Strategies for Multiplication Sequence strategy cards in order from least sophisticated to most List to the side the knowledge, skills and understandings a student would need to implement the strategy There is one too many cards to fit on one poster paper. Tape two pieces of poster paper together so that they overlap and give enough length for one additional card. (or remove one card from the sort.)

Gallery Walk Allow participants to briefly look at other’s work and discuss.

Levels in Problem representation and solution Briefly review the Levels in the progressions document starting on page 25 Level 1 – Making and counting all Level 2 – Repeated counting by a given number Level 3 – Use properties and relational thinking CC & OA Progression document – pages 25-26

Levels in Problem representation and solution For each level, identify at least 1 strategy card as a “clear example” of that level. How does your group’s card sequence align with the levels described? Do you wish to move any of your strategy cards? CC & OA Progression document – pages 25-26

Connecting to the Red Book Intro

the Development of Mathematical Knowledge (For MD) Sophistication of Unitizing Numbers Distancing the Setting Knowledge of Multiples and Sequences of Multiples Development of Non-counting Strategies Refer teachers to 143 of red book. Might give teachers some time to re-read this section or re-read parts as you go. Basically, on this slide we are summarizing the aspects that come together that let a student move from solving using the lowest level strategies (or not solving at all) to using high level strategies. Focus on painting a holistic picture. Do not get bogged down in distinguishing one aspect from another. Refer regularly back to their progressions to illustrate the ideas.

the Development of Mathematical Knowledge (For MD) Sophistication of Unitizing Numbers Distancing the Setting Knowledge of Multiples and Sequences of Multiples Development of Non-counting Strategies Refer teachers to 143 of red book. Might give teachers some time to re-read this section or re-read parts as you go. Basically, on this slide we are summarizing the aspects that come together that let a student move from solving using the lowest level strategies (or not solving at all) to using high level strategies. Focus on painting a holistic picture. Do not get bogged down in distinguishing one aspect from another. Refer regularly back to their progressions to illustrate the ideas.

Sophistication of Unitizing Number For example: 6 x 8 Composite unit of composite units Individual items 8 1 slide summary of unitizing… go from seeing only a “1” as a unit (so 48 is simply 48 ones) to being able to decompose a unit of units (i.e. 6 groups of 8) to two composite units (5 groups of 8 and one 8)… This is intended to be surface. If you prefer to go deep, the previously created slides are kept in the archives. Look in folder “Archive old versions” for the day 8 PowerPoint (last saved October 24) 5 x 8 6 1 x 8

Four Aspects of the Development of Mathematical Knowledge Sophistication of Unitizing Numbers Distancing the Setting 7x3 Shift gears here… revisit the familiar topic of Distancing the setting. 7 x 3

Four Aspects of the Development of Mathematical Knowledge Sophistication of Unitizing Numbers Distancing the Setting Knowledge of Multiples and Sequences of Multiples Starts page 143 of red book – see next slide for an expansion of these ideas

Four Aspects of the Development of Mathematical Knowledge Sophistication of Unitizing Numbers Distancing the Setting Knowledge of Multiples and Sequences of Multiples Development of Non-counting Strategies

Connecting Progressions to the Red Book Level 3: Properties and relational thinking Level 1: Making & counting all Level 2: Repeated counting 4 aspects of Mathematical Knowledge Sophistication of Unitizing Numbers Distancing the Setting Knowledge of Multiples and Sequences of Multiples Development of Non-counting Strategies After the brief reminder/review of the 4 aspects, ask: “what connections do you see?” “How would implementing the red book ideas support moving children through these level?” Or go specific with questions like “how would knowing multiples and skip counting sequences move a student through these levels?” or “What role does distancing the setting have?” Or “Why is the development of non-counting STRATEGIES important” (note – having non-counting strategies is not the same as being able to answer without counting. Remember our 12x12 girl? She answered without counting by no strategies!)… “Without getting bogged down in the sublevels part a on page 144… what’s the big picture… how does it align with the levels?” (Not sure how to ask this one.) Acknowledge the details for unitizing are likely fuzzy, but stress the progression from a “by ones only” kid (that doesn’t see or use groups) to a kid that sees and uses the structure and can decompose a group into groups. Some Connections that may be discussed Sophistication of Unitizing numbers… a growth process through which the student recognizes and uses groups… eventually seeing groups inside of groups. Distancing the setting … start with all items visible, gradually having only abstract representations to no representations at all Knowledge of multiples… Students need this to master level 2, skip counting without starting from the first multiple important at level 3 Development of non-counting strategies – describes level 3, doesn’t it!!! Ask…. Why is Level 2 important? Why not have kids “Make & count all” then just target level 3? Or “Does a kid need to spend time at level 2”?

Instructional Progression Level 3: Properties and relational thinking Level 1: Making & counting all Level 2: Repeated counting Why is it important for a child to “spend time” working at level 2? How quickly can we expect a child to “move through” these levels? How can we craft a classroom environment that support kids at their mathematical level AND encourages growth? Ask…. Why is Level 2 important? Why not have kids “Make & count all” then just target level 3? Or “Does a kid need to spend time at level 2”?

Strategies & Knowledge Students think multiplicatively rather than additively (i.e. seeing limited to seeing multiplication as only repeated addition). Students use prior knowledge to work out unknown. For example, Solve 6x9 by 6x6 = 36, 6x3 = 18…. 36+18 = 36+20-2 = 56-2 = 54. (From book) OR 6x9 = 60, take away 6… 54 (Learning to think article) (some of this is a repeat of ideas from the unitizing section – for obvious reasons. Talk about this with your teachers when it seems to work) Having strategies is closely related to the aspect of unitizing number and I’m not sure I really understand or explain the difference. Developing non-counting strategies is more about the behavior… having and using strategies. The unitizing aspect is having that fundamental understanding that we can work with units/groups and have groups of groups. But since we can’t actually get into a kids head, we determine that a kid has the understanding if they are doing the strategies. (If someone has a different or clearer understanding or way to say it… please feel free to add to the comments here or discuss it at one of our meetings!!!) For our participants, I don’t think we need or want them to be able to distinguish. Rather, they are all aspects of a whole and we only need to have an understanding of the whole, not what makes up each aspect. With this slide… we see the role in developing strategies as empowering the development of knowledge (and the reverse). Use link to go to website with and show diagram at it’s original source if possible. The diagram shows that there is a cycle of using strategies to solve will extend knowledge, and the more knowledge will expand the opportunity to use strategies. If internet not available, use next slide (currently hidden) to show the diagram. This comes from the online professional development provided on the NZ Maths site which we’ve reference before. There are online Numeracy PD modules. http://www2.nzmaths.co.nz/Numeracy/ONPD/M1/07.aspx

The relationship between Knowledge and Strategy Use link to show diagram at it’s original source if possible. If internet not available, use this slide to show the diagram. This comes from the online professional development provided on the NZ Maths site which we’ve reference before. There are online Numeracy PD modules.