– A product of Pearson (so is the EnVision Series)

– Reading, Language arts, and Math assessments.

– Two math assessments

– M-CAP Concepts and application probes (5th-grade ex. attached)

– M-Comp Computation (5th-grade ex. attached)

– 8-minute tests assessed multiple times throughout school year.

– Students start with question 1 and move forward until the time is up.

– They complete what they can, and skip what they do not know.

– Focused on showing what the students know, and not what they don’t know.

– Data throughout the year should show grow as students conquer more and more skills.

– Operation/Skill each question assesses does not change. Question one will always assess the same skill, however, the situation (word problem) and values used will change.

– Questions they have answered correctly before, they should get right again…

– Compare to national data, students in bottom 25% move down one grade level until they reach their instructional level.

i-Ready

(Math and reading diagnostic test and intervention)

– Diagnostic test is “Adaptive”

– Adjusts the level of difficulty of the question, while still assessing the same skill, to accurately assess the students grade level ability through understanding what the students can be successful at.

– Great for diving into student data, and understanding student grade level equivalency for the skills (Fluency – mid4th grade or early5th)

– Students must take test seriously, to receive credible data

The learning Odyssey/Odyssey Math/Compass Learning

(All names for the same thing)

– I believe I heard my principal say it cost 3,000 for our teachers at our school (pretty small school (15 teachers?))

– Provides each skill with cartoon instructional/situational videos, and multiple choice problems.

– Lets students learn through educational games.

– Gives students brain breaks, and mini games every 15 minutes or so, to break up work and increase stamina.

– As a teacher, you can “Build a Pathway” to mastery of standards

– All subjects available

– https://www.youtube.com/watch?v=waRfH0_nKog

– https://vimeo.com/40091821

Khan Academy

– I am a huge advocate of this program,

– There is no diagnostic test

– Each grade level is broken up into a series of skills (100-120 per grade) derived from the common core standards and sequenced in order according to math strand (Number sense, measurement, geometry…)

– You can create an account for each student, track their progress as they practice, level up, and finally master skills.

– Each question is accompanied by a video and hints if needed.

– The data piece is great

– Can look at class or individual data

– see what skills students have mastered, which they need work in, look at how many problems they have attempted in each skill, time they took took to answer, and percent correct.

– Free!Chase’s materials-1d43cms

A Model of Early Number Development

Kaye Treacy

Department of Education and Training, WA

<Kaye.Treacy2@eddept.wa.edu.au>

Sue Willis

Monash University

<Sue.Willis@Education.monash.edu.au>

- The research overviewed in this paper suggests that children will not develop an understanding of number as a representation of quantity through counting alone. This development is a complicated process that involves the interaction of a number of different quantitative aspects of a child’s daily life. From the research discussed above, a model (see Figure 1) is proposed of how these different components interact to contribute to a child’s developing understanding of numbers. The placement of the different components within the model indicates the interaction of the various components through time, not at particular times. Each is discussed briefly below.
- Protoquantitive comparison: Children, from about two years of age, become able to associate relational words with their innate ability to compare two amounts. They are able to say which is bigger or which amount has more or which has the most. Resnick calls this protoquantitive knowledge, though other researchers do not use this term, instead talking about relational knowledge.
- Children’s ability to subitize small amounts seems to develop out of their early ability to compare quantities. Children learn to associate a particular number word with a particular quantity. Starkey and Cooper (1995) found that by age two most children in their study could subitize one, two and three items, at three and a half years of age children subitize up to four items and by five they subitize up to five items. Sophian, Wood and Vong (1995), however, suggested that three and four year olds could subitize up to six items.
- At about two years of age, children begin to learn some of the principles of counting. They initially learn the first few words in the number sequence and to use one to one correspondence. Later they learn to give emphasis to the last word in the count. Children seem to initially learn these things as part of their socialization and may not link them with the idea of finding out how many. Even when children learn to repeat the last word of the number sequence in response to the how many question, they may not link this with the idea of quantity. Use counting to get. Counting and subitizing initially exist along side one another in a child’s mind as distinctly different processes. Children then begin to link the list of count words with the quantities that they know through subitizing and begin to understand that the last word said at the end of the count is telling them how many items in a collection. Children thus learn the quantitative significance of the number words in the counting process. According to Fuson (1988) (cited by Nunes & Bryant, 1996) children, at about five years of age are able to use counting to quantify single sets and to get an amount of items when asked.
- After children understand numbers words and counting as a means of quantifying a single set, they develop a trust in their counting processes and learn that no matter which way they count a collection they must always get the same result. As a consequence they “trust the count” and choose to use it to solve relational problems such as to make equivalent sets.
- Children learn to connect the number words and quantity. This allows them to develop an understanding of the part whole relationships of numbers attached to particular quantities. They can see, for example, that five fish could be made up of a group of three fish and a group of two fish. This helps them to develop a more robust understanding of the numbers they use in counting. They come to trust that no matter which way a collection is arranged or partitioned, the quantity of the set will always remain the same
- Children’s understanding of number from counting, subitizing, and part whole situations comes together so that they becom able to think about numbers as representations of quantity. They are able to disembed the number from the situation and so become able to think of any five items as “five”. The number becomes a conceptual entity in it’s own right. They understand the additive composition of number and so can think of numbers as compositions of other numbers, the number five, for example, can be thought of as three and two. They can work with numbers alone without having to refer to a quantity of materials.
- Caveat: This model has been used as the basis for an investigation involving 25 children with learning disabilities in a Western Australian school (Treacy, 2001). Tasks were developed for each of the components listed above and these were used to individually interview the children. It was found that the children showed understandings similar to those suggested by the model above. For example, there were some children who showed no evidence of understanding the quantity aspect of counting and yet could subitize to three or more. The teachers in this school found the model and the associated tasks particularly helpful in working out what their students knew and what they needed to know in order to develop a deeper understanding of number. Further research is needed however, to establish whether this model would be helpful for teachers working with children within the ‘normal’ range of intellectual ability.
- There is a visual of this process :visual-23js5zf

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I offered to continue our conversation so please feel free to email me at kls54@psu.edu. I’d be happy to help with lesson ideas and even implementation if it is on a Thursday!

We moved into some more investigation of the percentage problem by looking at the work of several students. And ended with our Bungie Barbie activity.

It was great working with all of you! Have a great summer.

I will include another blog with some information from Chase and more on the research on mathematical developmental stages.

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Building addition and subtraction through counting by ones;

• Building addition and subtraction through grouping;

• Building multiplication and division through equal counting

and grouping;

• Building place value through grouping;

• Forward number word sequences;

• Backward number word sequences;

• Number word sequences by 10s and 100s; and

• Numeral identification.

A couple of other sources I found were:

https://www.merga.net.au/documents/RR_treacy.pdf

https://books.google.com/books?id=JBmOAgAAQBAJ&printsec=frontcover#v=onepage&q&f=false

And Fran Arbaugh from University Park recommended the books:

https://www.amazon.com/Cognition-Based-Assessment-Teaching-Place-Value/dp/0325043434/ref=sr_1_1?ie=UTF8&qid=1502380549&sr=8-1&keywords=battista+mathematics

We had some great discussion about linear relationships that were both proportional and not proportional and non-linear relationships. We followed up with some challenging questions from Block 4.

The end of our day consisted of a discussion about “Smarter than We Think” and a video about Cena whose understanding of place value might be a bit more fragile than her teacher thought. Here are the ideas I shared about inverse and number sense that you could use in a few extra minutes of class.

“Tell me another way”

- I ask for two number whose product is 40 (or whose sum is 8).
- Then ask to find two other ones.
- After listing several look for a pattern.
- This can even lead in to negative numbers or fractions.

- Functions
- 6→∎→10 what could have happened in the box?
- 6→∎→10→∎→6 what could have happened in the boxes?

□→black box that multiplies by 2 and adds 4→ △ fill in the square and triangle with values.

- “What’s the process? – function of 2 variables”
- I put 2 and 3 in the black box and get out 5, what’s the black box doing?
- You can build on that with all sorts of processes.
- You can then have another box and have it undo the process of the first.

- Other ideas:
- I am starting with the number 7. I add 3 and then divide by 2 to get ______. How do I get back to the number 7 using the same number of steps I used?
- I am starting with the number 100, subtract 50, divide by 2, take square root. I end up with ______. How do I get back to 100 using the same number of steps I used?

https://www.youtube.com/watch?v=KdxEAt91D7k

We came up with these ideas about the different types of questions:

Funneling:

Goal: singular – particular answer

“Leading” because you already know the answer

Investigating background knowledge

Can build confidence for students who struggle to articulate thoughts

Ok when scaffolding

Help students learn what questions to ask themselves in solving

Might be better for English Language learners and L.S.

With word problems the reading levels might not match the math ability

They might be hard for students with processing issues

If a student has background knowledge of a different strategy it might be counterproductive

Focusing:

Goal: Open ended, allowing multiple pathways, check for understanding

Allows for explaining the thinking

Higher level process conversation

Build stamina or frustration

Opens up the time for commenting

The students direct the path with guidance from the teacher

You have to be careful you don’t go too far down the right path

You must listen carefully

It takes more planning and foresight on the teachers part who must have thought about what might happen.

There is a website where you can share and comment on mistakes students have made in math class – http://mathmistakes.org/

Charles began our math discussion by looking at the work of a student on the hourglass problem. There was some good discussion about the work and the process and the accuracy of the mathematical representation.

We ended the day with a pedagogy lesson involving making posters to define proportion, y intercept, x intercept, linear relationship and slope and we did a gallery walk for these posters.

**Here is a funny video Bernadette shared about slope:**

https://www.youtube.com/watch?v=KdxEAt91D7k

Homework: Read “Never Say Anything a Kid Can Say.” Record any question, “yea, buts” or comments in your journal.

]]>I found some articles about “the math brain” and mathematical thinking that I will share below:

‘Not a Math Person’: How to Remove Obstacles to Learning Math

http://www.dailymail.co.uk/sciencetech/article-2024570/Cant-add-We-born-mathematical-brain-not.html

https://www.scientificamerican.com/article/how-does-a-mathematician-s-brain-differ-from-that-of-a-mere-mortal/

A couple of quotes from these:

- “Most of the things that parents and kids believe about math learning are wrong,” said Dr. Boaler, who is the co-founder of
__Youcubed__, a website that argues for a revolution in math teaching for all children, and offers resources to teachers, students and parents. In fact, maybe what everyone needs — girls and boys both — is a different kind of math teaching, with much less emphasis on timed tests, and more attention to teaching math as a visual subject, and as a place for creativity. - “The lovely thing is when you change math education and make it more about deep conceptual understanding, the gender differences disappear,” Dr. Boaler said. “Boys and girls both do well.”
- “Recently, a colleague’s 7-year-old came home from school and announced he didn’t like math anymore. His mom asked why and he said, ‘math is too much answering and not enough learning.’ This story demonstrates how clearly kids understand that unlike their other courses, math is a performative subject, where their job is to come up with answers quickly. Boaler says that if this approach doesn’t change, the U.S. will always have weak math education.”
Another burning question was to think more about problem 3c on page 7 and I suggested using Excel to create the arrays for the division and the remainder.

Charles began with the mathematical content on page 10 of block 1. There was good discussion about the inverse and how a function can be its own inverse.

We spent some time practicing our questioning techniques and trying to use focusing questions to dig into the thinking processes of our students instated of funneling questions that lead to a certain type of thought process.

Pedagogy homework:

Read the article “Questioning our Pattern of Questioning” and think about the types of questions we asked in our pairings. Also percolate on the question – is it ever ok to use funneling questions? Tomorrow we will talk more about it.

]]>We began with a burning question from last week connecting scale factor and unit rate in solving proportions.

We also discussed the norms we are trying to establish in the workshop which included – everyone being heard, no fear of making errors, and respect for each others’ ideas.

Because of a glitch in the materials we then moved into discussing the first two standards for the common core and an explanation of these was found in the website: https://www.scholastic.com/teachers/blog-posts/meghan-everette/guide-8-mathematical-practice-standards/

There are also some good videos of what the standards could look like found at: http://www.insidemathematics.org/common-core-resources/mathematical-practice-standards/standard-2-reason-abstractly-quantitatively

Charles then led us into Block 1 with the sliding snail question. We realized we could use a piece-wise function to express the height at any integer time value.

There was discussion about the value of generic rules for expressing even and odd numbers. We expressed an even number as 2n and an odd number as either 2n + 1 or 2n – 1.

We had some good discussion about whether a remainder is negative or positive when dividing a negative number by a positive. We found that thinking of it as positive made the problem of comparing 3/4 and (-24/4) and (27/4).

We talked about graphing the walk to the bus stop and showed that the steepness of the line can show the speed of my travel.

HW for tonight:

Charles recommended the going further problems on page 9.

For pedagogy

- Read Principle’s to Action pp 35-41 and 53-57
- In your notebook write a response to two prompts:
- In questioning in small groups of students working on a problem, a teacher noticed that when she asked a “focusing” question, the students continued to look at their work and continued to engage in their own dialogue. When she asked a “funneling” question, the students looked up at the teacher. Comment on these observations.
- Look at Maddie’s work and Gabe’s work, shown in figure 22 on page 55. How could Ms. Lewis leverage the student’s representations to develop Maddie’s understanding of the problem?

Here are the websites for different ways to show multiplying two negative numbers with manipulatives and the number line:

- One way (around 3:51)
- ice cube and box idea
- Flipping over the origin (3:10)
- walking forwards and backwards (2:10)

We also reviewed that dividing by zero never gives a numerical answer. 6/0 is undefined. 0/0 is indeterminate.

Charles answered some multiplication questions and then headed into the division.

There was some good discussion about the difference between partitive and quotative division.

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http://www.realclearscience.com/video/2017/03/03/the_uncomfortable_effort_of_thinking.html

There was great discussion about how difficult it can be to take this time for ourselves and our own growth without constantly trying to figure how it can help our students. Teachers don’t always have this opportunity – even during in-service.

We talked a lot about positive and negative numbers and using the area model for multiplying fractions – we agreed that the model has its limitations for multiplying mixed numbers. We thought using the distributive property helped with that – for example – 4 1/2 x 2 1/3 – you could do 4 x 2 + 4 x 1/3 + 2 x 1/2 + 1/3 x 1/2 and use the area model for 1/2 times 1/3.

We watched a video on “My favorite No” – https://www.teachingchannel.org/videos/class-warm-up-routine

We also discussed the ideas that were evident in Ms. Flahive’s and Ms. Ramirez’s classrooms from Principles to Action. And then worked on coming up with rich problems for use in our classroom to set the tone for the new year. We will continue this idea tomorrow as some of us are ending the workshop then.

Homework:

Read the reference material on pages 11 and 13 of Block 3.

Read the message entitled “Upside Down Teaching” and in your notebook, complete respond to the discussion prompts for teachers at the end of the message (on p. 94). Try to make connections about what we’ve read about and discussed so far in PMI.

Thanks for a great day.

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