbing word associations (e.g., words that go with triangle or decimal), or other interactive vocabulary tasks (e.g., writing and solving math vocabulary riddles) provide ongoing exposure so that students can internalize math vocabulary.

Through the questions we ask, the ways in which we generate math talk in our classrooms, and our intentional focus on math vocabulary, we support students in talking frequently, precisely, and productively about their mathematical thinking. Unlike the silent classrooms in which we learned math, these language-rich classrooms allow our students to make sense of mathematical ideas through their communication.

Sue O'Connell is a former elementary teacher, reading specialist, and math coach. Now a nationally known speaker and education consultant who directs Quality Teacher Development, an organization committed to providing outstanding math professional development for schools and districts across the country, Sue is also the author of Math in Practice: A Guide for Teachers (Heinemann, 2016) and Introduction to Communication (Heinemann, 2007). Connect with Sue on Twitter: @SueOConnellMath

ASCD Express, Vol. 13, No. 22. Copyright 2018 by ASCD. All rights reserved. Visit www.ascd.org/ascdexpress.

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Questioning and Vocabulary Supports That Inspire Language-Rich Mathematics

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Questioning and Vocabulary Supports That Inspire Language-Rich Mathematics Sue O'Connell Most of us remember learning mathematics in quiet classrooms. Teachers demonstrated procedures, students silently practiced with worksheets and workbooks, and answers were quickly assessed as right or wrong.

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Ensuring that the technology we bring into math classes fosters active engagement is key.

Incorporating technology in mathematics classrooms enables educators to craft powerful collaborative learning experiences that support problem solving and flexible thinking. With strategic integration of both content-specific and content-neutral technology, students and teachers can construct their learning together in authentic ways that elevate mathematics learning.

Until recently, one of educators' primary concerns around educational technology was the lack of access that existed in many American schools. That gap has decreased, but a new digital divide has emerged: The updated 2017 National Education Technology Plan explains that in today's classrooms many students are using technology as a tool for passive learning rather than engaging in active learning experiences that promote student agency.

The Importance of Content-Specific Pedagogy

In order to create technology-infused experiences that support active mathematics learning, educators must of course have pedagogical content knowledge (PCK)-an understanding of best practices specific to mathematics.

One method a teacher can use to analyze the effectiveness of technology integration is the Technological Pedagogical Content Knowledge (TPACK) framework. This tool supports careful educator reflection on pedagogy, content, and technology not only as separate entities but as overlapping and intersecting domains.
Reproduced by permission of the publisher, ©2012 by tpack.org
For example, when planning to integrate technology into a lesson, educators can take into account the technology knowledge the students will need, the mathematics content knowledge they'll need, and the best practices for teaching both the technology and the math. This process is extremely important because without it, the technology may be integrated in a way that is pedagogically inappropriate for mathematics instruction.

Using PCK to Spot Unhealthy Apps and Websites

Teachers start from their understanding of PCK, or best practices for mathematics instruction, in choosing effective technology tools for the mathematics classroom. We know that mathematics should not be focused on speed or quick answer finding. And timed fact testing is a known trigger of math anxiety, which can lead to low mathematics achievement and mathematics avoidance. Yet math apps and websites that focus on speed and rote memorization are readily available and widely used.

This use of technology can promote fear and stress, and it also sends inaccurate messages about the purpose of mathematics. Math is about thinking deeply, discovering patterns, and making connections. Automaticity with math facts and math skills is critical, but how we get students to automaticity matters. A focus on memorization without understanding promotes a joyless, nonsensical form of mathematics that requires remembering a large amount of seemingly disconnected information.

In addition, technology that simply transfers a gradual release-the "I do, we do, you do" structure-to an online format is a form of passive learning that strips math of student agency and rigor. Although gradually releasing responsibility is an effective model in other content areas, in mathematics this model is best flipped to give students the agency to decide what strategies they want to use and how they might solve a problem.

That's because students should productively struggle with math. We need them to problem solve rather than learn to repeat a specific list of procedures given by the teacher. Problem solving skills are more valuable than memorization, and they're the true work of mathematicians. If we're integrating technology into our classrooms to engage students in real-world experiences, our students must be given opportunities to do real mathematics.

Technology That Fosters Deep Mathematical Thinking

When used appropriately, both content-specific and content-neutral technology can be effective in the math classroom. Research indicates that content-specific apps and websites that focus on math learning with the use of virtual manipulatives are highly effective, and in some cases more efficient than physical manipulatives.

The Math Learning Center, for example, provides several manipulative options, such as rekenreks, geoboards, number lines, and number frames. Apps and websites that provide these types of virtual tools are easy to use, support students with conceptual understanding, and increase student access to math tools.

Content-neutral technology includes tools such as virtual whiteboards, handheld clickers, and student collaboration apps. Virtual whiteboard and websites, such as Explain Everything, promote self reflection, enable students to make their learning visible and share and connect ideas, and have been linked with high-level student thinking and teacher questioning.

But content-neutral technology that promotes fast answering, such as handheld clickers, is associated with decreased cognitive demand, most likely due to the likelihood of being used with minimal student discourse.

Being a Critical Consumer of Technology

The educational technology market is flooded with new apps, tech tools, and gadgets, and in some instances, teachers are commended for increased technology use whether it supports healthy math learning or not. Technology can have a truly positive impact on student learning, but it should not replace teaching or ignore research-based best practices for math instruction.

If we believe that students of mathematics need opportunities for discussing math, creating and connecting visuals, analyzing models, discovering patterns, and making generalizations, the technology that we introduce into our classrooms should match those values.

There was a time when simply getting technology into the hands of our students was a goal, but the time has come to slow down and plan for technology integration that truly supports healthy and productive mathematics learning.

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Ensuring that the technology we bring into math classes fosters active engagement is key. Incorporating technology in mathematics classrooms enables educators to craft powerful collaborative learning experiences that support problem solving and flexible thinking. With strategic integration of both content-specific and content-neutral technology, students and teachers can construct their learning together in authentic ways that elevate mathematics learning.

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"I see that now when I give tests back, they're continuing to learn," Alcala said.

The students get their grade a day later on PowerSchool, and they're given opportunities to retake the test after they've digested their mistakes.

Alcala also projects "favorite mistakes" on the board that they talk about as a class. And students get time to look at their own mistakes and figure out where they went wrong. The other advantage of highlighting is that she can call attention to things that she won't necessarily take points off for, but that she wants students to notice. For example, she might highlight that they didn't put the correct units in a word problem. They got the math correct, so Alcala is not worried they won't be able to move forward, but she wants to remind them that units are important.

She grades in two go-rounds. First she reads the test from top to bottom, looking for the moment when the mistake gets made in each problem. Sometimes she sees what she calls "flow through mistakes," where the student made a mistake early on, thus got the wrong answer, but all the operations after that were done correctly. Other times a mistake was made early on, but more mistakes were made after that. Those two students would receive different grades on that problem, even though they technically both got it wrong.

After the first pass of the test, Alcala looks at the test as a whole for themes in the kinds of mistakes the child is making. Is she making the same mistake over and over? Or are there lots of different types of mistakes?

"It doesn't take longer to grade tests this way," Alcala said. "I think that was a big fear. It is a similar amount of time and it's far more enjoyable."

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A Grading Strategy That Puts the Focus on Learning From Mistakes

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"I see that now when I give tests back, they're continuing to learn," Alcala said. The students get their grade a day later on PowerSchool, and they're given opportunities to retake the test after they've digested their mistakes. Alcala also projects "favorite mistakes" on the board that they talk about as a class.

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June used it the next day. It did not go well. A forest of arms immediately shot up, and June moved frantically around the room answering questions. Many students gave up quickly, so June also spent much effort trying to motivate them to keep going. In general, there was some work attempted when June was close by and encouraging the students, but as soon as she left the trying stopped. This continued for the whole period.

The following day I was back with a new problem. The results were as abysmal as they had been on the first day. The same was true the third day. Over the course of three 40-minute classes, we had seen little improvement in the students' efforts to solve the problems, and no improvements in their abilities to do so. So June decided it was time to give up.

I wanted to understand why the results had been so poor, so I stayed to observe June and her students in their normal routines. After three full days of observation, I began to discern a pattern. That the students were lacking in effort was immediately obvious, but what took time for me to realize was that the students were not thinking. More alarming was the realization that June's teaching was predicated on an assumption that the students either could not or would not think.

Once I realized this, I proceeded to visit 40 other mathematics classes in a number of schools. In each class, I saw the same thing-an assumption, implicit in the teaching, that the students either could not or would not think. Under such conditions it was unreasonable to expect that students were going to be able to spontaneously engage in problem solving.

This motivated me to find a way to build, within these same classrooms, a culture of thinking. I wanted to build what I now call a thinking classroom-one that's not only conducive to thinking but also occasions thinking, a space inhabited by thinking individuals as well as individuals thinking collectively, learning together, and constructing knowledge and understanding through activity and discussion.

Over 14 years, and with the help of over 400 K-12 teachers, I've been engaged in a massive design-based research project to identify the variables that determine the degree to which a classroom is a thinking or non-thinking one, and to identify the pedagogies that maximize the effect of each of these variables in building thinking classrooms. From this research emerged a collection of 14 variables and corresponding optimal pedagogies that offer a prescriptive framework for teachers to build a thinking classroom.

1. The type of tasks used: Lessons should begin with good problem solving tasks. In the beginning of the school year, these tasks need to be highly engaging, non-curricular tasks. Later these are gradually replaced with curricular problem solving tasks that then permeate the entirety of the lesson.

2. How tasks are given to students: As much as possible, tasks should be given verbally. If there are data, diagrams, or long expressions in the task, these can be written or projected on a wall, but instructions should still be given verbally.

3. How groups are formed: At the beginning of every class, a visibly random method should be used to create groups of three students who will work together for the duration of the class.

4. Student work space: Groups should stand and work on vertical non-permanent surfaces such as whiteboards, blackboards, or windows. This makes the work visible to the teacher and other groups.

5. Room organization: The classroom should be de-fronted, with desks placed in a random configuration around the room-away from the walls-and the teacher addressing the class from a variety of locations within the room.

6. How questions are answered: Students ask only three types of questions: proximity questions, asked when the teacher is close; "stop thinking" questions-like "Is this right?" or "Will this be on the test?"; and "keep thinking" questions-ones that students ask in order to be able to get back to work. The teacher should answer only the third type of question.

7. How hints and extensions are used: The teacher should maintain student engagement through a judicious and timely use of hints and extensions to maintain a balance between the challenge of the task and the abilities of the students working on it.

8. Student autonomy: Students should interact with other groups frequently, for the purposes of both extending their work and getting help. As much as possible, the teacher should encourage this interaction by directing students toward other groups when they're stuck or need an extension.

9. When and how a teacher levels their classroom: When every group has passed a minimum threshold, the teacher should pull the students together to debrief what they have been doing. This should begin at a level that every student in the room can participate in.

10. Student notes: Students should write thoughtful notes to their future selves. They should have autonomy as to what goes in the notes and how they're formatted. The notes should be based on the work already on the boards done by their own group, another group, or a combination.

11. Practice questions: Students should be assigned four to six questions to check their understanding. They should have freedom to work on these questions in self-selected groups or on their own, and on the vertical non-permanent surfaces or at their desks. The questions should not be marked or checked for completeness-they're for the students' self-evaluation.

12. Formative assessment: Formative assessment should be focused primarily on informing students about where they are and where they're going in their learning. This will require a number of different activities, from observation to check-your-understanding questions to unmarked quizzes where the teacher helps students decode their demonstrated understandings.

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Over more than a decade, the author has developed a 14-point plan for encouraging students to engage deeply with math content.

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Create a thinking classroom for your students

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Spread the loveThinking classrooms are something for which many teachers strive. After all, wouldn't it be great for students to tackle a problem eagerly, explore variables, and collaboratively come up with a solution? That's not always what happens in classrooms, though. Students may attempt to solve problems but give up after a few failed attempts.

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Part 1
Part 2

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Post Talk Number Talk Part 4 Number Talk Part 3 Number Talk Part 2
MIA BULJAN: Who has, who wants to tell us their way that they did it? Can I get a boy? Angel.

STUDENT: Umm...

MIA BULJAN: Which answer are you defending first of all?

STUDENT: 56.

MIA BULJAN: 56. Okay. So Angel, how did you get 56?

STUDENT: Because...I just put the fourteens together like double.

MIA BULJAN: Okay. So what did you do? You did...

STUDENT: 14.

MIA BULJAN: And...

STUDENT: Plus 14.

MIA BULJAN: So this was one double and what was your answer? Rogelio, I'm going to ask you to stop that sweetheart.

STUDENT: 28.

MIA BULJAN: All right.

STUDENT: I did solve the other fourteens, there was like 3 more left so I took up the other one and then I did the same thing with the other ones.

MIA BULJAN: So if we're looking at Marlene's picture to help us understand his way, here are the two 14's that he put together. How many fourteens does he have left?

STUDENTS: Three.

MIA BULJAN: Three. So he thought if I didn't have this one what would he have here?

STUDENT: Two.

MIA BULJAN: Two 14's. He could double 14 again and he would get how much?

STUDENT: 28.

MIA BULJAN: So he has a double double. And then what did you do Angel? Adele, stop.

STUDENT: Um, I put the 14's together...first I did it with the 28's. I got the two 10's.

MIA BULJAN: Okay. So you did, you wanted to put these together, so you thought of these two 10's. How much is that?

STUDENT: Uh, 20.

MIA BULJAN: Uh-huh. And then you thought of these two 10's. And what would that give you?

STUDENT: Um, 40.

MIA BULJAN: Okay. So this became 40 and then what happened?

STUDENT: The 8 I put together with the 40 and got 48.

MIA BULJAN: Okay. So now you put the 40 and the 8 together and got 48.

STUDENT: And with the other 8 I just took away the 6 and put 2 together and

STUDENT: I got the 56 and then...

MIA BULJAN: Why would he do that? He said inside of here I know there's a 2 and a 6. Why would he pick 2 and a 6? Hold on, hold on. I want you to think about the question. I think you have an idea but I want everyone to think about this question. 8 can be made lots of ways. It can be made with a 1 and a 7, and a 2 and a 6, and a 3 and a 5, and a 4 and a 4, but he didn't pick those ways, he only picked the 2 and the 6. Why would he do that? Why would this be helpful for him? Monique.

STUDENT: Because he might be, like adding it together to make another number, so he would make the 40 and the, and the, um, 8 to make 48, to make the 8 different.

MIA BULJAN: Make it different. And what did he do Rogelio, to make it different?

STUDENT: Um, for like the 48 into a 50.

MIA BULJAN: Ahh, he used this 2 to make it different into a 50, and that felt friendly, so all you had to do now was put your 50 and your 6 into 56. You guys do you see all the way he's breaking apart his numbers and putting them back together in ways that are easier and friendlier to deal with?

STUDENT: Yeah.

MIA BULJAN: 28+28 feels hard to do in our heads but when we think of those 20's, double 20 is so easy.

STUDENT: 40.

MIA BULJAN: And then 40+8 is so easy and adding another 8 doesn't feel easy. He didn't count on his fingers though. He could've. But he broke it apart to make it friendly. Uh-huh. Oh, hold on, hold on, hold on because he might not be done. So Angel, when you see this here do you feel like your problem is done?

STUDENT: No because I forgot the other 14.

MIA BULJAN: So this was his original answer. How many people think that this answer is correct?

STUDENT: No.

MIA BULJAN: You're all convinced that 70 is the right answer?

STUDENT: Yes.

MIA BULJAN: Marlene that was a very convincing argument. Good job. Okay. And now Angel says that as he explains this what did you notice Angel?

STUDENT: So like I forgot that there was like 5 and...

MIA BULJAN: Ahh, that there are 5 of them and you only have 1, 2, 3, 4. 4 of them. So what else do you need to do?

STUDENT: So I just have another 14 and then I just remember that it's 70 because if you add the 10 with the 5 it will equal 15 and if you add like 6+4=10.

MIA BULJAN: And you put that 60 and that 10 back together to get 70. You like that way?

STUDENT: Yeah.

MIA BULJAN: Okay. So your new answer, so you want to change this answer to 70. Does everyone understand what Angel did?

STUDENTS: Uh-huh.

MIA BULJAN: Do you understand how he made a mistake? He thought he did 4 double doubles and forgot that other 14.

STUDENT: Ms. B?

MIA BULJAN: Yep.

STUDENT: What is Angel's, um, name you would call?

MIA BULJAN: So Angel did something kind of interesting to me. He did...

STUDENT: Breaking apart?

MIA BULJAN: He used a lot of breaking apart. Didn't he use a lot of Marlene's decompose strategies? But he also did something called using a known fact. He didn't start counting from 1, he used what he knew. He knew the doubles, he knew the doubles, and then his job was to just keep track of how many times he had to put those together. So we call this using a known fact. Okay? All right, so who has one more way that we can do?

STUDENT: Ms. B.

MIA BULJAN: I know there are so many good ones.

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classroom videos /number talks /3rd grade math: one-digit by two-digit multiplication /number talk part 2 /

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Post Talk Number Talk Part 4 Number Talk Part 3 Number Talk Part 2 MIA BULJAN: Who has, who wants to tell us their way that they did it? Can I get a boy? Angel. STUDENT: Umm... MIA BULJAN: Which answer are you defending first of all? STUDENT: 56.

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Educational Research Newsletter and Webinars

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Japanese methods of teaching mathematics have been intensively studied in the United States since the Third International Mathematics and Science Study (TIMSS) revealed that Japanese students score significantly higher than students in the United States at both fourth and eighth grades. John Woodward, University of Puget Sound, and Ymiko Ono, Naruto University of Education, summarize Japanese methods for teaching mathematics and describe how the educational system addresses student diversity. There have been contradictory accounts of Japanese education. Woodward and Ono draw on a range of research conducted by both American and Japanese researchers to clarify Japan's math curricula and its approach to teaching students at different skill levels.

Curricular Materials

U.S. mathematics texts typically present a great number of topics superficially and repeat the same topics over several grade levels with only minor increases in the difficulty of the concepts. In contrast, Japanese math texts follow strict guidelines established by the Ministry of Education and contain far fewer topics. Each grade level provides greater depth on a topic. Once students reach 6th grade in Japan, the number of topics covered per year decreases greatly. By seventh grade Japanese texts cover less than half as many topics as U.S. texts. By grade 8, Japanese curricula have moved students into relatively complex mathematics while U.S. curricula still emphasizes arithmetic. Japanese texts are slim volumes with few practice problems. It is assumed that well-established pedagogical practices will compensate for what is not presented in the text.

The TIMSS data shows that by eighth grade almost half of U.S. students were in schools that offered them three or more diff e rent math classes, while 99 percent of Japanese eighth-grade students were taught the same curriculum.

Pedagogical Practices

TIMSS reveals that teachers throughout the world tend to use a common set of teaching practices. They review previous lessons, go over homework, provide instruction on new concepts, have students work on exercises, and allow time for homework. But the major difference between Japan and the United States is the amount of time set aside for each activity. U.S. teachers, especially at the secondary level, spend more time going over homework and allow more time for homework in class. They spend less time explaining new concepts and allocate far less time for students to work on interactive classroom activities or thought-provoking problems.

Studies of Japanese elementary mathematics teaching describe carefully implemented practices that are passed on from experienced to novice teachers t h rough extensive apprenticeships. Math lessons begin slowly and build methodically in an attempt to engage students in a challenging problem. Japanese teachers place a much greater emphasis on thinking about the problem than on quickly coming up with a solution. Students are directed to work in small groups on one or two problems. At the elementary level, students use mathematics kits that contain visual aids and manipulatives. These are considered important elements in re p resenting and discussing mathematical solutions. Japanese students are four times more likely than U.S. students to work on exercises or challenging problems for 15 minutes or more each class. Japanese teachers interrupt small-group work to discuss diff e rent groups' solutions to problems. They examine and extend a group's solution regardless of its correctness. In contrast, U.S. teachers are reluctant to present errors or incorrect thinking as the basis for class discussion. Japanese teachers move back and forth from whole-class to group work, using whole-class discussions to raise questions and re v e a l inconsistencies. Japanese teachers are three times more likely than U.S. teachers to discuss alternative perspectives on a problem. Japanese teachers give feedback during seatwork. U.S. teachers tend to allow students to do seatwork or homework for an extended period at the end of the lesson, with much less interactive instructional time.

Academic Diversity

Research has revealed no major differences in cognitive abilities between U.S. and Japanese students. In Japan as well as in the United States, there is a wide range of performance on the TIMSS at the fourth- and eighth-grade levels. However, the Japanese levels of performance are significantly higher. Japanese classes are often much larger than U.S. classes and provide few formal programs for low-achieving students or students with learning disabilities. Slightly more than 1 percent of school-age Japanese children are in special education. However, a report by the National Institute of Special Education indicated that 10 percent of fifth- and sixth-grade students were two or more years behind in mathematics. This disparity in performance is one of the reasons given by teachers who reported having difficulty controlling their classrooms and conducting lessons effectively.

Woodward and Ono looked at several factors they believe reveal how Japan manages academic diversity. Japan has one of the most highly developed preschool systems in the world. Seventy percent of all children attend preschool for three years. Young children are gradually trained in important school-related behavior, practicing routines that will be used throughout elementary school. Teachers delegate authority to children , and young students learn to work with others and identify with the group. Everyone is expected to participate in activities and discussions. Groups are composed of students with diverse abilities. Teachers see great benefit in mixing slow learners with faster learners. Low-achieving children learn by listening to more capable peers, and fast learners benefit from explaining their mathematical thinking to others.

Two traits pervade the culture and are taught by both parents and teachers: effort and persistence. In a study of a puzzle-like game with a series of increasingly difficult stages, Japanese students persisted significantly longer than U.S. students. Researchers in a previous study saw the emphasis on effort that permeates Japanese education as one of the reasons that academically low-achieving Japanese students do better in school than their U.S. peers. Students are encouraged to put in extra effort to keep up, and Japanese teachers in that study were three times more likely than U.S. teachers to work with struggling students outside of class.

Education changes significantly in Japan during middle school. Instruction intensifies, and an increasing number of tests assess mastery of content, leading up to the high school entrance test given in eighth grade. Enrollment in after-school tutoring programs increases. These programs provide the remedial assistance and individualized instruction not found in Japanese schools. The three years of high school are not mandatory, but 97 percent of students attend high school and between 91 and 99 percent graduate. Struggling students can attend three alternative types of high school: evening, correspondence or special-education high schools. However, less than 5 percent of students attend an alternative school.

Trial efforts to serve students with learning disabilities began in 1990 in Japan. Current educational reform in Japan is focused on individual differences. A method called Suido appears to be effective for students struggling in mathematics. Similar to the national curricula, the Suido method focuses on the relationship between mental and written calculations; between the concept of number and the four basic operations; and the principles that should guide the calculation system. This method uses tiles to make quantities explicit. By working with tiles, children associate quantity with place value and written symbols. Once students have been shown the interrelationship between the tiles, place value, and the written symbol, they can be shown each operation on the number. The Suido method underscores the importance of teaching the four operations separately and demonstrates that tiles can be an effective mechanism for showing basic concepts such as uniting, taking away, and distributing. There is a significant emphasis on visual representations and on the careful analysis of mathematical problems. Unlike the strong emphasis on procedural fluency in arithmetic found in U.S. curricula for students with learning disabilities, in Japan significant attention is given to important concepts such as place value.
"Mathematics and Academic Diversity in Japan", Journal of Learning Disabilities, Volume 37, Number 1, February 2004, pp. 74-82. Published in ERN March 2004 Volume 17 Number 3

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Japanese methods of teaching mathematics have been intensively studied in the United States since the Third International Mathematics and Science Study (TIMSS) revealed that Japanese students score significantly higher than students in the United States at both fourth and eighth grades.

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8:00-8:30 Introductions

8:30-10:00 Session 1

10:00 -10:15 Break

10:15 - 11:45 Session 2

11:45 - 12:45 Lunch

12:45 - 2:15 Session 3

2:15 - 2:30 Break

2:30 - 4:00 Session 4

4:00 End of Day

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8:30-10:00 Session 5

10:00-10:15 Break

10:15-11:45 Session 6

11:45-12:45 Lunch

12:45-2:15 Session 7

2:15-2:30 Break

2:30-4:00 Session 8

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In the PYP, mathematics provides a vehicle for exploring, consolidating and transferring understanding to authentic contexts.

In a transdisciplinary programme, learning mathematics happens through a balance of strategies.

The PYP collaborative planning process supports planning for transdisciplinary inquiry-based learning and teaching of mathematics to build conceptual understanding.

Fostering an assessment culture involves the development of assessment capability among all members of the learning community.

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Please read "What is an IB Education?" with the following four themes in mind. Annotate or take notes under each:

International-mindedness

The IB Learner Profile

Curriculum Design (Broad, Balanced, Conceptual, Connected)

Approaches to Teaching and Learning (ATL)

Answer the following prompts in a digital document or hand-written notes. We will discuss our reflections during the workshop.

1. International-mindedness

In your own words, how do you interpret "international-mindedness"?

How might this influence the way math is taught or discussed in your classroom?

2. Learner Profile

Choose two attributes that feel most relevant or challenging to foster in math instruction.

How might you help students demonstrate or grow in these areas?

3. Curriculum Design

What does a “conceptual and connected” math curriculum look like in your context?

Share an example of how math can be linked to real-world, interdisciplinary, or global themes.

4. Approaches to Teaching and Learning (ATL)

Which approach (inquiry-based, conceptual, contextual, collaborative, inclusive, assessment-informed) most aligns with your teaching philosophy?

Which approach would you like to strengthen?

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Making the PYP Happen: How Are Math Practices Changing?

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Making the PYP Happen: How Are Math Practices Changing?

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https://resources.ibo.org/data/p_0_pypxx_gui_2412_1_e.pdf

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The progression videos are a series of videos that show how math concepts build and grow through the grade levels. Here is a post that gives a "behind the scenes" of what it takes to make the videos.

Click on the image below to watch the progression video and download the resources.

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https://bstockus.wordpress.com/numberless-word-problems/

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From Principles To Practice: Transdisciplinary POI 

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From Principles To Practice: Transdisciplinary POI 

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Example units of inquiry with math integrated

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Example units of inquiry with math integrated